Download Generation of Spin Squeezing in an Ensemble of Cold

Generation of Spin Squeezing in an
Ensemble of Cold Rubidium 87
Marco Koschorreck
Dissertation
at the
Institute of Photonic Sciences (ICFO)
in the group of
Prof. Dr. Morgan W. Mitchell
and Univeristat Politècnica de Catalunya
Thesis advisor: Prof. Dr. Morgan W. Mitchell
co-advisor: Prof. Dr. Jürgen Eschner
The work described in this thesis was carried out at
ICFO - The Institute of Photonic Sciences
Mediterranean Technology Park
Av. Canal Olimpic s/n 08860 Castelldefels (Barcelona), Spain
c
Marco
Koschorreck 2010
To my parents...
Abstract
At the beginning of the 20th century measurement precision was related exclusively
to the capabilities of the experimenter and the apparatus. Any imprecision was attributed to imperfect devices for the measurement. Later on, when the quantumness
of objects like photons, electrons, or atoms, was recognized, it became clear that the
measurement and the object to be measured could no longer be separated. Both
follow the laws of quantum physics which impose limits on the accuracy of any measurement. In the last three decades, a lot of research has been dedicated to find
strategies to overcome these limitations. This new field of quantum metrology is
where this work is placed.
This thesis describes the generation of inter-particle entanglement in form of a
squeezed spin state in a cloud of cold 87 Rb atoms by performing non-destructive
spin measurements. Towards this goal, we have implemented a shot-noise limited
polarization probing and detection system, atom counting by means of absorption
imaging, spin state preparation via optical pumping and non-destructive spin state
detection. The quantum noise limited spin read-out with polarized light is characterized in two complementary ways in terms of sensitivity and all possible quantum
and classical noise sources are measured.
To address the rich opportunities provided in the multilevel structure of 87 Rb with
dispersive probing, we have developed and implemented a frequency offset-lock which
allows a large tuning range of several GHz for the dispersive probing.
Working with a magnetically sensitive atomic system triggered the development
of a magnetic field imaging technique. This applies the elongated atomic sample and
internal atomic states as a way of measuring magnetic fields over several√millimeters
with micro-meter resolution and sensitivities down to a few tens of pT/ Hz.
We developed a new method for tomographic measurements of the atomic density
matrix. Based on our ability of shot noise limited detection of polarization rotations
we can in principle, characterize the whole atomic spin state. First, measurements
are shown and possible improvements are discussed.
An in depth characterization of the light-atom interaction between polarized pulses
of light and collective atomic states are the basis for measurements at the quantum
level. On one side, we apply classical measurements of spin polarized atoms to measure the strength of the light-atom interaction. On the other side, we use quantum
measurements of unpolarized atomic states to characterize all noise sources both
technical and quantum.
One of the major results of this thesis is the realization of quantum non-demolition
measurements in a large-spin system. We realized limitations to the performance
of the QND measurement implied by higher order light shifts in multilevel atoms,
which are often neglected for these kind of systems. We develop a technique which
suppresses the detrimental effects of higher order light shifts and recovers an ideal
QND measurement.
The thesis culminates in the demonstration of spin squeezing in the magnetically
sensitive ground state of 87 Rb. We achieve a quantum noise reduction by 3 dB, where
metrological sensitivity is improved by 2 dB beyond the standard quantum limit of
a coherent spin state with the same spin polarization.
We address the question how spatial and temporal inhomogeneities both in light
and atoms influence the preparation of non-classical states. An existing covariance
matrix approach is extended and we apply it to study effects of detector time resolution, spatial inhomogeneities and atomic motion.
Resumen
A principios de S.XX la precisión en la medición se asoció únicamente a la capacidad
del experimentador y el aparato. Cualquier imprecisión se atribuía a la imperfección
de los dispositivos de medición. Más tarde, cuando se se reconocieron las propiedades
cuánticas de objetos como los fotones, electrones o átomos, se puso de manifiesto que
la medición y el objeto a ser medido ya no podían ser separados. Ambos siguen las
leyes de la física cuántica que imponen límites a la exactitud de la medición. En
las últimas tres décadas, mucha investigación se ha dedicado a encontrar estrategias
para superar estas limitaciones. En este nuevo campo de la metrología cuántica es
donde este trabajo se encuentra.
Esta tesis describe la generación de entrelazamiento entre las partículas de una
nube de átomos fríos de 87 Rb en forma de un estado de espín apretado mediante la
realización de medidas de espín no destructivas. Con este objetivo, hemos implementado un sistema de sondeo y de detección de luz polarizado limitado solo por
’shot-noise’, contando átomos por medio de imágenes de absorción, preparación del
estado de espín a través de bombeo óptico y de detección no destructiva de estado
de espín. La detección con luz polarizada se caracteriza de dos maneras complementarias en términos de sensibilidad y de todos los posibles fuentes de ruido de origen
cuántico y clásico.
Para hacer frente a las oportunidades que proporciona la estructura de varios
niveles en 87 Rb en el sondaje con dispersión, hemos desarrollado e implementado un
offset-lock de frecuencia que permite un gran rango de configuración de varios GHz.
Trabajar con un sistema atómico magnéticamente sensible provocó el desarrollo de
una técnica de resonancia magnética de campo. Esto se aplica a la muestra alargada
atómica y a estados internos atómicos como una forma de medir los campos magnéticos sobre varios milímetros
√ con una resolución de micro-metros y la sensibilidad
a unas pocas décimas de pT/ Hz.
Hemos desarrollado un método para la medición tomográfica de la matriz de densidad atómica. Sobre la base de nuestra capacidad de detección de rotaciones de polarización que puede, en principio, caracterizar el estado atómico de espin. En primer
lugar, las mediciones se muestran y las posibles mejoras de estas comunidades.
Una caracterización en profundidad de la interacción entre los pulsos polarizados
de la luz con el estado colectivo de los atómos son la base para las mediciones a
nivel cuántico. Por un lado, aplicamos las medidas clásicas de los átomos de spin
polarizado para medir la magnitud de la interacción luz-átomo. Por otro, utilizamos
mediciones cuánticas de estados atómicos no polarizados para caracterizar todas las
fuentes de ruido tanto técnicas como cuánticas.
Uno de los principales resultados de esta tesis es la realización de las mediciones
cuánticas no destructivas en un sistema de gran espín. Nos dimos cuenta de las
limitaciones para la realización la medición QND implicadas por efectos de orden
superior en los sistemas con varios niveles, que a menudo son olvidadas para este
tipo de sistemas. Desarrollamos una técnica que elimina los efectos perjudiciales de
orden superior en al AC Stark shift y se recupera una medida QND ideal.
La tesis culmina en la manifestación del spin-squeezing en el estado fundamental
de 87 Rb, que es magnéticamente sensible. Logramos una reducción del ruido cuántico
en un 3 dB, donde se mejora la sensibilidad metrológica en un 2 dB por encima del
límite cuántico estándar de un estado de spin coherente.
Abordamos la cuestión de cómo heterogeneidades espaciales y temporales tanto
en la luz como en los átomos influyen en la preparación de los estados no-clásicos.
Un método de la matriz de covarianza existente se amplió y aplicó al estudio de
los efectos de la resolución del detector en tiempo, heterogeneidades espaciales y del
movimiento atómico.
Contents
Abstract
i
Contents
I
1. Introduction
1
2. Light-atom interfaces - background
2.1. Continuous variables for atoms . . . . . . .
2.2. Continuous variables for light . . . . . . . .
2.3. Light-atom interaction . . . . . . . . . . . .
2.4. Spin squeezing . . . . . . . . . . . . . . . .
2.5. Quantum non-demolition measurements . .
2.6. Simulation of collective quantum properties
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3. Experimental apparatus and techniques
3.1. Atomic trap . . . . . . . . . . . . . . . .
3.2. Atom number measurement . . . . . . .
3.3. Dipole trap characterization . . . . . . .
3.4. Spin state preparation . . . . . . . . . .
3.5. Probing atomic spins . . . . . . . . . . .
3.6. Shot-noise-limited polarization detection
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4. Dispersive Spin Measurements
87
4.1. Classical characterization of QND measurement . . . . . . . . . . . . 88
4.2. Measuring Zeeman coherences . . . . . . . . . . . . . . . . . . . . . . 94
4.3. Dispersive atom number measurement . . . . . . . . . . . . . . . . . 101
4.4. Relation between scattering cross-section and observed depolarization 102
5. Magnetic Field Measurements
105
5.1. Spatially integrating magnetic field measurements . . . . . . . . . . . 106
5.2. Spatially resolved magnetic field measurements . . . . . . . . . . . . 110
6. Spin State Tomography
121
6.1. Stern-Gerlach in free fall . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2. Measuring ground-state populations of trapped atomic sample . . . 124
7. Spin Squeezing
7.1. Calibration of spin detector . . . . . . . . . . . . . . . . . . . . . . .
7.2. QND measurement of large-spin ensembles by dynamical decoupling
7.3. Spin squeezing of magnetically sensitive atomic state . . . . . . . . .
133
134
143
152
8. Summary and Outlook
157
8.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.2. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 158
A. Appendix I: Gaussian state picture for light and atoms
A.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2. Physical system and mathematical model . . . . . . . . . . . . . . .
A.3. Unified description of physical processes . . . . . . . . . . . . . . . .
A.4. Instructive examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.6. Coherent and incoherent transport processes described by N-port unitaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
161
162
162
165
170
176
177
B. Appendix II: Numeric simulation of multi-level atoms
179
B.1. Scattering cross section for different hyperfine ground states . . . . . 180
B.2. AC Stark shifts and the limit of the rotating wave approximation . . 180
B.3. Numerical simulation techniques . . . . . . . . . . . . . . . . . . . . 183
Abbreviations
Bibliography
185
i
List of Publications
xv
Acknowledgements
xix
1. Introduction
The quest to perform more and more precise measurements has driven humans for
centuries and even millennia to improve the present technology towards a better
one. At the advent of quantum mechanics it was perceived that there exists, in
fact, a limitation implied by nature, in recent times referred to as standard quantum
limit [1]. Many decades later, it was realized that quantum mechanics itself gives a
solution to this problem. Quantum mechanical superpositions and entanglement are
concepts which provide strategies to go beyond classical limits. This gave birth to a
new class of ’Q-technologies’ like quantum information processing (QIP), quantum
communication (QC), and quantum metrology (QM).
Measurements performed with a system of N identically prepared particles, e.g.,
photons or atoms, will give a distribution of outcomes and show quantum noise,
√ −1
which scales as δN ∝ N . These fluctuations are not the fault of an unskilled
experimenter but rather a manifestation of the Heisenberg uncertainty principle and
will ultimately limit the accuracy in any precision measurement [2]. The standard
quantum limit is not fundamental in nature, but rather a result of an non-optimal
measurement strategy. Introducing non-classical, i.e., quantum, correlations between
the N particles, the measurement sensitivity can be improved towards the fundamental Heisenberg limit scaling as δN ∝ N −1 [3].
This thesis describes the generation of inter-particle entanglement in an ensemble of
atoms which results in reduced quantum fluctuations - below the standard quantum
limit - in one of its quantum variables. We use a measurement based approach,
by applying optical quantum non-demolition (QND) measurements, to reduce the
quantum uncertainty and generate inter-particle entanglement.
In quantum information processing and precision metrology atomic ensemble systems of meso- or macroscopic number of atoms have proven very useful. Key to
their utility, atomic ensembles are simultaneously macroscopic, with many atoms
contributing coherently to a given effect, and simple, having only a few relevant
quantum degrees of freedom. Seminal experiments with vapor cells demonstrated
light-induced entanglement between ensembles [4], quantum memory for light [5],
teleportation from light to matter [6], and projection noise limited magnetometry
[7].
We are exploring a new generation of a magnetically sensitive quantum light-atom interface - optically trapped cold atomic ensembles - suitable for high precision metrology and quantum information processing. Ensembles of laser cooled and optically
trapped 87 Rb atoms provide a very sensitive magnetic field sensor [8]. Generating a
non-classical state improves the sensitivity further towards more fundamental limits.
Cold atomic ensembles provide a number of advantages over room temperature vapor
2
1. Introduction
cells. Firstly, they reach the same interaction strength with considerably fewer particles, i.e., 106 compared to 1012 . This is important in terms of technical noise. This
allows us to work directly with rotation signals, without the modulation and demodulation typically used in the large-number vapor cell experiments. Secondly, because
our atoms are effectively stationary, we can have much faster interactions, without
need for averaging of thermal motion. Thirdly, non-classical states of light, e.g.,
single photons [9] or Schrödinger cat states [10, 11], or arbitrary two photon states
[12] have been generated with bandwidth compatible with microsecond interaction
times.
Macroscopic ensembles of atoms and light are conveniently described in terms of
collective spin operators. In Chapter 2, which collects the important theoretical
concepts, we give a summary of their basic properties which establishes the ’language’
for the rest of the work. Further we review the light-atom interaction and give a
description in terms of atomic spin and the optical Stokes operator which facilitates
the understanding of the process. We also review important calculation techniques,
e.g., density matrix and master equation, frequently used throughout the work to
understand and predict the behavior of the atoms and the light. The concept of spin
squeezing is explained and important criteria reviewed. Quantum non-demolition
measurements, as the cornerstone of the state preparation, are described at the end
of this chapter and there relation to spin squeezing is explained.
Chapter 3 summarizes the experimental apparatus and the essential improvements
over the previous setup. The optical trap as central tool to produce well-localized and
confined atomic clouds is characterized in terms of trap lifetime, atom temperature
and number, and loading characteristics. Absorption imaging as atom number counting technique important in the quantitative characterization of the QND interaction
is reviewed and its implementation discussed. The coherent spin state preparation
and probing are discussed and characterized. The polarization detection is measured
in terms of its noise characteristics.
The dispersive spin probing, which provides the quantum non-demolition spin measurement, is investigated in Chapter 4. The light-atom interaction strength is characterized with the help of polarized spin states and absorption imaging. We introduce a dispersive atom number measurement (DANM) which provides a way of
non-destructive atom counting necessary for measurements in later chapters.
In Chapter 5 we discuss efforts to produce a magnetic field free environment of
the atomic ensemble. This is important to achieve a long coherence time for the
coherent spin states and crucial for the spin squeezing experiments. Two techniques
to measure constant stray magnetic fields are presented and experimental results
shown. To measure magnetic field gradients and higher order moment we designed
a novel technique which uses the extended atomic sample as a way of measuring
magnetic field with spatial resolution. This magnetic field imaging technique is
applied to cancel the existing magnetic field gradient in the vicinity of the atomic
sample. The magnetic field imaging technique is published in [...].
In Chapter 6 we briefly review the concept of spin state tomography as characterization tool for the atomic state. Important requirements for a complete characterization of the full density matrix of a spin state are discussed. We propose a novel
3
tomographic technique based on state-selective population transfer and sensitive dispersive atom number measurement on a trapped atomic ensemble. We present first
results and discuss possible improvements.
In Chapter 7 we present measurements results of the preparation of a squeezed spin
state through measurement induced entanglement. We start by calibrating the lightatom interaction using a thermal spin state. In this way, a reliable characterization is
possible with a test state that shows with known quantum noise but which is robust
against technical noise and depolarization noise. The results of this subsection are
published in [13].
For the QND measurements of a quantum states in the hyperfine ground state of
rubidium 87 we identify an important obstacle which impedes the generation of spin
squeezing in such a system. A technique which surpasses this limitations based on
the concept of dynamical decoupling is developed and implemented. In this way we
are able to recover a pure QND measurement even in a large spin system. These
results are published in [14].
Finally, first results on spin squeezing in the F = 1 hyperfine ground state of 87 Rb
are shown. The quantum noise of the coherent input state is reduced by 3 dB on an
ensemble of 700000 atoms. The metrological improvement compared to the standard
quantum limit of the coherent spin state is 2 dB.
The atom light interaction between optically trapped atomic ensembles and a Gaussian laser beam is far from being uniform. In Appendix A we study the effects of
spatial and temporal inhomogeneities by applying the covariance matrix method.
The results of this chapter are published as [15].
2. Light-atom interfaces - background
Contents
2.1. Continuous variables for atoms . . . . . . . . . . . . . . .
2.1.1. Spin states in
87
Rb
6
. . . . . . . . . . . . . . . . . . . . . .
6
2.1.2. Collective atomic spin variables . . . . . . . . . . . . . . . .
8
2.2. Continuous variables for light . . . . . . . . . . . . . . . .
2.2.1. Polarized light . . . . . . . . . . . . . . . . . . . . . . . . .
9
9
2.2.2. Stokes operators . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3. Light-atom interaction . . . . . . . . . . . . . . . . . . . .
10
2.3.1. Dipole interaction . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2. Polarizability Hamiltonian
. . . . . . . . . . . . . . . . . . 12
2.3.3. Master equation . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4. Spin squeezing . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.4.1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2. Spin squeezing criteria
. . . . . . . . . . . . . . . . . . . . 18
2.5. Quantum non-demolition measurements
. . . . . . . . .
19
2.5.1. Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.2. Atomic quantum non-demolition measurements via dispersive probing . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.3. Non-destructive magnetization measurements in
87
Rb . . . 22
2.5.4. Squeezing via quantum non-demolition measurements . . . 23
2.6. Simulation of collective quantum properties . . . . . . .
27
6
2. Light-atom interfaces - background
The chapter summarizes the main theoretical framework to describe the interaction
of a collection of atoms and polarized light. We focus on the presentation of the
main models, ideas, and concepts and give reference to more detailed texts on the
subject.
We begin by introducing the concept of collective spin variables for an ensemble of
atoms and pulses of light. In this, a macroscopic number of particles, i.e., atoms or
photons, is described as a single large spin state. This will facilitate the description
of the state for atoms and light, which otherwise would be almost intractable.
We present the Hamiltonian for the light-atom interaction in the dipole approximation. For the special case of off-resonant light we derive the polarizability Hamiltonian which is the cornerstone in the description of the experiments of probing
the quantum noise of the atomic spin state. The master-equation, fundamental for
simulations during the course of the thesis is discussed.
The concept of spin squeezing is reviewed and the two main relevant spin squeezing
criteria are discussed. Quantum non-demolition measurements are introduced and
their implementation in the context of an ensemble of rubidium atoms is discussed.
In the end we sketch the main idea of the experiment to be implemented and give
the spin squeezing criteria in terms of measurable quantities.
2.1. Continuous variables for atoms
We review important aspects of the description of collective properties of atomic
ensembles. At the beginning we briefly recall important properties of single atom
angular momentum operators. Then we mention the hyperfine structure of 87 Rb and
the important transitions. At the end we introduce collective spin operators which
will be the basis of our description of the atomic state.
2.1.1. Spin states in
87
Rb
The D lines of Rubidium are perhaps the best studied transitions in atomic physics.
A very complete summary of the physical data are found in references [16, 17, 18, 19].
Here we present the spin and energy structure relevant to this thesis.
We are mainly concerned about rubidium 87 atoms (87 Rb) in their hyperfine ground
and excited states. The energy-eigenstate of atoms in free space are characterized
through a set of quantum numbers, both the ground and excited state of interest
have a principal quantum number n = 5 and a nuclear angular momentum quantum
number i = 3/2. Its single valence electron has an angular momentum quantum
number of l = {0, 1} = {S, P } for the ground and excited state, respectively, and
a spin quantum umber of s = 1/2. Through spin orbit coupling the total electronic
angular momentum is j = 1/2 for the ground and j = 3/2 for the excited state. In
spectroscopic notation we can write the ground and excited on the D2 transition as
as 5S1/2 and 5P3/2 .
2.1. Continuous variables for atoms
7
The coupling of nuclear spin, i, and total electronic spin, j, leads to the total angular
momentum f = j + i, reflected in the hyperfine splitting of ground and excited
states, respectively. Here and throughout, we use bold letters to indicate vector
quantities. In the following we will use atomic spin when we refer to the total angular
momentum of an atom. The projection of f onto a given direction, conventionally
called quantization axis, can take the following values mf = {−f, −f + 1, ..., f }. In
the following we are mainly concerned about quantum states and transitions which
are unambiguously specified by two numbers f and mf , and we write |f, mf i. The
multilevel structure with all hyperfine levels is shown in Fig. 2.1 and an almost
complete list of properties for 87 Rb can be found in [19].
Figure 2.1.: Level structure of D2
transition of 87 Rb.
Single spin operators
The above mentioned physical quantities, for example the total angular momentum
f , are represented by operators f̂ which we indicate with a hat. Spin operators have
certain properties which are briefly reviewed. More details can be found in the text
books by J.J. Sakurai [20] or A. Messiah [21]. The spatial components fˆx , fˆy and fˆz
of the total angular momentum f̂ obey the following commutation relations
[fˆλ , fˆµ ] = i~λ,µ,ν fˆν
(2.1)
where λ,µ,ν is the Levi-Civita symbol and the Greek symbols run over {x, y, z}.
As a result of the commutation relations Eq. (2.1) we find the following eigenvalue
equations for f̂ 2 and fˆz
f̂ 2 |f, mf i = f (f + 1)~2 |f, mf i
fˆz |f, mf i = mf ~ |f, mf i .
(2.2)
(2.3)
As quantum mechanical quantities, spin operators obey the Heisenberg uncertainty
principle expressed in the Robertson-Schrödinger relation
iE2
1 Dh
var(Â) var(B̂) ≥ Â, B̂ ,
(2.4)
4
8
2. Light-atom interfaces - background
where  and B̂ can be any of the three angular momentum operators along the
Cartesian coordinates x, y, and z. The variance of an operator Ô is denoted as
var(Ô) and calculated as var(Ô) = h Ô2 i − h Ô i2 , where h.i is the expectation value.
More properties can be found in [20] and references therein.
2.1.2. Collective atomic spin variables
For an ensemble of atoms we can define collective angular momentum operators and
states. If we assume symmetry under particle exchange we can write
F̂ ≡
NA
X
f̂ (i) .
(2.5)
i=1
Here we sum the individual single atom operators f̂ (i) over the number of atoms NA .
The total angular momentum of the collective state in the symmetric space1 will
be F = NA f . For the collective operators there exist the same commutation and
uncertainty relations as for single atom operators
[F̂λ , F̂µ ] =i~λ,µ,ν F̂ν
~2 D E2
var(F̂λ ) var(F̂µ ) ≥ F̂ν ,
4
(2.6)
(2.7)
where λ 6= µ 6= ν.
The eigenstates for the collective operators F̂2 and F̂z , along the quantization axis,
are called Dicke states and were introduced in the context of super-radiance [27].
Interestingly, the Dicke states have many similarities to Fock states in the optical
domain [28]. There exist a constructive definition [21] using the raising and lowering
operators defined as F̂+ = F̂x + iF̂y and F̂− = F̂x − iF̂y :
1
|F, MF i =
(MF + F )!
2F
MF + F
− 12
F̂+F +MF |F, −F i ,
(2.8)
which applies the raising operator F̂+ successively to generate a state with eigenvalue
MF from the lowest energy (ground) state |F, −F i defined through
F̂− |F, −F i = 0 .
(2.9)
Hence, any state |F, MF i has a well defined projection along z and a completely
undefined phase. Their uncertainty orthogonal to z is large for |MF | F . Of the
Dicke states, only those with |MF | = F are minimum uncertainty states in the sense
that their uncertainty orthogonal to z gives an equality sign in Eq. (2.7).
1
The limitation to the symmetric subspace is sufficient in many case where the individual particle
is not detectable. This was used in the past to describe many experiments [4, 22, 23, 24, 25, 26].
The dimension of the symmetric sub-space scales only linearly with the system size, compared
to the full Hilbert space, i.e., 2NA f + 1 (2f + 1)NA .
2.2. Continuous variables for light
9
There exist a class of states which is very similar to Glauber’s optical coherent
states, called spin coherent states [29] or atomic coherent states [28]. An intuitive
definition was given by Arecchi et al. [28]
|θ, ϕi = e−iϕF̂z e−iθF̂y |F, −F i .
(2.10)
The state |F, −F i is rotated about the y- and z-axis. The state |θ, ϕi is completely
characterized by the two angles θ and ϕ or equivalently by one complex number
ξ = 2θ e−iϕ , and the number of atoms NA . Note that the coherent spin state, as
defined above, is always minimum uncertainty for all angles θ and ϕ .
2.2. Continuous variables for light
Polarized light plays an important role in the manipulation and detection of atomic
spin states. For example, narrow-band on-resonant light is brought into interaction with atoms to prepare certain spin states, or off-resonant light to probe spin
states dispersively. It is, therefore, important to review some properties of polarized
light. The experiments use light from quiet frequency stabilized lasers, which for our
purposes can be described as coherent states.
2.2.1. Polarized light
For many practical purposes we are interested in this thesis, it is possible to neglect
the spatial properties of the light orthogonal to the propagation direction and describe the field as a plane wave. This one-dimensional description is valid if the beam
extension is much larger than the wavelength of the light.
A light field traveling along z with a single frequency ω and pure polarization is
described in classical terms [30, 31] as
E(z, t) = Ec
X
eq aq eikz−iωt + c.c.
(2.11)
q=−1,1
where Ec is the classical real electric field amplitude in SI units V/m, aq is a complex
dimensionless amplitude of polarization mode q, and eq are unit polarization vectors
[32]. The c.c. indicates the complex conjugate of the leading expression.
The electric field of a plane wave in second quantization is given by [30]
Ê(z, t) = Ê+ (z, t) + Ê− (z, t) ,
(2.12)
where
Ê+ (z, t) ≡ Eq
X
eq âq eikz−iωt
(2.13)
e∗q â†q e−ikz+iωt .
(2.14)
q=−1,1
−
Ê (z, t) ≡ Eq
X
q=−1,1
10
2. Light-atom interfaces - background
The operator Ê+/− is the positive/negative-frequency part of the field and contains only the annihilation/creation operators. The electric field amplitude is Eq =
p
~ω/2ε0 V defined in the quantization volume V , where ε0 is the permittivity of
free space. The complex mode amplitude of the classical description is replaced by
the annihilation operator âq of the polarization mode q.
2.2.2. Stokes operators
For the experiments on dispersive measurements of the atomic spin polarization
it is practical to describe the light in terms of its polarization properties through
the Stokes operators. For this, it is convenient to use a traveling-wave description
discussed in [33], and a wave-packet mode of duration τ of a paraxial beam with area
A [34], which defines the quantization volume V ≡ Aτ . We find
Ŝx =
~ †
â σ x â
2
Ŝy =
~ †
â σ y â
2
Ŝz =
~ †
â σ z â .
2
(2.15)
Here, â ≡ [â+ , â− ]T and â+ , â− are annihilation operators for left and right circular
polarization, respectively and σ x , σ y , and σ z are the Pauli matrices. The Stokes
operators obey angular momentum commutation relations: [Ŝx , Ŝy ] = i~Ŝz and
cyclic permutations. Bold indices are used for the components of the Stokes operators
to distinguish them from Cartesian coordinates. The expectation value of the Stokes
operators, and therefore polarized light, can be conveniently visualized as a point
on a sphere, called Poincarè sphere, which recalls the Bloch sphere for spin 1/2
particles.
We will usually work with highly-polarized states, in which one component dominates and for NL 1 we can substitute the operator Ŝx by its expectation value
Sx ≡ h Ŝx i. Thereafter, we will denote expectation values of operators as the same
symbol without hat. Only the orthogonal components Ŝy and Ŝz contain quantum
features and have to be treated as operator quantities. For example, with horizontally polarized laser light:
1
h Ŝx i = ~NL ≡ Sx
2
and
h Ŝy i , h Ŝz i = 0
(2.16)
where NL is the average number of photons. The variances are
1
var(Ŝx ) = var(Ŝy ) = var(Ŝz ) = ~2 NL .
4
(2.17)
The commutation relations are effectively [ Ŝy , Ŝz ] = i~Sx and [ Ŝx , Ŝy/z ] = 0.
2.3. Light-atom interaction
We briefly recall the dipole approximation and the resulting Hamiltonian as a starting
point of the discussion about light-atom interactions. More detailed treatments
can be found in many standard textbooks on quantum optics [30, 35]. Then we
2.3. Light-atom interaction
11
elaborate in more detail on the polarizability Hamiltonian which is at the heart of
the light-atom interaction in this thesis. Thereafter, we rewrite the dipole interaction
Hamiltonian in a semi-classical way, which serves as the basis for the numerical
simulations we carry out along the course of the thesis. The last point summarizes the
density matrix and master-equation formalism as working horse for the theoretical
understanding in this experimental work.
2.3.1. Dipole interaction
The interaction of a single atom with a single electromagnetic field mode can be
described in the so called dipole approximation as atoms being point-like emitters
compared to the wavelength of the light [36, 30]. We consider a single mode radiation
field with fixed polarization which has a wavelength much longer than the extension
of the atom. Usually, this is a safe assumption, knowing that atoms have normally
a size in the order of a few Ångstroms (10−10 m) and the light a wavelength of a
few hundreds of nanometers2 . We follow the approach described in [35, 30, 36].
There, the light is described by the quantum field of Eq. (2.12), and the atoms and
light-atom interaction are described by the Hamiltonian
Ĥ = ĤA + Ĥint .
(2.18)
The first term in Eq. (2.18) is the Hamiltonian of the unperturbed atom which can
be written [30] as
∞
X
ĤA = ~
ωi |Ψi i hΨi | ,
(2.19)
i=1
where ~ωi are energy-eigenvalues and |Ψi i the corresponding
energy-eigenstates.
Hy 0
perfine ground |gi = |f, mf i and excited state |ei = f , mf 0 on the D2 transition
are examples of such energy-eigenstates.
The second term in Eq. (2.18) describes the dipole interaction
Ĥint = −d̂ · Ê ,
(2.20)
where dipole operator d̂ in the Schrödinger picture is defined as
d̂ ≡
q
X
dij |Ψj i hΨi | .
(2.21)
i,j=1
The dipole matrix elements dij = e hΨi | r |Ψj i give the amplitude of a transition
between |Ψi i and |Ψj i.
2
Maybe things are different if we consider Rydberg atoms and microwave radiation.
12
2. Light-atom interfaces - background
2.3.2. Polarizability Hamiltonian
Now we discuss a very important way of representing the coherent part of the lightatom-interaction. The terminology we introduce here will be the basis for many
discussion in later chapters in connection with the spin squeezing experiment.
The multilevel structure of alkali metal atoms produces a light-atom-interaction with
tensorial character. That is, light interacting with an atom in a polarized atomic
state can change its polarization state, i.e., the medium is said to be optically active
[37]. This was recognized already a long time ago in connection with optical pumping
of atomic vapors [38, 34].
As shown in [38, 39], when second-order time-dependent perturbation is applied in
the limit of small excitation [39, 34, 40, 41, 42], with ĤA as the unperturbed Hamiltonian and Ĥint as the perturbation, the second-order contribution to the energy
is
(−) (+)
Ĥint,pol = −Ei Ej αij ,
(2.22)
where αij is the atomic dynamic polarizability tensor at the frequency of the laser
light ωL :
1 X P̂g d̂P̂e d̂† P̂g
α=−
,
(2.23)
~ g,e
∆eg
where P̂g = |f, mf i hf, mf | and P̂e = f 0 , mf 0 f 0 , mf 0 are projection operators
P
onto
P the ground and excited state subspace, respectively. The summation g,e ≡
f,f 0 ,m,m0 goes over all magnetic substates of both ground and excited states. The
detuning of the light from the atomic resonance is ∆eg = ωL − ωeg . The Eq. (2.22)
has the same form as the expression for the potential energy of a small polarizable
particle in an electric field oscillating or static [37].
The above expression (2.22) describes a light mediated connection of two states in
the ground state manifold via a state in the excited state. From a physical point of
view, the interaction will cause a change of the field through a spin-state dependent
refractive index and of the atoms through a polarization dependent light shift. To
see this effect more clearly it is practical to decompose the polarizability tensor into
its irreducible components
α̂ ≡ α̂(0) ⊕ α̂(1) ⊕ α̂(2) ,
(2.24)
where ⊕ is the tensor sum. After some considerable amount of math, we arrive at a
Hamiltonian that splits into different parts
Ĥint,pol = K̂0 Ŝ0 + K̂1 Ŝx + K̂2 Ŝy + K̂3 Ŝz .
(2.25)
where the operator K̂k are atomic observables [34].
K̂0 = g
2 (0)
α Îf + α(2)
3
NA
X
i
fˆz(i)
1
− f (f + 1)Îf
3
!!
,
(2.26)
2.3. Light-atom interaction
13
N
K̂1 ≡ gα(2)
A 1X
fˆx(i) fˆx(i) − fˆy(i) fˆy(i) ,
2
N
K̂2 ≡ gα(2)
i
A 1X
fˆx(i) fˆy(i) + fˆy(i) fˆx(i) ,
2
i
(2.27)
N
K̂3 = gα(1)
A
1X
fˆz(i) ,
2
(2.28)
i
α(k)
where
are the polarizability components [39, 40], Îf the identity on a space of
dimension 2f + 1, and g = ωL /(20 V ) is a constant which contains the interaction
volume V . Having casted the interaction into the form of Eq. (2.25) we can appreciate
the symmetry between atoms acting on the light and the light acting on the atomic
spin.
The individual terms of Eq. (2.25) have the following interpretation: The term proportional to S0 is the well known AC Stark shift. It is independent of the internal
atomic state and reveals no information about the atomic spin state nor introduces
spin rotations. Hence, we can neglect it for further considerations. The other terms,
from the perspective of the atoms, represent spin rotations proportional to the polarization components Ŝx , Ŝy , and Ŝz . For the quantum non-demolition measurement
P A ˆ(i)
fz is very important. It will prothe last term which is proportional to F̂z ≡ N
i
duce a polarization rotation of the light proportional to the spin polarization along
z. Likewise, the terms ∝ Ŝx and ∝ Ŝy can be used to measure the observables
P A ˆ(i) ˆ(i)
P A ˆ(i) ˆ(i)
(i) (i)
(i) (i)
fx fx − fˆy fˆy and F̂x F̂y + F̂y F̂x ≡ N
fx fy + fˆy fˆx . The
F̂x2 − F̂y2 ≡ N
i
i
final form of the interaction Hamiltonian is
Ĥpol =
i
~1
~1 h 2
G1 Ŝz F̂z +
G2 Ŝx F̂x − F̂y2 + Ŝy F̂x F̂y + F̂y F̂x
,
τ2
τ2
(2.29)
where G1 and G2 are light-atom coupling constants proportional to the vector α(1)
and tensor α(2) polarizability components, respectively. They can be interpreted as
the per-atom rotation of the incoming light polarization.
The coupling constants for atoms in F = 1 on the D2 line are
1 Γλ2
4
5
5
G1 (∆, A) ≡
−
−
+
,
A 16π
δ0 (∆) δ1 (∆) δ2 (∆)
1 Γλ2
4
5
1
G2 (∆, A) ≡
−
+
,
A 16π δ0 (∆) δ1 (∆) δ2 (∆)
(2.30)
(2.31)
where δf 0 = ∆ − ∆0f 0 for each hyperfine excited state f 0 , where ∆0f 0 is the frequency
difference between the lowest excited hyperfine level, i.e., f 0 = 0, and the level with
f 0 . The detuning from the resonance F = 1 → F 0 = 0 is ∆, where ∆ < 0 is referred
to as red detuning. We characterize the interaction of light and atoms by an effective
interaction area A. It is the transverse area over which we could spread atoms and
light homogenously and achieve the same interaction strength, i.e., Gi , as we actually
measure. The excited state linewidth is Γ.
14
2. Light-atom interfaces - background
|G1|, |G2|
ï6
10
ï4
10
ï8
10
ï6
10
ï1
ï0.5
ï0.05
6 [GHz]
ï8
10
ï10
10
ï2
ï1.5
ï1
ï0.5
6 [GHz]
0.5
1
Figure 2.2.: Coupling constants for vector and tensor light shift G1 (solid line) and
G2 (dashed line) for the interesting region of detunings ∆ from transition F = 1 →
F 0 = 0. The inset shows the characteristic scaling with detuning for G1 ∝ ∆−1 and
G2 ∝ ∆−2 . For the interaction area we chose A = 4 × 10−9 m2 , which is a realistic
value as we will see further below in Chap. 4.
The polarizability components α(1/2) and therefore the coupling constants G1/2 show
different scaling with detuning from resonance ∆, as we can see in Fig. 2.2. It is
possible to find spectral positions where one or the other vanishes, or where they
are equal. This spectral feature can be used to design the Hamiltonian at wish
for different applications. In reference [40] we discuss a number of proposals for
applications in quantum cloning, quantum memory and atom number measurement.
2.3.3. Master equation
Often it is important to model dissipative processes like absorption or spontaneous
emission or describe a system which is in a statistical mixture. The most appropriate
description is the density operator formalism [30] which can be used to develop a
master equation [43].
The density operator is
ρ̂ =
X
ρij |Ψi i hΨj |
(2.32)
i,j
and ρij is the density matrix [20]. The evolution without dissipation under Ĥ is
2.3. Light-atom interaction
15
described by the von Neumann equation
i
∂ ρ̂
i h
= − Ĥ, ρ̂
∂t
~
(2.33)
which is equivalent to the Schrödinger equation.
If we allow for Markovian dissipative processes, as for instance spontaneous emission,
the system described by ρ̂ is no longer closed. It is coupled to a reservoir, which,
for spontaneous emission, are a number of electromagnetic field modes. The master
equation can be used to describe its dynamics
i
∂ ρ̂
i h
(2.34)
= − Ĥ, ρ̂ + Ldiss (ρ̂) ,
∂t
~
where Ldiss (ρ̂) is a super-operator acting on ρ̂. Under the imposed Markovian approximation, the relaxation operator Ldiss is local in time, i.e., ∂∂tρ̂ only depends on
ρ̂ at the same time. All possible dissipative processes can be included in Ldiss . We
can write
i
1 Xh †
†
†
Ldiss (ρ̂) = −
Ĉm Ĉm ρ̂ + ρ̂Ĉm
Ĉm − 2Ĉm ρ̂Ĉm
,
(2.35)
2 m
with Ĉm being the Lindblad operators for different dissipative processes acting on
the system. The operator describing, for example, spontaneous emission has the
form
√
Cse = Γdrel, ge |Ψg i hΨe |
(2.36)
where Γ is the excited state decay rate and |Ψg i and |Ψe i are hyperfine ground and
excited states, respectively, and drel, ge is the branching ratio or relative oscillator
strength3 of the hyperfine transition |Ψe i → |Ψg i. We give the expression for drel, ge
in Appendix B.3.
The master equation (2.34) can be rewritten as a linear differential equation
∂ ρ̂
= Ltot (ρ̂) ,
∂t
(2.37)
where we can define another super-operator, sometimes called Liouvillian or Liouville
operator
Ltot (ρ̂) = −
i 1 Xh
i
i h
†
†
†
Ĥ, ρ̂ −
Ĉm
Ĉm ρ̂ + ρ̂Ĉm
Ĉm − 2Ĉm ρ̂Ĉm
.
~
2 m
(2.38)
It contains both the coherent and incoherent part of the evolution. Rewriting the
density matrix as a vector of density matrix elements, e.g., ρ
~ = (ρ11 , ..., ρqq ) and Ltot
as a matrix L, the master equation is
∂~
ρ
= L~
ρ,
∂t
where L has dimension q 2 × q 2 for a q-dimensional system.
3
In a simple two level atom this would be one.
(2.39)
16
2. Light-atom interfaces - background
In the case of a time-independent Hamiltonian Ĥ, the solution of Eq. (2.39) is a
simple exponential function of L
ρ
~(t) = eLt ρ
~(0) .
(2.40)
Hence, once the matrix L is determined, we can calculate the solution of the master
equation at any time from the initial state ρ
~(0). Throughout the thesis we use this
method to calculate the time behavior of the atomic state in processes like optical
puming atoms into a coherent spin state (cf. Sect. 3.4) or answer the question of
the de-polarization in dispersive spin measurements (cf. Sect. 4.4).
2.4. Spin squeezing
Spin squeezing is at the heart of this thesis and we give here a short review of
important spin squeezing criteria which are used to quantify the amount of noise
reduction and the implied multipartite entanglement.
2.4.1. General remarks
The concept of squeezing - the redistribution of quantum fluctuations between conjugate observable - was conceived when people started to think about the limitations for
an interferometric measurement with light [44, 45, 46] as for example in the detection
of gravitational waves. Caves [46] at the beginning of the 1980’s showed that, if the
secondary input port of a conventional interferometer, for example a Mach-Zehnder
interferometer, is fed with squeezed vacuum state, the random vacuum fluctuations
could be suppressed.
Short after, it was shown theoretically by Yurke et al. [47] that an interferometer, or
better any device having two input and output ports, i.e., a two mode device, can be
represented by the SU(2) group. This insight, seems to have inspired to ’think the
other way around’ and formulate the concept of spin squeezed states by Kitagawa
and Ueda [48] and at the same time by D. Wineland et al. [49]. That is, the two
modes in the optical case can be identified as the two ground states of a spin 1/2
half system.
We note, a decade later, the group around J.P. Dowling introduced the formal equivalence between a Mach-Zehnder interferometer, the Ramsey spectroscopy, and a
generic quantum logic circuit and called it the ’quantum rosetta stone”. They have
shown that there exist a close connection between interferometry and quantum computation [50]. Here, we are describing the correspondence between the Mach-Zehnder
interferometer and the Ramsey spectroscope containing N spin one-half particles.
This will facilitate the understanding of spins squeezing in general and the conditional spin squeezing we are going to generate.
In a classical interferometer as shown in Fig. 2.3 a) the input mode is coherently split
into the modes u and d by a first beam-splitter. This corresponds to the coherent rotation of the spin state |⇑i ≡ |↑i⊗N by π/2 about an axis lying in the equatorial plane.
2.4. Spin squeezing
17
Figure 2.3.: a) Correspondence between Mach-Zehnder interferometer and a spin onehalf system, here demonstrated in a Ramsey sequence. The single mode input of the
interferometer corresponds to a spin in one of the eigenstate, here|↑i. The first beamsplitter prepares a coherent superposition between path u and d, which is equivalent
to a coherent rotation of the spin state into a state in the equatorial plane. The lower
mode d experiences a phase shift ϕ which is a rotation in the spin picture about the
z axis. Finally, the two modes are over lapped on the second beams-splitter. At the
two output ports we receive a measurable signal proportional to the introduced phase
shift. In the spin picture we measure the population difference between |↑i and |↓i. b)
Phase sensitivity can be improved when the two modes u and d are entangled. In this
case the spin state is squeezed.
If we chose the axis to be along y, the state is rotated into the a coherent superposition
1
(|↑i + |↓i)⊗N . Both modes continue independently and one experiences a phase
2N/2
⊗N
1
|↑i + eiϕ |↓i
. The phase
shift ϕ which changes the superposition state into 2N/2
shift in the interferometer is equivalent to a rotation of the coherent spin state about
the z axis. The phase difference is read out by overlapping both modes on a second
beam splitter which results in a measurable signal m ≡ (Nu − Nd ) /2 = N2 sin ϕ,
where Nu and Nd are the population of the modesP
u and d, respectively. This has
its correspondence in the observable MJ of Jˆz = 21 i (|↑i h↑| − |↓i h↓|) which is the
difference in the number of spins pointing up and down.
∆m
,
The error in phase estimation according to estimation theory [51], ∆ϕ = ∂m/∂ϕ
is proportional to the error in population difference ∆m scaled by the slope of the
18
2. Light-atom interfaces - background
interferometer signal ∂m/∂ϕ. For N uncorrelated particles the uncertainty in the
population difference
is ruled by counting statistics for a collection of two state
√
systems ∆m = N /2 cos ϕ. The slope of the interferometer signal is ∂m/∂ϕ =
N/2
√ cos ϕ. Hence, the phase sensitivity for a classical coherent state state is ∆ϕc =
1/ N . Although it can be explained completely in terms of classical statistics [52],
it was given the name standard quantum limit by Braginsky and Vorontsov [1].
Introducing non-classical correlations between the N particles in the interferometer
allows to surpass the SQL and achieves a much better sensitivity which scales as
∆ϕ = 1/N . There exist a number of possibilities to reach a scaling which is better than the SQL by designing an appropriate input state to the interferometer or
changing the linear beam-splitter into a non-linear one as depicted in Fig. 2.3 b). A
comprehensive overview about appropriate input states in different kind of systems
can be found in [50]. The ’non-linear beam-splitter approach was proposed already
in the very beginning by Kitagawa and Ueda [48]. They showed theoretically that a
non-linear interaction of the form Jˆz2 will produce a spin squeezed state, where the
system evolves unitarily from a coherent into a squeezed spin state. This interaction
Hamiltonian is sometimes called “one-axis” twisting, referring to the transformation
of the quantum uncertainty from a circle into an ellipse. A third way of creating the
necessary non-classical correlations are measurement based strategies using quantum
non-demolition measurements. This technique which is applied in our experiment is
explained in more detail in Sect. 2.5.
2.4.2. Spin squeezing criteria
The direct measurement of quantum correlations, i.e., on a microscopic level, is
difficult. The non-classical correlations between a particle and any other of the N −1
particles in the system is small for collective states with large N . Fortunately, it is
sufficient to measure a finite set of macroscopic quantities to determine the degree
of squeezing and entanglement.
Metrological spin squeezing
The most stringent criterion, due to Wineland et al. [49] related the question of
squeezing to the improvement in measurement sensitivity. Any N -particle spin state
that provides a smaller phase estimation error than a coherent spin state of same
particle number is squeezed. The phase estimation error can be expressed in first
and second order moments of spin variables as
q
var(Jˆzs )
∆ϕs ≡
,
h Jˆxs i
(2.41)
where var(Jˆzs ) is the variance of the squeezed state and h Jˆxs iits expectation value.
Compared to the best classical case var(Jˆzc ) = h Jˆxc i /2 of a coherent spin state, the
2.5. Quantum non-demolition measurements
19
phase measurement is then improved by
ξm
q
N var(Jˆzs )
∆ϕs
≡
=
.
∆ϕc
h Jˆxs i
(2.42)
Note, most of the experiments that demonstrated spin squeezing will state the squeez2 which is readily computed from the variances var(Jˆs ).
ing in terms of ξm
z
c
ˆ
Later in the experimental section, we will measure var(Jz ) of the input state to verify
that we have a coherent spin state. This will serve as a reference level for the squeezed
state var(Jˆzs ). We will also describe how a QND measurement can reduce var(Jˆz )
from an initial, coherent state value var(Jˆzc ) to a reduced value var(Jˆzs ) < var(Jˆzc ).
In practice, the QND measurement will in addition reduce h Jˆx i from the coherent
(in)
(out)
(in)
state value h Jˆx i = NA /2 to h Jˆz
i = (1 − η) h Jˆx i (in the experiment, η will
be related to the fraction of atoms that suffer a spontaneous light-scattering event).
The operation results in a squeezed state, i.e ξm < 1 if
var(Jˆz )s < (1 − η)2 var(Jˆz )c
(2.43)
The above criterion implies non-classical correlations which was proven in reference [53]. The value of ξm can be strictly related to the depth of entanglement,
i.e., the minimum number of particles forming multi-particle entangled states in the
system. For decreasing ξm the number of particles involved in the entanglement is
increasing [54].
Kitagawa-Ueda-criterion
A weaker criterion than the above-mentioned was introduced by Kitagawa and Ueda
[48]. They define spin squeezing as the reduction of spin uncertainty below the
coherent state level
q
var(Jˆzs )
ξe = 2
(2.44)
h Jˆx i
or more closely related to our experiments
var(Jˆzs ) < (1 − η) var(Jˆzc ) .
(2.45)
Also this criterion implies entanglement.
2.5. Quantum non-demolition measurements
This section has the aim to review important aspects of quantum non-demolition
(QND) measurements. After a historical excursion we formally describe criteria for
a QND measurement. In Sect. 2.5.2 we explain the QND measurement strategy for
atomic spins via dispersive probing. The next section applies this idea to the case
of 87 Rb. At the end, we give the connection between spin squeezing and QND measurement and define the signal-to-noise ratio as central figure of merit in dispersive
QND measurements.
20
2. Light-atom interfaces - background
2.5.1. Concept
Interestingly the initial idea for quantum non-demolition (QND) measurements was
developed in the context of gravitational wave detection, i.e., the same field which
initiated the idea of squeezing, we discussed above. The first introducing the concept
was Bragisnky in the mid 1970’s [1]. The main idea is to generate an indirect measurement which gives information about a ’system’ using a ’meter’ without changing
the system variable. An ideal QND measurement can be viewed a realization of a
von-Neumann measurement. That is, the system is projected via the measurement
into one of its eigenstates corresponding to the measured eigenvalue. Any subsequent
measurement of the same variable will give the same results reflecting the fact that
the system, once projected, remains in the eigenstate [55, 56].
The QND measurement is an indirect way of gathering information about an observable  of the system S coupling it to a meter M. Before interaction, the combined
M + S system is in a separable state |ψS i ⊗ |ψM i, where |ψS i and |ψM i are the
respective eigenstates of S and M before interaction. The interaction changes the
combined quantum state from a separable into an entangled one
|ψS i ⊗ |ψM i → |ψSM i
(2.46)
At this stage, information about S is carried by M and vice versa. At the instant
when M is detected information about  is acquired which collapses the wavefunction of S according to the measurement outcome a.
There exist a necessary and sufficient condition [55] which ensures the QND character
of the measurement the observable Â
h
i
Â, Û |ψM i = 0 ,
(2.47)
where Û is the operator of the joint evolution of S and M. This condition is very
general and in many situations hard to verify and for many experimental situations
it can be replaced by a simpler one. Often the system Hamiltonian of S + M can be
written as a sum
Ĥ = ĤS + ĤM + Ĥint ,
(2.48)
where HS and HM are the Hamiltonian for the system and the meter and Hint
describes the interaction of both. With Eq. (2.48) two conditions about the QND
character of the variable  and the interaction Ĥint can be formulated [44]. The
observable  is called a QND variable if it is unchanged by the free evolution of the
system S.
h
i
ĤS , Â = 0
(2.49)
That means, Â is a constant of the free evolution of S. This condition ensures that
any excess noise which is introduced into the conjugate variable is not fed back into
Â. The interaction between M and S is called QND interaction if it does not perturb
the measured observable Â
h
i
Ĥint , Â = 0
(2.50)
We have a QND measurement, if both conditions (2.49) and (2.50) are fulfilled
simultaneously [57].
2.5. Quantum non-demolition measurements
21
2.5.2. Atomic quantum non-demolition measurements via dispersive probing
Dispersive atom-light interaction can be used to acquire information about atomic
properties in a non-destructive way. The concept relies on off-resonant interactions of
light with atomic quantum systems. In this, the population of certain atomic states
can be read out relative to each other by comparing the phase shifts acquired through
the dispersive interaction. This idea was introduced by Takahashi et al. [58] and
Kuzmich et al. [59] and assumes that we can couple two orthogonal electromagnetic
field modes u and d to two distinct atomic states |↑i and |↓i and their respective
excited states. As in the discussion about spin squeezing, the correspondence between
a two-mode oscillator and the elements of the SU(2) group is used here.
The properties of the proposal by Kuzmich et al. can be discussed without specifying
the details of the actual physical implementation of |↑i and |↓i. In an abstract
way, we can associate a pseudo-spin operator to the two spin states, which has the
components
ĵx =
~ †
b σxb
2
ĵy =
~ †
b σy b
2
ĵz =
~ †
b σz b ,
2
(2.51)
h
iT
where b ≡ b̂↑ , b̂↓ are bosonic operators with |↑ / ↓i = b̂†↑/↓ |0i. A collective spin
operator for an ensemble of NA atoms is defined as
Ĵ =
NA
X
ĵ(k) ,
k=1
where ĵ(k) is the spin operator for the kth atom. We are interested in the population
difference given by Jˆz . For all relevant cases we can assume that the population of
the hyperfine ground state is constant when the system evolves freely. Therefore, we
can call Jˆz a QND variable.
Also, the two electromagnetic field modes, represented by their photon annihilation
operators âu and âd , correspond to a spin angular momentum via the Schwinger
representation [60] of a two-mode oscillator
Ŝx =
~ †
a σxa
2
Ŝy =
~ †
a σy a
2
Ŝz =
~ †
a σz a ,
2
(2.52)
where a ≡ [âu , âd ]T . Each e.m. field mode âu/d interacts dispersively with one of the
spin states |↑ / ↓i which is described by a Hamiltonian of the form
Ĥ ∝ n̂u
NA
X
k=1
|↑ik h↑|k − n̂d
NA
X
|↓ik h↓|k = Ŝz Jˆz ,
(2.53)
k=1
where n̂u/d are the number operators for the modes u/d. The second criterion for a
QND measurement (2.50) is fulfilled for the observable Jˆz . Thereafter, we will call
any Hamiltonian of the form (2.53) a QND Hamiltonian, because it describes the
QND interaction we are interested in.
22
2. Light-atom interfaces - background
2.5.3. Non-destructive magnetization measurements in
87
Rb
Now, we focus on actual implementation of the Takahashi-Kuzmich proposal for the
measurement of the magnetization in the hyperfine ground state for alkali metal
atoms. We will restrict the discussion to the relevant case for this thesis which
are QND measurements in the F = 1 manifold of the 5S1/2 . The extension to
higher F manifolds is theoretically straightforward but might require slightly different
experimental realizations than discussed here.
To measure the magnetization h F̂z i in the F = 1 manifold, it is sufficient to limit
the description to the ground states with mF 6= 0. An effective spin one-half system
is defined by
|↑ / ↓i ≡ |1, ±1i .
This is physically reasonable, since atoms in |1, 0i are not contributing any magnetization along the quantization axis z, i.e., h1, 0| F̂z |1, 0i = 0. The set of pseudo-spin
operators for this case is
1
Jˆx =
2
i
Jˆy =
2
1
Jˆz =
2
NA
X
i=1
NA
X
i=1
NA
X
[|↓i h↑| + |↑i h↓|]i
(2.54)
[|↓i h↑| − |↑i h↓|]i
(2.55)
[|↑i h↑| − |↓i h↓|]i .
(2.56)
i=1
Linearly polarized off-resonant light propagating along z serves as a meter for the spin
magnetization Jˆz = F̂z /2. The left and right circular components of the linear probe
light play the role of the two orthogonal electromagnetic field modes we introduced
above. By identifying âu = â+ and âd = â− , we recognize that the abstract Schwinger
operators (2.52) are in this particular case the Stokes operators (2.15).
The off-resonant interaction of polarized light with alkali metal atoms was discussed
further above in Sect. 2.3.2. We found a very compact description in terms of irreducible tensor components (2.29) which can be restated here using the pseudo-spin
operators (2.54-2.56)
~
~
Ĥpol = G1 Ŝz Jˆz + G2 Ŝx Jˆx + Ŝy Jˆy .
(2.57)
τ
τ
If we neglect the tensor term for a moment and set G2 = 0, the Hamiltonian describes
a perfect QND interaction for the measurement of Jˆz . In Chapter 7, we return to
the full Hamiltonian and discuss the limitation of the assumption of vanishing tensor
polarizability. There, we propose and demonstrate a method to recover the pure
QND interaction ∝ Ŝz Jˆz , even in the case of G2 6= 0.
In the Heisenberg picture we can study the evolution of the spin operators for light
and atoms. For a first qualitative statement it is sufficient to study the evolution
2.5. Quantum non-demolition measurements
23
of the operators to lowest order in Ĥpol . This gives simple input-output relations
between operators before (in) and after (out) interaction. A more detailed model is
presented in Appendix A, which allows to study the exact evolution. For infinititsimal changes the operator after interaction can be written as
τ h (in) i
Ô(out) = Ô(in) +
Ô , Ĥ .
(2.58)
i~
After QND interaction Hpol light and atom operators are
 
  (in) 
(out)
(in)
Ŝx
Ŝx
1
−G1 Jˆz
0
 (out)  
  (in) 
(in)
 Ŝy
 =  G1 Jˆz
1
0   Ŝy 
(out)
(in)
0
0
1
Ŝz
Ŝz
(2.59)
  (in) 
 
(out)
(in)
Jˆx
Jˆx
1
−G1 Ŝz
0
  ˆ(in) 
 ˆ(out)  
(in)
 =  G1 Ŝz
 Jy
1
0   Jy 
(in)
(out)
ˆ
0
0
1
Jˆz
Jz
(2.60)


In the above representation the symmetry between light and atoms becomes apparent. Both systems contain information about the other part. If the input states are
pure, the interaction will entangle atoms and light. As requested, the z components
for both systems remain unchanged by the interaction, i.e., they are QND variables.
Information about the magnetization is transferred onto linear components of the
probe light, i.e, Ŝx and Ŝy .
2.5.4. Squeezing via quantum non-demolition measurements
We explain the role of QND measurement for the generation of spin squeezing and
introduce the notion of conditional variance following [61, 57]. The experimental
proposal of Kuzmich for our case is explained. We relate the theoretical spin squeezing criteria introduced above to experimental parameters we actually measure in the
experiment.
There exist a close connection between QND measurement and squeezing. An ideal
QND measurement, i.e., a projective measurement in the sense of von Neumann
20, will produce a post-measurement state which is known through the recorded
measurement outcome M1 on the meter system M. A second measurement M2 on
the same quantum system will be perfectly correlated with the first one, M2 = M1 .
Also in the realistic case of imperfect QND measurements, i.e., if the state is not
projected with certainty into an eigenstate, a second measurement will show some
degree of correlation with the first one.
More formally, the amount of correlation can be parametrized by a correlation coefficient, which can be measured directly and has the definition [57]
κ≡
cov(M1 , M2 )
,
var(M )
(2.61)
24
2. Light-atom interfaces - background
Figure 2.4.: Metrology scheme for conditional spin squeezed spin states. We show the
part of the interferometer after and before the initial and final beam-splitter, respectively. The coherent spin state is measured in a first QND measurements which gives
a result M1 . This state is now exposed to a field which introduces a rotation Ry (ϕ). A
second QND measurement gives a result M2 .
with cov(M1 , M2 ) = h M1 M2 i − h M1 i h M2 i, where M1/2 are the results of a first and
second measurement of the meter on the system S [61, 57] and we have assumed that
var(M ) ≡ var(M1 ) = var(M2 ). In the case of an ideal QND measurement we have
κ = 1 and less for imperfect QND measurements. If the first QND measurement is
omitted, i.e., M1 is unknown and no prior information about the system is recorded,
we have κ = 0.
Using the language of an interferometer as introduced in Sect. 2.4.1 the relation
between squeezing and QND measurement becomes apparent. We concentrate on the
part of the interferometer where the actual phase measurement takes place (cf. 2.3)
which is shown in Fig. 2.4. The coherent input state is probed in a first QND
measurement which converts it into a squeezed state with certain offset in Jˆz direction
∝ M1 . The system is now subjectto a field
which introduces a phase shift ϕ, which
ˆ
is in Fig. 2.4 a rotation R̂y = exp −iϕJy about y. The second QND measurement
will give a result M2 . Combining the first and second QND measurement outcomes,
the phase shift is determined better than the intrinsic quantum uncertainty of a
coherent spin state would allow.
The amount of metrological enhancement in the sense of Wineland et al. (2.42) can
be calculated directly from the statistics of the two measurement outcomes for ϕ ∼ 0
via the conditional variance var(M2 − κM1 )
2
ξm
=
var(M2 − κM1 )
= 1 − κ2 .
var(M2 )
(2.62)
We define a conditionally squeezed state as the one which shows experimentally
var(M2 − κM1 ) < var(M ) ,
(2.63)
where we have to assume that the measured variances contain mostly quantum fluctuations4 .
4
It is also important that var(M1/2 ) is governed mostly by quantum noise. Classical correlations
2.5. Quantum non-demolition measurements
25
Experimental proposal
For the experiments described later, we use linearly polarized light pulses with average number of photons NL that have the following characteristics before interaction
iE N
Dh
h
i N
L
L
Ŝx , Ŝy , Ŝz =
, 0, 0
var Ŝx , Ŝy , Ŝz =
[1, 1, 1] . (2.64)
2
4
The ensemble of NA atoms will be polarized in a coherent superposition state (|↑i + |↓i)⊗NA
which represents a pseudo-spin state polarized along x with
i N
iE N
h
Dh
A
A
ˆ
ˆ
ˆ
Jx , Jy , Jz =
, 0, 0
var Jˆx , Jˆy , Jˆz =
[0, 1, 1] . (2.65)
2
4
Note the difference between in Eq. (2.64) and (2.65) in the fluctuation along the
polarized direction for light and atoms. This is because the number of atoms in the
sample is a fixed quantity, which in principle could be measured precisely [28, 62, 29].
In experiments, of course, the number of atoms is also fluctuating due to technical
noise through the trap loading process. In contrast the photon number is in general
governed by quantum fluctuations.
Via the interaction, information about Jˆz is transferred to Ŝy , i.e., Ŝy is the meter for
Jˆz , which is measured many times on re-prepared atomic states, but otherwise under
the same conditions. If we compute the variance of the measurement outcomes, we
find
(out)
(in)
(in)
var(M1 ) ≡ var(Ŝy ) = var(Ŝy ) + G21 Sx2 var(Jˆz ) ,
(2.66)
where for large photon numbers we can assume Sx2 var(Ŝx ). The above equation
can be used to quantify the sensitivity of the QND measurement of Jˆz . The first
term is the shot noise of the probe light, which places an ultimate limit on the QND
measurement sensitivity. The second term contains the atomic quantum information
which is amplified by the product of the coupling constant and the number of photons.
The different scaling of the first and second term with photon number allows us to
“amplify” the quantum signal to an amount comparable or larger than the limiting
(in)
light shot noise var(Ŝy ).
We define a ratio of atomic and light quantum noise and call it the signal-to-noise
ratio of the QND measurement.
ς≡
(in)
G21 Sx2 var(Jˆz )
(in)
var(Ŝy )
= G21 NL
NA
,
4
(2.67)
where the second equality holds if all light and atomic input states are quantum
(in)
(in)
noise limited, i.e., var(Ŝy ) = N4L and var(Jˆz ) = N4A .
The state after the first QND measurement, conditioned on the measurement outcome M1 , will show a reduced quantum fluctuations as derived in [63]
(in)
var(Jˆz )
(out)
var(Jˆz |M1 ) =
,
1+ς
(2.68)
can easily lead to a large correlation coefficient and on the same time to large var(M1/2 ). This
issue will be discussed in the measurement of atomic quantum noise in Chapter 7.
26
2. Light-atom interfaces - background
which can be derived also by classical multivariate Gaussian statistics [64].
For the final judgment on the success of the spin squeezing generation via QND
measurements we have to introduce the conditional spin variance (2.71) into one of
the squeezing criteria (2.43) or (2.45) and we find spin squeezing in the respective
definitions by Wineland et al. or Kitagawa et al. iff
(out)
(in)
|M1 ) < (1 − η)2 var(Jˆz )
(2.69)
(in)
|M1 ) < (1 − η)var(Jˆz ) .
(2.70)
var(Jˆz
or
(out)
var(Jˆz
Experimentally, we can verify the spin squeezing in the first QND measurement
by performing a second QND measurement. This is in complete analogy to the
interferometric scheme we introduced above (cf. Fig. 2.4), except that we don’t
allow for a rotation of the state, i.e., ϕ = 0. More concretely, we perform a double
pulse experiment, i.e., two QND measurements, on a coherent spin state polarized
along h Jˆx i = NA /2, where the first and the second probe pulse interact in the same
way with the atomic sample. The experiment is repeated many times under the
same experimental conditions and we gather a list of measurement values {M1 , M2 }.
The conditional variance and the fluctuation of the input state, necessary for the
spin squeezing criteria (2.69) and (2.70), is calculated form the set measurement
outcomes by
(out)
var(Jˆz
|M1 ) =
(a)
(a)
(e)
var M2 − κM1
− var M2
G21 NL
and
(a)
(e)
var M1
− var M1
(in)
var(Jˆz ) =
G21 NL
,
,
(2.71)
(2.72)
where we indicate the presence of atoms with (a) and an empty trap with (e). The
number of photons NL is measured by photo-detection, the number of atoms NA by
absorption imaging, the coupling constant G1 by observing average rotation for a
known number of atoms in a polarized state, and the depolarization is characterized
by measuring the decay of the polarization as a function of probe photons. The
polarization of the atomic state after QND measurement is
(out)
Jx
(in)
= Jx
exp(1 − η) ,
(2.73)
where we can model the depolarization by a single parameter η [41, 63, 65]. It
describes the probability that an atom absorbs a photon out of the pulse of NL
incoming photons. In the limit of small destruction, we can write
η=
σ(∆)
NL ,
A
(2.74)
where σ(∆) is the off-resonant scattering cross section, A the effective interaction
area between light and atoms and NL the number of probe photons. The exact
2.6. Simulation of collective quantum properties
27
expression for σ(∆) is given in Appendix B.1. Both, σ(∆) and SNR are proportional
to NL ∆−2 . Thus, the ratio between them is constant and there exist no preferred
detuning for the probe laser.
2.6. Simulation of collective quantum properties
A technique which has proven to be very useful in the simulation of the evolution
of the quantum state during QND measurements is the covariance matrix approach
[66, 67, 68, 65, 69, 70, 71]. Much of the intuition gained during the work on spin
squeezing came through simulation of the system using this technique. Even so
this thesis is based on experiments, we were able to contribute some insight on
the influence of spatial and temporal inhomogenieties on the degree of spin squeezig.
The results are published in: “Unified description of inhomogeneities, dissipation and
transport in quantum light-atom interfaces” by M. K. and Morgan W. Mitchell in J.
Phys. B 42, 195502 (2009) [15]. In Appendix A we present the covariance matrix
technique and discuss the extension to incorporate inhomogenieties in a natural way.
We give examples which illuminate the basic questions as for instance the influence
of the detector time resolution or thermal motion to the degree of spin squeezing.
3. Experimental apparatus and techniques
Contents
3.1. Atomic trap . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.1.1. Trap overview . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.2. Cooling and trapping lasers . . . . . . . . . . . . . . . . . . 33
3.1.3. Preparation of cold atomic ensembles
. . . . . . . . . . . . 35
3.2. Atom number measurement . . . . . . . . . . . . . . . . .
39
3.2.1. Fluorescence methods . . . . . . . . . . . . . . . . . . . . . 39
3.2.2. Absorption imaging . . . . . . . . . . . . . . . . . . . . . . 40
3.3. Dipole trap characterization . . . . . . . . . . . . . . . . .
3.3.1. Trap depth and sample shape
47
. . . . . . . . . . . . . . . . 47
3.3.2. Loss through light scattering . . . . . . . . . . . . . . . . . 49
3.3.3. Trap lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.4. Atom temperature . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.5. Axial shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4. Spin state preparation . . . . . . . . . . . . . . . . . . . .
58
3.4.1. Optical pumping into dark states . . . . . . . . . . . . . . 58
3.4.2. Loss of spin polarization . . . . . . . . . . . . . . . . . . . 61
3.5. Probing atomic spins . . . . . . . . . . . . . . . . . . . . .
64
3.5.1. Polarization control and photon-number referencing . . . . 65
3.5.2. Frequency shifting and pulse generation . . . . . . . . . . . 66
3.5.3. Frequency Offset-lock . . . . . . . . . . . . . . . . . . . . . 67
3.5.4. Experimental details on probe lasers . . . . . . . . . . . . . 72
3.6. Shot-noise-limited polarization detection . . . . . . . . .
74
3.6.1. Polarimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6.2. Linear birefringence and optical losses in detection . . . . . 76
3.6.3. Operation Principle of balanced detector . . . . . . . . . . . 78
3.6.4. Detector Calibration . . . . . . . . . . . . . . . . . . . . . . 79
3.6.5. Noise of the detection system . . . . . . . . . . . . . . . . . 82
30
3. Experimental apparatus and techniques
This chapter summarizes the essential parts of the experimental apparatus and explains the important techniques. The cooling and trapping setup was described
several times in previous doctoral theses [72, 73, 74]. For this reason, we concentrate
on extensions and improvements of the existing setup which are important steps
towards a well controlled atom-light interface.
A variety of laser sources are the main tool to trap, cool, and manipulate external
and internal atomic properties, as for instance temperature and spin polarization.
In Fig. 3.1 we give a functional diagram of all involved laser light sources and their
interplay. This will facilitate the understanding of particular experimental details in
later chapters. Details of particular parts of the laser systems are explained in detail
in according chapters.
The central part is the atomic trap which we describe in Sect. 3.1. In the functional
overview we divided this into its three parts 2D-MOT, 3D-MOT and dipole trap.
Single lasers are often subdivided into different tasks which are marked as rounded
rectangles close to the laser. Manipulations of laser light in frequency and intensity
F=2 laser
Yb:YAG laser
F=2 depletion
cooling
seeding
referencing
imaging
F=1 laser
repumping
opt. pumping
transfer
referencing
tensor probe
vector-probe laser
h polarized
v polarized
F=2 slave
cooler
push beam
2D MOT
crossed beam
3D MOT
mode matching
single beam
FORT
Figure 3.1.: Functional diagram of laser sources for experiment. Dark red boxes are
lasers tuned near the D2 line of 87 Rb and the purple box is a far-detuned laser for optical
atom trapping. The lasers are subdivided into functions marked in white rounded
rectangles. Free space and fiber transfer of laser light is marked in dotted and solid
lines, respectively. For details see text.
3.1. Atomic trap
31
is done by means of acousto-optic modulators (AOM). The F=1 and F=2 lasers are
referenced to atomic transitions in 87 Rb.
All laser light is transferred to the atom trap apparatus by polarization-maintaining
(PM) single mode optical fibers. This ensures a controlled delivery of laser radiation
and stable working conditions on a daily basis. In addition it makes the trap itself
movable. In the future, experiments with a non-classical light source are planned. For
this purpose the trap and the non-classical light source would have to be combined
on a separate table.
3.1. Atomic trap
In the last 30 years, trapping and cooling of neutral atoms has become a standard
technique in atom and molecular physics. The first atom beam slowing experiments
were demonstrated in the 1980’s, followed by neutral atom traps in 1985 using an
optical molasses [75] and a purely magnetic trap [76]. The first cooling and trapping
in a magneto optical trap was reported 1987 AT&T Bell labs [77]. Only ten years
later the three pioneering groups of Steven Chu (Stanford), William D. Phillips
(NIST, Gaithersburg) and Claude Cohen-Tannoudji (Paris) were awarded with the
Nobel Prize in physics.
We describe the laser system for the atom trapping apparatus and the preparation
of cold atomic ensembles. Full details can be found in references [72, 73, 74]. We
assume a familiarity with basic concepts of trapping and cooling. Detailed explanations of cooling and trapping atoms can be found in several textbooks and review
publications, for example [78, 17, 79].
3.1.1. Trap overview
We use a double stage magneto-optical trap (MOT) setup depicted in Fig. 3.2.
Atoms are accumulated in a two-dimensional MOT which continuously feeds a threedimensional MOT. A single-beam optical dipole trap is overlapped with the 3D MOT.
After loading, the gradient fields and trapping beams of the 2D and 3D MOT are
turned off, leaving the atoms in an all-optical trap.
The two stage atom loading has several advantages. Using two MOT systems, separated by a differential pumping stage allows us to have a high pressure system for
fast loading and a low pressure system for long lifetimes. The 2D MOT is used to
load atoms from a background pressure of 10−8 mbar into an elongated cloud which is
only cooled and trapped in two dimensions allowing atoms to move freely in the third
dimension. This elongated cloud of atoms is constantly illuminated by a ’push-beam’
along the untrapped direction from top to bottom in order to assist the transfer of
atoms into the 3D MOT. In this way, we are able to transfer ∼ 1 × 107 atoms per
second. In the 3D MOT we have very pure vacuum of < 10−11 mbar background
pressure. We use a standard three dimensional MOT arrangement, which allows us
32
3. Experimental apparatus and techniques
Figure 3.2.: Magneto-optical and all-optical trap scheme: The upper 2D MOT work
at a background gas pressure of 10−8 mbar. Cooling and trapping beams (red arrows)
collect atoms into an elongated sample which is constantly transferred into the lower
chamber. The chambers are separated by a differential pumping stage which allows
a much smaller background pressure of only 10−11 mbar for the lower one. There,
atoms are collected into a 3D MOT formed by six counter-propagating laser beams. A
gently focused laser beam, the far off-resonant trap (FORT), is superimposed with the
atomic cloud of the 3D MOT. After extinguishing the quadrupole magnetic field and
the cooling lasers, atoms are held in the FORT for further experiments.
3.1. Atomic trap
33
to cool the atoms below the Doppler limit to about 25 µK. A schematic drawing of
the combined MOT setup is shown in Fig. 3.2.
For further manipulation we extinguish the MOT light beams and the quadrupole
magnetic field. Trapping atoms only by means of a focused laser beam offers a way
to store them in a state independent way. This is in strong contrast to magnetooptical trapping where atoms change constantly their internal state while absorbing
and re-emitting photons.
3.1.2. Cooling and trapping lasers
We summarize the most important features and parameters of the laser setup for
magneto-optical and optical trapping.
MOT lasers
To cool and trap alkali metal atoms in a magneto optical trap two laser fields are
needed. The first, often called cooler, is tuned slightly below the cycling transition
in F = 2 → F 0 = 3. It is responsible for the trapping force. The second, called
repumper, transfers atoms once they have escaped from the cycling transition back
into F = 2. This laser is tuned close to the resonance F = 1 → F 0 = 2.
The cooling light for the 2D and 3D MOT is supplied by two separate laser diodes.
This provides us with sufficient optical power for fast loading. The F=2 laser,
which feeds the 3D MOT, and all derived beams is schematically shown in 3.3. It
is a commercial ECDL1 that is stabilized to an atomic transition in 87 Rb by means
of saturated absorption spectroscopy (cf. Fig. 3.3 a)). We blue shift the laser by
60 MHz before it enters into the spectroscopy setup (cf. Fig. 3.3 b)) to have a large
range for the cooler detuning of many natural linewidths. In this way we effectively
lock the laser closer to F 0 = 2 and not as typically done at the crossover transition
between F 0 = 2 and F 0 = 3 2 . The large detuning range is important for subDoppler cooling and increased the number of atoms in the dipole trap by factor of
two compared to previous experiments [74].
The 2D MOT is supplied by the F=2 slave laser. This is a Fabry-Perot diode laser
which is injection locked to the F=2 laser. For injection we tune the light 10 MHz
to the red of the transition F = 2 → F 0 = 3. This light is a small fraction of the
light used for spectroscopy of the F=2 laser. It is shifted in a double-pass AOM
configuration by 2 × 62 MHz. Reliable injection is achieved at a power of 1.5 mW.
The output of the slave is split in two parts. One, the majority of the light, is used
1
2
The used model is DL100 from Toptica AG, Germany
At the crossover transition the signal is larger compared to the direct transitions. This is readily
explained by the fact that for a cross-over transition not the zero velocity class is selected but
two non-zero velocity classes. Atoms which co- or counter-propagate relative to the pump beam
with a velocity which bridges half of the hyperfine excited state splitting by means of Doppler
shift will absorb a photon on one transition and appear saturated for the other.
34
3. Experimental apparatus and techniques
Figure 3.3.: Schematic layout of F=2 laser. a) is the spectroscopy setup to perform
saturated absorption spectroscopy. b) to e) are acousto optical modulator setups to
shift the laser frequency. Details are given in the text. The output beams are labeled
from 1 to 5.
3.1. Atomic trap
35
for the 2D MOT beams. The other part is used for the vertically push-beam (cf.
Fig. 3.2)
We use a portion of the F=1 laser to provide the repumper light for both magnetooptical traps. The laser is a home built ECDL which is directly locked to the transition from F = 1 to the cross over transition between F 0 = 1 and F 0 = 2 . The light
is shifted by a single pass AOM to the resonance F = 1 → F 0 = 2.
Dipole trap laser
The laser for the far off-resonant optical trap is a diode pumped Yb:YAG thin disk
laser3 . It is capable of producing 20 W of continuous wave power in a single longitudinal mode. The center wavelength is 1030 nm at a linewidth of < 5 MHz. Single-mode
operation is provided by an intracavity etalon. We monitor the single mode operation constantly using a scanning Fabry-Perot cavity. The wavelength is adjustable
via an intracavity Lyot filter in a range between 1000 and 1060 nm. Both the etalon
and the Lyot filter are adjusted from time to time to ensure single mode operation at
1030 nm. The laser itself has proven to perform best in terms of single mode behavior
and long term power stability when optimized at high powers. Over long measurement times of more than 12 hours the relative change in wavelength is < 3 × 10−4
and in power < 4 × 10−2 .
As earlier stated, all light that arrives at the trapping setup is fiber coupled into
single mode fibers. Since normal single mode fibers can not handle tens of Watts of
optical power easily, we are using a large mode area photonic crystal fiber4 . It has a
solid core of 25 µm diameter. The single mode light propagation is enabled through
the photonic crystal structure surrounding the solid core in a hexagonal arrangement.
The photonic band-gap supports only a single spatial mode over a large wavelength
range. The intensity at the surface is reduced because the focus is translated into the
fiber-material. Nevertheless, greatest care has to be taken to avoid power damages
on the input facet. We included a mode cleaning spatial filter consisting of a pair of
lenses and a pinhole of 50 µm diameter in the focus. This spatial filter rejects all but
a single TEM00 mode. We achieve coupling efficiencies of over 60 %. At the location
of the MOT we have a maximum power of 6.0 W.
The laser power can be switched or changed continuously with an AOM. This allows
us to switch the trapping light in a computer controlled way.
3.1.3. Preparation of cold atomic ensembles
We load the optical trap in a standardized scheme which is the same for all experiments in this work. The procedure can be divided into three main steps, which
are illustrated in Fig. 3.4. First we produce cold atoms in a MOT. Second, we cool
atoms below the Doppler limit to increase the number of atoms in the trap. Third,
3
The company producing the product was originally called ELS. It has since been purchased by
another laser company called SLT.
4
The model is LMA-25 from Crystal Fibre (now part of NKT Photonics).
36
3. Experimental apparatus and techniques
Figure 3.4.: Dipole trap loading cycle. We show the three most important parameters
for the trap loading. The MOT loading phase can vary between a few hundreds of ms
to 4 s. During the thermalization period of 400 ms untrapped atoms will escape from
the trapping region and the atoms thermalize in the trap.
we allow untrapped atoms to escape from the trapping region by waiting a period of
400 ms before we start to use the laser cooled atoms.
The light of the dipole trap is present from the beginning. In our experience this
increases the number of atoms in the trap. Atoms which are cold enough already
in the MOT loading phase will stay in the dipole trap potential. There, the light
scattering from the MOT beams is reduced due to the AC Stark shift induced by
the trapping light. In this way we continuously start loading sufficiently cold atoms
from the MOT cloud. After an atomic cloud of ∼ 4 × 107 is produced in about
4 s, the sub-Doppler cooling phase starts. To reduce the temperatures below the
Doppler cooling temperature (TD = 144 µK for 87 Rb), we slowly ramp down the
magnetic field gradient and the start to reduce the light intensity of the cooling and
repumper light. At the same time, the detuning of the cooler light is increased.
The process which allows sub-Doppler temperatures is called polarization gradient
cooling. In brief, the main cooling mechanism is the differential scattering of light
from two counter-propagating laser beams. The differential scattering is caused by
the imbalance of the ground state population which results from its non-adiabatic
following to the local light polarization [80, 78]. The final phase of 400 ms is used to
allow untrapped atoms to escape from the trapping region and a thermalization of
the trapped atoms along the weakly confined direction with 4 Hz trapping frequency.
Several parameters are important in the loading process. First, we have to guarantee
enough light power over the whole detuning range of the cooling laser. In Fig. 3.5
we plot the measured total power into the MOT beams as a function of the cooler
detuning (set by the RF frequency on the double-pass AOM, cf. Fig. 3.29). By
tilting this AOM in the double-pass configuration (cf. Fig. 3.29) we can shift the
position of the maximum slightly and equalize the power which is used for cooling.
3.1. Atomic trap
37
Second, the amount of repumper light in the last 112 ms of the sub-Doppler cooling
phase has to be adjusted very carefully. Small differences can change the number of
atoms in the dipole trap by a factor of two. In this last phase we have ∼ 4.8 mW/m2
injected through all six MOT beams. Third, the position of the atomic cloud from
the MOT is fine-adjusted by correcting the compensation coil currents. We see a
strong dependence in x and y direction due to the small transverse dimension of the
dipole trap.
60
P [mW]
50
40
30
20
Detuning Range
10
0
ï100
ï80
ï60
ï40 6
cooler
[MHz] 0
Figure 3.5.: Measured power into the
3D MOT apparatus as a function of
detuning from the F = 2 → F 0 = 3
transition. For a trapping-beam waist
of ∼ 6 mm and a coupling efficiency of
75 the intensity in the trap center is
∼ 9 W/m2 per 1 mW. We indicate the
range over which the laser is detuned
during the sub-Doppler phase.
The sub-Doppler phase is adjusted for the best dipole trap loading. However, no
detailed study was carried out since we are able to load more than > 106 atoms in
about 2 s into the trap by adjusting the three major parameters described above.
The evolution of the number of atoms in the dipole trap is measured as a function of
the MOT loading time. In Fig. 3.6 we number of atoms measured with absorption
imaging, which we explain in Sect. 3.2. The data are fitted by the solution of simple
rate equation model for the transfer of atoms from the MOT into the dipole trap
NA (t) = NA,∞ (1 − exp (−kt)) ,
(3.1)
where NA,∞ is the final number of atoms in the dipole trap and k is the trap loading
rate. The gives NA,∞ = 1.21(5) × 106 and k = 0.85(9) s−1 .
38
3. Experimental apparatus and techniques
Figure 3.6.: Evolution of atom number NA in dipole trap as a function of
MOT loading time. The time constant
is k = 0.85 s−1
3.2. Atom number measurement
39
3.2. Atom number measurement
For the characterization of light-atom interactions it is important to have, in addition
to shot noise limited photon detectors, also a reliable measurement for atom number.
There are various ways to measure the number of atoms present in an ensemble. The
most primitive, used in atomic vapor cell experiments, is to use ideal gas theory and
calculate the number of atoms knowing temperature and pressure. This works well
for homogeneous distribution. For atoms in a potential of a focused laser beam,
things are a bit more difficult. The solution we prefer is absorption imaging. The
necessary considerations and the experimental setup are explained in Sect. 3.2.2.
Before, we present in Sect. 3.2.1 a brief summary of fluorescence techniques which
were used in the experiment to measure temperature (cf. Sect. 3.3.4) and trap
lifetime (cf. Sect. 3.3.3).
Figure 3.7.: Different atom number counting strategies. a) Fluorescence detection with
calibrated photo-diode or CCD camera. The amount of collected fluorescence photons
is determined by the numerical aperture of the imaging lenses L1 and L3, respectively.
b) The absorption image is taken with linearly polarized light. The atomic absorption
casts a shadow which is detected by the CCD camera.
3.2.1. Fluorescence methods
The principle of atom-detection with fluorescence is very simple. It relies on the
measurement of spontaneously emitted photons from previously excited atoms. It
is possible, at least in principle, to get a one-to-one correspondence between the
40
3. Experimental apparatus and techniques
number of detected photons and the number of atoms present. To find the relation
between atoms and photon-number, we have to know the following parameters:
1. The absorption cross section, σ(∆), for the atomic transition we want to
probe. For this, it is important to chose a transition which is closed. That
means, the atom cycles between one ground and one excited state. In many
situations this requirement is not fulfilled because of the complex structure in
multilevel atoms.
2. The detected solid angle which determines the part of the full solid we have
excess to. This can be estimated from the imaging optics used. A slight modification has to be taken into account if the emitted radiation has an anisotropic
radiation pattern. This is the case if we would observe, for instance, the emission of π-polarization under a small angle to the quantization axis.
3. The most critical parameter is the light intensity at the position of the atoms.
Since the amount of emitted light is directly related to the number of incident
photons we have to rely on a calibrated illumination. Light fluctuations would
be transferred immediately to the recorded signal.
For quantitative atom number measurements, all above parameters have to be characterized with good accuracy. For the first two, this poses no major obstacle. However,
the stability in light intensity is the major problem in quantitative atom detection
with fluorescence. There are experimental situations were fluorescence imaging is
the only and, at the same time, a very good method. The shelving technique [81]
in ion detection is a perfect tool which allows a high-fidelity state detection with
fluorescence imaging. All measurement of a small number of atoms can be carried
out with fluorescence techniques. There, atoms are detected by discrete changes in
the detected light. This allows simple calibration and a larger tolerance to intensity
fluctuations. [82, 83, 84, 85].
We use two methods of detecting atoms with fluorescence as depicted in Fig. 3.7 a).
One uses a calibrated photodiode which records the light collected by a lens close to
the atomic cloud. The other images the fluorescence light onto a CCD camera.
3.2.2. Absorption imaging
We use absorption imaging (AI) as the primary method of measuring the atom
number. The technique is widely used in the field of ultracold atomic physics and
there exist several in-depth treatments of the physical implementation and the data
analysis [86, 87]. Here we summarize the important concepts.
Principle
Absorption imaging could be seen as a spatially-resolved transmission measurement
of the atomic cloud. Resonant laser light is sent through an atomic cloud. Atoms
scatter photons out of the beam so that the transmission is reduced as sketched in
3.2. Atom number measurement
41
Fig. 3.7 b). The relation between incoming and outgoing light in the limit of no
saturation is given by Lambert-Beer’s law
I (out) (x, z) = I (in) (x, z)e−d0 (x,z)
(3.2)
The optical density, d0 , is directly related to the absorption cross section introduced
in App. B.1 and the atomic density distribution
ZL
d0 (x, z) = σ(∆)
n(x, y, z)dy = σ(∆)nc (x, z) ,
(3.3)
0
where we simplified the expression by using the column density nc . Measuring the
transmission as the ratio between output and input intensity reveals the column
density distribution of the atoms.
One advantage, compared to fluorescence detection, is that measuring transmission
in the low saturation regime is independent of the intensity used. Therefore, slight
intensity fluctuations can be neglected.
Realization
In the ideal situation it seems sufficient to take a single picture and calculate the
transmission. For high quality absorption images, however, it is necessary to acquire
three pictures. A shadow image Iatoms which contains the atomic cloud, a reference
image Ilight without atoms, but under the same conditions as the first, and a dark
image Idark with neither imaging beam nor atoms. There are several reasons for
that. First, calculating the transmission as a ratio greatly reduces the sensitivity to
intensity fluctuations. Second, the imaging light, I (in) (x, y), is not a smooth plane
wave or Gaussian distribution but is rather distorted, e.g., by diffraction effects from
dust particles etc. (cf. Fig. 3.8 b)). That is, we can not extrapolate the light
intensity from the values of I (in) in the vicinity of the atomic cloud. Third, the bias
image gives a calibration of the dark currents in the CCD array and accounts for
stray light which is not coming from the probe. It is important to take the three
pictures as close as possible in time to ensure the same conditions for all of them. In
Fig. 3.8 a), b) and c) we show the three images. The transmission and optical depth
are than calculated as
Tmeas = e−d0,meas =
Iatoms − Idark
.
Ilight − Idark
(3.4)
The result is shown in the right most image in Fig. 3.8.
The are two main systematic errors in absorption imaging. One is the saturation of
the atomic absorption by power broadening. However, to avoid this, we use imaging
intensities two orders of magnitude below saturation intensity.
The second, and more important systematic effect, is a limited observable optical
depth. Probe light that does not interact with the atomic cloud, but reaches the
42
3. Experimental apparatus and techniques
Figure 3.8.: a)-c) Shadow, Reference and Dark image. The dark image is scaled by a
factor of 100. Red color is maximum intensity. d) retrieved transmission image. Black
color is small transmission.
CCD, will reduce the apparent optical depth. Two reasons can be found. First,
light which is off-resonant will not experience any absorption. Sometimes, laser
diodes have a broad pedestal which is not part of the main frequency mode. This
light is produced by amplified spontaneous emission and observed as an incoherent
background. It has a very small power density (power per frequency window) of
. 10−4 compared to the main frequency mode, but a large spectrum over the whole
gain profile of the laser diode (up to 50 nm). Second, probe light, resonant or not,
which is scattered indirectly into the CCD and carries no information about the
atomic cloud.
We can find a correction for the finite observable optical density d0,max by applying
a simple model. The intensity of the light reaching the CCD chip is a sum of
light which experienced atomic absorption and a fraction ε which reached the CCD
without interaction. Both are proportional to the incident light intensity Ilight
Iatoms = (1 − ε)Ilight e−d0,real + Ilight ε .
(3.5)
We measure a transmission of
Tmeas =
Iatoms
= (1 − ε)e−d0,real + ε
Ilight
(3.6)
In the limit of large real optical density, d0,real → ∞, we would observe a remaining
minimum transmission of
Tmin = e−d0,max = ε
(3.7)
The real optical depth can be retrieved from the measured transmission via
1 − e−d0,max
.
d0,real = ln
Tmeas − e−d0,max
(3.8)
Practically, the maximum observable optical depth is measured by using a dense
cloud of atoms with a known density distribution.
Implementation
In the present experimental apparatus we implemented absorption imaging as shown
in Fig. 3.9. Owing to the large on-axis optical depth, we record absorption images
3.2. Atom number measurement
43
Figure 3.9.: a) Schematic of absorption imaging of an elongated atom cloud. b) Resonant light fields for absorption imaging.
only in the transverse direction. In this direction the maximum optical depth is
approximately one, and straightforward to measure.
For imaging we use linearly polarized light, resonant with the cycling transition
F = 2 → F 0 = 3. Atoms can be probed many times before they are lost from the
F = 2 manifold5 . The choice of linear light is not optimal. The polarization is defined
by the optical pumping light, which shares the same optical path. We describe the
optical pumping in Sect. 3.4. Compared to circular light, where atoms are driven in
a closed transition, we observe a smaller scattering cross section than the maximum
λ2
on the closed transition of 3 2π
. To determine the exact scattering cross section, we
have to use a model which includes optical pumping. It would not be sufficient to
calculate the scattering cross section assuming equal population in F = 2. During the
time the imaging light is on, atoms will be pumped into a steady state distribution
among the hyperfine ground states in F = 2. In Fig. 3.10 we plot the results of a
master-equation simulation. The left panel shows the evolution of the population in
F = 2. In the right panel we compare the number of scattered photons. The black is
the result of the ME and the red is calculated via the scattering cross section for this
λ2
' 1.356×10−13 m2 . The later would
transition assuming equal population, σ0 = 75 2π
overestimate the number of photons via Eq. (3.3). We find σ0 ' 1.587 × 10−13 m2
(Isat = 30.5 W/m2 ) which is ∼ 17 % larger than the former.
5
The probability to make a transition from F = 2 → F = 1 due to one scattered photon on the
imaging transition is < 1.4 × 10−5 . This number can be deduced from the calculations shown
in Fig. 3.10 knowing that 30 scattered photons cause 4 × 10−4 atoms to end up in F = 1.
44
3. Experimental apparatus and techniques
0.5
Populations
35
mF=0
Scattered Photons
30
ME calculations
Theory weak excitation
0.4
25
0.3
mF=ï1,+1
20
15
0.2
10
0.1
mF=ï2,+2
0
0
25
50
75
t [µs]
5
0
0
25
50
75
t [µs]
Figure 3.10.: Left: Population evolution during the imaging pulse. We plotted only
the population of the five F = 2 magnetic substates, because the total population in
F = 1 is only ∼ 4 × 10−4 . Right: Number of scattered photons as a function of time.
The black solid line is the result of the master-equation simulations and the red dashed
one the theoretical result for equal distribution of population in F = 2.
Geometrically, the imaging beam is a large Gaussian beam of ∼ 5 mm waist. It
is expanded in the horizontal direction by a one-axis telescope consisting of two
cylindrical lenses to ∼ 8 mm. In this way, the whole sample is covered with light.
The intensity drops from the center to the edges from ∼ 0.43 W/m2 to ∼ 0.14 W/m2 .
Any power broadening can be neglected since the intensity is only a few percent of
the saturation intensity over the whole sample.
The timing of the imaging light is as follows. For each picture we shoot a rectangular
pulse of constant power and a duration of 300 µs. The separation between the pictures
is 80 ms. This time is only limited by the frame rate of the camera. Before the
shadow image is taken we start to pump atoms from F = 1 to F = 2 by applying
the repumper laser for 100 µs. It is also kept on during the shadow image.
Characterization
Here we present the two main results for the characterization of the absorption
imaging. i.e., the maximum measurable optical depth and linewidth of the imaging
laser. Both are important to guarantee the quantitative aspect of the absorption
imaging.
Maximum OD
We produce a dense cloud of atoms by loading the 3D MOT for 5 s. Then we take
the absorption images as explained above. The raw data for a single experiment
3.2. Atom number measurement
45
2D Fit
Data
Residual
Figure 3.11.: Left: Transmission image of dense atomic cloud. Center: 2D fit using a
rotated Gaussian, where the rotation angle was a free parameter. Right: Residual of
the fit.
and the deduced transmission image are plotted in Fig. 3.8. This measurement is
repeated 20 times to acquire statistics. The results are shown in Fig. 3.11.
The atomic cloud has an optical depth which is larger than d0,max and, therefore
we see the saturation of the transmission signal. We deduce d0,max by assuming a
Gaussian density distribution of the atomic cloud
2 −2B(x−x
d0,real (x, y) = d0,real (0)e−A(x−x0 )
2
0 )(y−y0 )−C(y−y0 )
,
(3.9)
with
A =
cos2 θ sin2 θ
+
2σx2
2σy2
(3.10)
B =
sin(2θ) sin(2θ)
−
4σy2
4σx2
(3.11)
C =
sin2 θ cos2 θ
+
2σx2
2σy2
(3.12)
and express the observed transmission as
Tmeas (x, y) = e−d0,real (x,y) 1 − e−d0,max + e−d0,max .
(3.13)
For the Gaussian density distribution we assume a very general two-dimensional
distribution with different width in two orthogonal directions σx and σy which can
be rotated by an angle θ. The center plot in Fig. 3.11 shows the result of the twodimensional fitting. We find a maximum optical depth of d0,max = 3.10(4). This
value will be used throughout the rest of the thesis to correct absorption images. If
the imaging setup is changed, the calibration of d0,max has to be repeated.
Linewidth measurement
For the second essential characterization, we measure the linewidth and absolute
frequency of the imaging laser. The knowledge about the absolute frequency is
46
3. Experimental apparatus and techniques
1.5
Optical Depth
1
0.5
0
−10
−5
0
5
Detuning from F=2 −> F‘=3 [MHz]
10
Figure 3.12.: Linewidth measurement with atomic cloud. The real
optical depth d0,real is plotted and
we deduce a linewidth of Γmeas =
6.18(85) MHz and a peak position
at ∆0 = 0.24(25) MHz.
crucial for quantitative atom number measurement. Another important aspect is
dispersive effects in the absorption imaging. If the imaging laser is not exactly on
resonance, the atomic sample would act as lens6 . This can lead to image distortions,
which lead to systematic errors [88]. The linewidth is interesting as a measure on its
own. It will give important insight into the quality of the laser lock used.
For the experiment we load the 3D MOT as before, but stop after 2s. This ensures
a modest optical depth of ∼ 1, well below d0,max . The detuning of the imaging
beam is changed between −6 to 6 MHz around the estimated resonance and the
d0,real is determined by the three-image procedure as described above. The results
are plotted in Fig. 3.12. The linewidth and center position are deduced by fitting a
power broadened Lorentzian line to it. The saturation parameter s is deduced from
the local intensity which is measured directly with the reference images Ilight . The
average saturation parameter over the whole atomic ensemble is I/Isat = 0.014.
The measured line width is ∼ 110 kHz larger than the natural line width. We take
this amount as the linewidth of the imaging laser. The center-position is shifted
240 kHz to higher frequencies compared to the resonance-position we deduced form
absorption spectroscopy. All absorption imaging measurements reported in what
follows were made at this adjusted line center.
6
The atomic cloud effectively acts as a gradient-index lens.
3.3. Dipole trap characterization
47
3.3. Dipole trap characterization
We use a single strong far off-resonant laser which is gently focused to trap pre-cooled
atoms state-independently in a small spacial region. The concept of the AC Stark or
light shift can be found in many quantum optics textbooks [30, 36]. Here, we collect
a set of parameters which are important for later interpretation and understanding
of other experimental results.
3.3.1. Trap depth and sample shape
We use the simplest realization for a dipole trap, a gently focused laser beam. In the
case that the laser is red detuned and has a Gaussian profile, atoms are attracted to
the point of largest intensity. We will only discuss this case here, whereas other configurations of blue detuned light could be imagined using Laguerre-Gaussian beams
to trap atoms at an intensity minimum. We furthermore restrict the discussion to
linearly polarized light, for which all Zeeman substates stay degenerate. However,
later in Sect. 3.4.2, we will discuss the influence of circularly polarized trapping light
as a decoherence mechanism for spin polarized atoms.
Dipole potential
The shape of the dipole potential follows the intensity distribution, I(r), of the
focused Gaussian beam
2
exp − w2ρ
2 (z)
I(r) = I(ρ, z) = I0
(3.14)
2 .
1 + zzR
Where, ρ and z are transverse and longitudinal coordinates which both have their
origin in the focal point. The Rayleigh
range zR is a measure of the length of the
r
2
focus7 . The function w(z) = w0 1 + zzR
describes the waist size at position z
starting at the minimum waist w0 . The peak intensity for a Gaussian beam with
2P
given power, P , is I0 = πw
2 .
0
We write the potential as the product of the light shift at peak intensity and the
spatial characteristics of the Gaussian beam
2
exp − w2ρ
2 (z)
U (ρ, z) = ς(λ)I(ρ, z) = ς(λ)I0
(3.15)
2 .
z
1 + zR
In Appendix B.2 we have calculated ς(λ) for a large spectrum of frequencies. Here,
we are only interested at the value for λ = 1030 nm, which is ς = −3.4573 ×
7
More precisely, it measures the distance at which the intensity on axis drops to one half of the
intensity at the focus.
48
3. Experimental apparatus and techniques
10−3 Hzm2 /W. The dipole trap laser has a minimum waist in the focus of w0 =
52 µm and we usually use P = 6 W, which results in a maximum potential depth
U0 /h = U (0, 0)/h = −4.9 MHz. Expressed in thermal energy this corresponds to
trap depth of U0 /kB = 234 µK.
Sample shape and harmonic approximation
The shape of the sample, or better, the distribution of atoms in space, can be deduced
from the trap potential. Under the assumption of a thermalized atom cloud, we can
write the number density using the Boltzmann factor
!
Ũ (ρ, z)
n(ρ, z) = n0 exp −
,
(3.16)
kB T
e (ρ, z) = U (ρ, z) + |U0 | is the shifted trap potential which ensures zero powhere U
tential energy in the focus of the dipole beam. Equation (3.16) has two unknown
parameters, the peak density, n0 , and the temperature, T , of the atoms. If the atoms
have a temperature much smaller than the potential depth, i.e., kB T |U0 |, we can
e as a harmonic potential. Usually this is a good approximation for
approximate U
atomic samples with similar extension in all direction, as for instance in a magnetic
trap. However, great care has to be taken when the sample is very elongated and
trapped in a gently focused laser beam, as in our case. In axial direction, the shape
of the atomic sample is Lorentzian rather than Gaussian. The axial shape of the
sample is measured in Sect. 3.3.5. Nevertheless, for the radial direction we can make
the approximation of a harmonic potential and find
e2D harm. (ρ, z) = U0
U
2ρ2
w(z)2
1+
+
z
zR
z2
2
zR
2
.
(3.17)
It is interesting to compare the full and the approximated potential and the resulting
number density distribution. In Fig. 3.13 we plot the equipotential line with the
thermal energy for both models in blue. The resulting density distributions, drawn
as black lines, are rather similar. In the experiments of Sect. 3.3.4 we observe a
Gaussian density distribution in transverse direction.
Atomic Waist
It is useful to introduce a parameter ζ which relates the thermal energy of the atoms
to the potential depth. In literature this is also referred to as the filling factor .
ζ≡
kB T
.
U0
(3.18)
3.3. Dipole trap characterization
49
1.2
1.2
Non−trapping Region
Non−trapping Region
1
Trapping Region
ρ [w0]
0.8
0.6
1
0.8
Trapping Region
0.01
0.6
0.01
0.4
0.1
0.2
0.5
1
1.5
2
2.5
0
0.01
0
0
0.2
0.
5
0.01
5
0.
0.2
0.4
0.1
0.
2
0.5
1
1.5
2
0
2.5
z [zR]
z [zR]
Figure 3.13.: We show in black the normalized atomic density and in blue the equipotential line for the trapping potential when it equals the thermal energy. The left plot is
for the full potential from the Gaussian beam (3.15) and the right for the 2D harmonic
approximation (3.17).
On one hand, we have a comparison of energy scales, which is always handy. On
the other hand, we can use ζ to verify the approximation of a harmonic potential.
For the verification we introduce ζ into the expression for the number density (3.16),
under the harmonic approximation (3.17), and at position z = 0:
1 2ρ2
.
(3.19)
n(ρ, 0) = n0 exp −
ζ w02
For atomic sample we find now a characteristic extension, which is similar to the
waist of a Gaussian beam. In the future we will call this the atomic waist 8 given by
p
wa ≡ ζw0 .
(3.20)
3.3.2. Loss through light scattering
In order to estimate the influence of photon scattering on the trap lifetime, we
calculate the scattering rate and from this the heating rate for our trap parameters.
The scattering rate for a two level system is calculated as (see for example [17])
Rsc, 2 level =
1 2
Γ
2Ω
,
1
2
2 ∆ + 2 Ω2 + 14 Γ2
(3.21)
where Ω is the Rabi frequency, Γ the excited state line width, and ∆ = ωL − ωge the
detuning from resonance. The laser frequency is ωL and the resonance of the atom
two level atom is ωge . In the limit of large detuning, ∆ Ω, Γ, the denominator can
8
The relation to the full width at half maximum (FWHM) is given by wFWHM =
1.18wa .
√
2 ln 2wa '
50
3. Experimental apparatus and techniques
be approximated as ∆2 . For very large detunings, as we find them in our dipole trap,
the RWA is no longer applicable and we have to modify the detuning. Therefore, we
use expression (B.8) introduced in Appendix B.2. In addition, a correction for the
DC limit, i.e., ωL → 0, has to be introduced, which ensures that the scattering rate
approaches zero. Similar to the argument in Appendix B.2, the classical Lorentz
oscillator can be applied and gives a prefactor of (ωL /ωge )3 . For the multilevel case
we sum over all excited states
X ωL 3 Γf Ω2if
Rsc =
,
˜2
ωif
4∆
f 6=i
if
˜ if is defined in Eq. (B.8).
where i and f label the initial and final states, and ∆
For the Yb:YAG laser we use in the experiment, the scattering rate as a function of
intensity is Rsc = I × 2π × 1.14 × 10−10 s−1 m2 /W, where I is the intensity of the
laser.
The atomic sample is heated due to absorption and re-emission of photons from the
trap laser. Absorption occurs in the propagation direction of the laser and is therefore
anisotropic. Emission happens isotropically if we average over all polarizations.
If
we sum up, the sample is heated in longitudinal direction ~ ωrec + 31 ωrec and in the
two transverse directions each with 31 ~ωrec , where ~ωrec = ~2π × 3.6325 kHz is the
energy of one recoil, or expressed in temperature units 348.66 nK.. In total, each
scattered photon leaves a mean energy of 2~ωrec to the atom. Hence, we can express
the heating rate in term of the scattering rate by
Rheat = 2~ωrec Rsc .
For our trap parameters we get a heating rate of 112 nK/s at the center of the trap.
This is negligible compared to the trap depth of 234 µK and implies a loss rate due
to photon scattering of 112 nKs−1 /234 µK ' 4.8 × 10−4 s−1 . This number should
be the same in the whole trap, since scattering rate and trap depth have the same
scaling with intensity.
Here we summarize some auxiliary measurements which are standard tools in any
atom trap laboratory. They are performed after major changes in the trap setup, as
for instance after changing the single mode fiber for the high power trapping laser
or optimization of the MOT-loading.
3.3.3. Trap lifetime
Measuring the lifetime of optically trapped atoms not only gives an idea how much
time the experimenter has to ask his questions, but also reveals import parameters
like the background pressure of the vacuum setup. There are different mechanism
which limit the lifetime. One can categorize them by their scaling with atom number.
Independent of particle number are collisions of trapped atoms with background gas
atoms or molecules. Equally, inelastic scattering of the trapping light is an atom-
6
Number of Recaptured Atoms [x10 ]
3.3. Dipole trap characterization
51
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
20
40
60
Trapping Time [s]
80
100
Figure 3.14.: Recaptured
number of atoms as a
function of trapping
time. We fit the data
with Eq.
(3.23) plus
an offset Noffset .
The
fit
parameters
are
N0 = 7.6(1) × 105 ,
α = 4 × 10−6 s−1 (upper
bound 3.3 × 10−4 s−1 ),
β = 7.9(6) s−1 , and
an offset of Noffset =
1.3 × 103 (±1.3 × 104 ).
number independent loss mechanism9 . Density dependent loss mechanism are twoand three-body collisions between trapped atoms. In general, we can describe the
atom number dynamics with a rate equation with different terms
dN
= −αN − βN 2 − γ N 3 .
dt
(3.22)
The coefficient α , β and γ describe the back-ground, two-body and three-body losses.
To measure the trap loss, we load atoms for about three seconds as described in 3.1.3.
After a variable trapping time t we measure the remaining number of atoms in the
trap by recapturing them into a MOT. Fluorescence measurements from MOT atoms
give a signal proportional to the number of trapped atoms, while fluorescence measurements of absolute atom number are difficult to calibrate, here we are interested
in the relative atom number. To gather enough statistics, we repeat the trap loading
several dozen times. For every other measurement, we inhibit the MOT loading and
and dipole trapping to measure the background fluorescence signal.
In Fig. 3.14 we plot the number of recaptured atoms as a function of trapping
time. We extract the loss parameters by fitting the solution of Eq. (3.22) to the
data points, neglecting three-body collisions, i.e., γ = 0. For our atom densities of
n ∼ 1011 cm−3 this is a safe assumption . The solution for arbitrary α and β is
β
1
β −1
N (t) = eαt
+
−
.
(3.23)
α N0
α
In the limit of short times or dominant density dependent losses, α → 0, this reduces
to
N0
N (t) =
.
(3.24)
1 + N0 βt
9
One can imagine that at very high atomic densities the re-absorption makes light scattering
density dependent. However, for the atomic densities and the elongated geometry, we do not
expect this to be the case.
52
3. Experimental apparatus and techniques
To out surprise, we see negligible linear losses and almost no exponential decay. The
loss of atoms from the trap can be described solely with two body collisions. The
solid line in Fig. 3.14 uses Eq. (3.23) with the parameters mentioned in the caption.
The linear loss coefficient is tiny and it would be possible to fit the data points only
with Eq. (3.24). If we compare the loss rate α = 4 × 10−6 s−1 to the estimated loss
due to inelastic photon scattering from the dipole laser of 4.8 × 10−4 s−1 , calculated
in Sect. 3.3.2, it becomes clear that it is hard to adjoin any physical meaning to the
found α. Nevertheless, we can use the upper bound of the error in the fitted value
αmax = 3.3 × 10−4 s−1 to estimate an upper limit for the back ground pressure in
the 3D MOT. At very low pressure the background gas contains mostly H2 and He.
Helium has a lower collision-rate with 87 Rb than hydrogen. Therefore, we take the
collisional rates calculated in [89] and estimate an upper bound for the back ground
gas pressure. We find that the background pressure is as low as 0.7 × 10−11 mbar.
3.3.4. Atom temperature
Another interesting parameter for us is the temperature of the dipole trapped atoms.
To measure the temperature we use the time-of-flight (TOF) technique [86]. It
relies on the very definition of temperature for a thermalized ensemble of particles,
i.e., the kinetic energy. When atoms are released from the trap, i.e., the trapping
potential is removed, they will expand ballistically depending on its initial position
and momentum at time t = 0. Theoretically, one can show that the characteristic
size of the ensemble is described by [86]
r
4kB T 2
wa (t) = wa2 (0) +
t .
(3.25)
m
This expression contains at most two free parameters: the initial extension of the
ensemble wa (0) and its temperature T . We will limit ourselves to the measurement
of the transverse extension wa of the atom cloud. In longitudinal direction the measurement is more difficult, since atoms at micro Kelvin temperatures, expand very
little compared to the sample length of several millimeters. Strictly speaking, the
measured temperature is primarily connected to the kinetic energy in transverse direction. However, by assuming that atoms were loaded from an isotropically cooled
MOT cloud and are not further cooled in the dipole trap, both transverse and longitudinal dimensions have the same kinetic energy.
To measure the extension directly, we have to image the atom cloud while it is
expanding. Usually, the measurement is done by imaging the atoms, either in fluorescence or absorption. Both of these techniques are in general destructive or have
a slow frame rate. Therefore, we have to repeat the experiment several times for
different expansion times.
For the experiment we load atoms into the dipole trap, as described in 3.1.3. We use
6 W of trapping light which produces a maximum trap depth of 234 µK. After some
thermalization time of 1.2 s we switch the trapping laser off and take a fluorescence
image after time t. For the fluorescence image, the atoms are illuminated by the
3.3. Dipole trap characterization
53
MOT beams for a short period of a few tens of microseconds. The imaging system
collects the fluorescence light onto a charge-coupled device (CCD).
Figure 3.15.: Fluorescence images of time-of-flight measurement. The atomic cloud is
released after 1.2 s of thermalization. Expansion times are indicated in the lower right
frame.
In Fig. 3.15 we plot the fluorescence images for different expansions times from 0 to
4 ms. Each frame is an average of 10 individual images. The vertical stripes, mainly
visible for longer expansion times, are an artifact from the video electronics. For
the analysis, we integrate each frame along the longitudinal direction (horizontal)
and determine the center and the characteristic extension of the Gaussian profile. In
Fig. 3.16 we plot the profiles in a semi-logarithmic plot. The distribution is clearly
Gaussian with expanding width and moving center.
Free falling cloud
The center position of the free falling cloud, can be used to check the imaging magnification by measuring the gravitational acceleration g. In the left of Figure 3.17 we
plot the center positions as a function of expansion time. We determine the gravitational acceleration g by fitting s(t) = 12 g (t − t0 )2 + y0 to the data points, where t0
Integrated Fluorescence Signal [arb. units]
54
3. Experimental apparatus and techniques
3
10
2
10
ï500
ï400
ï300
ï200
ï100
0
100
Vertical Position [µm]
200
300
400
Figure 3.16.: Integrated fluorescence profiles. Note the
transverse dimension is measured from top to bottom, i.e.,
increasing time of flight.
is the time delay for switching off the dipole trap, and y0 the exact initial position.
We find g to be 9.98(+0.39/ − 0.40) ms−2 . This is less than two percent of error to
the accepted value of 9.81 ms−2 . For the delay we find 41(+78/ − 78) µs and for the
initial position y0 = 0.15(+57/ − 58) µm. The error in g can be mainly attributed
to the calibration of the imaging system.
Expanding cloud
The expanding width of the atomic cloud contains the information about the temperature of the atoms in the trap. The characteristic extension of the atomic cloud wa
is plotted in the right of Figure 3.17 as a function of time. The data points are fitted
by expression (3.25), where we substituted t by t − t0 , to account for the uncertainty
of the trap opening. As initial waist we use the waist measured at times close to zero
wa (0) = 16.65(7) µm as a fixed parameter. For the initial delay we find 38(±27) µs,
which is very close to the value we deduced from the center-of-mass trajectory. The
temperature of the atomic sample is 25.7(6) µK 10 .
From the temperature we can calculate the filling ratio of the optical trap, which we
defined in Eq. (3.18) comparing the thermal energy kB T and the potential depth
U0 , which is calculated from the laser power and the measured waist of the trapping
laser w0 . We find that the filling ratio is ζ = 11.0(3) % of the total trap depth.
Note, we have repeated the same measurement for a shorter thermalization time of
only 0.3 s. The temperature we find is slightly smaller at 24.3(10) µK. Since both
values are similar we have not investigated the evolution of the atom temperature as
a function of trapping time. For all experiments we will perform on the atoms the
trapping times will be between 0.3 and 1.2 seconds.
10
If we use wa (0) as free parameter, the fitting gives very similar results. We find t0 = 35(+34/ −
33) µs, wa (0) = 16.3(+4.4/ − 4.5) µm, and T = 25.6(7) µK.
3.3. Dipole trap characterization
55
450
0
400
−10
350
Atomic Waist wa(t) [µm]
Vertical Position [µm]
−20
−30
−40
−50
−60
300
250
200
150
−70
100
−80
50
−90
0
1
2
3
Time of Flight [ms]
0
0
4
1
2
3
Time of flight [ms]
4
Figure 3.17.: Left: Center-of-mass position of the expanding cloud. Right: Characteristic width of the expanding cloud.
3.3.5. Axial shape
Through the imaging of the atomic cloud, either with fluorescence or absorption
imaging, we are able to deduce important parameters, as for example temperature
and transverse dimension of the atomic ensemble. The axial shape of the atomic
ensemble does not carry as much information as the transverse dimension, but it is
still interesting to have a closer look.
In Fig. 3.18 we plot the integrated fluorescence signal over the transverse dimension
ρ of the sample. The black solid line is a fit with a Lorentzian of the form
L(z) =
A
,
0 2
1 + ( z−z
za )
where A is the amplitude of the observed signal, z0 is the position of the peak density
and za is the half width at half maximum (HWHM) of the sample. We define the
length of the atomic sample
√ as La = 2za . We expect a relation between Rayleigh
range and La of La = 2 ζzR , similar to the one we found for the beam waist w0
and the atomic waist wa in Eq. (3.20).
Alternatively, we use the trapping potential in its 2D harmonic approximation (3.17)
to calculate an analytic expression of the signal from the axial sample shape, which
is shown as the orange dashed line. the imaging integrates over the two transverse
dimensions. The observed signal is proportional to the integral of the atomic density
(3.16) over ρ
Z
∞
S(z) = 2π
ρn(ρ, z)dρ .
(3.26)
0
The atomic density n(ρ, z) is calculated from Eq. (3.16) the trapping potential in
56
3. Experimental apparatus and techniques
Fluorescence [norm.]
1
0.75
0.5
0.25
0
0
2
4
6
8
z [mm]
10
Figure 3.18.: Axial shape of the trapped atomic ensemble measured with fluorescence
imaging. The black solid line is a Lorentzian fit and the dashed line a fit with the
analytic solution of the integrated atomic density.
the 2D harmonic approximation (3.17)


 1
1 z2

n2D (ρ, z) = n0 exp − 2
exp
−
2
ζ z + zR
 ζ

2ρ2

.
2 2 

z
2
w0 1 + zR
(3.27)
This leads to the measurable signal
π
S(z) = ζ w02
2
1+
z
zR
2 !2
1 z2
exp − 2
2
ζ z + zR
.
(3.28)
For the two fits in Figs. 3.18 we get the following result. On one hand, the simple
Lorentzian, shown as solid black line in Fig. 3.18,
√ gives a sample length (FWHM)
of La = 3.0(3) mm, where we would calculate 2 0.11 × 8.2 mm = 5.4 mm from the
filling ratio ζ = 0.11 and the Rayleigh range of the trapping beam (zR = 8.2 mm).
Going the opposite direction by calculating the trapping beam properties from the
0 = 4.8(1) mm for a
measured signal S(z) gives a hypothetical Rayleigh range of zR
fixed filling ratio ofζ = 0.11.
We measure the longitudinal distribution in the trap. Unlike the transverse dimension, the imaging shows a distribution which is more compact than the predicted
Boltzmann distribution. This is plausible, since the MOT which loads the trap is
smaller than the Boltzmann distribution and because the trap frequency is low. If
we would allow the sample to thermalize longer it would probably lead to a more
extended sample, which, at the same time, will have less atoms due to the trap loss.
3.3. Dipole trap characterization
57
We note that the actual longitudinal sample shape has no major consequences for
the subsequent experiments. In Chap. 4, for example, we measure an optical depth
which is among the largest for these kind of system. However, in the future it might
be interesting to explore possibilities of loading even more atoms into the trap. This
could be achieved by translating the MOT cloud along the dipole trap potential to
load atoms over a larger spatial range.
58
3. Experimental apparatus and techniques
3.4. Spin state preparation
We discuss now the possibilities of creating polarized spin states via optical pumping. A ’basis’ set of polarized spin states, with either orientation or alignment, can
be created through the proper choice of polarization, frequency and propagation direction of the light field used for optical pumping. In Sect. 3.4.1 the principle of
optical pumping is explained and the evolution of the single atom density matrix is
simulated for realistic experimental conditions. At the end we discuss polarization
errors in the dipole as a possible decoherence mechanism.
3.4.1. Optical pumping into dark states
For given F , F 0 , and given polarization, there may be states in the F manifold that
are not coupled to any state in the F 0 by the given polarization. These do not
scatter light and are called dark states. For example, for F = 1 and F 0 = 1, with
σ + polarization, by angular momentum addition rules, the state with mF = 1 would
be coupled to the state F 0 = 1, mF 0 = mF + 1 = 2. But there is no such state, so
mF = 1 is coupled to no F 0 = 1 state by σ + polarization and is thus a dark state. If
there is only one dark state for a given polarization it is possible to produce highly
polarized spin states using optical pumping which is an incoherent processes.
If we limit ourselves to the polarization basis states of σ + , σ − , and π light, we can
give necessary conditions for the existence of a dark state in the ground state |F, mF i
of a particular multilevel atom which is coupled to an excited state manifold with
F 0 . For circular polarization σ ± , mF is a dark state, if
F 0 < 1 ∓ mF .
(3.29)
In the case of linear polarization along the polarization axis, i.e., π-light , a dark
state exists, if
mF = 0 and F = F 0 .
(3.30)
The two states we are interested in are the following ones: First, the z-polarized state:
|Φi ≡ |1, ±1i, which has the maximum projection on F̂z . From criterion (3.29) we see
that |Φi is only a dark state if the laser is resonant to the transition
F = 1 → F 0 = 1.
√
Second, the x-polarized state: |Ξi ≡ {|1, −1i − |1, +1i} / 2, which is polarized
along the pseudo-spin direction Jˆx . The latter state is an example of a dark state for
linear polarization. If we define x as the quantization axis and use π light resonant
to F = 1 → F 0 = 1, the dark state is |1, 0ix , where the subscript indicates the choice
of quantization axis. Changing into the ’normal’ frame of reference, i.e., with the
quantization axis points along z, we find Ry (−π/2) |1, 0ix = |Ξi, where Ry (−π/2)
is a rotation of the system about y by an angle of −90◦ .
Experimentally, we apply pumping light on the F = 1 → F 0 = 1 transition either
along the sample axis in σ polarization or orthogonal to the sample axis in linear
polarization. At the same time we apply light resonant to the transition F = 2 →
F 0 = 2. This transfers atoms back into F = 1 which were scattered into F = 2. We
3.4. Spin state preparation
59
Figure 3.19.: a) Geometrical configuration for x and z optical pumping. The light
for pumping atoms into the state |Φi(see text) is circularly polarized and sent along
the longitudinal direction. For the state|Ξi (see text) the light comes from the side
orthogonal to the long sample axis. b) The pumping light is resonant to transition
F = 1 → F 0 = 1 and the light to deplete F = 2 is resonant to F = 2 → F 0 = 2.
call the laser F=2 depletion. It is applied through the same optical path as the beams
for the MOT. This ensures two things. First, the illumination from six different
directions will create a polarization pattern which has a sub-wavelength structure.
This is very important to avoid a pumping into a dark state of F = 2, which exist
for the case of linear polarization, as we can see in Eq. (3.30). Thermal motion or
oscillations of atoms in the trap will produce an effectively unpolarized light field.
Atoms will move randomly through the polarization pattern and experience all kinds
of polarizations in a random order. If the motion of atoms is fast compared to the
inverse of the scattering rate from the F=2 depletion laser, atoms never enter into
a dark state. Even if an atom was pumped into a dark state, it will be repumped
an instant later when it has changed place and the local polarization is different.
Second, the mechanical action of absorption and re-emission is balanced and atoms
are not moved on average.
In Fig. 3.19 we show the experimental implementation of the two pumping schemes.
The z-polarized state is generated by sending circularly polarized light along the long
sample axis. For the coherent superposition state we send linearly polarized light
from the side.
60
3. Experimental apparatus and techniques
Simulations
We perform numerical simulations of the pumping dynamics from a completely mixed
state in F = 1 into a dark state without and with the F=2 depletion laser. We
are interested in the time-scale of the pumping process for realistic experimental
parameters, the number of scattered photons and the maximum fidelity which can
be reached. The main tool for the simulations are explained in Appendix B.3. As a
figure of merit we use the mixed state fidelity which is introduced in Appendix B.3
in Eq. (B.13). The general dynamics will be similar for the states |Φi and |Ξi and
we concentrate on the case of x-polarized state |Ξi.
The parameters for the simulation are chosen close to the experimental conditions.
The intensities for pumping and F=2 depletion are Ipump = 0.44 W/m2 and Idepl. =
3.6 W/m2 which correspond to the measured values. The unpolarized depletion laser
is modeled by changing the polarization randomly between two orthogonal states,
which are in our case x and y polarization. The actual choice of the depletion
polarization in unimportant for the simulations. A small bias field of Bx = 0.5 G is
included as well.
Population
0.5
mF=±1
0.4
0.3
0.2
0.5
mF=0
0.1
Re(l) 0
0
Error
ï0.5
ï1
0.1
F=1 0+1
ï2
0.01
ï1
F=2
0.001
0
20
40
60
80 Time [µs]
0
+1
+2
ï1 0
+1 +2
ï1 0
F=2
+1 ï2
F=1
Figure 3.20.: Left-Top: Evolution of hyperfine ground state population during optical
pumping into the dark state |Ξi starting from a completely mixed state in F = 1. No
F = 2 depletion laser is applied. Left-Bottom: Error in state preparation calculated for
all eight hyperfine ground states using a mixed state fidelity [90]. Right: Final density
matrix (real part) of the hyperfine ground state. The Blue and dark red parts are the
blocks for the two hyperfine manifolds F = 1 and F = 2, respectively.
In Fig. 3.20 we plot the evolution of the populations of the eight hyperfine ground
states, the fidelity, and show the final density matrix in the case of no F = 2 depletion
laser. We see that we can reach a maximum fidelity of ∼ 80 % . The missing ∼ 20 %
reflect directly the amount of atoms which are pumped into F = 2. If we instead
apply light for the depletion of F = 2 at the same time as the optical pumping light,
the situation changes considerably. In Fig. 3.21 we can see that the error in the
state preparation is reduced to ∼ 10−3 . In the main graph of Fig. 3.22 we plot the
number of scattered photons for a single atoms. We can see that an atom reaches
3.4. Spin state preparation
61
the desired state by scattering fewer than two photons on average. That means, the
energy deposited in the atomic cloud is rather small and won’t heat atoms too much
while pumping them. As we calculated in Sect. 3.3.2 the scattering of a photon will
increase the thermal energy of an atom by two recoil energies (Erec /kB = 0.35 µK),
which is negligible compared to the trap depth.
Population
0.5
mF=±1
0.4
0.3
0.2
0.5
mF=0
0.1
Re(l) 0
0
Error
ï0.5
ï1
0.1
F=1 0+1
ï2
0.01
ï1
F=2
0.001
0
20
40
60
80 Time [µs]
0
+1
+2
ï1 0
+1 +2
ï1 0
F=2
+1 ï2
F=1
Figure 3.21.: Left-Top: Evolution of hyperfine ground state population during the
optical pumping into the dark state |Ξi starting from a completely mixed state in
F = 1 The F = 2-depletion light field is applied simultaneously tot the pumping light
F = 1 → F 0 = 1. Left-Bottom: Error in state preparation calculated for all eight
hyperfine ground states using a mixed state fidelity [90]. Right: Final density matrix
(real part) of the hyperfine ground state. The Blue and dark red parts are the blocks
for the two hyperfine manifolds F = 1 and F = 2, respectively.
The amount of photon scattering, once the atoms are pumped into the dark state, is
reduced by two orders of magnitude as we can see in the inset of 3.22. Off-resonant
absorption is responsible for the remaining scattering, which could be reduced if the
optical pumping would be performed on the D1 transition of 87 Rb, where the excited
state hyperfine splitting is with 814 MHz, much larger than on the presently used D2
transition.
3.4.2. Loss of spin polarization
The spin polarization can be reduced due to decoherence and dephasing mechanisms.
Here, we discuss some of the main mechanisms for the loss of spin polarization which
could be relevant for our system.
One important mechanism for the loss of spin polarization are inhomogeneous magnetic fields. This issue will be discussed in great detail in Chap. 5.
62
3. Experimental apparatus and techniques
No. scatt. photons
2
0
1.5
20
40
60
80 t [µs]
5
10
R
sc
[sï1]
1
4
10
0.5
3
10
0
0
20
40
60
80 Time [µs]
Figure 3.22.: Average
number
of
photons
scattered by an atom
in the course of optical
pumping as simulated
in Fig. 3.21. The inset
shows the scattering
rates of pumping (dotted
black line) and F = 2
depletion laser (dahsed
blue line).
Number
of scattered photons
and scattering rate are
calculated as described
in Appendix B.3.
Dephasing due to elliptical trapping light
Any elliptically polarized trapping light, which couples differently to individual magnetic substates mF , will break the degeneracy of the hyperfine ground states. The
coherent superposition state |Ξi will start to precess about z. The same effect is
described by the QND part of the Hamiltonian (2.53), which is ∝ G1 Ŝz Jˆz . The coupling constant is proportional to the vectorial part of the polarizability tensor, which
describes the AC Stark effect. We can make an estimate of the effect by calculating
the energy shifts of all hyperfine ground states for a completely circularly polarized
dipole trap. In Fig. 3.23 we plot the results for the ground state as the difference
compared to the light shift from linear trapping light. We can see that the energy
difference between |1, −1i and |1, +1i is about 100 kHz in the focus of the dipole
trap. The state |Ξi would precess at this frequency about the z axis. We get the
same result if we would calculate the coupling constant G1 for from the interaction
of the trapping laser on the D1 and D2 lines.
We use trapping light which is mostly linear, with < 10−5 of circular contributions.
This is estimated from the specifications of the polarization dependent components
in the setup and the output of the laser. A maximum precession frequency of 10−5 ×
100 kHz = 1 Hz could be introduced by this amount of circular light. We can conclude
that the effect of hypothetical circular polarization of the trapping light will not cause
a major problem for the polarized spin state.
Decoherence via spin-exchange collisions
A decoherence mechanism for spin-polarized atoms are spin changing collisions [91].
In a binary collision event atoms with initial magnetic quantum number mF,1 and
mF,2 are transferred into a final state characterized by the quantum numbers mF,3
and mF,4 . The total spin projection on the quantization axis is a constant, i.e.,
3.4. Spin state preparation
63
6 Em+ [kHz]
100
0
ï100
F=2
100
F=1
0
ï100
ï2
ï1
0
1
2
mF
Figure 3.23.: Level shifts of
magentic substates in the two
hyperfine manifolds at the focus of a completely circular
dipole trap.
mF,1 + mF,2 = mF,3 + mF,4 . For example we can have |1, 0i + |1, 0i → |1, −1i +
|1, +1i . This leads to a decoherence of the initially polarized spin state 11 .
From the cross-section of
exchange as [94]
87 Rb-87 Rb
collision we can calculate the rate of spin -
ΓSE = nσSE v̄ ,
where n ∼ 1011 cm−3 is the atomic density, σSE = 1.9 × 10−14 cm−2 [95], and v̄ is
the average relative velocity of atoms. For our temperatures of T = 25 µK, we find
v̄ = 2vrms = 16 cm s−1 . The expected spin exchange rate for our systems is therefore,
ΓSE = 0.03 s−1 .
11
In contrast, condensed ultracold quantum gases can show coherent collisions both in an extended
BEC [92] and in the Mott insulator phase [93].
64
3. Experimental apparatus and techniques
3.5. Probing atomic spins
An essential part of the experiment are dispersive spin measurements. In Chap. 2 we
discussed the spectral properties of the vectorial and tensorial polarizabilities, which
are the main handles to get information about the atomic spin state. To address
them individually we have to have very good control over the polarization properties
of the light and the possibility to tune the laser within a large range around the
atomic resonances in on the D2 -line.
There are two interesting spectral regions for us. In Fig, 3.24 we marked them blue
and green. To measure the coherent spin state polarization along the pseudo-spin
direction Jˆx (blue region) we use the tensorial part of the effective Hamiltonian
(A.17). The tensorial polarizability very quickly decays away from the resonances
as ∆−2 . Therefore, we want to tune the laser only a few hundred MHz to the red
of the F = 1 → F 0 = 0 transition. Hereafter, we call the laser which probes the
Jˆx -component the tensor-probe. The QND measurement of the Jˆz -component relies
on the vectorial part of the polarizability tensor. Since, the relative strength of
vectorial to tensorial part increases linearly with the detuning, we want to put the
probe laser rather far away from resonance at a distance of 0.5 to 2.5 GHz to the red
of F = 1 → F 0 = 0 (green region). We call this probe the vector-probe.
Figure 3.24.: Spectral probe region for tensor- and vector-probe marked in blue and
green related to the level scheme of 85 Rb and 87 Rb. The gray area indicates the spectral
region accessible by locking to F = 2 → F 0 transition of 85 Rb. The four black Gaussian
peaks show the Doppler-broadened absorption spectrum for 87 Rb and 85 Rb.
3.5. Probing atomic spins
65
3.5.1. Polarization control and photon-number referencing
The heart of the spin measurement is a polarization rotation measurement on light.
Therefore, it is of utmost importance to have precise control over the polarization
state of the probe laser. At the same time, we also need a reliable measurement
of the number of photons that reach the atomic sample. The following description
applies for both the tensor- and the vector-probe beams. In Fig. 3.25 we show the
essential parts of the polarization control and photon-number referencing setup. It
is separated into five parts, which are explained now.
Figure 3.25.: Polarization cleaning and preparation scheme for vector- and tensorprobe beams. a) Linearly polarized light is coupled through a polarizing beams splitter
cube into a polarization-maintaining single mode fiber. b) At the output side the light
is projected into a well defined linear polarization by transmitting it through a second
polarizing beam-splitter. c) To monitor the photon-number we install a reference detector which is measuring a portion of the probe light. The splitting ratio can be adjusted
freely with a half wave-plate and a PBS. d) The polarization is cleaned a last time
and a combination of half and quarter waveplate can be used to compensate optical
elements which come farther down stream. e) Finally, the probe light is mode matched
over two mirrors onto the dipole trap beam and overlapped on a dichroic beam-splitter
cube (DCC).
In the very beginning of this chapter we mentioned that all light that reaches the
atoms is coupled with polarization-maintaining (PM) single mode optical fibers. This
is of huge advantage and makes the setup stable and flexible. As we learned over
time, it is also important to use the PM fibers in the right way. For all fibers we
install a PBS in front of the input coupler and align its axis either orthogonal or
parallel to the slow axis of the PM fiber. This is shown in Fig. 3.25a. Usually
we measure extinction ratios of more than 30 dB with a fiber coupled polarimeter
(Thorlabs PAX5710IR1). Fibers prepared in this way show very little polarization
distortion when put under stress by bending.
66
3. Experimental apparatus and techniques
When the light leaves the PM fiber we clean it with a second PBS, shown in Fig.
3.25b. This is to project the light into a well defined linear polarization state of high
purity.
For the photon-number referencing we use a combination of half-wave-plate (HWP)
and PBS to split the beam into two modes (cf. Fig. 3.25c). One part propagates
towards the atomic cloud and the other serves as a reference. We record it with a fast
photo-detector 12 and save the output on a digital storage oscilloscope. The waveplate-PBS combination allows a flexible ratio between reference and probe beam.
The calibration is performed on a regular basis by measuring the amount of light
close to the position of the atoms and relating it to the voltage reading of the photodetector. We see very little change over the course of many weeks, which we ascribe
(from previous experience) to the elaborate polarization control before and after the
fiber.
Before the light can interact with the atomic ensemble, it is overlapped with the
trapping light beam on a dichroic beam-splitter cube shown in Fig. 3.25e. This
allows us to use the same optics for the dipole trapping and for the probing. In
this way, we achieve a well defined mode overlap of atoms and light. However, the
dichroic cube and other components in the beam path, change the polarization state
of the probe light. For this reason we have to manipulate the polarization of the
probe light beforehand to account for the changes down-stream. This is done with a
half- and a quarter wave-plate. In this combination we can produce any polarization
state we wish. In Fig. 3.25d we see that an additional PBS is used to project the
light once again into a pure linear polarization. For this projection we use a thin-film
polarizer cube which has an extinction ratio of 105 : 1.
The compensation wave-plates in Fig. 3.25d are adjusted according to the measurement of the polarization state of the light right after it has passed the atomic cloud
and left the glass cell again. The polarization state is determined in two ways. First,
with a commercial polarimeter (Thorlabs PAX5710IR1), which has an accuracy of
0.2◦ in the Poincaré sphere. Second, with the polarimeter we built to measure the
atomic quantum noise which is much more precise. It will be explained in Sect 3.6.1.
3.5.2. Frequency shifting and pulse generation
We make broad use of acousto-optic modulators (AOM)13 , both for frequency shifting
in the range of 30 to 200 MHz and for pulse generation. Since, we almost have a
dozen AOMs in the experimental setup, it is worth spending a few lines to explain
the way we are using them. We were able to double the number of trapped atoms
in the dipole trap by changing to the configuration described below.
From the electronic point of view we use them in the following configuration. A
voltage controlled oscillator (VCO) serves as radio frequency (RF) source and is
controlled via a fixed DC voltage or a variable voltage coming from one of the data
12
13
We use PDA10A or DET36A from Thorlabs.
More correctly these devices are frequency shifters. However, depending on the provider terms
are interchanged.
3.5. Probing atomic spins
67
acquisition cards (DAC). The RF signal passes through a voltage controlled attenuator (VCA) which is used to select the amount of RF power that reaches the AOM.
After the VCA we send the RF through a switch which generates RF pulses with
rising times of a few nanoseconds. It is driven with the digital output of the DAC
or the output of a pulse generator. Before, the RF reaches the AOM, it is amplified
to power-levels of 1 − 2 W. In this way we can control the amount of light sent to
a certain part of the experiment and the can generate pulses down to a few tens of
nanoseconds.
Double-pass configuration
In the case we need tunability, the AOMs are used in a double-pass (cat’s eye)
configuration [96]. This ensures the maximum in conversion efficiency and perfect
alignment over the whole detuning range. In Fig. 3.29 (further below Sect. 3.5.4) we
show the typical configuration we use in many places. As a first step we reduce the
size of the laser beam with a telescope to fit the aperture size of the AOM crystal.
We use a Galilean telescope to save space. It consists of a convex lens of 100 mm focal
length and a concave lens of 50 or 75 mm focal length, depending on the required
de-magnification. The laser is then sent through the crystal of the AOM where it
is diffracted by the traveling density wave. To optimize the diffraction efficiency we
mount the AOM on a kinematic mount to change its angle about a vertical and
horizontal axis (both orthogonal to the propagation direction of the laser beam).
We always optimize the AOM for the single pass efficiency. The diffracted light is
then collected by a focal lens and focused onto a mirror. The mirror and AOM are
both placed one focal length away from the lens. This is the optimal configuration
to ensure maximal double-pass efficiency and best possible injection efficiency into a
single mode fiber. The double-pass efficiency is optimized by adjusting the mirror.
We reach efficiencies of above 70 % for the center frequency.
The double-pass scheme is also very flexible. If we, for example, decide to change the
AOM frequency or the diffraction order we want to use, i.e., from +1 to −1, almost
no realignment is necessary.
Even if AOMs are fantastic devices which are easy to use and give a reliable frequency
tuning, their bandwidth is limited. The rule of thumb is that the bandwidth is at
maximum half the center frequency, e.g., for a model with center frequency 80 MHz,
we can have at most shifts between 60 and 100 MHz. Moreover, the conversion
efficiency is not constant over the band-width, but decreases rather strongly towards
the edges. Therefore, we describe the technique we are using to detune the laser over
a large range of up to several GHz.
3.5.3. Frequency Offset-lock
To overcome the limitations of the AOMs there are different techniques to achieve
a large detuning from a given reference, e.g., an atomic resonance. Starting from a
laser which is referenced to an atomic resonance, which is called master, there are
different ways to frequency lock a second laser, called slave. One is for example,
68
3. Experimental apparatus and techniques
to use the discrete frequency spectrum of a Fabry-Perot cavity to separate master
and slave by an integer of the free-spectral range of the cavity [97, 98, 99, 100].
The technique we use, which is better suited for for frequency differences between
a few hundred MHz and several GHz, is to interfere both lasers on a beam splitter
and use the beat-note signal detected by a fast photo-detector. The upper limit for
the frequency-difference is set only by the bandwidth of the photo-detector used to
measure the beat-note.
Analog offset-locks
The beat-note signal, which can go up to several GHz, is usually reduced to a smaller
frequency in the MHz-range by mixing it with a local oscillator of known frequency.
Then there exist several ways to convert this frequency information into a measurable
signal, e.g., voltage. One approach proposed by Rutt [101] and realized for external
cavity diode lasers by Schünemann et al. [102] uses an electronic delay-line, with
delay time τ , to produce a frequency dependent phase-shift Φ = (ωbeat − ωV CO ) τ .
This is converted into an amplitude signal by an electronic phase detector. Despite
its simplicity it has some serious drawbacks. First, there are multiple locking points
separated by the inverse of the delay time τ . Second the frequency precision and the
capture range are not independently variable.
Another approach by Ritt et al. [103] used a sharp electronic RF high pass filter
as frequency-to-amplitude converter. There, the frequency is fixed by locking to a
certain attenuation given by the frequency response of the high pass filter. This
approach has the advantage to have only one locking point, i.e., capture ranges of
several hundreds of MHz are possible. Furthermore, the precision is given only by
the steepness of the high pass filter, which can be arbitrarily large.
Digital offset-locks
A very simple frequency offset-lock using digital components is described by Stace et
al. [104]. They use a digital frequency-to-voltage converter (FVC). In their design
they have to digitize the beat-note signal using a voltage comparator. Furthermore,
it is necessary to divide the resulting TTL signal down to the narrow band-width of
the FVC of only 500 kHz. To generate an error-signal with zero crossing, a negative
control voltage is added to the output voltage of the FVC. In this way, the locking
point can be chosen by selecting an appropriate control voltage. Limitations of this
design are the finite gain and drifts in the output voltage of the FVC.
Implemented technique
We apply an idea by Günter [105] for a digital offset-lock. It is based on a commercial
digital phase-locked loop (DPLL) synthesizer, which is in-expensive and straight
forward to use. Usually, the DPLL is designed for frequency- and phase-locking of
a VCO in GSM devices and works up to several GHz. If the VCO signal is now
3.5. Probing atomic spins
69
substituted for the beat-note of the master and slave laser, we are able to control
the frequency- and the phase-difference of both lasers.
The whole digital offset lock can be divided into three parts. First, a fast photodetector is needed with a bandwidth larger than the frequency difference we want
to establish between slave and master. Second, the above mentioned digital PLL.
Third, a servo controller to act on the grating and the current of the slave laser.
Figure 3.26.: Functional block diagram of digital offset lock using a digital phase-locked
loop.
Fast photo-detector
As a fast photo-detector we use a fiber coupled photo-receiver which has a bandwidth
of 9.5 GHz (model PT10GC from Bookham14 ) usually used in telecommunication
applications. It consists of an InGaAs-PIN photo-diode which is pre-amplified on
chip and connectorized to a single-mode fiber pigtail.
Digital PLL
The DPLL contains several functional blocks which we show schematically in Fig.
3.26. The beating signal runs through a pre-scaler which is a fractional N-divider that
reduces the beat-note frequency to something close to the reference frequency of the
local oscillator. The local oscillator is either provided by a precision crystal oscillator
on the used evaluation board or by an external frequency source up to 250 MHz. For
the presented experiments, we use the temperature compensated crystal oscillator
of 10 MHz from the evaluation board. The reference frequency can be optionally
doubled. Both the local oscillator and the pre-scaled beat signal are fed into the
“heart” of the DPPL, the phase frequency detector (PFD), which generates a signal
proportional to the frequency- and phase-difference.
The PFD is an electronic element, which can have various implementations [106, 107].
In our case, this is a circuit with two D-type flip flops U1 and U2, shown in Fig. 3.27.
A D-type flip flop is a 1-bit memory. If it is clocked in rising edges, then it changes
its output state Q from low to high, if the input IN made a transition from low to
14
www.bookham.com
70
3. Experimental apparatus and techniques
high. The output is reset to low if the CLR input receives high. The feedback over
the AND gate (U3) brings both outputs Q1 and Q2 back to low if they were both
high. To understand the principle better, we plot the output for two case. First,
neither frequency nor phase are locked. Second, the frequency of beat signal and
reference are the same but the phase is not.
The output of the PFD are current pulses with a maximum current of 4.375 mA,
where the voltage goes from V − = 0 V to V + = 4.88 V. The maximum repetition
rate is limited by the local oscillator frequency, which is the maximum comparison
frequency.
Figure 3.27.: a) Phase frequency detector as used in the DPLL. U1 and U2 are D-type
flip flops, U3 an AND gate. The two outputs Q1 and Q2 control the current on the
output OUT. b) Example of different frequencies on the IN- and IN+ which result in a
large output signal. c) Example of same frequency but different phase. The output is
a periodic signal which eventually drives the two oscillator in phase. All three figures
are taken from [106].
To illustrate the principle of the frequency offset lock, we record the integrated error
signal and compare it to the simultaneously recorded saturated absorption spectrum
with open feedback loop. We lock the reference laser to the cross-over transition
F = 1 → 20 |10 (see figure) and set the offset lock frequency to 1 GHz. In Fig. 3.28 we
show the error signal after integration (monitor output of PID). The measurement
is taken with the laser scanning over the F = 1 → F 0 transitions of 87 Rb. We fit the
spectrum including the Doppler-broadened profile and the saturated lines.
Closing the feedback loop
The frequency of an external cavity diode laser can be controlled in various ways.
One is to change the temperature of the laser diode, which will change the cavity
length of the diode. It is very slow and usually not used to stabilize a laser to
an external reference. The second way is to change the external cavity feedback.
If the external cavity is built by a diffraction grating than its angle is changed
3.5. Probing atomic spins
71
Figure 3.28.: Top: Error signal after low pass filtering in the PID controller of the probe
laser driver. The white area is the forward scanning direction of the PZT, the gray
one the backward direction. Bottom: Transmission signal (blue dots) of the saturated
absorption spectroscopy. The solid black line is a fit, see text for details. In the inset
we show a close-up of the hyperfine transitions F = 1 and their cross-over transitions.
The spectrum is grayed for the backward movement of the PZT.
[108]. Alternative designs use frequency selective elements in the external cavity
[109, 110]. Having moving mechanical parts limits the bandwidth of this feedback
by the resonance frequency of the mechanical components. The typical bandwidth
is a few kHz, mostly limited by mechanical resonances of the grating holder. The
fastest way of applying feedback is on the current of the laser diode. Here one can
achieve bandwidth of several MHz.
The output of the PFD is split into two branches. For fast feedback we connect
the PFD directly to the FET input of the laser diode. The laser diode has a large
bandwidth and allows, in principle, to use the locking system to phase-lock the master
and the slave. Unfortunately, we were not yet able to realize this. For slow feedback
we send the second branch into the PID controller of the laser system which actuates
on the piezo. The piezo feedback is intrinsically slow and limited to the mechanical
resonances. The maximum frequency is in the order of few kHz. With this we can
correct for acoustic vibrations and slow drifts introduced by temperature.
72
3. Experimental apparatus and techniques
3.5.4. Experimental details on probe lasers
Before we continue, we close the section by explaining how the probe lasers are locked
in the experiment and give relevant details.
Vector probe
The probe laser for the QND measurements of Jˆz is an external cavity diode laser
of the type DL100 from Toptica. In Fig. 3.29 we can see that it is split into different
beam paths. After a Faraday isolator, to prevent optical feedback, we use a wedge
and split off two small portions of 3−4 %. One part, which corresponds to ∼ 2.5 mW,
goes to a non-polarizing beam splitter cube and is overlapped with light from either
the F = 1 or the F = 2 laser. Both beams are then coupled into the same single
mode fiber which sends them to the offset lock box. The other portion is attenuated
and sent into a saturated absorption setup, which allows to have a control over the
position of the laser before it is locked.
The continuing portion of light (∼ 56 mW) has a frequency of νm − νbn , where νm
is the frequency of the master laser15 and νbn is the beat-note frequency we want
to lock to. The light propagates now through a telescope to a double-pass AOM
configuration, described above. We use the double-pass configuration also here, even
if we never scan the AOM frequency. This allows us to compensate the lower locking
frequency of the beat-note lock, which is νbn.min = 300 MHz and achieve a continuous
tuning range for the vector probe laser resonance F = 1 → F 0 = 0 down to more
than 2.5 GHz red detuning. The final vector probe detuning as a function of the
beat-note frequency is
∆vect = νm − νF =1 + 2νAOM − νbn = −(0...2.7) GHz .
As we will see further below, for most of the experiments we are working with trains
of probe pulses. It is fundamental to change have access to the pulse duration,
repetition rate, number of pulses and light power. The later is adjustable with the
RF amplitude of the AOM. Where the former are dialed with a function generator
which is pre-programmed or alternatively can be remotely programmed to define the
pulse width and repetition time. The number of pulses is determined by the duration
of the pulse train, which could be one to many pulses. The timing precision of the
function generator, which has a resolution of 1 ns and a jitter of < 400 ps is very
small.
Tensor probe
For the measurements of Jˆx we use only a double-pass AOM configuration and no
offset-lock, because we probe much closer to resonance (cf. Fig. 3.24). The light
15
The master laser could be F=1 or the F=2 laser. The former is locked to the cross-over transition
between F 0 = 1 and F 0 = 2, and has a frequency of νm = νF 0 =0 + 150.7 MHz − νF =1 . The
latter is locked 60 MHz below the cross-over between F 0 = 2 and F 0 = 3, and has a frequency
νm = νF 0 =2 + 73.3 MHz − νF =2 .
3.5. Probing atomic spins
73
Figure 3.29.: Vector probe setup: a) Spectroscopy setup with rubidium vapor cell b)
laser light from the vector probe and from either F = 1 or F = 2 laser are overlapped
on a beam splitter and one output port is coupled into a single mode fiber which goes
to the frequency off-set lock box. c) the pulse generation is achieved by means of an
AOM in double-pass configuration. The output light is coupled into a single mode fiber
and sent to the atom trap.
from the F=1 laser is shifted by an AOM with a center frequency of 200 MHz and
bandwidth ∼ 80 MHz. This gives us a range of detunings from the F = 1 → F 0 = 0
transition of
∆tens = 150.7 MHz − 2νAOM = −(170...330) MHz .
The pulses are also generated with a pulse generator in gated trigger mode.
74
3. Experimental apparatus and techniques
3.6. Shot-noise-limited polarization detection
We measure atomic spin polarization stroboscopically by sending pulses of polarized light and record their polarization rotation in a polarimeter. For the range of
probe light detunings we are working in (c.f. previous section), the pulsed interogation is necessary for technical reasons. On one side, the photon flux we can use
to gather information about the atomic polarization is limited through off-resonant
light scattering which destroys the atomic state. On the other side, intrinsic noise of
the detection electronic and low frequency noise (intensity fluctuations) of the probe
laser can be rejected by using pulses instead of a continuous wave.
We measure polarization rotations of light with a pulsed polarimeter down to very
small angles below 10−4 rad. For this we set up a polarimeter that measures the
45◦ -component of the Stokes operator, i.e., Ŝy . Here, we explain the main optical
elements for the polarimeter and discuss the influence of polarization dependent losses
in Sect. 3.6.1 and 3.6.2. Further, we describe the operation principle of the balanced
detector (3.6.3), discuss the calibration in the time domain (3.6.4) and determine its
performance at the quantum level (3.6.5).
3.6.1. Polarimeter
We use a simple polarimeter consisting of a polarizing beams-splitter (PBS) (Wollaston or normal cube), a HWP, and a differential photodetector, shown in the right
part of Fig. 3.31. In this way we are sensitive to any Stokes component on the equator of the Poincaré sphere, depending on the angle of the HWP. This polarimeter
setup is most sensitive when the photo-detector is working in the balanced regime,
i.e., when the differential photo-current is zero. Any rotation of the input polarization about the Ŝz axis of the Poincaré sphere will cause an imbalance which is
recorded as an electrical signal. The Stokes component we are most interested in is
Ŝy . Hence, we balance the two output ports of the PBS by rotating the incoming
light into the 45◦ -basis.
The balanced regime is not only the most sensitive, but also suppresses unwanted
technical noise in the probe light, e.g. from intensity fluctuations. This can be seen
if we write down the intensity of the light emerging from the two output ports of
Figure 3.30.:
Polarization rotation
of a) electric field
vector in real space,
and b) Stokes vector
in the Poincaré
sphere. The rotation angles have the
relation ϕ = 2α.
3.6. Shot-noise-limited polarization detection
75
◦
the PBS. As shown in Fig. 3.30 a), the
in
electric field components
the +45 and
π
π
◦
−45 basis are E+45◦ = E0 cos 4 + α and E−45◦ = E0 sin 4 + α , respectively,
where E0 is the envelope of the electric field and α the angle between the electric
field vector and the +45◦ axis. The intensities at the two output ports are
I+45◦
I−45◦
I0
(1 + sin(2α))
2
I0
= |E−45◦ |2 =
(1 − sin(2α)) ,
2
= |E+45◦ |2 =
(3.31)
(3.32)
where we write the intensity at the input as I0 = |E0 |2 . We can separate the intensity
I0 into a common mean intensity I 0 , and into fluctuating parts δI±45◦ = δIc +
δIu,±45◦ which are different for the two output ports. They consist of a correlated δIc
and an uncorrelated term δIu,±45◦ . The differential photo-detector will measure a
signal which is proportional to the difference of the two intensities (3.31) and (3.32)
and we get
D∝
1
δIu,+45◦ − δIu,−45◦ + 2I 0 + 2δIc + δIu,+45◦ + δIu,−45◦ sin(2α) , (3.33)
2
where we assume equal gain of the two photo-diodes. The signal of the polarimeter
(3.33) is directly proportional to the rotation α of the electric field vector.
For the balanced case, i.e., α = 0, the mean value of the polarimeter output is zero
h D i = 0 and the uncertainty is
δD = δIu ,
(3.34)
where we assume that the uncorrelated fluctuations are equal in magnitude for both
output ports δIu ≡ δIu,+45◦ = δIu,−45◦ . If the uncorrelated fluctuations are completely quantum, the polarimeter is said to be shot noise limited [111]. For an imbalanced polarimeter, i.e., α 6= 0, the uncertainty is increased and correlated noise,
e.g., intensity fluctuations, enter into the signal
δD = δIu + sin(2α) (δIu + δIc ) ,
(3.35)
and the polarimeter is limited by classical fluctuations.
In Sect. 3.6.5 we measure the noise behavior for a large range of photon-number and
identify the individual noise contributions by their respective scaling with photonnumber.
In the experiment, as shown in Fig. 3.31, we measure the rotation angle of the electric
field α, or more precisely, of the Stokes vector, which is ϕ = 2α. If the analyzer waveplate is set to 22.5◦ , we measure Ŝy . For vertically polarized light entering the setup,
(in)
i.e., h Ŝx i = NL /2, the rotation angle of the Stokes vector can be calculated by
!
(meas)
h Ŝy
i
ϕ ≡ arcsin
.
(3.36)
(in)
h Ŝx i
(meas)
where h Ŝy
i is the measured mean photon-number imbalance.
76
3. Experimental apparatus and techniques
Figure 3.31.: Polarization probing of atomic spin. The dipole trapping and probing
beams are overlapped to maximize atom-light interaction. The probe beam has acquired a polarization rotation after passing through the sample. The birefringence
of the dichroic mirror (DCM) is corrected by a quarter- and a half-waveplate. The
polarization is analyzed in the 45◦ -basis by a differential photo detector.
3.6.2. Linear birefringence and optical losses in detection
In our experimental setup probing and trapping light are overlapped/separated on
dichroic optical elements. This permits a complete overlap of the atomic cloud
and probing light, but at the same time introduces problems of birefringence and
polarization dependent losses.
Birefringence
The dichroic mirror (DCM) which is used to separate the dipole from the probing
light (cf. Fig. 3.31) shows (linear) birefringence. Combined with a second DCM
of the same kind, which compensates the walk-off introduced by the first, they act
similar to a quarter wave-plate16 . To compensate this, we use a combination of
quarter- and half-waveplate before the light enters into the polarimeter setup (cf.
Fig. 3.31).
To make a full polarimetry and to determine the Müller matrix [112] of the optical
component between the atomic cloud and the Ŝy -measurement is rather difficult in
terms of available space. Instead, we use crossed polarizers which are placed around
the DCMs and the compensation wave-plates. The first polarizer in front of the DCM
will define a pure linear polarization (v or h). A second crossed polarized after the
combination of quarter and half wave-plate serves as an analyzer. Both wave-plates
are adjusted until the transmitted amount of light is minimized. In this way, we can
guarantee the compensation of the DCMs by the wave-plates.
16
Before building them in the optical properties of the dichroic mirrors (DCM) were measured
separately outside. They have shown to behave very similarly to a quarter waveplate.
3.6. Shot-noise-limited polarization detection
77
Polarization dependent losses
In the presence of polarization dependent losses (PDL) the relation between observed
(meas)
photon-number difference h Ŝy
i and the rotation angle of the Stokes vector ϕ is
slightly more complicated than in Eq. (3.36). We have to take into account that the
presence of PDL can change the angle of the electric field vector.
To model the influence of PDL, we measure the attenuation of horizontally and
vertically polarized light propagating through the DCMs and compensation waveplates. We found transmission of vertical (s-polarized) and horizontal (p-polarized)
light of Tv = 74.5(2) % and Th = 86.6(2) %. The large difference can be explained
with the usual observation that a dielectric mirror has a better reflection for s than
for p polarization.
The relation between the actual and the measured polarization rotation is derived in
the following way. The incoming light is vertically polarized and receives a rotation
through the circular birefringence. Neglecting the time and spatial behavior we can
write for the input
1
Ein = √ E0 {e+ − e− } ,
(3.37)
2
where e± are the spherical basis vectors, defined in the textbook by Edmonds on
angular momentum algebra 32. The output reads
1
Eout = √ E0 eiϕ+ e+ − eiϕ− e− ,
(3.38)
2
where ϕ± are phase shifts on the left- and right-circular light. Transforming from
circular into the linear polarization basis (see for instance 32) we find
1
1
Eout =
E0 eiϕ− ei∆ϕ + 1 ex + E0 ieiϕ− ei∆ϕ − 1 ey ,
2
2
= Av ex + Ah ey
(3.39)
where ∆ϕ = ϕ+ − ϕ− it the phase difference between the two circular polarization
components, which is the measure of interest. Note that this quantity is twice the
rotation angle of the electric field. Av and Ah carry the amplitude of the vertical
(horizontal) polarization component. The reduction of the vertical (x) and horizontal
(y) amplitudes changes the electric field into
p
p
E0out = Tv Av ex + Th Ah ey .
(3.40)
Since the polarimeter measures the Ŝy component we have to transform
the above
√
expression into the +45◦ / − 45◦ basis using e±45◦ = (ex ± ey ) / 2
!
!
r
r
r
r
Tv
Th
Tv
Th
0
Eout =
Av +
Ah e+45◦ +
Av −
Ah e−45◦ .
(3.41)
2
2
2
2
We find a similar expression for rotation angle as in the ideal case, which has corrections for the introduced loss
!
(meas)
Sy
.
(3.42)
∆ϕ = arcsin
(in) √
Sx
Tv Th
78
3. Experimental apparatus and techniques
3.6.3. Operation Principle of balanced detector
The heart of our detection system is a differential, integrating photo-detector. It is
designed to detect pulses between 10 ns and 200 µs with a very low electronic noise.
The version of the detector we use was designed and made in the group of Eugene
Polzik at the NBI in Copenhagen by Patrick Windpassinger and Jöreg Helge Müller.
The design is adapted from [113]. Previous detectors following the same design where
developed together in collaboration between ICFO and NBI [74].
Since we are only user of the device, we limit the explanation to the operation principle to get an idea of the main concept and refer the reader to [114, 115]. Nevertheless,
to fully understand the functionality of the device and use it appropriately in the
experiment, a series of characterizations is necessary.
The detector is built to sense the difference photo charge of two photo-diodes (HAMAMATSU S388317 ) which are connected in series. This signal will correspond to the
imbalance in photon-number between the two detection modes. We bias the photodiodes in reverse direction to improve their time response by lowering the photo-diode
capacitance. The differential signal is AC coupled to the first essential part of the
detector, the integrator. It consists of a operational amplifier which has in its feedback branch a capacitor Ci and a (discharge) resistor Ri . Together they determine
the time constant τ = Ri Ci of the integrator. An input charge signal which is faster
than τ will be converted into a voltage output, where the gain is set by Ci−1 . For this
reason these devices are also sometimes called charge sensitive amplifiers. In Fig.
3.32 we see how a train of squared charge pulses are converted into a step-like voltage
signal with linear ramps connecting the steps. The integrated signal will decay exponentially with a time constant τ. The integrated voltage signal is fed into the second
important part, the pulse shaper. Here, a combination of high- and low-pass filter
with identical time-constant τd = Rd Cd converts the step-like signal into √
a pulse of
almost Gaussian shape. The output pulse has a duration (FWHM) of 2 2 ln 2 τd .
The combination of filters can also be interpreted as an active differentiator followed
by a low pass filter. It is sometimes called Gaussian shaping amplifier.
One additional refinement is necessary to make the detector work properly. The
fact that the output of the integrator is not a pure step function, but decays with
τ to zero, makes the pulse shaper output have an undershoot. The reason is that
the transfer function of the integrator stage has a pole. A zero in the transfer
function of the high-pass filter can compensate this pole. This is achieved by adding
a resistor RP/Z in parallel to the capacitor Cd . The value has to be chosen so that
RP/Z Cd = τ = Ri Ci .
In the present detector the integrator and the pulse shaper are self contained components provided by Cremat, Inc. 18 The integrator CR-110 is a module which contains
the input FET and the feedback network. For the shaping amplifier a series of modules with fixed shaping time τd is available. The detector is equipped with module
CR-200 with 250 ns shaping time. Only the resistor for the pole-zero-cancellation
17
18
The diodes have a quantum efficiency of 92 % at 780 nm.
http://cremat.com
3.6. Shot-noise-limited polarization detection
79
Figure 3.32.: Schematic of detector circuit [114]. From left to right: Photon number
τ light
= Ri reaching
Ci
differences between
upper or lower photo diode is measured as the differential photo current in between the two diodes. The (differential) photo electron pulses
RP/ZFor further processing the steps are converted into
are integrated to a step-like signal.
a pulsed signal of fixed pulse width and a height proportional to the number of detected
photons.
has to be added externally. An additional AC coupled buffer amplifier was added
between integrator and pulse shaper to add more gain to the detector.
3.6.4. Detector Calibration
For the characterization of the detector we limit ourselves to the time domain. We
will measure the response for short pulses below 25 ns, compare the output when
only one photo-diode is illuminated at a time, and measure the gain, integration
window, and shot-noise performance for different pulse lengths between 250 ns and
5 µs.
“Delta” pulse response
τ = 1.46
As a first test we measure the intrinsic response of the detector. That means we send
a pulse which is much shorter than the pulse shaping time of the shaping amplifier,
which is in our case is 250 ns. From the specifications we should expect to observe a
Gaussian pulse of 590 ns (FWHM) out of the shaping amplifier. Using an AOM we
can produce pulses with durations of 15 ns (FWHM). To create an observable signal
we illuminate only one of the photo-diodes to create a differential photo current.
In Fig. 3.33a) we plot the input to the balanced detector recorded with a 150 MHz
band-width amplified silicon reference photo-detector (Thorlabs PDA10A). In the
inset we show the zoom around t = 0, which has almost a Gaussian shape with a
duration of 15 ns. Compared to the pulse shaping time the input pulse represents a
good approximation of a delta input. The response of the balanced photo-detector
plotted in Fig. τ3.33b)
has
a duration
of
= Ri C
Ri of 587.5 ns. This is the intrinsic pulseCshape
i
i
the shaping amplifier and corresponds to the specified 590 ns for this module.
We present this measurement here, because it shows the principle of the detector in
a nice way. An input pulse is recorded and an output pulse, with different shape, is
τ
produced.
The numberCof
i detected charges is, as we will see later, proportional to
80
3. Experimental apparatus and techniques
the area of the output pulse. The difference between input and output pulse shape
requires that we calibrate the detector for each pulse length we want to use. In the
following we will present the results of this calibration for a set of pulse lengths.
Figure 3.33.: Short pulse response of balanced detector. a) Input pulse with 25 ns
FWHM. b) Response of balanced detector. The gray area is the integration window
over which we average the signal. c) The delta pulse response as given in the spec
sheet.
Integration window
Before we calibrate the gain and characterize the shot noise behavior of the detector,
we should study the relation between input- and output-pulse length. This is important because the signal we are interested in, is the pulse area, which is proportional
to the photon-number difference. Later, we will use the detector always together
with a digital storage oscilloscope. On this, we record the pulse shape and integrate
it numerically during the data analysis. To ensure a good low noise performance,
the integration window has to be chosen depending on the pulse length τpulse used.
We chose the integration window to contain the whole pulse shape. It seems to be
an obvious choice, which is at the same time rather conservative. In Fig. 3.34 we
plot the integration windows for pulse shapes between 250 ns and 5 µs. We find that
the integration window follows
Tint (τpulse ) = T0 + τpulse ,
(3.43)
3.6. Shot-noise-limited polarization detection
81
where T0 is the initial integration windows due to the shaping time of the pulse
shaping amplifier. For our module with 250 ns shaping time we have T0 = 1.3 µs. If
we compare T0 to the pulse shape in Fig. 3.33 for the delta pulse response, we see
that the integration window would cover the whole pulse.
7
Integration window [µs]
6
5
4
3
2
1
0
0
1
2
3
4
5
6
Pulse length [µs]
Figure 3.34.: Manually chosen integration
windows for different pulse lengths. The
solid black line is directly proportional to
the pulse length and offset by 1.3 µs.
Calibration of photon-number gain
We define the photon-number gain of the balanced detector as the conversion between
observed voltage U (t), averaged over the integration window Tint ,
Z t0 +Tint
1
Uavg =
U (t)dt .
(3.44)
Tint t0
and the photon-number difference ∆NL , which caused the signal:
Gγ−V ≡
∆NL
,
Uavg
(3.45)
where the integration window starts at t0 . In this way, we can relate the observed averaged voltage signal to the photon-number imbalance and, hence, to the observable
of Ŝy .
2 h Ŝy i ≡ ∆NL = Gγ−V Uavg
(3.46)
The gain as a function of pulse length is measured by sending pulses of known photonnumber ∆NL = NL onto one of the photo-diodes and integrating the output signal
over the corresponding integration window.
In the left panel of Fig. 3.35 we plot the averaged output signal as a function of
photon-number difference ∆NL . As expected, we observe a linear relation between
photon-number imbalance and output signal. The inverse of the linear slopes, i.e.,
Gγ−V , are plotted in the right panel of Fig. 3.35 for different pulse lengths. The
82
3. Experimental apparatus and techniques
1.2
1
−1
U
avg
0.25 µs
0.50 µs
1 µs
1.50 µs
2 µs
3 µs
4 µs
5 µs
−2
10
−3
10
3
10
4
5
10
10
Photon number imbalance ∆NL
Gγ−V [Photon/µV]
[V]
10
0.8
0.6
0.4
0.2
0
0
2
4
Pulse length [µs]
6
Figure 3.35.: Left: Averaged voltage over the integration window for different input
pulse lengths as a function of pulse energy in number of photons NL . Right: The
photon-number gain Gγ−V deduced as the slope of measurement shown in the left
panel. The blue line is a linear fit with a slope of 0.151(7) photons/(µVµs) and a zero
at t0 = 1.5(1) s. This time is very close to T0 .
almost linear relation between integration window Tint and Gγ−V , is related to definition (3.45). By fitting a line we get a constant ratio
∆NL
photons
.
= 0.151(7)
R t0 +Tint
µV µs
U (t)dt
t0
(3.47)
We didn’t expect this fixed ratio between photon-number and integrated voltage, in
the beginning, and see that it also changes for pulses shorter than 500 ns. Hence, we
decided to keep the definition of Gγ−V as introduced, throughout the remainder of
the thesis. In practice, we will always work with a constant pulse length of 1 µs, for
reasons which become clear in the next section.
3.6.5. Noise of the detection system
Now, we characterize the capability of the detector to measure quantum noise of the
light. For the preparation of non-classical spin states via QND measurements it is
of utmost importance to have a detection system which is only limited by quantum
noise. Therefore, we will measure now the smallest number of photons we have to
send in a coherent state, so that their shot noise equals the intrinsic electronic noise
of the device [111]. We call this number the noise equivalent photon-number (NEPN)
NL,SN . From this we can deduce the equivalent noise charge (ENC) as
pthe number
of photo-electrons necessary to overcome the electronic noise qSN = ηQ NL,SN . The
quantum efficiency of the photo-diodes in the detector is ηQ ∼ 0.92.
As a reference we use coherent states of light, which is a natural choice to calibrate a
photo-detector because of its well defined quantum properties. Here, we measure the
3.6. Shot-noise-limited polarization detection
83
quantum noise in the Ŝy component, because we use the polarimeter setup in this
way for later measurements. In order to suppress any classical intensity fluctuations
we chose as input state light which is polarized vertically, i.e.,h Ŝx i = NL /2 and
h Ŝy i = 0 and var(Ŝy ) = NL /4.
Data analysis
The way we analyze the data for this calibration measurement is the same as later
for the quantum noise measurements. Therefore, we will explain it in more detail
here and refer later to this section.
First, we always convert the measured voltage into an photon-number by means of
Gγ−V . This photon-number will correspond to the observable for the 45◦ -component
of the Stokes operator, i.e., Ŝy . We can write the y Stokes component for an individual pulse
Z t0 +Tint
Gγ−V 1
sy,i =
U (t)dt ,
(3.48)
2 Tint t0
where t = 0 at the moment when the light pulse is triggered. The shift t0 is a delay
to place the integration window right. Delays occur due to the electronics for the RF
generation and the finite speed of sound in the AOM crystal. Once, t0 is measured, it
will stay the same until something essential in the experiment has changed. Usually,
t0 is in the order of a few hundreds of nanoseconds.
Second, to suppress low frequency fluctuations, e.g., a drifting baseline due to imperfectly compensated base line pulling, we integrate also about a region before the
pulse and subtract this from the signal. This is very similar to the correlated double sampling used in charge coupled devices (CCD) to determine the added charge
more precisely. In this way we ensure that a wrong baseline, which is a kind of
memory effect from previous pulses, does not lead to classical correlations between
adjacent pulses. The expression for the individual measurement outcome in the case
of correlated double sampling is
scds
y,i
Gγ−V 1
=
2 Tint
Z
t0 +Tint
Z
tb +Tint
U (t)dt −
t0
U (t)dt ,
(3.49)
tb
where tb is the starting point for the background window, which has the same width,
Tint , as the actual integration windows for the pulse.
From the set of scds
y,i we calculate the variance as
n
n
i=1
i=1
1 X cds 1 X cds
var(Ŝy ) =
sy,i −
sy,i
n−1
n
where n is the number of samples.
!2
,
(3.50)
84
3. Experimental apparatus and techniques
Performance
We send a series of probe pulse with pulse lengths between 250 ns and 5 µs and a
period of 20 µs into the polarimeter and vary the number of photons per pulse. For
each pulse length and photon number, we send a train of 500 pulses and evaluate
var(Ŝy ).
In Fig. 3.36 we plot the variance as a function of photon-number for a pulse length of
1 µs. We clearly can distinguish two different scalings. The first one for small photonnumber below NL = 104 shows no dependence on photon-number and represents the
electronic noise level of the detector. The value we can read off Fig. 3.36 as the
noise equivalent photon-number is NL,SN = 3 × 105 . This will be referred to as the
photon-number at which the detector starts to be shot-noise-limited.
We have to notice that the correlated double sample we perform by subtracting a
base line from√
each pulse (3.49) adds one unit of electronic noise. The resulting NEC
is qSN = 0.92 0.5 ∗ 3 × 105 ' 360. This coincides with the noise estimation given
by the Cremat, Inc. for a similar combination of CSA and pulse shaping amplifier.
For the detector it is also important to stay shot-noise-limited for large photonnumbers. We can see in Fig. 3.36 that this is fulfilled for whole measured range
which represents the range of interesting photon-numbers for all later experiments.
We repeated this calibration also for other pulse lengths and plot the result for the
NL,SN in the left panel of Fig. 3.37. We can see that there exist an optimal pulse
length in terms of shot-noise-limit. For increasing pulse lengths one could expect a
better performance if the noise of the detector would be white noise. An average
over a longer time would decrease the electronic noise proportional to the integration
time [116]. This behavior is nicely visible in the right panel of Fig. 3.37, where we
plot the electronic noise versus pulse length. In the conversion from voltage to
photon-number via Gγ−V , the noise in terms of photon-number will increase linearly
9
10
8
4Var(Sy)
10
7
10
6
10
5
10 3
10
4
10
5
10
6
10
Photon number NL
7
10
8
10
9
10
Figure 3.36.: Variance
of detector signal as
function of photonnumber. The pulse
length is 1 µs.
3.6. Shot-noise-limited polarization detection
85
5
5
x 10
4
Electronic noise [x10−6 V2]
3.5
NL,SN [photons]
4
3
2
1
3
2.5
2
1.5
1
0.5
0
0
2
4
Pulse length [µs]
6
0
0
2
4
6
Pulse length [µs]
Figure 3.37.: Left: Noise equivalent photon-number as function of input pulse length.
The black line is not a fit, but the product of the fit to the electronic noise and the
photon-number gain (squared). Right: Electronic noise in integration window as function of pulse length. The solid black line is a fit ∝ (τpulse − τ0 )−1 , where τ0 ∼ 230 ns is
a small offset.
with pulse length, since Gγ−V is a linear function in time visible in Fig. 3.35.
The minimum results from the fact that the electronic noise is proportional to the
pulse length (up to a shift of 230 ns) and the photon-number gain to the integration
window. If we multiply fit functions for the electronic gain and the square of Gγ−V ,
we get the black line shown in the left panel of Fig. 3.37.
4. Dispersive Spin Measurements
Contents
4.1. Classical characterization of QND measurement . . . . .
88
4.1.1. Interaction strength . . . . . . . . . . . . . . . . . . . . . . 88
4.1.2. Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1.3. Spin resolution . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2. Measuring Zeeman coherences
. . . . . . . . . . . . . . .
94
4.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.2. How to measure Zeeman coherences in the presence of vector light shifts? . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.3. Calibration of G2 . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.4. Optical pumping dynamics . . . . . . . . . . . . . . . . . . 99
4.3. Dispersive atom number measurement
. . . . . . . . . . 101
4.4. Relation between scattering cross-section and observed
depolarization . . . . . . . . . . . . . . . . . . . . . . . . . 102
88
4. Dispersive Spin Measurements
In this chapter we characterize the atom-light interaction for the vector- and the
tensor-probe, which are used to measure the observables Jˆz and Jˆx , respectively.
Both coupling constants G1 and G2 are measured over a range of probe light detunings. For this we prepare atoms in the according spin state, i.e., along Jˆz or
Jˆx , via optical pumping as explained in Sect. 3.4. Simultaneous measurement of
macroscopic rotations and the number of atoms in the sample allows us to obtain
an absolute calibration of G1 and G2 . The effective interaction area for both probe
beams and atomic sample is determined using the theoretical expressions for G1
and G1 , as given in Sect. 2.3.2. At the end we explain how we use the dispersive
measurements to count the number of atoms in the atomic ensemble.
4.1. Classical characterization of QND measurement
We measure the per-atom rotation for the QND interaction ∼ Ŝz Jˆz and the depolarization introduced. Based on this classical measurement we estimate the sensitivity
of the quantum non-demolition measurement for Jˆz .
4.1.1. Interaction strength
We can measure the atom-light interaction for the QND measurement of Jˆz given
by the coupling constant G1 in the effective interaction Hamiltonian (2.22). The
coupling constant G1 can also be viewed as a per-atom rotation of the probe light
polarization. We use a completely polarized atomic spin state along the propagation
direction of the probe light. In terms of atomic operators this corresponds to h Jˆz i =
NA
1
2 h F̂z i = 2 . A pulse of off-resonant polarized light will experience a large rotation
of its polarization vector, compared to its quantum components. If the rotation angle
is small, i.e., φ 1, the input-output relation for the 45◦ -component of the Stokes
operator is
(out)
(in)
(in) (in)
(4.1)
Ŝy
= Ŝy + G1 Ŝx Jˆz .
The per-atom rotation is calculated from the measured rotation angle and the number
of atoms by
2φ
G1 ≡
,
(4.2)
NA
(out)
where the rotation angle is defined as φ ≡ h Ŝy
(in)
i / h Ŝx
i = G1 N2A , assuming the
(in)
input light is polarized completely along the vertical direction, i.e., h Ŝx i = NL /2
(in)
and h Ŝy i = 0. The number of atoms is determined using absorption imaging as
explained in Sect. 3.2.
For the experiments, we load atoms into the dipole trap as described in Sect. 3.1.3.
The ensemble is polarized by illuminating atoms along the quantization axis with
circularly polarized light, which is resonant to the transition F = 1 → F 0 = 1. A
small guiding magnetic field along z defines the quantization axis and prevents the
depolarization of the state through stray magnetic fields. To avoid the accumulation
of atoms in the F = 2 manifold, we apply light resonant to the transition F = 2 →
4.1. Classical characterization of QND measurement
89
ï7
G1×10
2
1
0.5
ï2
ï1
ï0.5 6 [GHz]
Figure 4.1.: Per-atom
rotation for different detunings calculated from
the rotation angle and
atom number. The black
solid line is a fit using the
theoretical expression for
G1 (∆, A) and leaving the
interaction area as only
free parameter. We find
A = 3.7(1) × 10−9 m2 ,
where the given error
indicates one standard
deviation.
F 0 = 2 via the MOT beams. Optical pumping into dark states is avoided through the
thermal motion of the atoms in the sub-wavelength polarization pattern produced
by the six MOT beams.
For the dispersive probing we send pulses of vertically polarized light, i.e., h Ŝx i =
NL /2, along the trap axis and record their polarization rotation, determined in
Sect. 3.6.1 according to Eq. (3.36). The detuning of the probe laser is changed in
various steps to measure the spectral dependence of the atom-light coupling.
In Fig. 4.1 we plot the measured coupling constants as a function of detuning ∆ from
the transition F = 1 → F 0 = 0. We can see the ∆−1 dependence of the interaction
strength, which is typical for the vector polarizability. The solid black line is a fit
using the theoretical expression for G1 (∆, A) as defined in Sect. 2.3.2 in Eq. (2.30).
The only free parameter is the effective interaction area A between light and atoms.
We find a value of 3.7(1) × 10−9 m2 . The concept of the effective interaction area
is very useful to simplify the inhomogeneous interaction of the Gaussian density
of atoms and photons. Assuming a top
p hat distribution of light and atoms we
can associate a hypothetical radius of A/π = 34 µm to the measured effective
interaction area A.
Optical depth
There is an alternative way of characterizing the amount of interaction strength
[63, 117, 118]. To state the amount of interaction independent on the detuning
used, we can define the on-resonant optical depth d0 . In a multilevel atom ’onresonance’ is ill-defined and needs some explanation. In our case, we have three
90
4. Dispersive Spin Measurements
relevant excited state hyperfine levels which are important for the interaction, which
are F = 1 → F 0 = 0, 1, 2. In the limit of large detuning, the main transition for the
dispersive interaction is F = 1 → F 0 = 0. The other two give contributions to the
vector polarizability of same magnitude and opposite sign. their sum vanishes for
detunings much larger than the excited state hyperfine splitting. The absorptive part
of the interaction is described in terms of the scattering cross section. In Appendix
B.1 we derive the scattering cross section in terms of atomic quantities as a function
of detuning [17]. We find the on-resonance scattering cross-section for atoms in F = 1
2
of σ0 = λπ , where we assume that the excited states are degenerate and interact with
linearly polarized light.
We define the effective on-resonance optical depth d0 in analogy with the usual
definition for a homogeneous medium:
d0 =
σ0
NA .
A
(4.3)
Note that for our system, which is far from homogeneous, and in addition has three
resolved spectral lines, the effective optical depth does not describe the transmission
characteristics of the medium. It is an important figure of merit, however, for comparing dispersive effects, and allows direct comparison with other atom geometries.
If we use the parameters we deduced from the per-atom rotation measurement above,
we can calculate the on-resonance optical depth for our atomic ensemble. We find
OD = 57(3) (we used NA = 1.09(3) × 106 , A = 3.7(1) × 10−9 m2 , and σ0 = 1.94 ×
10−13 m2 ). This optical depth is among the largest observed for such systems. Other
groups working with cold atomic samples have measured OD ∼ 16 [119] or OD ∼ 4
[120]. Higher optical depth of above 1000 have been observed in BEC.
4.1.2. Depolarization
Probing a polarized atomic state consecutively with a large number of pulses allows
us to monitor the decrease of spin polarization as a function of the number of probe
photons. In the simplest case the spin polarization is exponentially decaying as
(out)
h Jˆz
(in)
(NL ) i = h Jˆz
i e−ηγ NL ,
(4.4)
where ηγ is the depolarization per probe photon. We can define a total damage for
a number of photons NL as
η ≡ ηγ NL .
(4.5)
We measure the depolarization for the same range of detunings as in Fig. 4.1. For
this, we send a train of 900 pulses, each containing ∼ 2.5 × 106 photons, and record
the rotation signal. The left plot of Fig. 4.2 shows the measured ϕ for all detunings.
For the simulations, we start, as in the experiment, with a polarized atomic state
in F = 1 that is probed by a train of pulses. The shown traces are the rotation
signals calculated as the sum of the rotation signal from atoms in F = 1 and F = 2,
i.e., the signal we actually measure. The results are plotted in the right of Fig. 4.2
4.1. Classical characterization of QND measurement
q [mrad]
q [mrad]
100
100
80
80
60
60
40
40
20
20
0
0
91
200
400
600
Number of Pulses
800
0
0
200
400
600
Number of Pulses
800
Figure 4.2.: Angle of polarization rotation as function of probe pulses for different
probe detunings between −490 and −1990 MHz (from top to bottom) in steps of
100 MHz. Left: Measured rotation signal. Right: Numerical simulation of rotation
signal for NA = 1.1 × 106 , A = 3.7 × 10−9 m2 and NL = 3.1 × 106 per pulse.
The number of atoms NA ∼ 1.1 × 106 and the interaction area A = 3.7 × 10−9 m2
are fixed to the experimental values and the photon number was adjusted to achieve
comparable decay constants. This fitting finds a photon number larger by ∼ 24 %
compared to the photon-number in the experiment, when measured after the vacuum
cell. A small systematic error in the measurement of the actual number of photons,
interacting with the atomic cloud, is present due to the additional reflection of probe
light on the window of the vacuum cell. Although, this effect can only account for ∼
4% of difference. It is also possible that spontaneously emitted light is reabsorbed in
the atomic ensemble, which is often referred to as radiation trapping. The simulations
is not taking such effects into account.
The measured and the simulated signal both show a behavior which is not a single
exponential we have postulated in Eq. (4.4), but has some offset. It becomes apparent in the crossing of the curves at larger times. This can be understood as the
transfer of atoms from F = 1 into F = 2 by the probing light. Since, atoms where
polarized in F = 1 they remain partly polarized in F = 2. Atoms in F = 2 will not
be completely polarized, but a large population imbalance is still present. The red
detuned probe beam for the transition F = 1 → F 0 = 0, will be blue detuned for
the transition F = 2 → F 0 = 0. For increasing red detuning, we reduce the blue
detuning on the other transition. This is the reason why we see the almost flat traces
for the larges detuning of ∆ = −1990 MHz. This corresponds to a blue detuning of
+4850 MHz for atoms in F = 2.
The depolarization can be extracted from the measured decay of the signal by assuming the following model: Atoms contribute some signal from each hyperfine manifold.
The polarization in F = 1 will exponentially decay and a polarization in F = 2 will
be built up at the same rate. The measurable signal is then described as the sum of
92
4. Dispersive Spin Measurements
d [×10ï4]
10
5
2
depolar. in F=1
polar. in F=2
modified decay
sgl. exp. decay
1
ï2
ï1
ï0.5 6 [GHz]
Figure 4.3.: Results of the simulations of the decay of atomic polarization in F = 1 as
a function of the probe detuning. Conditions are the same as for the simulations in
the right plot of Fig. 4.2. The black circles show the depolarization of atoms in F = 1.
The red squares indicate the growth of polarization in F = 2. The blue crosses are
determined from the modified decay of the rotation signal as described in the text. The
black dots are the result of fitting a simple single exponential decay for small photon
numbers. For more details see text.
the decaying and growing signals of different amplitude
ϕ(p) = A1 exp (−ηγ NL ) + A2 (1 − exp (−ηγ NL ))
= A2 + (A1 − A2 ) exp (−ηγ NL ) .
(4.6)
To gain some intuition, we apply the above model for the modified decay on the
simulated data. There we have access to the depolarization of atoms in F = 1
and the build-up of polarization in F = 2. In Fig. 4.3 we plot the extracted total
depolarization η = ηγ NL using the above model and the loss of polarization in F = 1
for the simulated signals. The decay of the polarization in F = 1 is plotted as black
circles and the growth of the polarization in F = 2 as red squares. For detunings
close to resonance, i.e., small |∆|, we can measure the depolarization directly from
the exponentially decaying signal shown as black dots. This starts to fail for larger
detunings, where the depolarization in F = 1 is hidden by the growing polarization
in F = 2.
This model neglects the loss of trap atoms which is negligible at this photon number.
For another experiment we measured as rate of ∼ 3 × 10−12 per photon.
4.1. Classical characterization of QND measurement
93
d [×10ï4]
10
5
2
1
expected
measured
ï2
ï1
ï0.5 6 [GHz]
Figure 4.4.: Measured and expected polarization decay of the data shown in the left
plot of Fig. 4.2.
The depolarization rate from the measurements (cf. left of Fig. 4.2) are shown in
Fig. 4.4. We used the single exponential decay of the first part of the signal. This
gives results which are close to the depolarization rate calculated with the master
equations for the range of detunings we are interested in, i.e. −1.5 < ∆ < −0.5 GHz.
Fitting with the modified decay model (4.6) gives results with large error bars for
large detunings, where the signal is almost flat.
4.1.3. Spin resolution
Once we have characterized the atom-light interaction in terms of the per-atom
rotation and the destruction, we can estimate the sensitivity of the QND probing
from this measurement of mean values, which we call a classical measurement. For a
dispersive spin measurement the sensitivity is in the best case limited by the quantum
noise of the probe light. The minimum angular resolution we get from a coherent
polarization state, as opposed to a squeezed polarization state, is
∆φmin = √
1
.
NL
(4.7)
94
4. Dispersive Spin Measurements
Introducing this expression into (4.2), we can find the smallest number of spins that
can be distinguished from the light shot noise of the dispersive probing 1 .
∆NA = √
2
.
NL G1
(4.8)
A QND measurement is considered sensitive if it allows to resolve the quantum
noise of the measured state. In terms of the signal-to-noise ratio, introduced in the
theory chapter in Eq.
√ (2.67), this corresponds to ζ = 1.√A coherent spin state has a
quantum noise of NA . Hence, starting from ∆NA = NA the QND measurement
reaches an appreciable sensitivity. However, this definition only makes sense if we
keep depolarization effects in mind. Therefore, it is important at which level of
destruction η the QND measurement starts to be sensitive.
In Fig. 4.5 we plot ∆NA as a function of the depolarization η, changing the number
of probe photons. For each point we calculate the ∆NA from the number of photons
NL sent via Eq. (4.8). The depolarization is computed from the amount of decay
observed in the rotation signal (cf. Fig. 4.2) for this particular number of photons.
For example, for the data at ∆ = −490 MHz (black circles) we can have a sensitivity
of ∆NA ' 900 atoms at NL ' 1.5 × 108 photons, which cause a damage of 6 %.
We can derive a relation between η and ∆NA by introducing Eq. (4.8) into Eq. (4.5)
and get
4
η(∆NA ) = ηγ
.
(4.9)
(G1 ∆NA )2
In Fig. 4.5 we show η(∆NA ) for detuning of −490 and −590 MHz. In addition we
−2
plot a curve with η ∝ d−1
0 (∆NA ) , where the black line is for d0 = 57, the red and
the blue for d0 = 25 and d0 = 100, respectively.
For our atomic ensemble of NA = 1 × 106 we indicate the projection noise level with
a vertical gray line at ∆NA = 1 × 103 . This level of sensitivity can be achieved
with only ∼ 5 % of destruction. As we have seen in Appendix A by simulating the
quantum noise behavior for our experimental parameters, a damage of about 20 %
is the optimum trade-off between the information gain and state destruction for the
best spin squeezing. The measurements here confirm that we have enough dispersive
interaction at a reasonable amount of absorptive interaction to perform sensitive
QND measurements. Nevertheless, this experiment is not yet proof of having a
spin detector which is quantum noise limited and allows us to produce a squeezed
spin state and the associated macroscopic entanglement. This characterization is
performed in the Sect. 7.1.
4.2. Measuring Zeeman coherences
In this section we present measurements of the per-atom rotation caused by the
tensorial term of the polarizability tensor G2 . We will use this interaction to measure
1
For a photon detector this would be the noise equivalent photon number to overcome the electronic
noise, which is in our case the light shot noise.
4.2. Measuring Zeeman coherences
95
Figure 4.5.: Sensitivity:
measured
depolarization versus minimum
detectable number of
spins ∆NA for detunings close to resonance.
The circles and the
squares show the data
for −490 and −590 MHz,
respectively. The black
line has the expected
∆NA−2 -dependence
for
the measured optical
depth of 57(3). As comparison we plot lines for
the case of smaller (25)
and larger (100) optical
depth as compared to
the present situation.
the spin polarization of the coherent spin state we are using for the spin squeezing
measurements in Chap. 7.
4.2.1. Introduction
Using the tensorial term in the dipole interaction Hamiltonian is the same as using the
vectorial, with the difference that it allows to measure the coherences between the two
hyperfine ground states |1, −1i and |1, +1i. In the case of the Jˆz measurements, the
physical effect was circular birefringence or paramagnetic Faraday rotations. Now,
we want to use the effect of linear birefringence as in a wave-retarder, for instance. A
spin state along Jˆz has a symmetry axis along z. The coherent superposition state,
we want to measure now, has a symmetry axis which is orthogonal to the z in a
direction which depends on the way we prepared the superposition (cf. Sect. 3.4).
Experimentally, we can make very sensitive measurements of Ŝy , so it is convenient
(in)
(in)
to use circular light for the probing, i.e., h Ŝz i = NL /2 and h Ŝy i = 0. This
circular light will acquire an Ŝy -component proportional to the Jˆx component of the
state. In terms of Stokes vectors, we can write the input-output relation of the light
entering and leaving the atomic cloud as
(out)
Ŝy
(in)
= Ŝy
(in) (in)
(in) (in)
+ G1 Ŝx Jˆz + G2 Ŝz Jˆx ,
(4.10)
(in)
where the second term for the case of h Jˆx i = NA /2 and circular polarization is
much bigger than the first. Also, for a detuning where G1 = 0 we can neglect the
first term. The measured signal is then directly proportional to Jˆx and we can write
(out)
Ŝy
(in)
= Ŝy
(in) ˆ(in)
Jx
+ G2 Ŝz
,
(4.11)
Analogous to the case of the vector probe we can measure the per-atom rotation or
coupling constant G2 by recording both the rotation angle of the light polarization
96
4. Dispersive Spin Measurements
0
s [rad]
3
s [rad]
6
s [mrad]
5
ï0.05
2
4
ï0.1
1
ï0.15
0
3
2
1
ï0.2
ï500 ï400 ï300 ï200 ï100
6 [MHz]
ï1
0
100
200
6 [MHz]
300
400
0
432
462
6 [MHz]
492
Figure 4.6.: Spectrum of tensor interaction. The solid black line is calculated through
the tensor polarizability α(2) and the blue circles are numerically calculated values for
the linear-birefringence of the atomic sample. Both show a reasonably good agreement
which ensures that the approximations applied in deriving α(i) are correct. We plot the
polarization rotation obtainable with realistic experimental parameters of NA = 1×106
and A = 4 × 10−9 m2 . Left: Region we use for the experiments which has reasonable
dispersive interaction. The large green circle at −200 MHz marks the point where
we simulated the time behavior in Fig. 4.7 Center: Region between the excited state
hyperfine levels. The point, marked with the square, is the position where G1 = 0.
There, dispersive effects due to G2 are strong but so is absorption. Right: Region
around the point of vanishing G1 (∆ = 461.7 MHz). This would be ideal place to probe
Jˆx with a purely tensorial interaction. Unfortunately, the dispersive interaction very
small. Note, the difference in the scale for the rotation angles.
and the number of atoms in the ensemble. The rotation angle is now defined as the
(out)
(in)
ratio of Ŝy - and Ŝz -components: ψ ≡ h Ŝy i / h Ŝz i = G2 N2A .
Ideally, we would like to measure at a spectral position where the vector contribution
is vanishing. There exist two points in the spectrum where we have G1 = 0 and
G2 6= 0. One of them is located between the resonance F = 1 → F 0 = 0 and
F = 1 → F 0 = 1 at ∆ = 35.84 MHz. The other is situated at ∆ = 461.7 MHz. In the
two right plots of Fig. 4.6 we show the spectrum of G2 around these two points. For
the first one at 35.84 MHz the dispersive interaction is very large. Simultaneously,
the vicinity of the resonances implies a large absorptive interaction. Hence, we
cannot use this point for a non-destructive measurement of the state. For the point
at 461.7 MHz the situation is better in terms of absorption, but also the dispersive
interaction is particularly small. The reason is that also G2 has a zero at about
501.8 MHz. Much better is the region below the resonance F = 1 → F 0 = 0. This is
shown in the left panel of Fig. 4.6. There, an appreciable dispersive interaction can
be obtained at reasonably low photon scattering rates. The price we have to pay is
that the vector interaction is not vanishing and we get contributions from the term
∼ Ŝz Jˆz . We explain how we can overcome this problem in the next section.
4.2. Measuring Zeeman coherences
97
4.2.2. How to measure Zeeman coherences in the presence of vector light
shifts?
A circularly polarized probe pulse would cause a precession of the atomic state about
Jˆz which interchanges Jˆx and Jˆy . Physically, we can understand this precession as
the result of a different AC Stark shift of the magnetic substates in the ground state
from the circular probe light. They lift the degeneracy of the two components of the
coherent superposition state, i.e., |1, −1i and |1, +1i, and the state starts to precess
at a rate related to this energy splitting. We could assign a fictitious magnetic field
to the energy shifts and interpret the interaction as being purely magnetic [121].
One possibility to avoid this problem is to probe the atomic state successively with
pulses of left and right circular light. In effect this will “balance” the spin state along
Jˆx and no effective precession is taking place. Though very elegant, the solution
demands a separate control of an additional light field with exactly orthogonal polarization. Later, when we measure the projection noise of a the coherent spin state
in presence of non-vanishing tensor light shifts, we will apply this strategy. Now,
since we are interested only in the magnitude of the large spin polarization, it is
much simpler to apply a magnetic field along the symmetry axis of the state.
A magnetic field which is large enough to dominate the fictitious magnetic field will
stabilize the prepared coherent superposition while we measure its magnitude Jˆx .
We simulate both situations, i.e., with and without a magnetic field, and compare
their results. For the simulations we start with a coherent superposition of |1, −1i
and |1, +1i which has a mean value of h Jˆx i = NA /2, where we assume NA = 1 × 106
and an interaction area A = 4 × 10−9 m2 , as in the calculations shown in Fig. 4.6.
In the left of Fig. 4.7 we plot the rotation signal as a function of probing time. In
the case of no magnetic field, the state precesses due to the vectorial light shifts.
With the magnetic field, we suppress the influence of the vector light shifts and can
measure the signal over a long time scale. Both situations are equivalent in terms
of the information deduced. The decay we observe is the depolarization introduced
through the probe light. In the right plot, we show the evolution of the pseudo-spin
projections. In the case of no magnetic field we see the expected precession of the
state about the z axis. The Jˆz component which is increasing over time is the result
of optical pumping during the probing process with circularly polarized light. In the
experiment we probe atoms over much shorter timescales to avoid the reduction of
the atomic polarization. Here, we show the evolution for longer time to see all the
processes.
The precession of the state due to the influence of the vectorial polarizability can
also be used to produce a state along any direction in the Bloch sphere. For this, one
would have to change the illumination, in order to have a constant intensity over the
whole sample. For our geometry this might be a difficult undertaking. In smaller
ensembles this manipulation is already demonstrated [122, 123].
98
4. Dispersive Spin Measurements
40
s [mrad]
0.5
Jx [NA]
30
20
10
0
0
ï10
ï20
Jx
ï30
ï40
0
Jy
Time [ms]
0.25
ï0.5
0
Jz
0.25
Time [ms]
Figure 4.7.: Simulation of rotation signal ψ and pseudo-spin projections as a function
of time (photon number) with and without guiding magnetic field B = (0, 0.5, 0) mG.
The tensor-probe light is tuned ∆ = −200 MHz away form the F = 1 → F 0 = 0
transition. Left: The green solid and the blue dotted line are the respective signals
with and without magnetic field along y. Right: Projection along the pseudo-spin
directions Jˆx , Jˆy , and Jˆz for the case of no bias field. The blue dotted line is the
projection along Jˆx when the magnetic field is applied. In this case the magnitude of
Jˆy ∼ Jˆz ∼ 0 and not shown in the plot.
4.2.3. Calibration of G2
Now we measure G2 by polarizing the atomic sample along Jˆx in a guiding magnetic
field. The circular probe light detuning is changed over one order of magnitude.
The number of atoms is measured via absorption imaging. In this way we can get a
precise measure of G2 as the per-atom rotation.
The atomic state is prepared as explained in Sect. 3.4 by optical pumping atoms with
x-polarized light propagating orthogonal to the sample axis. The light is resonant to
F = 1 → F 0 = 1 and transfers atoms optically into the hyperfine ground state |1, 0ix
which is a dark state for this polarization and detuning. The index x indicates the
quantization axis is along x, parallel to the light polarization. In the usual notation, i.e., with the quantization axis along z, the state is the coherent superposition
between |1, +1i and |1, −1i
|Ξi =
1
2NA /2
[|1, −1i − |1, +1i]⊗NA
(4.12)
In terms of pseudo spin operators we find hΞ| Jˆx |Ξi = −NA /2.
After loading the atomic ensemble we switch on a small magnetic field of a Bx ∼
200 mG along x and send linearly polarized light through the imaging optics to the
atomic cloud. We choose a pumping time of 300 µs, which is sufficiently long to
ensure a high degree of optical pumping. Once the coherent spin state is prepared,
4.2. Measuring Zeeman coherences
99
we send a train of circular probe pulses along the atomic ensemble and record their
polarization rotation ψ. The number of atoms is measured by absorption imaging
after the probing. With Eq. (4.11) we can calculate the coupling constant G2 , which
is plotted in Fig. 4.8 for all measured detunings.
G [×107]
2
10
1
0.1
ï500
ï300
ï200
ï100
ï70
6 [MHz]
Figure 4.8.: Measured tensorial coupling strenth G2 as
a function of probe detuning.
The black line is a fit using expression (4.13). The
free parameters are interaction area A = 3.0(3) × 10−9 , a
frequency off-set due to light
shifts ∆0 = 4(6) MHz, and
an initial imbalance of the polarimeter G2,offset = 6(2) ×
10−9 .
Using Eq. (4.11) we can construct the fitting function
G2,meas = G2 (∆ − ∆0 , A) + G2,offset ,
(4.13)
where G2 (∆, A) is the theoretical expression of the tensorial interaction strength
given in Eq. (2.31), with an effective interaction area A, a frequency offset ∆0 , and
(in)
an offset G2,offset caused by an initial imbalance of the polarimeter h Ŝy i =
6 0. The
frequency offset ∆0 might be caused by the AC Stark shift of the trapping light.
The result of the fitting reveals an interaction area A = 3.0(3) × 10−9 m2 which is
comparable to the one we measured for the vector probe. The shift in the detuning
is ∆0 = 4(6) MHz. This in good agreement with the light shifts, we measured for
example in the state tomography in the trap (cf. Chap. 6) or the optical pumping
dynamics in the next section. The initial imbalance of the polarimeter corresponds
to an interaction strength of G2,offset = 6(2) × 10−9 .
The data points follow over the whole range of detunings the typical ∆−2 behavior of the tensorial polarizability. We conclude that the experiment successfully
shows a measurement of the pseudo-spin component Jˆx or equivalently the coherences between magnetic substates by using the tensor term in the dipole interaction
Hamiltonian.
4.2.4. Optical pumping dynamics
We apply the measurement of spin state polarization to monitor the evolution of
the optical pumping into the coherent superposition state. For this we vary the
100
4. Dispersive Spin Measurements
duration of the optical pumping light and measure the amount of spin polarization.
As already mentioned, we are able to record the pumping light with a calibrated
CCD camera which gives a measure of the local intensity. The time evolution of the
spin polarization together with numerical simulations can be used to estimate the
degree of optical pumping.
The measurement proceeds in the same way as before, except we stop the optical
pumping after a variable time t and measure the signal of the tensor probe.
We model the pumping process by a simple three level system where we pump atoms
from an initial into a final state by exciting an intermediate level, which decays into
both the final and the initial levels. This gives an exponentially saturating growth
of spin polarization
ψ(t) = ψ∞ (1 − exp (−k(t − t0 ))) ,
(4.14)
where k is the rate of optical pumping, t0 an offset in pumping time, and ψ∞ the
final amplitude of the signal. In the left plot of Fig. 4.9 we show the measured spin
polarization as a rotation signal of the tensor probe and a fit to the data points using
Eq. (4.14). We find the following parameters: k = 4.7(3) × 104 s−1 , ψ∞ = 17.1(4),
and t0 = 1.7(5) µs. The offset in pumping time is an experimental artifact of the
preparation of the pumping light, i.e., a delay in the switching of the AOM used.
After ∼ 100 µs the spin polarization reaches a steady state.
20
s [arb. units]
J [norm.]
x
1
15
0.8
0.6
10
0.4
6=0MHz
6=ï1MHz
6=ï2MHz
6=ï3MHz
6=ï6MHz
norm. data
5
0.2
0
0
50
100
150
t [µs]
0
0
50
100
150
t [µs]
Figure 4.9.: Left: Measured rotation signal from tensor probe as a function of optical
pumping time t. Right: Master equation simulations of the optical pumping for different
detunings of the pumping laser from resonance. We used: Ipump = 0.44 Wm−2 and
Ideplete = 3.5 Wm−2 .
For a better understanding of the pumping process, we use our knowledge of the local
light intensity as an input parameter for numerical master equation calculations. In
this way, we can estimate the amount of optical pumping from the shape of the
4.3. Dispersive atom number measurement
101
measured rotation signal. We also have access to the amount of light used for the
depletion of F = 2 during the pumping. The only free parameter in the calculation
is the detuning of the pumping laser. This is only known within a few MHz since we
prepare atoms in the dipole trap, where they experience an AC Stark shift. In the
right of Fig. 4.9, we show the results of the simulations for a set of detunings. We
find the best overlap of simulation and normalized rotation signal for a detuning of
−3 MHz of the pumping light.
The good agreement between simulations and measured data is a first indication
of a pumping efficiency close to unity for the atoms in F = 1. However, there are
other effects which can not be revealed by this measurement. Our main concern is
the depletion laser of F = 2 which could pump atoms into a dark state in F = 2.
This effect should be small since the depletion light is applied from six different
directions and forms a polarization pattern on the sub-wavelength scale. Atoms will
move through this pattern due to their thermal energy. If we estimate the rate of
light scattering for atoms in F = 2 we get values of a few percent of the natural
linewidth Γ = 2π × 6.066 MHz. If this is large compared to the time an atom needs
to travel a distance to experience a different local polarization, e.g., ∼ 100 nm, the
atom cannot adiabatically follow. For our temperatures, we estimate this time to be
around 1 µs. From this we conclude that the thermal motion of atoms is sufficient to
largely suppress the pumping into dark states. For an order of magnitude estimate,
it is important to know that in the worst case, i.e., a single fixed polarization for the
depletion laser, at maximum of 10 % of the atoms are pumped into a dark state in
F = 2. In summary, we conclude that the effect of ’trapping’ atoms in a dark state
in F = 2 is a small effect of a few percent of the total number of atoms. Hence, the
measured degree of optical pumping reflects realistic values close to unity.
4.3. Dispersive atom number measurement
Throughout the work we will use paramagnetic Faraday rotation measurements as
a fast and nondestructive way of determining the number of atoms. We will call
this technique dispersive atom number measurement (DANM). For this we measure
the per-atom rotation G1 as described above, through the macroscopic rotation from
a spin polarized atomic state and the atom number by absorption imaging. This
allows us to have a quick and precise reference of the number of atoms in the atomic
ensemble without the destructiveness of absorption imaging.
In a single trap loading, we are able to perform many experiments and measure the
number of atoms present at each of them. Before or after any large experimental
run, we repeat the calibration to account for daily fluctuations of light intensities
and alignments.
The DANM has the following pulse sequence: the sample is spin-polarized along z
by on-axis optical pumping with a 50 µs pulse of circularly-polarized light tuned to
the F = 1 → F 0 = 1 transition. At the same time, light resonant to the F = 2 →
F 0 = 2 transition (via the MOT beams) prevents accumulation of atoms in F = 2.
We define a quantization axis by applying a small bias field of ∼ 100 mG along z.
102
4. Dispersive Spin Measurements
Probe pulses with NL = 1 × 105 − 5 × 105 tuned 500 − 1000 MHz to the red of the
F = 1 → F 0 = 0 transition are used to measure the rotation angle φ = NA G1 . No
appreciable atomic depolarization could be observed experimentally after 20 probe
pulses, which is supported by calculations that give a destruction of 0.08 % for this
number of pulses.
4.4. Relation between scattering cross-section and observed
depolarization
Sometimes in the literature [63, 124, 125, 117, 118, 126] the depolarization η, we
have measured above in Sect. 4.1.2, is related to the atomic scattering cross-section
σ(∆) and an effective interaction area A . The relation used is
η=
σ(∆)
NL ,
A
(4.15)
where we can relate the depolarization per photon to the ratio of scattering crosssection and effective interaction area
ηγ =
σ(∆)
.
A
(4.16)
Making this straight connection is not always valid. A scattering event will not
inevitably destroy the coherence of the state and reduce the polarization. Using
the master equation we simulate polarization loss ηME of atoms in F = 1 from
light scattering out of a probe pulse with NL photons at a detuning ∆ and with
an effective interaction area A. This we compare to the value η which is calculated
from expression (4.15) for the same set of parameters. The ratio between the value
(in)
(out)
for the master equation simulations ηME ≡ ln h Jˆi i − ln h Jˆi
i, for small NL and
different i = x, z and Eq. (4.15) are shown in Fig. 4.10 for a polarized spin state
along Jˆz , i.e., |1, 1i, and along Jˆx , probed with linearly and circularly polarized
light, respectively.
We see that there is a large difference between the estimated depolarization from
the scattering cross-section and the actual value. In the case of the polarized state
along z we see that only every third scattering event will lead to a polarization
loss of the atomic state. In order to use the scattering cross section to predict the
depolarization of an atomic ensemble one has to apply a correction. The correction
factor is the ratio κ ≡ ηME /η we plot in Fig. 4.10. The corrected expression for the
depolarization is
σ(∆)
η = κ(∆)
NL ,
A
where the correction has to be calculated depending on the spin state. Also the
scattering cross-section σ(∆) is different for different states.
4.4. Relation between scattering cross-section and observed depolarization
1
103
g
0.8
0.6
0.4
0.2
0
ï2
ï1.5
ï1
ï0.5
6 [GHz]
Figure 4.10.: Ratio of depolarization rates ηME and η for master
equation simulations and using the
scattering cross section. The upper and the lower curve show the
ratio for a spin state along Jˆx and
Jˆz , respectively.
5. Magnetic Field Measurements
Contents
5.1. Spatially integrating magnetic field measurements . . . 106
5.1.1. Magnetic field measurement via MOT position . . . . . . . 106
5.1.2. Magnetic field measurements via collective Larmor precession107
5.2. Spatially resolved magnetic field measurements . . . . . 110
5.2.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2.2. Spatially-resolved Larmor precession . . . . . . . . . . . . . 111
5.2.3. Measurements
. . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.4. Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
106
5. Magnetic Field Measurements
Atoms are highly sensitive to magnetic fields. This can be useful, because an atomic
cloud can be used as a highly sensitive magnetometer. It also means that experiments
such as spin squeezing require good control over the magnetic field.
We use three different methods to determine the magnetic field around the atomic
ensemble. All of them use the atomic sample as magnetic field probe, but in different
ways. We start with a very simple measurement of the MOT position as a function
of applied bias fields in Sect. 5.1.1. This method gives an accuracy of tens of mG. In
Sect. 5.1.2 we use Larmor precession, measured via paramagnetic Faraday rotations.
This increases the sensitivity dramatically to sub-mG precision. Finally, in Sect. 5.2
we develop a √
method to measure the field spatially resolved with sensitivities down
to ∼ 150 nT/ Hz and at a resolution of 50 µm.
5.1. Spatially integrating magnetic field measurements
5.1.1. Magnetic field measurement via MOT position
The main idea of this approach is to measure uniform stray magnetic fields from the
position of the atomic cloud in the MOT. One example of a stray field is the earth
magnetic field, which we have to compensate for. In this, we assume that the stray
magnetic field and the compensation field, we apply, are both uniform over the size
of the atomic cloud 1 . Furthermore, we assume that the magnetic field, produced
by the anti-Helmholtz coils, has a constant gradient, i.e., grows linearly going away
from the center in any spatial direction. The light fields of the MOT are suppossed
to produce equal radiation pressure in all directions. Thus, no systematic error in
the MOT position due to the light fields is present.2 We can write for the field along
a single direction
∂Bz
Bz (z) =
(z − z0 ) + Bz,0 ,
(5.1)
∂z
z
where ∂B
∂z is the gradient of Bz along z, Bz,0 a constant offset, and z0 the displacement from the geometrical center of the quadrupole coils. If the light fields of the
MOT beams are well balanced, then the position of the atomic cloud in a MOT
potential is the place of vanishing magnetic field. We can write
zM OT = z0 −
Bz,0
∂Bz
∂z
.
(5.2)
The offset field can be expressed as the sum of a stray-field Bz,s , we want to determine, and a known contribution Bz,c from the compensation coils, Bz,0 = Bz,s +Bz,c .
z
Hence, measuring the MOT position as a function of the gradient ∂B
∂z and the applied compensation field Bz,c , will reveal the unknown offset field. At Bz,0 = 0 the
position of the atomic cloud becomes independent of the applied gradient.
1
We also have to ensure that the center of the quadrupole coils and the dipole trap are close
together. If not, we just measure the stray field at the geometric center of the quadrupole coils.
2
This assumption was tested epxerimentally by optimizing the MOT alignment for a stable optical
molasse.
5.1. Spatially integrating magnetic field measurements
107
Figure 5.1.: Fluorescence images of atomic cloud position in MOT for different gradients (indicated as current in the MOT coils in the left upper corner). Horizontal and
vertical axes show the z and x direction in pixel, where one pixel= 9.4 µm.
We plot in Fig. 5.1 fluorescence images taken along the y-direction for different
applied gradient fields, to illustrate the method. The atomic cloud position depends
on gradient current due to an uncompensated x field. The magnetic field gradient is
reduced from left to right by 50 % of its initial value.
To study the effect in more detail, we take a series of measurements for different
gradients and scan the compensation fields along x and z. For values of the gradient
coil current between 3 and 6 A, we plot the results in Fig. 5.2. As expected from
Eq. (5.2) the center of the atomic cloud is a function of applied field. The slope is
inversely proportional to the applied gradient.
We make an estimate on the accuracy of this method from Fig. 5.2. For our setup,
the accuracy of this method is limited by the precision of the current in the coils.
With uncertainty of about 10 mA for both coils. This converts into a magnetic field
sensitivity of several tens of mG. In summary, we see that this method can only give
a coarse estimate of the bias field amplitude.
5.1.2. Magnetic field measurements via collective Larmor precession
More precision can be gained by using internal atomic states. This, will be the first
application, in this thesis, of the atomic ensemble as a sensitive magnetic field probe
- a magnetometer. For this purpose we use our ability to polarize the atomic ensemble along the quantization axis z, described in Sect. 3.4, and the non-destructive
measurement of spin polarization along z, described in Chap. 4.
This gives us a method to determine the fields orthogonal to the quantization axis
z with high accuracy. The compensation field Bk,c , necessary to cancel the existing
stray field Bk,s for k = {x, y}, can be found by scanning the coil current (compensation field) and measure the Larmor frequency as a function of Ik,c . For this, we
polarize the atomic sample along z and send a train of hundreds of far detuned probe
pulse through the cloud, as demonstrated in Chap. 4. The rotation signal will be
modulated in time due to the Larmor precession of the polarized spin state about
the magnetic field axis.
108
5. Magnetic Field Measurements
525
335
330
475
Z MOT Center [pixel]
X MOT Center [pixel]
500
450
425
400
320
315
310
375
350
−30
325
0
30
60
X Coil Current [mA]
90
305
−30
0
30
60
Z Coil Current [mA]
90
Figure 5.2.: MOT position as function of compensation coil current for x-compensation
coils (left) and z-compensation coils (right). The observation direction is y. A single
pixel on the CCD corresponds to 9.4 µm at the position of the atoms. The MOT coil
current is changed from 3 A (circle), 4 A (square), 5 A (cross) and 6 A (triangle), which
corresponds to a change from 21 G/cm to 42 G/cm. Exemplary, we show the linear fit
for the case of 6 A in the black solid line.
The light parameters are chosen in a way that we minimize the damage of the atomic
spin polarization, but probe rather non-destructively. This allows to observe the spindynamics in real time. Practically, we us NL ∼ 106 at a detuning |∆| > 1 GHz as in
Chap. 4. For the optical puming we use a rather short pulse of ∼ 10 µs to have an
instantaneous preparation on the time scale of the precession frequency. The amount
of optical pumping is of minor importance for this kind of measurement.
The expectation value of the spin component along the quantization axis will describe
a damped oscillation
Bz2 + Bx2 + By2 cos (γB t) e−ηt
h F̂z i = NA
,
(5.3)
B2
where Bk are the field amplitudes along the k-direction, γ is the gyromagnetic ratio3 ,
and η describes the loss of collective spin polarization. The collective spin polarization could be reduced due to inhomogeneous magnetic fields which would dephase
different parts of the atomic sample to each other.
The observed signal can be written as
h Ŝy i = h Ŝy i∞ + h Ŝy iosc cos (γB t) e−ηt ,
(5.4)
where h Ŝy i∞ ∝ Bz2 /B 2 is the rotation signal after the oscillations have ceased and
h Ŝy iosc ∝ (Bx2 + By2 )/B 2 is the amplitude of the oscillation at t = 0.
3
For atoms in the F = 1 manifold γ = 2π700 kHz/G.
5.1. Spatially integrating magnetic field measurements
109
Precession Frequency [kHz]
12
10
0
8
6
4
0
2
0
35
37
39
41
Y Coil Current [mA]
43
45
0
0.1
0.2 0.3 0.4
Time [ms]
0.5
Figure 5.3.: Left: Larmor precession frequency measurement for the y compensation
coil. We deduce a conversion between current and magnetic field for
q the y coil of
2 + B2
3.62(5) G/A. For the stray field we find By,s = 145.6(2) mG and Bx,s
z,s =
+1.2
1.2 −1.1 mG. Right: Time traces of Larmor precession for two different compensation
fields along y. Here, they serve only for illustration and were not taken at the same
time as the data for the left figure. Nevertheless, the bias field components By are
the same as used for the corresponding points in the left plot, indicated by the colored
squares. The deduced z bias field is therefore slightly different than what is shown in
the left plot. We find Bz,s = 4.3(4) mG and Bz,s = 3.1(3) mG for the upper and lower
plot using Eq. (5.6).
At the right side of Fig. 5.3, we plot two examples of the Larmor precession signals
when applying a compensation field along y. The fast decay of the oscillation contrast
is the result of magnetic field inhomogeneity over the length of the atomic sample.
One reason could be the inhomogeneity produced by the compensation coil itself
which are far from being in a Helmholtz configuration. Another reason could be
the existence of gradients from external sources, which will be discussed in the next
section in greater detail. The different phases in the observed signal, i.e., the red
data points start from above and the green from below, are due to the different delay
time between the preparation and the probing.
From the measurement of the Larmor frequency as a function of the compensation
fields, as plotted in the left panel of Fig. 5.3, we get several pieces of information. To
understand them better, we can write the expression for the frequency as function
of field components, determined from (5.3)
q
γ Bx2 + By2 + Bz2
fL =
.
(5.5)
2π
First, we can deduce the relation between applied current and resulting magnetic field
from the linear slope on either arm of the “V”. The slope is linear in a region where
110
5. Magnetic Field Measurements
By Bx , Bz , which gives a direct measures of By . Second, the compensation field
for the scanned direction, i.e., By = By,c + By,s = 0, can be found from the position
of the minimum in fL . Third, we can estimate the uncompensated
p field in the plane
orthogonal to the scanned direction, i.e., in the present case Bx2 + Bz2 . By fitting
the data with the theoretical expression (5.5), we can deduce all three quantities.
In Fig. 5.3 we show the resulting fitted function for i) the case of no remaining
orthogonal components (Bx2 + Bz2 = 0) and ii) the exact expression, in the solid and
the dotted lines, respectively. The precision in the determined compensation field in
i) and ii) is 0.1 %, or in absolute terms, 200 µG for the presented data in Fig. 5.3.
The orthogonal components are less well resolved. The experimental errorp
in measuring very low frequencies is large and prevents a reliable measurement of Bx2 + Bz2 .
For this reason, we use this method in an iterative manner, compensating x and y
several times. To determine the remaining z component we compute
s
h Ŝy i∞
Bz = Bx
,
(5.6)
h Ŝy iosc
where we assumed Bx2 By2 . The accuracy of this method is also rather poor,
compared to the compensation field scan and measurement the fL . The stray fields
we found for the example in Fig. 5.3 are given in its caption.
In summary, we have seen that Larmor precession measurement gives a precise measure of the magnetic field. We can reach sub-mG sensitivities in the field component
which is scanned. Furthermore, we are still limited by magnetic field inhomogeneities
which are not easily resolvable with this method. A possible way to measure the inhomogeneity of the field would be a Fourier analysis of the precession signal as proposed
in [127]. Instead, we are using another method to measure the magnetic fields with
spatial resolution, which will be explained in the next section.
5.2. Spatially resolved magnetic field measurements
We have talked about the inhomogeneity of the magnetic field several times in the
course of this chapter. On our journey to the realization of spin squeezing (Chap. 7)
we had to learn that the magnetic field environment of the trap apparatus is more
complicated than expected and contains spatially changing magnetic fields. Our
atomic sample has a length of ∆z ∼ 10 mm and dephasing happens at a rate4 of
dφ/dt == ∆z2γ∂B/∂z, where γ is the gyromagnetic ratio. Practically, our experiments will take about 100 µs, so that it will be sensitive to gradients of the order of
∂B/∂z = 2π(100 µs 1cm 2 2π700kHz/G)−1 ∼ 7 mG/cm, where we assumed a complete dephasing for a rate of dφ/dt = 2π/100 µs. This demands a well controlled
magnetic field environment in terms of spatial field properties. Otherwise, a spin
polarized atomic state would dephase very fast and the time for its use, e.g., as a
4
The above expression for the dephasing rate assumes an extended sample spread over a length
∆z which is exposed to a magnetic field gradient of ∂B/∂z. The phase difference between the
beginning and the end of the atomic sample after a time t is φ = 2ωL t = 2γ∆z∂B/∂z t, where
ωL is the Larmor precession frequency.
5.2. Spatially resolved magnetic field measurements
111
magnetometer, is limited to unnecessarily short periods. At the same time, the extended atomic ensemble is an attractive system for measuring magnetic field with
high spatial resolution over a large spatial range. Here, we present a technique which
can measure magnetic fields down to µG over several millimeter with micrometer resolution.
In this section we present a method which uses the atomic cloud to map the magnetic
field along the sample. We further show measurements in which we apply this method
to cancel the magnetic field gradient present in the trapping setup. Part of this
chapter is published in 128.
5.2.1. Background
There exist a number of approaches to measure magnetic fields with high sensitivity and spatial resolution. Scanning Hall probes can reach sub-micron resolution
with µT sensitivity [129, 130, 131]. On the other hand, superconducting quantum
interference devices (SQUID) reach sensitivities down to 0.1 pT but are limited in
spatial resolution to tens of microns [132, 133, 134]. Ultra-cold atomic ensembles
seem to bridge the gap by offering high magnetic field sensitivity (10−10 − 10−11 T)
together with micron size spatial resolution over length scales of a few hundred microns [135, 136, 137]. The most sensitive magnetometers today use room temperature
vapor cells and measure
√ Larmor precession of spin polarized atoms. They can reach
sensitivities of 0.5 fT/ Hz at a very poor spatial resolution of tens of millimeters. A
more complete overview can be found in [138] .
5.2.2. Spatially-resolved Larmor precession
Spatially varying magnetic fields are detected measuring Larmor precession by means
of sensitive absorption imaging. Similar to room temperature vapor cell magnetometers [139], Larmor precession give a sensitive and direct measure of local magnetic
fields. However, micro Kelvin cold atoms held in a dipole trap, compared to hot vapor cells, are much better localized (< 20 µm) and their thermal motion is reduced,
which allows for spatial resolution within the atomic ensemble.
Locally, the measurement is similar to a Ramsey spectroscopy sequence: The initial
state for a single atom ρo ≡ |Ψd i hΨd | is a magnetically sensitive coherent superposition of hyperfine ground states. It is produced via optically pumping atoms into
a dark state |Ψd i. The density matrix ρ(t) evolves under Ĥmag = γB · F, where γ
is the gyromagnetic factor, and possible decoherence mechanisms, including atomic
motion and collisions, for a time t. Atoms which leave the state |Ψd i are detected by
state-selective absorption imaging. For an extended atomic sample with the density
n(~r) we find the absorption imaging signal to be
Z ∞
S(x, z, t) ≡ K
n(~r) (1 − Tr [ρ0 ρ(t)]) dy ,
(5.7)
−∞
where y is the imaging direction and K is a constant. For elongated thin atomic
samples, as ours, there is little information in x-direction. Essentially, we are left
112
5. Magnetic Field Measurements
with a 1+1 dimensional map of the magnetic field
Z
S(z, t) ≡ S(x, z, t) dx
(5.8)
along the sample and in time. Here we integrate only over the region of the signal
S(x, z, t) which contains atomic information. In the experiment, this will be a few
lines of the CCD image.
For alkali metal atoms the magnetic Ramsey sequence can be implemented in the
following way. In Fig. 5.4 we show the sequence for the special case of 87 Rb and the
D2 line. The existence of two hyperfine ground states with total angular momentum
F± = I ± 1/2, where I is the nuclear spin quantum number, allows us to split the
task of spin state preparation and detection. An initial state is prepared in the
F− manifold and evolves for some time t under the influence of Ĥmag . Atoms are
detected by state-selectively transfer to F+ from where they can be counted by means
of absorption imaging, which is highly sensitive if performed on a cycling transition
with F+ → F 0 = F+ + 1. The state preparation on the F− → F 0 = F− transition
with π-polarized light produces an initial state by pumping atoms into the dark-state
|Ψd i ≡ |F− , 0i. This state has a symmetry axis s which is parallel to the quantization
axis, i.e., the electric field direction of the pump light in this case [140]. For the case
of F− = 1, and no decoherence, the signal can be directly calculated using
2
Tr[ρ0 ρ(t)] = hψd | exp[−iĤmag t/~] |ψd i
(5.9)
2
S(z, t) = 1 − 1 + sin2 θ (cos (ωL t) − 1) ,
(5.10)
to get
where cos θ ≡ s · B(z)/ |B(z)| and ωL ≡ γB(z) is the Larmor frequency.
Vector Magnetometer
For one initial state |Ψd i we acquire full information about the magnitude and some
information about the direction through θ of the magnetic field. To identify the
magnetic field direction unambiguously, it is sufficient to repeat the magnetic field
imaging with another dark state |Ψ0d i along a different symmetry axis s0 ∦ s. The
process should be supplemented by applying small bias fields along three orthogonal
direction to distinguish between Bk and −Bk , for k = x, y, z.
In more detail, we can implement the vector field measurement protocol as illustrated
in Fig. 5.5: a) A single magnetic field image (MFI) would reveal the the angle θ up
to a phase of π. That means that a single measurement would reduce the possible
direction of the magnetic field from 4π to two cones with an opening angle of θ and
θ + π. b+c) Applying a magnetic field parallel or anti-parallel to the symmetry axis
s, we find out in which hemisphere the magnetic field is pointing. d) A second dark
state with symmetry axis s0 is used. Experimentally, it is desirable to use another
5.2. Spatially resolved magnetic field measurements
113
Figure 5.4.: Magnetic Ramsey sequence: a) The atoms are prepared into a dark state
for linearly polarized light. resonant laser light empties the F = 2 manifold. b) Local
magnetic field will rotate the state. c) The state selective transfer is performed with
pumping light. d) Absorption imaging is used to count the number of transferred
atoms.
linear polarization orthogonal to the first one. This prepares a dark state which
has its symmetry axis 90◦ flipped. We are still left with four possibilities for the
magnetic field vector. e) As for the first dark state we apply a magnetic field along
its symmetry axis s0 g) To distinguish between the last two possibilities we apply a
magnetic field along the remaining axis which is orthogonal to s and s0 .
Spatial Information
The magnetic field imaging technique provides a convenient way to measure gradients
and higher derivatives, which often give better signal to noise in the presence of
fluctuating backgrounds [141]. If we write
B(z) = B0 + B1 z +
θ(z) = θ0 + θ1 z +
B2 2
z + O(z 3 ) ,
2
θ2 2
z + O(z 3 ) ,
2
(5.11)
(5.12)
where Bn = ∂ n B(z0 )/∂z n and θn = ∂ n θ(z0 )/∂z n , we see the gradients cause a spatial
modulation of the magnetic field image S(z) at any given time τ . In addition, B2 ,
the magnetic field curvature will produce a linear chirp of the spatial frequency.
114
5. Magnetic Field Measurements
Figure 5.5.: Visualization of vector magnetometer protocol. a) Magnetic imaging with
dark state along x. b) Discrimination between θx and θx + π, by application of small
field along x. c) Magnetic field is pointing in upper hemisphere. d) Magnetic imaging
with dark state along y. e) Discrimination between θy and θy + π, by application
of small field along y. e) The magnetic field has positive values for Bx and By . g)
Discrimination between positive and negative value for Bz , by application of small field
along z. h) We measured the amplitude and the direction of the unknown magnetic
field unambiguously.
More quantitative information can be deduced from (spatial) Fourier transformation
of the measured signals S̃(k) ≡ F (S(z)) at any given time t = τ , where k is the
spatial frequency. We also assume the angle of the magentic field is a constant
over the length of the sample.
shape of the atom distribution along z, i.e., the
R The
~
atomic line-density n(z) = n(r) dxdy, will influence the Fourier spectrum of the
√
signal. For a Gaussian distribution, n(z) ≡ n0 exp −z 2 /σ 2 /(σ π), with a FWHM√
length 2 ln 2σ. Expanding the trigonometric function cos(ωL t) in Eq. (5.10) into
complex exponential function the measurable signal reads


2
X
S(z, t) = n(z) 
Ap exp −itγp(B1 z + B2 z 2 /2) + c.c. ,
p=0
where A0 = sin2 θ − 43 sin4 θ, A1 = sin4 θ − sin2 θ, and A2 = − 14 sin4 θ. We can
identify three parts Sp of the signal, which have the Fourier transformations
(k−pB1 τ γ)2
exp − 4/σ
2 +i2pB τ γ
2
S̃p (k) = Ap p
,
(5.13)
2π + iπB2 pτ γσ 2
where we only write the positive frequency part for k > 0. Later, we compute the
power spectrum S̃p S̃p∗ of the Fourier transformation from the measured data. This
5.2. Spatially resolved magnetic field measurements
115
can contain up to three Gaussian peaks at
kp = pB1 τ γ
(5.14)
with a FWHM of
r
√
2
B2
∆kp = 2 ln 2
+ 2 σ 2 τ 2 γ 2 p2 ,
(5.15)
2
σ
2
where we observe only two in the case of θ = π/2 because A0 = 1/4, A1 = 0, and
A2 = −1/4. No peaks are observed for the trivial case of θ = 0.
5.2.3. Measurements
We present two sets of measurements. The first one shows the main features of the
technique by following the spatially modulated signal over time for an inhomogeneous
background magnetic field. The second illustrates the use of the method for in-situ
monitoring of magnetic field changes. We show how a gradient field is canceled and
how the method can serve as a quantitative characterization tool.
Figure 5.6.: a) Laser frequency for pump-, transfer-, imaging-, and depletion-lasers. b)
The atomic sample is held in a dipole trap and interrogated with linear pump, transfer
and imaging light from the side. The depletion laser is applied through the MOT
beams.
Proof of principle
The measurement proceeds as follows: we load the atomic trap for about 4 s and
wait 400 ms for untrapped atoms to escape from the trapping region. Then the dark
116
5. Magnetic Field Measurements
state is prepared by sending a 10 µs pulse of x-polarized light into the direction of
the camera. At the same time, F = 2 is depleted by resonant light applied through
the MOT beams. After a variable waiting time ∆t of 50 to 1000 µs we send a second
10 µs pulse to transfer atoms from the bright state to F = 2. The absorption image
is taken with 100 µs of light resonant to the F = 2 → F 0 = 3 transition.
At each waiting time we acquire 40 images under the same conditions, averaged them
and integrate over four horizontal lines of the CCD which contain the atomic absorption signal. Knowing the on-resonant scattering cross-section (cf. Appendix B.1),
we convert the signals from relative absorption into atom number. To suppress high
frequency noise in the images, we apply a moving average over 7 adjacent pixels.
The sensitivity in atom number per pixel is better than hundred atoms, as we can
see from Fig. 5.8.
A small background, due to atoms that remain in F = 2 during the preparation, is
subtracted using a reference image taken immediately after preparation. Atoms that
remained F = 2 do not influence the measurement of the magnetic field amplitude.
However, it is an indication that the used preparation time of 10 µs or the applied
depletion laser are not sufficient for effective state transfer. The measured signals
for all delay times are plotted in Fig. 5.7.
The model for fitting the data is:
A(z, t) = n(z) [p (1 − C(z, t)) + C(z, t)S(z, t)] ,
(5.16)
where 1 − p ≡ Tr [ρtherm ρ0 ] = 1/(2F + 1) is the overlap of an unpolarized (thermal)
R ~
spin state and the dark state and n(z) = n(r)
dxdy is the atomic line density
which has approximately Lorentzian shape as we measured in Sect. 3.3.5. The local
coherence is modeled by
C(z, t) = exp −ηL2 (z)t
(5.17)
where L(z) is a Lorentzian function to account for density dependent variations in
the coherence times. The entire data set, shown in Fig. 5.7, is fitted simultaneously
to find parameter values given in the caption.
The origin of the additional decoherence in the trap center was not investigated
further. One possible explanation are spin changing collisions which cause a rapid
decoherence of the spin state [38, 142]. In Sect. 3.4 we calculated the rate of spinchanging collisions to be ∼ 0.03 s−1 for realistic densities of n = 1011 cm−3 . This
is too slow to explain the rapid decoherence within ∼ 900 s−1 . Another possibility
are hyperfine changing collisions between the remaining atoms in F = 2. Further
experiments are necessary to resolve this issue and are beyond the scope of the
presented work.
To see temporal Larmor precession, we plot the number of transferred atoms versus
time for specific positions along the sample in Fig. 5.8. We chose two points which
are located on either side of the trap center. There, we see that the local coherence
time of the spin state is many milliseconds. The influence of the angle θ between
the magnetic field and symmetry axis of the dark state is visible in the bimodal
5.2. Spatially resolved magnetic field measurements
117
Figure 5.7.: Spatial distribution of transferred population into F = 2. The horizontal
axis covers a field of view of ∆z = 13.2 mm. Delay times are indicated for each frame.
The solid yellow line is the fitted profile using the expression discussed in the text. The
deduced parameters are for z0 = 6.6 mm: B0 = 5.7(2) mG, ∂B/∂z = 19.4(4) mG/cm,
∂ 2 B/∂z 2 = 7.9(8) mG/cm2 , θ = 81(2)◦ . The additional decoherence in the trap center
has a FWHM ∼ 1.3(3) mm.
structure of the oscillation signal. The observation of the long local coherence time
shows that we would be able to produce coherent spin states with ms coherence
times, by removing the magnetic field inhomogeneities.
In-situ magnetic field gradient compensation
The method described above permits us to shape the magnetic field environment.
Our main goal is to compensate all magnetic field gradients along the sample axis,
∂By
∂Bz
x
i.e., ∂B
∂z , ∂z , and ∂z . We have seen that the field is sufficiently inhomogeneous
to de-phase the collective polarization of a spin state in less than 50 µs (cf. first
frame of Fig. 5.7). In the spin squeezing measurement we conduct in Chap. 7, it
will be important to nullify the magnetic field over the whole sample length of more
than 10 mm and increase the coherence time to > 100 µs, which is the necessary
measurement time.
The main sources for the gradient observed above are the permanent magnets built
into the two ion-getter pumps of the vacuum system. They are mounted at a distance
of 60 − 70 cm to the atomic trap and not magnetically shielded. At the time of the
experiments, we were not able to provide a good magnetic shielding. The method
for the gradient compensation is simple but laborious. We mount a pair of magnetic
field sensors close to the trap apparatus at a distance of about 5 − 8 cm to each side
of the trap along the sample axis. In this way, we monitor to first order the gradient
∂By
∂z . To compensate this gradient, we use a permanent magnet which was placed
118
5. Magnetic Field Measurements
NA [per pixel]
200
0
NA [per pixel]
200
0
0
0.2
0.4
0.6
Time [ms]
1
Figure 5.8.: Larmor precession at fixed position. The upper trace is taken at z =
0.04 mm and the lower trace at z = 2.89 mm. The black solid line is a least-square fit
of a damped oscillation. The fitted angles are θ = 80◦ (20◦ ) and θ = 80◦ (6◦ ), for uper
and lower trace, respectively. The exponential decay of the contrast corresponds to a
local coherence time ∼ 5 ms. Errorbars indicate statistical fluctuations in the sample
of 40 measurements.
in a ’strategic’ position5 to counteract the magnetic field gradient produced by the
pump magnets. The permanent magnet is mounted on a rotation stage which allows
to change the magnetic field at the position of the atoms in controlled way. With
∂B
x
this method we compensate ∂zy and partly ∂B
∂z by turning the permanent magnet.
∂Bz
For the remaining gradient ∂z we use the compensation coils along z. For this,
we independently adjust the current through each of the z compensation coils. The
resulting magnetic field is the sum of a constant and a gradient field. For the currents
in the left and right coil Il and Ir we can write:.
Bz = Kc
Il + Ir
Il − Ir
+ Kg
z,
2
2
(5.18)
where Kc (Kg ) is the conversion factor between applied current and constant (gradient) magnetic field, which has units of G/A (G/cm/A).
For the in-situ gradient cancellation at the position of the atoms, we fix the delay
time to τ = 500 µs and record the spatial profile as an externally-applied field is
adjusted stepwise by changing Il and Ir . In Fig. 5.9 a) and b) we plot the oscillating
part of the signal, i.e., the Lorentzian atom distribution is subtracted, before and
after compensation. In Fig. 5.9 c) we plot the power spectrum of the Fourier
transformation of the signals for the individual adjustment steps. The oscillations
5
The atomic sample, the permanent and the pump magnet form a isosceles triangle, where the
two magnets are placed symmetrically.
5.2. Spatially resolved magnetic field measurements
119
Figure 5.9.: Fourier analysis of gradient compensation. In a) and b) we plot the differential transmission ∆T ∝ S before and after the compensation, respectively. The
constant atom number distribution n(z) was subtracted. In plot c) the FFT signal
shows a successive reduction in the gradient from ∼ 20 mG/cm down to < 5 mG/cm.
The width of the resonant peaks suggests a gradient resolution of ∼ 2 mG/cm.
120
5. Magnetic Field Measurements
2πk
frequency is converted into a field gradient by ∂B
∂z = 2γτ , where k the frequency of
the FFT. We start at ∼ 20 mG/cm and reduce the gradient down to < 5 mG/cm.
The width of the resonant peaks suggests a gradient resolution of ∼ 2 mG/cm.
5.2.4. Sensitivity
Motivated by recent experiments demonstrating projection noise limited atom number counting by absorption imaging [52, 143], we estimate the magnetic field sensitivity in the projection noise limit following Ref. [8]. The sensitivity is
δB =
−1
1p
zres nτ T
γ
(5.19)
, where zres is the spatial resolution, n the line density, T the total measurement time.
The coherent evolution time τ is either the duration of the measurement τm , limited
by atomic motion, or the spin coherence time τc , depending on the context. In our
conditions, the measurement time is limited by atomic motion through τm = zres /v,
where v is the mean atomic velocity along z. For
√ zres = 50 µm v = 0.08 m/s, and
8
n = 10 atoms/m we find a sensitivity ∼ 15 pT/ Hz. In optical lattices the thermal
motion along z is suppressed. This improves the sensitivity since we are limited
only by the spin coherence time, which can reach 60 s [144] or about 105 times the
τm considered above [145] giving√an improvement by ∼ 300. This would lead to a
sensitivity of the order of 50 fT/ Hz.
6. Spin State Tomography
Contents
6.1. Stern-Gerlach in free fall . . . . . . . . . . . . . . . . . . . 122
6.1.1. Falling atom clouds
. . . . . . . . . . . . . . . . . . . . . . 123
6.1.2. Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2. Measuring ground-state populations of trapped atomic
sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.1. Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.2. State selective transfer . . . . . . . . . . . . . . . . . . . . . 125
6.2.3. Re-distribution and dispersive probing . . . . . . . . . . . . 127
6.2.4. Measurements
. . . . . . . . . . . . . . . . . . . . . . . . . 128
122
6. Spin State Tomography
There exist an asymmetry in quantum physics. Given a quantum state, we can, at
least in principle, easily find the expectation value of a given operator. However,
going the other way around to determine the quantum state from a set of measurement outcomes is non-trivial. Wolfgang Pauli has recognized this already in the early
years of quantum mechanics [146] and since, we refer to it as the “Pauli problem”.
In the advent of the quantum computation and information era the problem has again
gained considerable importance. Even so, there exist no general solution to it, many
fields in quantum physics have developed their own techniques. The “drosophila”
in this respect is probably the homodyne detection of quantum states of continuous
light [147]. It allows one to measure the phase space distribution and therefore a
complete reconstruction of the state. For other quantum systems, as for example
polarization-entangled photon pairs [148], center-of-mass motion in ion traps [149],
ions [150], molecular vibrations [151], and atomic beams [152] techniques have been
demonstrated.
In the field of atomic ensemble experiments there has been a lot of pioneering work
in the last decade, especially in the group of P. Jessen with theoretical input from
I. Deutsch [34]. Complete characterization of angular momentum states of arbitrary
magnitude, ranging from destructive Stern-Gerlach type of measurements [153] to
non-destructive dispersive measurements [122, 123].
In this chapter we discuss the possibility to apply a Stern-Gerlach technique, demonstrated by Klose et al. [153], with our setup in Sect. 6.1. In Sect. 6.2, we introduce
an alternative approach for spin state tomography which translates spatial SternGerlach into the time domain. We report first results and discuss improvements to
elevate the technique onto a practical level.
6.1. Stern-Gerlach in free fall
We study the possibility to perform a Stern-Gerlach experiment (SGE) of a free
falling atomic cloud in a gradient field. In this, atoms in different magnetic substates
are separated during the free-fall by application of a magnetic gradient. We follow
a demonstration of this technique for thermal atomic clouds by Klose et al. [153].
They have measured the ground state density matrix for angular momentum F = 4
in a cold (T = 3.5 µK) cloud of Cesium. This technique is rather temping because
it gives a direct measurement of the diagonal elements as 2F + 1 atomic clouds, so
to say ’ad oculos’ 1 . We concentrate on the question of the SGE alone and make a
feasibility calculation for our experimental conditions.
The SGE in Ref. [153] is performed in a time-of-flight (TOF) measurement. On their
free fall atoms pass through a light-sheet which is placed below the atomic cloud and
produce a time dependent absorption or fluorescence signal. The temporal shape
will have, in the ideal case, 2F + 1 maxima, which represent the populations of the
different diagonal elements of the density matrix.
1
ad oculos = for the eyes; This expression was used by Arnold Sommerfeld in 1922 in the revised
version of his classic book “Atomic Structure and Spectral Lines ” in reaction to Stern and
Gerlach’s experiment [154].
6.1. Stern-Gerlach in free fall
123
In previous chapters we characterized all relevant experimental parameters of our
system which are necessary for the SGE experiment.
6.1.1. Falling atom clouds
Performing a SGE on a falling atomic cloud in a gradient field relies on the different
acceleration of atoms with different spin quantum numbers mF due to the magnetic
interaction. The force an a free atom in a gradient field is given by
F = mg +
∂B
~γmF ,
∂x
where g is the gravitational acceleration (which we measured in Chap. 3) and m is
the mass of an atom. If the gradient is homogeneous over the region of the atoms, we
can describe the movement of the whole cloud by their distribution. In Sect. 3.3.4
we measured the radial distribution in TOF and approximated it by a Gaussian of
the form (cf. Eq. 3.19)
2x2
n(x) = n0 exp − 2
,
(6.1)
wa
where n0 is the peak density and wa the atomic waist. After releasing the atomic
cloud from the trap it will expand. We measured the trap waist as a function of time
using (cf. Eq. 3.25)
r
4kB T 2
wa (t) = wa2 (0) +
t ,
(6.2)
m
from which we deduced the temperature T = 25 µK and the initial waist wa (0) =
17 µm. The center of mass of the atomic cloud would describe a parabola with a
trajectory of
1
∂Bx ~γmF
x(t) = x0 −
g+
t2
(6.3)
2
∂x m
We can write a normalized version of Eq. (6.1) where we substitute x → x(t) and
wa → wa (t)
2 (t)
exp − 2x
w2 (t)
ñ(x, t) = p π a
.
(6.4)
2 wa (t)
6.1.2. Simulations
The relevant experimental quantities are the magnetic field gradient we can apply
and the distance atoms can fall freely, i.e., the space below the dipole trap. From the
measurements in Sect. 5.1, when we characterized the magnetic field environment via
the position of the MOT, we deduce a gradient of ∂B
∂x = 4 G/cm for 1 A in the MOT
∂B
coils. This results in ∂x max = 24 G/cm if we apply the maximum current of 6 A.
For the maximum distance the atomic cloud can fall, we assume ∆xmax = 15 mm.
If we use these parameters we can simulate the signal shape, which are shown in
Fig. 6.1. We see three distinct peaks which are almost not overlapping. By postprocessing the individual populations could be retrieved with good precision even
124
6. Spin State Tomography
in the case of a slight overlap. Hence, we can measure the (2F + 1) populations in
a single experimental run. In this way, atom number fluctuations, e.g., from trap
loading, will not influence the accuracy of the results.
SGE signal [arb. units]
mF=ï1
m =0
F
m =+1
F
0
0.05
0.1
Time [s]
0.2
Figure 6.1.: Simulations of SternGerlach experiment in gradient of
quadrupole field of MOT coils. The
used parameters are: ∂B
∂x = 2400 G/m,
xmax = 15 mm, T = 25 µK, and
wa (0) = 17 µm. Including realistic
beams diameters of ∼ 1 mm renders
virtually the same result.
6.2. Measuring ground-state populations of trapped atomic
sample
We have the ability to measure atomic polarization with high precision as we have
seen in Chap. 4. This triggered the idea to use dispersive spin measurements also
as the detection mechanism in a tomography experiment. The non-destructiveness
of the dispersive probing would allow to measure all (2F + 1) populations of the
magnetic substate |F, mF i on a single trapped atomic ensemble.
We discuss the general idea which is applicable to all alkali-metal atoms. Three possible implementations are briefly presented. One of these is examined in more detail
and first measurements are presented. At the end we discuss possible improvements.
6.2.1. Proposal
The main idea is to transfer the population of each magnetic substate |F− , mF i
individually into the hyperfine manifold F+ = I + 1/2, by applying a homogeneous
magnetic field to lift the ground state degeneracy. To this process we will refer in the
future as state-selective transfer . Once transferred, the population is then measured
non-destructively via paramagnetic Faraday measurements. Measuring in different
bases, by manipulating the state with magnetic field prior to the measurement as
demonstrated in Ref. [153], would extend this method into a full tomographic technique.
The complete experimental sequence, displayed in Fig. 6.2, would consist of 2F + 1
blocks, each having the following three steps:
1. T: State-selective transfer of atoms from |F = 1, mF i into F = 2.
6.2. Measuring ground-state populations of trapped atomic sample
125
2. S: Re-distribution (shuffling) of atoms in F = 2 to guarantee an asymmetric
population.
3. P: Probing the transferred population with paramagnetic Faraday rotations
from F = 2.
Figure 6.2.: General scheme of transfer, shuffle, and probing to perform a
Stern-Gerlach-like measurement on alkali metal atoms. Small magnetic fields
allows state selectivity. A single block
has three steps and is repeated 2F + 1
times.
All three steps are discussed in detail in the next three sub-sections.
6.2.2. State selective transfer
For the state-selective transfer (SST) we can imagine three different scenarios. Each
of them has a different implementation of the transfer mechanism and a different
minimal requirements on the magnetic fields applied:
• For small magnetic fields starting from a few mG we
could use micro-waves
(MW) to transfer atoms directly from a specific state F− , mF− into F+ , mF+ .
Small Zeeman shifts of a few tens of kHz are sufficient since hyperfine transitions are magnetic dipole transitions with narrow natural linewidth less than
kHz. In Fig. 6.3 a) we show the transfer with linearly polarized microwaves
from the F = 1 into F = 2 manifold.
• For magnetic fields starting from a few hundreds of mG we can use a stimulated
Raman processes. For this, two phase-locked light fields separated in frequency
by the hyperfine plus the Zeeman
splitting
of the specific initial state F− , mF−
and the specific final state F+ , mF+ state transfer the population, as shown
in Fig. 6.3 b). Different Raman schemes exist, which can achieve a complete
transfer in the case of stimulated adiabatic Raman passage (STIRAP) [155,
156, 157], or a partial transfer ( ≤ 50 %) in a single-pulse scheme [158, 159].
• For magnetic fields starting from a few tens of Gauss, when the induced Zeeman
shift is larger than the excited state linewidth, simple optical pumping can be
used to transfer population from a specific |F− , mF i into the F = 2 manifold.
This is possible because the Landé factor for the ground state F = 1 and the
excited states F 0 have different signs. In this way we can address transitions
from |1, mF i → |F 0 , mF 0 i individually.
126
6. Spin State Tomography
We pursue the third option of transferring atoms incoherently via optical pumping
in fields of a few tens of Gauss. We note, the following derivation can be applied
to all other alkali metal atoms. In general, it is favorable to use the D2 transition,
because the Zeeman shifts of the excited states are larger as compared to the D1
excited states [19, 160, 161, 162]. We give here numbers for the D2 transition in
87 Rb .
The approach takes advantage of the fact that the magnetic moment of ground state
hyperfine manifold F− = I − 1/2 has a different sign than the excited state magnetic
moments. On the D2 transition in 87 Rb we have gF =1 = − 12 and gF 0 = 23 . This
situation is depicted in Fig. 6.4 a). Transitions from a specific ground state |F− , mF i
to an excited state |F 0 , mF 0 i become energetically separated. For polarized light in
one of the basis states σ ± or π, the transitions we have a spacing of (gF 0 − gF ) µB B =
2π 67 1.4 MHz/G. Already for some tens of Gauss we reach the situation where the
energy splitting is larger than the natural line width and off-resonant excitations
Rsc (∆) ∝ Γ2 / 4∆2 + Γ2 are suppressed quadratically in the detuning ∆.
For the transfer we will use circularly polarized light which ensures that we can
transfer atoms out of one |F− , mF i state without interfering with the population in
the other ones. In Fig. 6.4 b) we show the decay channels of a number of excited
states which get populated during the transfer. There we can see that it is possible to
measure the ground state population in each |F− , mF i state when transferring first
the state with the largest magnetic quantum number mF in the case of σ + light and
vice versa. By calibrating the system with a known reference state, we can account
for atoms which are not transferred into F = 2.
Figure 6.3.: Possible alternative transfer schemes: a) Transfer of population directly
from F = 1 to F = 2 by micro-waves. b) Raman transitions driven by a pump and
a Stokes laser field Ωp and Ωs . The detuning for the single and two-photon transition
are ∆ and δ, respectively. Other frequency-polarization combinations are also possible.
6.2. Measuring ground-state populations of trapped atomic sample
127
Figure 6.4.: a) Transfer of atoms with σ + light in a magnetic field. The different sign of
the magnetic moment for F = 1 manifold is visible as the opposite slope to the rest of
the magnetic substates. b) Different decay channel for σ + transfer light. The numbers
in the gray boxes indicate the line strengths ×12.
6.2.3. Re-distribution and dispersive probing
For the paramagnetic Faraday rotation measurement, we need an asymmetric distribution of atoms among the magnetic substates in F = 2, i.e., h F̂z i =
6 0. For some
decay channels, for example from mF 0 = 0, this is not given and we get h F̂z i = 0 (cf.
Fig. 6.4 b)). For this reason, we apply another laser field which re-distributes atoms
in F = 2 before we measure the rotation signal. Furthermore, the re-distribution
equalizes the measurement strength for all mF substates and guarantees a state
independent detection.
The re-distribution light ought to be tuned to the cycling transition F = 2 → F 0 = 3
to avoid scattering atoms back into F = 1. One possibility is to apply σ ± -light
which keeps atoms in a closed transitions and moves them to mF = ±2. This requires
sending light along the sample axis, which is the direction of maximum optical depth.
To avoid problems of shadowing and light trapping, another possibility is to apply
linearly polarized light perpendicular to the sample axis. If we chose the quantization
axis along the magnetic field and electric field vector orthogonal to it, the atoms
experience a coherent superposition of σ − and σ + light. The transition frequencies
for left- and right circular light, which are degenerate for zero magnetic field, will
experience different shifts in the presence of a field. In the left graph of Fig. 6.5
we show the shift of the transition frequencies from |2, mF i to their corresponding
excited states for σ ± light. Taking advantage of this non-degeneracy, we are able to
use linear light to pump atoms into a state where we have h F̂z i =
6 0 for F = 2.
The situation is slightly more complicated when the light shift of the dipole trap is
taken into account. Ground and excited states experience an additional shift. For
reasons we discuss in Appendix B.2 the excited states experience a different light
128
30
6. Spin State Tomography
6i±(mF) [MHz]
40
20
30
10
20
0
10
ï10
0
ï20
mï ï10
6i±(mF) [MHz]
mï
+
m+
m
ï30
ï2
ï1
mF
+1
+2
ï20
ï2
ï1
mF
+1
+2
Figure 6.5.: Effect of Zeeman shifts and AC-Stark shifts on transition frequencies for
the transition F = 2 → F 0 = 3. Left: Transition frequency shifts ∆ν for σ ± light as a
function of mF in F = 2 for a magnetic field of B = 20 G. Right: Transition frequency
shifts including Zeeman and AC-Stark shifts. The light intensity corresponds to the
value we find in the focus of the dipole laser at 6 W optical power.
shifts depending on mF . In the right of Fig. 6.5 we plot the energy shifts from
magnetic field and the light shift due to the trapping laser.
For the dispersive paramagnetic Faraday rotation measurement from F = 2 we use
light which is blue detuned compared to the transition F = 2 → F 0 = 3 to maximize
the vector polarizability2 . It is sent along the sample axis and detected in the same
way as in Chap. 4.
6.2.4. Measurements
As a first calibration we measure the spectrum of the transfer operation by scanning
the transfer laser over the three expected resonances and measure the rotation signal.
This is important to locate the exact position of the transfer resonances for the
tomography. at the same time it gives us the strength of the magnetic field, which
is actually applied, and the amount of light shift the atoms experience in the dipole
trap.
In the measurement we load atoms in the dipole trap as described in Sect. 3.1.3. Then
we ramp up an additional magnetic field of 15 − 20 G along z. This provides the field
strength to split the atomic resonances sufficiently to be measured individually. For
the transfer, we use light from the F = 1 laser (cf. Chap. 3) by sending it through
2
When probing atoms in F = 2 the spectrum of the vector polarizability is qualitative the same as
for F = 1, when interchanging red for blue detuning. That means, we have the same behavior
for blue detuned light on F = 2 as for red detuned light on F = 1. See for instance Fig. 2.2.
6.2. Measuring ground-state populations of trapped atomic sample
Transfer Signal [norm.]
1000 a
3000 a
1
0.8
0.6
0.4
0.2
0
ï60
ï40
ï20
6i [MHz]
20
40
129
Figure 6.6.: Transfer
spectrum: normalized rotation
signal vs.
transfer-laser
detuning. The transfer laser
is scanned from blue to red
detuning.
We change the
number of photons in the
transfer pulse between 1000
and 3000 and renormalize the
rotation signal accordingly.
Note, we flipped the sign of
the rotation signal. Usually,
the signal is negative for
the whole spectrum since
we observe rotations from as
state with mF > 0 and the
laser is blue detuned.
two AOMs. This gives us the possibility to scan over all expected resonances in a
range of −68.5..31.5 MHz around the field-free resonance F = 1 → F 0 = 2. For the
off-resonant probing we frequency off-set lock a laser to the F = 2-laser and shift it
above the resonance F = 2 → F 0 = 0.
The whole spectrum is measured on the same atomic ensemble in a single run.
For this we interrogated with blocks of three pulses, which consists of transfer-,
probe- and depletion light, at different detunings of the transfer laser. The transfer
laser has right circular polarization, i.e., σ − -light, and is scanned in steps of 2 MHz.
The magnetic field is pointing along the positive z-axis. The probe laser is locked
+570 MHz to the blue of the transition F = 2 → F 0 = 0. We send 130 probe pulses
with NL = 6 × 106 per pulse which are 1 µs long probe pulse and have a period of
10 µs. The large number of probe photons and the relatively close detuning make
the application of a re-distribution laser unnecessary. The depletion light is resonant
from F = 2 → F 0 = 3.
A single block of transfer-, probe-, and depletion light will do the following: The
transfer laser pumps a fraction of the population from F = 1 into F = 2, depending
on its spectral position. The probe will re-distribute the atoms into a steady state
distribution among the magnetic substates in F = 2. We take only the last pulses of
the train of probe pulses which measure the steady state. After the probing, we send
light which heats the atoms in F = 2 out of the dipole trap to reset the rotation to
zero.
A background trace is taken under the same conditions as for the transfer but without
applying transfer light. We recognized that this is necessary since there are atoms
transferred from F = 1 due to the probe laser. Even for the large detuning of more
than 6 GHz, we get off-resonant scattering and the depletion laser is not removing
these atoms completely.
In Fig. 6.6 we show the result of the experiment for two different intensities of the
transfer laser. For the trace with blue circles we used about 1000 photons per transfer
pulse. Increasing the number of photons by a factor of three gives proportionally
130
6. Spin State Tomography
more signal as we can see in the green squares, which were rescaled by 1/3. The
scanning direction for both traces is from blue to red detuning.
We fitted the two curves by a the sum of three Voigt functions. This function
is often used in spectroscopy for structures which are convolutions of Gaussian and
Lorentzian signals. As analytic expression for the Voigt function we used the PearsonVII function, which approximates the Voigt function well [163]. For comparison, we
also show the best fit for a sum of three power broadened Lorentzian as thin solid
lines.
We interpret the shift of the peak position for higher photon numbers as a depletion
effect from mF = −1 and mF = 0. While the transfer laser slowly scans over
the resonances too many atoms are transferred into F = 2 , which empties the
according magnetic substates. A maximum in transfer is achieved when a substantial
depletion has taken place. This is not necessarily the position of the actual transition
resonance, as we can best appreciate in the left-most peak in Fig. 6.6.
The general broadening of the transitions is most probably related to the distribution
of atoms in the dipole trap. Depending on the position, atoms see smaller or larger
AC Stark shifts. Quantitatively, we can estimate the effect by computing the atom
number distribution as a function of energy. For a 3D harmonic potential, we find
nA () =
2
1
,
2~3 ωρ2 ωz exp −µ − 1
kB T
where the first term is the density of states for the 3D harmonic potential and the
second the Bose distribution [164]. The energy in the harmonic potential is , ωρ
and ωz are the radial and axial trap frequency and µ the chemical potential. The
temperature is measured in Chap. 3, the trap frequencies can be calculated from the
trap parameter and the chemical potential can be found
R ∞ by normalizing the density
to the total number of atoms in the sample NA = 0 dnA (). In Fig. 6.7 we see
that for our atomic sample we have a rather broad distribution of energies in the
range of ∼ 8 MHz. This is consistent with the observed linewidth of the transfer
spectrum.
In summary, we see that it is possible to measure individual ground state populations by incoherent atom transfer via optical pumping. The observed spectra show
the expected three peak structure as we would assume from an unpolarized atomic
ground state. Nevertheless, the peak position slightly depends on the amount of
transfer. On the other hand, if we could use other transfer techniques like microwaves or Raman processes, the position in the dipole trap would not lead to such
large broadening of the transition.
6.2. Measuring ground-state populations of trapped atomic sample
131
2.5
n(¡)
___
NA
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
U0
Figure 6.7.: The atom number
distribution as a function of energy in the trap is calculated for
NA = 1 × 106 , T = 25 µK, and
U0 /h = 4.9 MHz. To get the whole
frequency brodening one has to
multiple the U0 with a factor of
three to get the shift of the transition frequency F = 1 → F 0 = 2
7. Spin Squeezing
Contents
7.1. Calibration of spin detector . . . . . . . . . . . . . . . . . 134
7.1.1. Learning from photons . . . . . . . . . . . . . . . . . . . . . 134
7.1.2. Noise scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.1.3. Experimental details . . . . . . . . . . . . . . . . . . . . . . 136
7.1.4. Trap loss in thermal state preparation . . . . . . . . . . . . 139
7.1.5. Projection noise limited spin detection . . . . . . . . . . . . 140
7.2. QND measurement of large-spin ensembles by dynamical decoupling . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2.1. Convenient fiction . . . . . . . . . . . . . . . . . . . . . . . 143
7.2.2. Dynamical decoupling in QND measurements . . . . . . . . 146
7.2.3. Two-polarization meta pulses . . . . . . . . . . . . . . . . . 148
7.2.4. Experimental verification of dynamical decoupling idea . . . 149
7.3. Spin squeezing of magnetically sensitive atomic state
. 152
7.3.1. Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3.2. Spin squeezing pulse sequence . . . . . . . . . . . . . . . . . 153
134
7. Spin Squeezing
In this chapter we present measurements of spin squeezing in a magnetically sensitive
atomic system. In the first part we calibrate our spin detection using a thermal spin
states. Analogous to the calibration of a photo detector, we use thermal spin states
as robust noise references to determine the amount quantum and technical noise
sources in the measurement.
In the second part we present a novel technique which allows us to perform quantum
non-demolition measurements in large-spin atoms. For atoms with spins larger than
1/2 the dipole interaction is more complicated than the simplified paramagnetic
Faraday rotation ∝ Ŝz Jˆz and involves coherent Raman processes (also referred to as
tensor light shifts, alignment effects [165], and non-linear effects [166]). The desired
QND character of the light atom interaction interaction is hindered. We propose a
technique to overcome this fundamental limitation and present results verifying our
expectation.
The latter parts lead directly to the first results on the generation of spin squeezing
in a magnetically sensitive state.
7.1. Calibration of spin detector
An important step towards creation of measurement induced squeezing is the thorough characterization of the spin detection method. Measuring the sensitivity of the
spin detector as a function of the introduced damage, as we did in Chap. 4, is one
part. But the ultimate characterization of any detector, i.e., for atoms or photons, is
the measurement of all contributing noise sources. In the past, there has been a big
controversy about the demonstration of spin squeezing in a system like ours, because
a proper calibration of the measurement at the quantum level was not provided and
two publications [167, 168] from high profile journals had to be retracted [169]. We
think, therefore, that it is of utmost importance to ensure a proper calibration of the
spin measurement, before using the spin detector.
Here, we demonstrate sub-projection-noise sensitivity of our spin detection system.
The calibration procedure employs as known reference state, the maximum-entropy
or ‘‘thermal’’ spin state, and quantitative imaging-based atom counting to identify
electronic, quantum, and technical noise in both the probe and spin system. The
measurement achieves a sensitivity 2.8 dB better than thermal state quantum noise.
Most of the content in this section is published as "Sub-projection-noise sensitivity
in broadband atomic magnetometry" in Phys. Rev. Lett. 104, 093602 (2010) [13]
7.1.1. Learning from photons
In quantum optics, the use of noise reference states for the calibration of the detection
system is a well established method [111]. There, thermal, vacuum, or coherent
states of light are used as robust references. They can be produced in a reliable and
reproducible way and their noise properties are not influenced by the measurement
itself. The characteristic scaling of the quantum noise as a function of the input size
7.1. Calibration of spin detector
135
of the state, i.e., the photon number for a coherent or thermal state of light, can be
used as benchmarks. The quantum and classical (including technical) noise sources
can be distinguished by their respective scaling with the size of the system. In Sect.
3.6 we applied this strategy already for the calibration of the balanced photo-detector
used in the polarimeter. There, we could distinguish between the electronic noise and
the light shot noise due to their distinct scaling for photon number of the coherent
input state.
Thermal spin states
To characterize our spin detector, we use thermal spin states (TSS). For NA atoms
with spin quantum number F , their density matrix is given by
A
ρTSS = ρ⊗N
cms ,
(7.1)
where ρcms = (2F + 1)−1 I2F +1 is the completely mixed state of dimension 2F + 1
and Im the identity of dimension m . In terms of collective (real) spin F̂ the TSS
has zero average value, and a noise of
var(F̂n ) = V1 NA ,
(7.2)
where F̂n is any spin component and V1 = F (F + 1)/3 is the variance of a single
atom.
Thermal spin states provide a robust tool to characterize optical QND measurements
and have several advantages compared to coherent spin states. First, photon scattering during the dispersive probing leaves their quantum noise properties largely
unchanged due to the complete mixedness. This is very important and distinguishes
thermal from coherent spin states. The latter are very vulnerable to any photon
scattering, as we have seen in Sect. 4.1.2. Second, TSS are less prone to systematic
errors, e.g., due to imperfect spin polarization. Third, the TSS are less susceptible
to magnetic field perturbations because of their rotational symmetry (cf. Eqs. (7.1)
and (7.2)).
7.1.2. Noise scaling
The expected noise behavior of the measurement signal can be calculated from the
effective interaction Hamiltonian (A.17), written in terms of the collective real spin
operator F̂
e1
~G
Ĥ =
Ŝz F̂z ,
(7.3)
τ
where we neglect the terms proportional to the tensor part of the polarizability tensor.
In the case of a completely un-polarized atomic state, their effect is negligible. For
e 1 = G1 /2, where G1 is the coupling
convenience, we define the coupling constant as G
constant defined in Eq. (2.30).
136
7. Spin Squeezing
(in)
(in)
(in)
For linearly polarized probe pulses h Ŝx i = NL /2 and h Ŝy i = h Ŝz i = 0 without
(in)
(in)
initial correlation between Ŝy and F̂z , we find the variance of the light signal
after interaction
(out)
var(Ŝy
2
(in)
e 1 NL var(F̂z(in) ) .
) = var(Ŝy ) + G
4
(7.4)
The first term, the input optical polarization, in general has a variance
(in)
var(Ŝy ) =
NL
+ αNL2 ,
4
(7.5)
where the first part is light shot noise and the second technical noise due to variations
in the optical state preparation of strength α. Similarly, the second term of (7.4)
contributes a variance related to atomic quantum and technical noise, respectively,
i.e.,
var(F̂z(in) ) = NA V1 + βNA2 V1 ,
where β is the amount of technical noise. Finally, we must add a constant “electronic
noise” VE from the detector, and arrive to the measurable signal
(meas)
var(Ŝy
) = VE +
N2
N2
NL
+ αNL2 + G2 V1 L NA + βG2 V1 L NA2 .
4
4
4
(7.6)
Equation (7.6) contains the essential elements of the calibration technique. All terms
have distinct scaling with photon and atom number, and can thus be separately iden(meas)
tified if var(Ŝy
) is measured as a function of NL and NA . The terms proportional
2
to NL and NL NA correspond to quantum noise of light and atoms, respectively. Together they provide an absolute calibration of the gain of the detection system and
e 1 . The remaining terms represent various noise sources.
the atom-light coupling G
Only if these are simultaneously small relative to the atomic quantum noise, quantum
signals will be detectable. The spin detector is said to be “projection-noise limited”
if the atomic projection noise is the dominant contribution in the observed signal,
i.e., bigger than the sum of the remaining parts.
7.1.3. Experimental details
Measuring the quantum noise requires that we prepare and measure the system
many times and compute the variance of the measurement results var(Ŝy ). Each
spin measurement has to be performed on a freshly prepared atomic state, which
has a random F̂z . To observe the scaling with photon and atom number we have to
change NL and NA and repeat the spin measurement for each pair (NL , NA ).
A single experiment can be divided into three blocks which are illustrated in Fig. 7.1.
1. Preparation of thermal spin state
2. QND probing with NL linearly polarized photons
3. measurement of number of atoms NA in the thermal spin state
7.1. Calibration of spin detector
137
Figure 7.1.: Three major blocks in the spin projection noise measurement pulse sequence. Details are given in the corresponding sections.
For the atom number measurement we use the dispersive atom number measurement,
explained in Sect. 4.3. In this way, we can take advantage of the long atomic sample
lifetime by repeating the experiment on the same trap loading several times. The
dispersive atom number measurement is non-destructive and leaves the number of
atoms in the trap unchanged.
The intrinsic loss rate of the optical trap, measured in Sect. 3.3.3, is too low to
sample in a practical way many different atom numbers. Therefore, we remove
atoms by heating them out with resonant light or they are lost ’naturally’ during the
preparation stage. Here, in the calibration measurement, we have a non-negligible
loss rate due to the way we prepare the thermal spin state. Later, in the spin
squeezing experiments, we remove atoms by shining resonant light to heat them out.
We repeat experiments on the same atomic sample up to 20 times. With a reasonable
loss rate of a few tens of percent we sample about an order of magnitude in atom
numbers.
Meta pulse concept - simultaneous probing of different photon numbers
In order to measure the noise behavior as a function of photon number one could
change the power or the duration of the probe pulse. A more convenient way is
to sum the signal from multiple pulses in a metapulse, containing a larger total
number of photons. As we are in the linear regime, a metapulse will have the same
information as a single higher-energy pulse.
If the outcome of a single probe pulse with nL photons is
(out)
ŝy,j
(in)
= ŝy,j + G1
nL (in)
F̂
,
2 z
(7.7)
we can define the result of combining p pulses with NL = pnL photons as
M=
p
X
(out)
ŝy,j
(7.8)
j=1
(in)
If the individual Stokes components ŝy,j were not correlated before interaction with
(in)
(in)
the atomic system, i.e., cov(ŝy,j , F̂z
) = 0, and there is no correlation between light
138
7. Spin Squeezing
(in)
(in)
pulses, i.e., cov(ŝy,j , ŝy,k ) = 0, the variance of the metapulse is
var(M) =
p
X
(in)
var(ŝy,j ) + p2 G21
j=1
=p
n2L
var(F̂z(in) ) .
4
n2
nL
+ p2 G21 L var(F̂z(in) )
4
4
(7.9)
(7.10)
For a single pulse with NL = pnL photons we find
(out)
var(Ŝy
(in)
) = var(Ŝy ) + G21
=
NL2
var(F̂z(in) )
4
NL
(pnL )2
+ G21
var(F̂z(in) )
4
4
We see that the information in a single and a metapulse is identical.
In the experiment we send a train of 1 µs long pulses with 10 µs period to the atoms.
Each pulse contains 25 × 106 photons, vertically polarized and tuned 800 MHz to the
red of the F = 1 → F 0 = 0.
Forced thermalization
It could be expected that atoms which are loaded into the optical trap are unpolarized. To our surprise, this is not the case for the loading cycle we apply (cf. Sect.
3.1.3). A small net polarization was present after loading the trap and a thermalization period of 400 ms. Most probably, this is due to an unbalanced light field in
the last sub-Doppler cooling phase. The optical beams for the MOT are all retroreflected in order to recycle the light. That is, the light passes twice through the
atomic cloud with opposite circular polarization (but same helicity). The imbalance
can be either caused by static losses on optical elements or due to atoms. The latter
is occurs because we retro-reflect the MOT beams once thy have passed the atomic
cloud. Hence, the second pass will, in general, have less intensity than the first one.
We actively produce thermal spin states in F = 1 by repeatedly optically pumping
atoms from F = 1 to F = 2 and back, using lasers tuned to the F = 1 → F 0 = 2 and
F = 2 → F 0 = 2 transitions, and applied via the MOT beams (cf. Fig. 7.2). Each
pumping cycle takes 300 µs. To avoid any residual polarization, we apply bias fields
of a few hundreds of mG along z, y, and x during the three back-and-forth cycles.
Finally, the F = 2 manifold is further depleted with a 100 µs pulse of resonant light
on the F = 2 → F 0 = 2 transition with zero magnetic field. After these steps, no remaining mean polarization along z is observed. This procedure is designed to transfer
disorder from the thermalized center-of-mass degrees of freedom to the spin state.
Illumination from six directions produces a polarization field with sub-wavelength
structure, in which the atoms are randomly distributed. Possible net imbalances
in the pump polarizations are scrambled by the application of different bias fields.
The success of the thermalization procedure is directly measurable through the noise
scaling and will be discussed further below.
7.1. Calibration of spin detector
139
Figure 7.2.: (a) Atomic ensemble with probing, pumping, and imaging light fields.
The polarimeter measures in the 45 basis, i.e., the Stokes component Ŝy . (b) Atomic
transitions for probing, preparation, and imaging light fields.
Full pulse sequence
For the calibration measurement we prepare atomic ensembles of 106 atoms as explained in Sect. 3.1.3. The full experimental sequence is illustrated in Fig. 7.3. After
loading the trap we perform 20 experiments which we call blocks. In each block we
prepare a thermal spin state, perform the QND probing of the spin state, and measure the atom number NA via DANM, as described above and in Fig. 7.3. The atom
number from one block to the next is reduced by ∼ 15 %. After removing the atoms
from the trap at the end, we measure the bias of the polarimeter by sending a train
of probe pulses. We load the dipole trap for more than 500 times to acquire enough
statistics.
Figure 7.3.: Full pulse sequence for projection noise measurement of thermal spin state.
a) Atom trap loading, b) 20 repetition of the spin state measurements, c) atoms are
released from dipole trap, and d) bias of polarimeter is measured. This whole cycle is
repeated more than 500 times.
7.1.4. Trap loss in thermal state preparation
The loss of atoms from the dipole trap during the state preparation can be estimated
from the evolution of the atom number as a function of number of blocks. In Fig. 7.4
we plot the measured number of atoms. For this we convert the rotation signal
measured in the DANM at the end of each block with the independently measured
140
7. Spin Squeezing
N
A
6
10
5
10
1
6
11
16
[Block #]
Figure 7.4.: Atom number for different blocks measured in the
DANM after the spin measurement. The single exponential decay rate is k = 0.169(2)/block.
That is, we lose 15.5 % in each
state-preparation/probing block.
Error bars (within the circles) indicate the 95 % confidence interval
of the fitted atom number.
e 1,DANM = 6.6(2) × 10−8 . This was determined from measuring
coupling constant G
the rotation signal of a fully polarized atomic sample and for a known atom number.
For more details see Sect. 4.3. The data points follow a single exponential decay
NA (p) = NA,0 exp (−k p), where k = 0.169(2) is the decay constant. The loss per
block can be calculated from 1 − e−k and is 15.5(2) %. The loss of atoms from one
block to the other can be attributed almost completely to the preparation of the
thermal state. The other loss mechanism where measured by repeating the cycle
without thermalization and found to be negligible.
7.1.5. Projection noise limited spin detection
Experimental data for atom numbers between 4×104 and 8×105 and photon numbers
up to 109 are shown in Fig. 7.5. The data are fitted with the theoretical expression
e 1 = 6.65(3) ×
(7.6) which is shown as a surface. The deduced coupling constant is G
−8
5
10 and the electronic noise level is VE = 4.9×10 . The coefficients for the technical
noise are α = 4.3(1)×10−11 and β = 3.1(7)×10−7 . Atomic quantum noise dominates
over other technical and quantum noise sources for a large range of NL and NA , as
seen in the vertical panels of Fig. 7.5.
For the maximum number of photons NL = 1 × 109 , the noise scaling with atom
number is highlighted in Fig. 7.6. For the largest atom number measured, i.e.,
NA = 7.6 × 105 , the light shot noise, atomic technical noise, light technical noise
and electronic noise are respectively 3.5, 6.3, 11.2, and 30 dB below the quantum
noise level of the thermal reference state. From Fig. 7.6 we can read a spin readout
noise which is factor of ∼ 1.9 (2.8 dB) better than the spin noise for the thermal
spin state at NA = 7.6 × 105 with spin F = 1. Comparing to the coherent spin
state, which defines the projection noise level, we measure a reduction of 1.6 dB. In
absolute terms, this corresponds to a sensitivity of δNA = 515 spins.
The light technical noise may be due to small imbalance of the polarization analyzer
7.1. Calibration of spin detector
141
Figure 7.5.: Measured variance of Ŝy plotted as black dots and a fit to the data using
Eq.(7.6) as colored surface. The distance of the data points from the fitted surface
is indicated as vertical gray lines. The left plot shows the atom-number-scaling for
NL = 109 . See Fig. 7.6 for more details. The right plot shows the photon-numberscaling for NA = 7.6 × 105 . In the left and right plot curves indicate (from top to
bottom): a) total noise, b) quantum spin noise plus light noise, c) light shot and
technical noise, and d) light shot noise.
142
7. Spin Squeezing
and thermal birefringence produced by the dipole laser. Active stabilization of the
balancing could improve and reduce the light technical noise considerably. Atomic
technical noise may come from classical fluctuations in the lasers during optical
pumping.
Extrapolating the technical noise of atoms and light, both remain below their respective quantum noise terms up to NA,qn ≡ β −1 = 3.2 × 106 and NL,qn ≡ (4α)−1 =
5.8 × 109 , respectively. It would thus be possible to increase the number of atoms in
the trap while remaining projection-noise limited.
Figure 7.6.: Measured variance of Ŝy for NL = 109 as
a function of atom-number.
The errobars indicate the
statistical uncertainty in Ŝy
and the atom-number fluctuations. Dashed curve: Fit
using expression 7.6. Solid
Line: Pure spin quantum
noise. Dotted Line: Shot
noise and technical light
noise, Thin solid line: Light
shot noise. The electronic
noise is not plotted because it
is negligible for this number
of photons.
7.2. QND measurement of large-spin ensembles by dynamical decoupling
143
7.2. QND measurement of large-spin ensembles by dynamical
decoupling
Now we present a theoretical proposal and its experimental verification for a technique which allows QND measurement with polarized light of large-spin systems.
In Sect. 4.1 and the previous section, we have calibrated the spin detector with a
classical measurement on a polarized spin state, and a quantum measurement on a
completely unpolarized state. For both calibration techniques we were able to neglect
the influence of the tensorial part of the interaction Hamiltonian ∝ G2 (Ŝx Jˆx + Ŝy Jˆy ).
This is different for measurements of the quantum properties of polarized spin states,
e.g., h Jˆx i = NA /2 and h Jˆy i = h Jˆz i = 0.
First, we discuss the a problem that arises in optical QND measurements on a multilevel system. It is a direct result of the multilevel structure of the atomic system
used and caused by an accumulative built up of Raman coherences. We present the
problem in detail, propose a solution and demonstrate its viability.
This section is partly published as “Quantum Nondemolition measurement of largespin ensembles by dynamical decoupling” in Phys. Rev. Lett. 105, 093602 (2010).
[14]
7.2.1. Convenient fiction
Spin QND measurements in the realm of the Takahashi-Kuzmich proposal [58, 59]
on atomic systems has proven to be a powerful tool in the last decade. In the original
proposals by Takahashi and Kuzmich the authors are concerned with real or effective
spin 1/2 systems. The influence of the multilevel structure present in, for example,
alkali metal atoms is neglected using an argument of large detuning [58, 59]. The
argument goes as follows: The different scalings for vector and tensor polarizability,
i.e., G1 ∝ ∆−1 and G2 ∝ ∆−2 , seems to suggest that for detunings large compared
to the excited hyperfine splitting, the full polarizability Hamiltonian
Ĥpol =
~
~
G1 Ŝz Jˆz + G2 Ŝx Jˆx + Ŝy Jˆy
τ
τ
(7.11)
can be reduced to the ideal QND Hamiltonian
ĤQND =
~
G1 Ŝz Jˆz .
τ
(7.12)
We argue that in experiments that intrinsically rely on large spin ensembles, i.e.,
F > 1/2, as the one presented here, or novel proposals for quantum polarization
spectroscopy in spinor gases and the detection of exotic quantum phases [125, 170,
171], the large-detuning argument fails. In Fig. 7.7 we illustrate the effect that
each of the Hamiltonian terms has on a polarized spin state h Jˆx i = Jx = NA /2 for
h Ŝx i = NL /2.
144
7. Spin Squeezing
Figure 7.7.: Polarized probe light h Ŝx i = NL /2 interacts with a polarized spin state
along Jˆx . Left: The QND interaction introduces noise along Jˆy through the quantum
fluctuations in the probe light var(Ŝz ) = NL /4. Center: The tensorial term ∝ Ŝx Jˆx
rotates the introduced noise into Jˆz . Right: the tensorial term ∝ Ŝy Jˆy introduces light
quantum noise var(Ŝy ) = NL /4 into Jˆz . The last effect is much smaller than the first
two, because for our choice of light and atom polarization h Ŝy i = h Jˆy i = 0.
It becomes most obvious that the detuning argument breaks down if we study the
input-output relations for the QND variable Jˆz and its conjugate Jˆy after interaction.
To lowest order in Ĥpol we have
Jˆz0 = Jˆz + G2 Ŝx Jˆy − G2 Ŝy Jˆx .
Jˆy0 = Jˆy − G1 Ŝz Jˆx − G2 Ŝx Jˆz ,
(out)
Ŝy
=
(in)
Ŝy
+
(in)
G1 Ŝx Jˆz
−
(in)
G2 Ŝz Jˆx
(7.13)
(7.14)
(7.15)
and the variance of the output variable is
var(Jˆz0 ) = var(Jˆz ) + G22 Sx2 var(Jˆy ) + G22 var(Ŝy )Jx2 ,
var(Jˆy0 ) = var(Jˆy ) + G21 Jx2 var(Ŝz ) + G22 Sx2 var(Jˆz )
(7.16)
(7.17)
where we assumed Jy = Jz = 0 and Sy = Sz = 0. The quantum fluctuations
of both variables Jˆy and Jˆz are coupled. We get a first order estimation of the
tensorial effects if we compare the pure QND measurement dynamics of var(Jˆz0 )QND
and the additional noise in Jˆz that is introduced. In the ideal case of G2 = 0 and no
decoherence effects the uncertainty in Jˆz is
var(Jˆz0 )QND =
var(Jˆz )
,
1+ς
(7.18)
where ς = G21 Sx2 NA /4 is the signal-to-noise ratio of the QND measurement defined
in Sect. 2.5.4. The conjugate variable receives the equivalent amount of quantum
noise and reads
var(Jˆy0 )QND = (1 + ς) var(Jˆy ) .
(7.19)
7.2. QND measurement of large-spin ensembles by dynamical decoupling
1.2
1
Standard Quantum Limit
var(Jz)
0.8
0.6
0.4
0.2
0
ï2
covariance matrix
first order estimate
ï1.5
ï1
Detuning 6 [GHz]
ï0.5
145
Figure 7.8.: Simulation
results for optimal squeezing
in the presence of tensorial
effects.
The solid line is
the minimal variance from
a full covariance matrix
calculation and the dotted
the simple first order estimation Eq.˜(7.23). The first
order model overestimates
the noise reduction, because
we introduce NL,opt into
var(Jˆz )QND and not in the
exact solution. The number
of atoms in the simulations
is 106 and the interaction
area A = 4 × 10−9 m2
We estimate the additional noise in Jˆz due to tensorial effects using Eq. (7.16) and
Eq. (7.19)
var(Jˆz0 )add ≡ var(Jˆz0 ) − var(Jˆz0 )QND
= G2 S 2 (1 + ς) var(Jˆy ) ,
2 x
(7.20)
(7.21)
where the much smaller contribution G22 var(Ŝy )Jx2 is neglected.
We can estimate the maximum squeezing in the presence of tensorial effects, if we
assume there exist an optimal interaction time, or equivalently number of probe
photons NL,opt , when var(Jˆz0 )add = var(Jˆz0 )QND . From this we find the optimum
number of photons
p
G22 + 2G21 G2 NA − G2
2
(7.22)
NL,opt =
G21 G2 NA
The minimum uncertainty in Jˆz in this simple first order estimation is
var(Jˆz )min =
2var(Jˆz )
p
.
1 + 1 + 2G21 NA /G2
(7.23)
Now, it becomes evident why the detuning argument is not valid. The factor G21 /G2
in the denominator is in good approximation independent of the detuning, i.e., a
constant. To verify this simple model, we use the covariance matrix calculation
technique of Appendix A and apply the full polarizability Hamiltonian Ĥpol . The
time steps are chosen sufficiently short to simulate the time evolution realistically. In
Fig. 7.8 we plot the simple and the covariance calculation for a set of typical system
parameters in a range of interesting detunings.
146
7. Spin Squeezing
There is another effect which is related to the second tensorial term ∝ Ŝy Jˆy that
is responsible for the last term in Eq. (7.16). It introduces quantum noise from Ŝy
into Jˆz which is amplified by the coherent spin state polarization Jx . Fortunately,
the effect is small compared to the initial variance of the atomic state. For realistic
experimental parameters we can compute the additional noise introduced to the
atomic state, which is G22 Jx2 var(Ŝy )/var(Jˆz ) ∼ 10−2 . For this term the detuning
argument works, and we could suppress its contribution even further by using larger
detuning.
Another important aspect is technical noise. Imperfectly prepared atomic or light
states for instance with Jy 6= 0 or Sy 6= 0, will lead to technical noise in Jˆz . The
additional coupling through the tensor terms will cause the technical noise be redistributed among all atomic spin components, e.g., Eq. (7.13).
We note that there are experiments in multilevel alkali metal atoms, which are affected little by the tensorial effects. For effective spin 1/2 systems on clock transition
the discussed effects are not present [172, 173, 119]. Others experiments circumvent
the problem “naturally” in using composite systems of two oppositely polarized ensembles and study joint properties of the system [4]. The tensor terms can also
be used for creating non-linear single atom interactions as demonstrated in Ref.
[174, 22].
7.2.2. Dynamical decoupling in QND measurements
We approach this problem using the methods of dynamical decoupling [175, 176, 177],
which allow us to effectively cancel the non-QND terms in the Hamiltonian while
retaining the QND term. To our knowledge, this is the first application of this
method to quantum non-demolition measurements. Dynamical decoupling has been
extensively applied in magnetic resonances [178, 179], used to suppress collisional
decoherence in a thermal vapor [180], to extend coherence times in solids [181], in
Rydberg atoms [182], and with photon polarization [183].
The main idea behind dynamical decoupling (DD) is that unwanted properties of
open quantum systems (OQS), e.g., decoherence through interaction with a bath,
can be controlled with time-varying control fields which act on the system on time
scales shorter than the memory time of the environment [175]. The method allows
one to engineer many effective OQS evolutions which are, in principle, immune to
noise and decoherence. In our case, where we are interested in suppressing certain
parts of the light-atom Hamiltonian, this strategy seems very promising.
The open quantum system is the system variable Jˆz which is coupled to a degree of
freedom, Jˆy , which is neither system nor meter in the QND measurement and can
be considered the bath. This coupling introduces noise into the system variable, and
decoherence into the state of the ensemble. To remove the decoherence associated
with this coupling G2 Ŝx Jˆy , we adopt the strategy of “bang-bang” dynamical decoupling [175, 176, 177]. In this method, a unitary Ûb and its inverse Ûb† are alternately
and periodically applied to the system p times during the evolution, so that the total
evolution is
Ûtot = [Ûb† ÛH (t/2p)Ûb ÛH (t/2p)]p ,
(7.24)
7.2. QND measurement of large-spin ensembles by dynamical decoupling
147
where ÛH (t) describes unitary evolution under Ĥpol for a time t. With this evolution,
those system variables that are unchanged by Ûb continue to evolve under Ĥpol , while
others are rapidly switched from one value to another, preventing coherent evolution.
For large p, the system evolves under a modified Hamiltonian
Ĥ 0 = P̂ Ĥ ,
(7.25)
where P̂ projects onto the commutant of (i.e., the set of operators which commute
with) {Ûb , Ûb† } [177].
To eliminate G2 (Ŝx Jˆx +Ŝy Jˆy ), while keeping G1 Ŝz Jˆz we choose a Ûb which commutes
with Jˆz , but not with Jˆx or Jˆy , namely a π rotation about Jˆz , Ûb = exp[iπ Jˆz ]. This
leaves Jˆz unchanged, but inverts Jˆx and Jˆy . By the symmetry of Ĥpol , this is
equivalent to inverting Ŝx and Ŝy , which suggests a practical implementation: probe
with pulses of alternating Ŝx , and define a ‘meter’ variable taking into account the
inversion of Ŝy .
Two-polarization probing
(1)
We consider sequential interaction of the ensemble with a pair of pulses, with Ŝx =
(2)
(diff)
(1)
(2)
−Ŝx = NL /4p. We define also the new ‘meter’ variable Sy
≡ Ŝy − Ŝy .
We describe the atomic variables before, between, and after the two pulses with
superscripts (in), (mid), (out), respectively. We apply Equations (7.13) and (7.14)
to find:
(mid)
(in)
(1) (in)
Jˆz
= Jˆz + G2 Ŝx Jˆy
(1) (in)
(mid)
(in)
(1,in) ˆ
Jx − G2 Ŝx Jˆz
Jˆy
= Jˆy − G1 Ŝz
(1,out)
(1,in)
(1) (in)
Ŝy
= Ŝy
+ G1 Ŝx Jˆz
(7.26)
(7.27)
(7.28)
and
(out)
(in)
Jˆz
= Jˆz
(diff,out)
Ŝy
=
(diff,in)
Ŝy
(7.29)
+
(1) (in)
2G1 Ŝx Jˆz
(7.30)
plus terms in G1 G2 Ŝx Ŝz Jˆx , G22 Ŝx2 Jˆz and G1 G2 Ŝx2 Jˆy which become negligible in the
limit of large p. The ideal QND form is recovered by the dynamical decoupling.
We verify the method in a more realistic modeling using the covariance matrix.
There also the influence of incoherent effects, i.e., photon scattering are taken into
account. A homogeneous atomic ensemble is probed by a single pulse or by a train
of p alternately polarized pulses. In Fig. 7.9 we show the results for the covariance
matrix simulation (cf. Appendix A) applying the technique to realistic experimental
parameters, which are given in the caption. We plot the amount of spin squeezing
−ξe2 (cf. Eq. (2.44)) as a function of normalized interaction time t0 , which is proportional to the probe photon number. The black solid line is the simulation with
148
5
7. Spin Squeezing
j2 [dB]
K
4
3
2
1
0
0
2
4
6
8 Time [t ]
0
Figure 7.9.: Amount of spin squeezing
as a function of probing time in units
of normalized interaction time t0 (cf.
Chap. A). The black line shows the
case of G2 = 0, i.e., no tensor effects.
The red dotted line and the blue circles
show is the case of the full interaction
Hamiltonian for a single and alternating polarization, respectively. For the
simulations we used ∆ = −600 MHz,
G1 = 1.27×−7 , G2 = −6.86 × 10−9
and NA = 1 × 106 . Note, simulations
included decoherence effects from photon scattering, which is the reason for
the finite squeezing of ∼ 4.5 dB. The
oscillatory behavior for G2 6= 0 shows
the rotation of the uncertainty ellipse
about Jˆx .
Hamiltonian (7.11) with G2 = 0. The red dotted line and the blue circles are the
results for the full interaction Hamiltonian Ĥpol probed with a single polarization
and alternating pulses, respectively. In the simulation the normalized interaction
time t0 was subdivided into 200 time steps.
In theory, the method can restore the spin squeezing and effectively compensate the
effect of the tensorial terms in the interaction Hamiltonian. The oscillations of the
curve for the full interaction without DD show nicely the character of the Ŝx Jˆx term
in the interaction Hamiltonian, which leads to a precession-like deformation of the
uncertainty ellipse about Jˆx .
It is important to note that the two polarization technique naturally removes also
inhomogeneous effects of the unwanted effects due to the tensorial terms. All effects
we are concerned about, are proportional to either the number of photons or atoms
in the sample. In a dipole trapped atomic sample interrogated with a Gaussian laser
beam the tensorial effects are different over the size of the sample. This would, if
not compensated by reversing the dynamics, lead to an dephasing of the spin state.
In this aspect, the TPP method is similar to a spin echo technique.
7.2.3. Two-polarization meta pulses
We adapt the concept of meta-pulses to the case of alternating probe pulses. This is
important to account for experimental limitations. In theory, the coupling constant
G1 is the same for vertical and horizontal probe light, if the spatial properties of
both probe beams and the coupling to the atoms are identical. Experimentally,
there can be slight differences due to alignment issues. Furthermore, probe pulses
for vertical and horizontal polarization can contain different photon numbers. All
these experimental issues can be taken into account by defining the measurement
7.2. QND measurement of large-spin ensembles by dynamical decoupling
outcome as
M ≡ (NL /4)
−1/2
2p
X
(out)
(−1)j+1 sy,j
,
149
(7.31)
j
(out)
where sy,j is the polarimeter output for j vertical (odd j) and horizontal (even j)
probe pulses and p is the number of pairs of v − h probe pulses. The total number of
photon sent is NL = NL,v + N L,h , where NL,v ≡ pnL,v . The normalization constant
in front of the above sum was chosen so that we get var(M) = 1.
In the presence of atoms and for the absence of technical noise in light and atoms
we expect (cf. (2.66)) the following outcome for the variance in the measurement
results
var(M) = 1 + G21 NL var(Jˆz ) ,
(7.32)
where the second term is the signal-to-noise ratio ς. In the case of arbitrary number
of probe photons per pulse nL,v (nL,h ) and interaction strength G1,v (G1,h ) for vertical
(horizontal) polarization, we get
var(M) = 1 +
4
(G1,v NL,h + G1,h NL,h )2 var(Jˆz ) .
NL
(7.33)
If we compare Eqs. (7.32) and (7.33) we find the relation between the individual and
the effective interaction strength
G1 = 2
G1,v NL,v + G1,h NL,h
.
NL
(7.34)
In the case of equal photon number for both polarizations we get G1 = G1,v + G1,h ,
as we should expect.
7.2.4. Experimental verification of dynamical decoupling idea
The experimental sequence is shown schematically in Fig. 7.10. In each measurement
cycle the atom number NA is first measured by a dispersive atom-number measurement (DANM) as introduced in Chap. 4. A Jˆx -polarized coherent spin state (CSS)
is then prepared and probed with pulses of alternating polarization. Immediately
after, h Jˆx i is measured to quantify depolarization of the sample and any atoms having made transitions to the F = 2 manifold are removed from the trap, reducing NA
for the next cycle and allowing a range of NA to be probed on a single loading. This
sequence of state preparation and probing is repeated ten times for each loading of
the trap. The trap is loaded 350 times to acquire statistics.
In Fig. 7.11 we plot the measured noise versus atom number, which confirms the
linear scaling characteristic of the QND measurement. The red squares indicate
the variance var(Ŝy ) normalized to the optical polarization noise, i.e., var(M).
Independent measurements confirm the polarimetry is shot-noise limited in this
regime. The black solid line is the expected projection noise scaling 4var(Ŝy )/NL =
150
7. Spin Squeezing
Figure 7.10.: Experimental sequence for projection noise measurement. The CSS is
prepared once and its magnitude h Jˆx i is measured. This serves as a measure of the
spin polarization prior to the QND probing. We prepare the CSS a second time and
assume it has the same spin polarization as in the first preparation. The state is probed
with a train of pulses of alternating polarization. The complete train of probe pulses
scatters about 16 % of atoms into F = 2, which are removed from the trap with resonant
light in order to reduce the number of atoms in the trap. The whole cycle is repeated
10 times during one trap loading. For more details see text.
1 + G21 NL var(Jˆz ), calculated from the independently measured interaction strength
G1 = 1.27(5) × 10−7 and number of probe photons NL = 8 × 108 .
Also shown are results of covariance matrix calculations simulating the DD technique,
including loss and photon scattering. The scenarios considered include the naive
QND measurement, i.e., with a single polarization, and the “bang-bang” or twopolarization QND measurement, with p = 1, 2, 5. These show a rapid decrease in the
quadratic component with increasing p. This confirms the removal of G2 due to the
dynamical decoupling. Also included in these simulations is the term Ŝy Jˆy which
introduces noise into Jˆz proportional to G22 var(Ŝy ) h Jˆx i2 . For our experimental
parameters this term leads to an increase of var(Jˆz ) of less then 2 % and as noted
above could be reduced with increased detuning.
A measurement without applying the DD technique is plotted in the blue circles.
The data points show a quadratic scaling which exceeds the predicted one from the
covariance matrix simulations (solid gray line). Classical noise is responsible for
the large amount of additional noise. This successfully confirms the strength of the
method to suppress not only the unwanted quantum noise redistribution but also
classical noise sources coupled via G2 .
At the largest atom number of NA = 8.8 × 105 , the QND measurement achieves
projection-noise limited sensitivity, i.e., the measurement noise is 5.7(6) dB below
the projection noise. Within the precision of the measurement, no technical noise
from atoms or light is observed.
7.2. QND measurement of large-spin ensembles by dynamical decoupling
151
Figure 7.11.: Variance of polarimeter signal as a function of atom number, comparing
naive probing, i.e., a single input polarization, to “bang-bang” dynamically-decoupled
probing of different orders p. Grey curves indicate simulation results for: naive probing (solid), and decoupled probing with p = 1 (widely dashed), p = 2 (dashed), and
p = 5 (dotted). The black solid line shows the expected projection noise for p → ∞, or
the ideal QND interaction G2 = 0. All curves are calculated using the independently
measured interaction strength G1 = 1.27(5) × 10−7 and have no free parameters. Red
squares are measured data using dynamical decoupling with p = 5. Error bars indicate
statistical uncertainties: plus/minus one standard deviation of atom number (horizontal) or of var(Ŝy ) (vertical). Blue circles are measured data with naive probing, which
show a quadratic scaling (dash-dotted line). Technical noise from laboratory fields dominates the naive probing results, and pushes them above the theoretical curve, while
technical noise is suppressed in the two-polarization probing.
152
7. Spin Squeezing
7.3. Spin squeezing of magnetically sensitive atomic state
Having devised a technique to perform spin QND measurements on large-spin systems we are prepared for the generation of (conditional) spin squeezing in a magnetically sensitive spin state on the F = 1 hyperfine ground state of 87 Rb.
At the beginning we will characterize the de-polarization of the coherent spin state as
a function of photon number in the probing pulse. Then the experimental sequence
is detailed and finally the experimental results are presented and discussed.
7.3.1. Depolarization
To claim spin squeezing it is not only important to measure the reduction of spin
noise, but also the influence of the QND measurement on the state, i.e., its polarization Jˆx . The experimental spin squeezing criteria, presented in Chap. 2 (2.69) and
(2.70), require that we measure the polarization of the coherent spin state after the
QND interaction.
Here, we measure the reduction in spin polarization Jx as a function of number of
photons NL in the vector probe (cf. Sect. 3.5.4). The experimental sequence, shown
in Fig. 7.12, consists of two spin polarization measurements with the tensor probe
(cf. Sect. 3.5.4). In the first, we measure the spin polarization Jx (0) immediately
after preparation. In the second, Jx (NL ) is measured after a certain number of probe
photons NL was sent to the atomic sample.
The coherent spin state is prepared as explained in Sect. 3.4. During the whole
measurement we apply a small bias field along x of Bx ∼ 100 mG. The guiding
field has two purposes. First, it stabilizes the state against un-compensated stray
magnetic fields. Second, it prevents precession of the spin state induced by the
vector part of the interaction Hamiltonian (cf. Sect 4.2.2). The tensor probe is
tuned 190 MHz to the red of F = 1 → F 0 = 0, where the vectorial term has a large
coupling strength of G1 ∼ 6 × 10−7 .
The vector probe is tuned 800 MHz to the red of the transition F = 1 → F 0 = 0.
This detuning is in the commonly used range for our experiments. Later we will
tune the laser slightly closer, to 600 MHz. We send a train of pulses which have a
duration of 1 µs and a period of 10 µs. Each pulse has a fixed number of photons and
we change the total number of photons, NL , by varying the number of probe pulses.
Figure 7.12.: Scheme for characterization of depolarization of coherent spin
state. "Pump Jx ": atoms are prepared in the coherent spin state via optical pumping, "Probe Jx ": atoms are
disperisvely probed with circular light,
and the blue pulses indicate the QND
probing with NL photons.
7.3. Spin squeezing of magnetically sensitive atomic state
153
In Fig. 7.13 we plot the normalized spin polarization as a function of photons NL .
The black line is a single exponential fit
Jx (NL ) = Jx (0) exp(−ηγ NL ) ,
(7.35)
where ηγ is the depolarization probability for a single probe photon. We see that the
spin polarization is well described by an exponential decay. The depolarization per
probe photon is ηγ = 1.69(3) × 10−10 . In later experiments we determine ηγ from a
two-point measurement. For this we prepare the atomic ensemble twice and measure
its polarization. The first time the atoms are not probed and the second time we
send NL0 vector probe photons. From Eq. (7.35) we can compute ηγ by
1
Jx (0)
ηγ = 0 ln
.
(7.36)
NL
Jx (NL0 )
7.3.2. Spin squeezing pulse sequence
Atoms are loaded into the dipole trap as described in Sect. 3.1.3. We start the
experiment with NA ∼ 106 cold atoms in the dipole trap. Before the QND measurement, we measure the photon scattering probability per probe photon ηγ in a two
point measurement as explained above.
In Fig. 7.14 we show the following step in the experimental sequence. For the QND
measurements, we prepare the coherent spin state twice. The first time, we apply a
small guiding magnetic field Bx ∼ 100 mG along x and measure the spin polariza(in)
tion. This value serves as the initial spin polarization value Jx for this particular
measurement. The second time, for the actual QND measurement, the coherent
spin state is prepared in magnetic-field-free conditions. Immediately after, we send
Jx(NL) [normalized]
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
2
4
6 N [×109]
L
Figure 7.13.: Decaying spin polarization as a result of photons scattering.
The black line is a fit to the data
points (blue squares) with Jx (NL ) =
Jx (0) exp(−ηγ NL ). We find the decay
constant ηγ = 1.69(3) × 10−10 . The
error bars indicate one standard deviation.
154
7. Spin Squeezing
a train of 10 pairs of x − y-polarized probe pulses. Each of them is 1 µs long and
carries nL ∼ 8.84(2) × 107 photons. The probe laser is tuned 600 MHz to the red of
the F = 1 → F 0 = 0 resonance.
To sample different atom numbers in a single trap loading, we remove atoms from the
trap after the QND measurement. This is possible since a non-negligible fraction of
atoms is transferred into F = 2 by the probing light. From simulations we estimate
that in ∼ 40 % of the scattering events that lead to a depolarization of the CSS the
atom is transferred into F = 2. We apply resonant light from F = 2 → F 0 = 3,
which should heat out atoms that are F = 2 without scattering them back into
F = 1. The light is applied for 5 µs through MOT beams.
The preparation, probing and cleaning steps are repeated ten times on the same trap
loading. After that, we remove all remaining atoms from the trap by switching the
trapping light off for a few tens of milliseconds. A last train of probe pulse is sent to
measure the shot noise of the polarimeter in absence of atoms, i.e., var(M (e) ). The
whole sequence is repeated ∼ 350 times to acquire statistics.
Characterization of depolarization and coupling constants
In the calibration measurement of ηγ right after trap loading we measure a depolarization of 41(2) % for a total of NL = 20 × nL = 1.77 × 109 photons. From this,
we calculate the depolarization per probe photon ηγ = 3.0(1) × 10−10 . Note, we are
working slightly closer to resonance compared to the measurement of the depolarization presented in Sect. 7.3.1. If we calculate the ratio of the scattering cross section
Figure 7.14.: Important steps in the experimental spin squeezing sequence. After loading the dipole trap and measurement of the depolarization rate per photon ηγ , which
are not shown here, we send the following pulse sequence: “Pump Jx ” and “Probe Jx ”
determines the number of spin polarized atoms. The atoms are re-prepared in the
second “Pump Jx ” without bias magnetic field and the QND measurements M1/2 are
performed. Different atom numbers are sampled by removing atoms which have been
scattered into F = 2 with resonant light (“clean F = 2”). The sequence is repeated ten
times before the trap is emptied completely to measure the light shot noise.
7.3. Spin squeezing of magnetically sensitive atomic state
155
according to expression (B.2) for −800 MHz and −600 MHz, we get
σ (−600 MHz)
' 1.665 ,
σ (−800 MHz)
(7.37)
which is close to the ratio between
ηγ (−600 MHz)
∼ 1.775
ηγ (−800 MHz)
(7.38)
we measure. For the calculation of the amount of spin squeezing, we will use
ηγ (−600 MHz) and compute the depolarization for meta-pulses of different photon
number NL .
The heating of atoms out of the trap after the QND measurement removes 12.8(4) %
in each cycle. Knowing the depolarization of 41(2) % (measured) and the transfer
probability of 40 % (calculated) we would expect ∼ 16 % to be transferred to F = 2.
The slight difference may be due to insufficient optical power. However, the reduction
by almost 13 % is sufficient for the purpose of reducing the atom number in the trap
to scan a large range of NA .
The coupling constants for the two probe polarizations are measured independently.
For this we polarized the atomic ensemble along z in a small bias field of Bz ∼ 100 mG
and sent linearly polarized vector probe light with the same detuning as in the QND
measurements. The number of atoms is measured via absorption imaging. We find
slightly different results for G1,v = 1.34(5) × 10−7 and G1,h = 1.20(4) × 10−7 . We
explain this by a different mode matching for both probe beams. Using Eq. (7.34),
we can compute the effective coupling strength G1 = 1.27(5) × 10−7 .
Observation of spin squeezing
The detected signals of the train of probe pulses are converted into meta-pulses M
as explained in Sect. 7.2.3. We combine two pairs of v and h polarized pulses into
one meta-pulse for the first and second QND measurement M1 and M2 , respectively.
(in)
Then, we group results according to measurement of Jx . For each set of atom
(in)
(out)
numbers we compute var(Jˆz ) and var(Jˆz |M1 ) via Eq. (2.72) and (2.71), respectively. In Fig. 7.15 we plot the variance of the initial coherent spin state as
blue squares. The solid line is the projection noise level for the coherent spin state,
(in)
i.e., var(Jˆz ) = NA /4. The linear scaling of var(Jˆz ) with NA along the theoretical
prediction, indicates that we are measuring exclusively the projection noise of the
coherent spin state.
(out)
In the red squares we plot the conditional variance var(Jˆz |M1 ), calculated via
Eq. (2.71). At an atom number of NA = 6.7 × 105 the spin noise is reduced by
2.9(+1.3
−1.0 ) dB compared to the projection noise level.
The amount of metrologically relevant spin squeezing [49] is calculated from the
amount of depolarization during the squeezing interaction via
(out)
2
ξm
var(Jˆz |M1 )
=
,
(in)
var(Jˆz )(1 − η)2
(7.39)
156
7. Spin Squeezing
var(J ) [x105]
z
2
1.5
1
0.5
0
0
2
4
N [x105]
6
A
(in)
Figure 7.15.: Measured variance of coherent spin state var(Jˆz ) (blue dots) and conditional variance var(Jˆz |M1 ) (red squares) as a function of atom number. The solid
line is the expected spin projection noise calculated from the independently measured
coupling constant G1 . The dashed line is the projection noise level of the depolarized
(in)
spin state calculated as (1 − η)2 var(Jˆz ). In the dotted line we show the conditional
variance expected from the measured coupling strength for the ideal situation, i.e.,
(in)
var(Jˆz )/(1 + ζ). The errorbars in horizontal direction indicate one standard deviation of the atom number fluctuations and in the vertical direction the statistical
uncertainty of the variance.
where η = 1−exp(−ηγ NL ). For the combination of four individual probe pulses (two
2 = −2.0(+1.3 ) dB
v and two h) the depolarization is η = 9.9(4) %. This leads to ξm
−1.0
of metrologically relevant spin squeezing. As described in Sect. 2.4.2, this implies
entanglement between atoms in the ensemble [53, 54].
We conclude, that the dynamical decoupling via two polarization probing is restoring
the QND measurement and can be used to produce measurement induced squeezing.
8. Summary and Outlook
8.1. Summary
Quantum entanglement is one of the most surprising discoveries of the last 100 years,
and physicists are still trying to understand both its meaning and the practical effects
it may have. In recent years entanglement has mostly been studied in small numbers
of particles, and for purposes of information processing. In contrast, this thesis
describes experiments to generate entanglement of millions of atoms, and aims at
practical application in precision measurement.
The measurement induced generation of entanglement, i.e., spin squeezing, relies on
the preparation and detection of quantum states of light and atoms with a minimum of classical noise. Furthermore, it is important to have sufficient interaction to
surpass the inevitable light shot noise of the dispersive spin read out. We characterized the light-atom interaction by measuring macroscopic rotations from a polarized
atomic ensemble. These gave, in connection with quantitative absorption imaging
a “classical” estimate of the QND interaction strength. From this we established a
technique which substitutes the quantitative, but destructive absorption imaging, by
dispersive measurement atom number measurement (DANM). In this way, we are
able to perform many experiments on a single atomic cloud by re-preparation and
measurement without the need for reloading the trap every time. Further, we extended this method towards the measurement of Zeeman coherence via tensor light
shifts, i.e., pseudo spin components Jˆx or Jˆy by measuring of polarization rotation
of circularly polarized light.
Working with a magnetically sensitive system demanded the thorough characterization of the magnetic field environment at the position of the atomic cloud. We
developed and demonstrated a method which uses the atomic cloud as a sensitive
magnetometer with spatial resolution in the order of a few tens of microns. Furthermore, the magnetic field imaging technique potentially provides a full vector-field
measurement. We performed in-situ gradient control with 2 mG/cm resolution. Further reduction of atomic motion,
√ for instance in an optical lattice, could in principle
give sensitivities of ∼ 50 fT/ Hz.
It is paramount for the generation of squeezing to have a knowledge of the input
state. We proposed a spin tomography technique for alkali metal atoms which is
based on dispersive paramagnetic Faraday rotation measurements. First promising
results were presented. Combined with better magnetic field control this technique
could serve to measure the ground state density matrix.
Trying to change the quantum properties of a spin state has the pre-requisite to
have knowledge about all of the noise contribution to the measurement signal of the
158
8. Summary and Outlook
spin detection method. We use noise scaling and a thermal spin state to obtain
an absolute quantification of the measurement noise. The results are confirmed
by independent quantification of the QND measurement gain, i.e., the atom-light
interaction strength. The method detects different noise sources, i.e., atomic and
light quantum and technical noise, and the electronic noise floor, by their respective
scaling with atom and photon number. In this measurement we were able to show
that the sensitivity of the spin detection is ∼ 500 spins at a moderate destruction of
20 %.
Working in the hyperfine ground state of an alkali-metal atoms, like 87 Rb, offers a lot
of interesting physics. At the same time, it is quite different from the simple two-level
atoms of many quantum optics textbooks. We apply dynamical decoupling to remove
some of this complexity and perform optical quantum non-demolition measurement
in a large-spin system with F = 1. We first identify an often-overlooked impediment
to this goal: the tensorial polarizability causes decoherence of the measured variable
and prevents (pure) QND measurement except in the limit of large optical depth.
We then design an appropriate two-polarization probing strategy to cancel the tensorial components of the effective Hamiltonian. The dynamically-decoupled QND
measurement achieves a sensitivity 5.7(6) dB better than the projection noise level.
Finally, all the previous steps were leading us to the successful generation of spin
squeezing in an ensemble of 7 × 105 laser cooled and optically trapped 87 Rb atoms.
A spin noise reduction of 2.9 dB was observed at a destruction of < 10 %. The
noise reduction and the destruction corresponds to an improvement in measurement
sensitivity by 2 dB compared to a coherent state of the same spin polarization.
8.2. Conclusions and Outlook
The interaction of light and matter is, on one side, an everyday problem which, for
instance, allows us to read this text printed on the sheet of paper in our hands in
from of us. One the other side, it is one of the most fundamental and intriguing
phenomena which still provides interesting insights and large range possibilities for
new developments.
In this thesis, I presented results on the interplay between light and matter as quantum mechanical systems and the manipulation of the state of an ensemble of atoms
through measurements. This research was motivated from two directions. On one
end, the manipulation of the collective quantum state of atomic system that might
serve as a very sensitive magnetic field sensor can potentially improve measurements
which can give new insights in fundamental and applied problems. A magnetometer
reaching sensitivities beyond current technologies might open new ways to the direct
detection of magnetic fields from the human heart and brain, or might allow tests of
the fundamental symmetries of nature, to name only a few. On the other end, the
controlled interaction of atoms with light is a fundamental pre-requisite in the world
of quantum information processing (QIP) and quantum computation (QC). There,
the light as transmitter of information and matter (in a well controlled quantum
state) as the stationary memory.
8.2. Conclusions and Outlook
159
Controlling both systems in a way that they behave as quantum objects was one
of the main goals for this thesis. Trapping atoms in the focus of a strong laser is
one way to approach this goal. It allows to trap atoms in a state-independent way
with well protected from the environment. On the other side, we interrogate the
atomic spin system in a dispersive measurement which provides enough interaction
to observe quantum effects in the atomic system, but at the same time, introduces
only little disturbance.
Managing the magnetic field environment is another important issue in a system
which is intrinsically highly sensitive to magnetic fields. For this purpose, we implemented a technique that makes direct use of the atomic sample to measure the
magnetic field at the position of the atoms with high spatial resolution. It allowed us
to identify one impediment to produce long-lived atomic spin state due to magnetic
field gradients. The technique itself has the potential to be applied also in other
contexts, as for example, imaging of biomedical fields.
Making predictions about the future is always hard and mostly wrong. But the
system has the potential to allow for the investigation of interesting questions in the
near future.
The spin squeezing could be extended from the present pseudo- to a “real”-spin system
for spin-polarized atomic states along F̂x or F̂y which makes it a magnetometer in
the common sense [8].
The ability to perform QND measurements on large-spin atoms using dynamical
decoupling techniques will enable the experimental realization of novel theoretical
proposals for quantum polarization spectroscopy in spinor gases and the detection of
exotic quantum phases that intrinsically rely on large-spin systems [125, 170, 171].
Furthermore, the implementation of quantum memory protocols requires strong atom
light interaction. The demonstration of projection noise limited spin detection is one
step towards this goal.
Spin squeezing via QND measurements is also interesting for unpolarized atomic
states. In contrast to polarized ensembles there is no significant measurement backaction, which allows simultaneous squeezing of all spin components. The resulting
state will approach a macroscopic singlet. Spin squeezing in such systems implies
entanglement of a macroscopic number of atoms with arbitrary spin.
Macroscopic singlets are interesting in different areas of physics. First, they are interesting for condensed matter physics since they are the solution to fundamental spin
models. Second, they play an important role in quantum information processing.
Their invariance under rotations makes them interesting to encode quantum information in decoherence free subspaces [184] and for sending information independent
of a reference direction [185]. Third, magnetic field gradients or fluctuations can be
detected with high spatial resolution [186].
A. Appendix I: Gaussian state picture for
light and atoms
Contents
A.1. Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . 162
A.2. Physical system and mathematical model . . . . . . . . . 162
A.2.1. Linearized continuous variables and covariance matrix . . . 163
A.2.2. Spatially and temporally extended systems . . . . . . . . . 164
A.3. Unified description of physical processes . . . . . . . . . 165
A.3.1. Coherent effects
. . . . . . . . . . . . . . . . . . . . . . . . 166
A.3.2. Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.3.3. Transport processes . . . . . . . . . . . . . . . . . . . . . . 169
A.3.4. Projective Measurement . . . . . . . . . . . . . . . . . . . . 169
A.3.5. Combining effects
. . . . . . . . . . . . . . . . . . . . . . . 170
A.4. Instructive examples
. . . . . . . . . . . . . . . . . . . . . 170
A.4.1. Detector time-resolution . . . . . . . . . . . . . . . . . . . . 171
A.4.2. Spatial inhomogeneities . . . . . . . . . . . . . . . . . . . . 173
A.4.3. Atomic motion . . . . . . . . . . . . . . . . . . . . . . . . . 174
A.5. Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . 176
A.6. Coherent and incoherent transport processes described
by N-port unitaries . . . . . . . . . . . . . . . . . . . . . . 177
A.6.1. Transport processes . . . . . . . . . . . . . . . . . . . . . . 177
162
A. Appendix I: Gaussian state picture for light and atoms
This chapter extends the covariance matrix model for collective light-atom interfaces,
developed in [70, 71, 187, 188, 65, 189, 63], to describe the quantum properties of
continuous variable states of light and atoms as Gaussian states. The main interest
is to model real world effects such as the finite temporal resolution of the detector,
spatial inhomogeneities in light-atom interaction and atomic mixing. Furthermore,
we introduce the tools to describe the evolution of entanglement in the presence of
tensor light shifts present in multilevel atoms.
The content of this chapter is published as: “Unified description of inhomogeneities,
dissipation and transport in quantum light-atom interfaces” by M. K. and Morgan
W. Mitchell in J. Phys. B 42, 195502 (2009) [15]
A.1. Introduction
A variety of atomic ensemble systems for spin squeezing and continuous variable
quantum memories have been proposed or demonstrated, including hot atoms in
cells and cold atoms in magneto-optical traps and in optical dipole traps. The
conditions vary greatly from system to system, notably in the number of atoms,
from ∼ 1012 in cells to ∼ 106 in dipole traps, in temperature, from ∼ 300 K in cells
to ∼ 30 µK in atom traps, and in the time-scale of the interactions, from milliseconds
in cells to microseconds in cold atoms. At the same time, a variety of other physical
effects, such as loss and decoherence of atoms, scattering and diffraction of light,
and inhomogeneous coupling of the light and atomic variables are present to varying
degrees in these systems.
Modeling of quantum interfaces requires addressing this diversity of scales and physical effects without sacrificing the simplicity that motivates the collective variables
approach. Fortunately, many important problems involve Gaussian states [190],
which allows the quantum variables to be described very simply in terms of covariance matrices. The covariance matrix formalism also naturally handles partitioning
of an ensemble into sub-ensembles, a technique pioneered by Madsen et al. [65],
which we use to describe both temporal and spatial inhomogeneities.
Previous work on inhomogeneity has included mode matching [191] and introduction
of weighted variables [192] in order to describe their influence. In some cases decoherence effects due to inhomogeneous coupling have been identified [193]. To extend
the model of Madsen et al. [65], we work with physical variables, for example the
Stokes (S) or atomic spin (J) operators in their extrinsic (as opposed to canonical)
form. This way, we show that transport and dissipation can also be treated via the
covariance matrix formalism.
A.2. Physical system and mathematical model
We describe atomic ensemble systems and continuous-wave light fields in terms of
Gaussian continuous-variables [63, 189]. This treatment is a well established method
A.2. Physical system and mathematical model
163
and applies to the majority of operations used in experiments, e.g., light atom interaction, homodyne detection etc. [68]. Furthermore, it allows us to describe nonclassical properties, e.g., squeezing, entanglement, etc. [194, 195, 196].
A.2.1. Linearized continuous variables and covariance matrix
Mesoscopic ensembles of atoms and photons are described to good approximation
by Gaussian states, which are characterized completely by their first and second
moments [190]. While first moments describe the expectation values of the system,
the second moments contain the quantum properties of the state and classical fluctuations. The theoretical framework of covariance matrices used to describe many
experimental situations, e.g., coherent and incoherent light-atom interaction and
measurement induced state preparation is well developed [187, 65, 15, and references
therein]. However, all of the treatments rely on well polarized systems. In this limit
the description can be reduced from a three dimensional phase space, for SU(2), to
the harmonic oscillator phase space. Formally, this is referred to as the contraction
from SU(2) to the Heisenberg-Weyl group [28]. In this picture the quantum and the
classical variables are separated and can be treated independently.
For a general description, the restriction to a subspace is not satisfactory. The twodimensional approximation is not capable of treating, for instance, noise in the main
polarization or cases where no main polarization exists at all, i.e., for a thermal
atomic state.
Here, we will extend the covariance matrix formalism to arbitrary spin systems by
linearizing their dynamics [111]. We split the angular momentum operator into a
mean value and a fluctuating term
Jˆi = Ji + K̂i
and Ŝj = Sj + T̂j ,
(A.1)
where Ji ≡ h Jˆi i (Sj ≡ h Ŝj i) is a real number representing the mean value, and K̂i
(T̂j ) describe both quantum and classical fluctuations of Jˆi (Ŝj ). Further, we combine
variables for atoms and light into a single vector
v̂ = (Jˆx , Jˆy , Jˆz , Ŝx , Ŝy , Ŝz )T ,
(A.2)
sometimes referred to as displacement vector , which splits up into a vector of mean
values and fluctuating variables
v̂ ≡ u + ŵ ,
(A.3)
u ≡ h v̂ i = (Jx , Jy , Jz , Sx , Sy , Sz )T ,
(A.4)
where
and the fluctuating term is naturally defined as
ŵ = v̂ − u = (K̂x , K̂y , K̂z , T̂x , T̂y , T̂z )T
The covariance matrix is defined as
E
1D
γ=
ŵ ∧ ŵ + (ŵ ∧ ŵ)T − hŵi ∧ hŵi .
2
(A.5)
(A.6)
164
A. Appendix I: Gaussian state picture for light and atoms
where ∧ is the outer product between two vectors [71, 68]. For our purpose of
examining entanglement and squeezing, only the second moments are of interest.
We can write the covariance matrix of the joint atom-light system as
A C
γ=
,
(A.7)
CT L
where C describes correlation between atoms (A) and light (L).
A.2.2. Spatially and temporally extended systems
A central goal of present chapter is to include spatial inhomogeneities in a description
of the light-atom interaction. In cell experiments, the atomic ensemble has a constant
number density while a cold trapped sample can be highly inhomogeneous. In almost
all experiments, the light distribution is also inhomogeneous, e.g. from a Gaussian
beam.
The idea is to subdivide the physical system into locally homogeneous parts, which
are described by a single set of continuous variables. We partition the inhomogeneous
ensembles of atoms and light into several segments. That is, we define angular
0
momentum variables for the atom segments Ĵ(k,l) , and for the light segments Ŝ(k,l ) ,
with
(k,l)
(k,l)
(m,n)
(A.8)
[Jˆλ , Jˆµ
] = i~λµν Jˆν δkm δln
and
(k,l)
[Ŝλ
(m,n)
, Ŝµ
(k,l)
] = i~λµν Ŝν
δkm δln .
(A.9)
Segments orthogonal to the direction of light propagation are called transverse segments (first index) and along the direction of light propagation, longitudinal segments
(second index) (cf. Fig. A.1). The segmentation of the light in the transverse direction is a major difference to [65] and the pre-requisite to include any inhomogeneity
of light and/or atoms which would appear in realistic experiments. Note that in experiments the destruction of the atomic state will always be non-negligible. Hence,
any light inhomogeneity will be mapped onto the atoms even if they were evenly distributed at the beginning. This implies that we need to model the spatial dependence
of both atoms and light.
Here, we describe the segmentation in two dimensions only; the extension to the
third is straightforward. The continuous variable description remains valid so long
^ (1;l0 )
S
:::
^ (1;2)
S
^ (1;1)
S
^ (2;1)
S
..
.
..
.
:::
^ (1;2) : : : J
^ (1;l)
^ (1;1) J
J
:::
^ (2;1)
J
^ (k;l0 )
S
..
.
:::
S^µ
:::
:::
..
.
channel 1
^ (k;1)
S
..
.
^ (k;1)
J
..
.
..
.
:::
^ (k;l)
J
channel k
Figure A.1.: Schematic of partitioned atom-light interface. See text for details.
A.3. Unified description of physical processes
165
(k,l)
(k,l0 )
as the particle number in each segment is itself large, i.e., NA 1 and NL
1.
The total angular momenta for atoms and light are
X
X
0
Ĵ =
Ĵ(k,l)
and
Ŝ =
Ŝ(k,l ) .
(A.10)
k,l0
k,l
Further, we define a channel as the set of segments (light and atoms) which have the
same transverse index k.
We note that by using the physical angular momenta, we are working with extensive
quantities, as opposed to the canonical variables, which can be defined for
√ the conˆy / ~Jx , P̂ ≡
tracted
systems
in
the
limit
of
large
classical
components,
e.g.,
X̂
≡
J
√
Jˆz / ~Jx . That is, sums such as those given above are also angular momenta with
the expected quantum properties. This greatly facilitates the partitioning, and also
makes it possible to study situations in which Jx changes, for example through loss,
decoherence, or transport.
In a common phase space for atoms and light, we define an overall phase space vector
in terms of angular momentum operators as
h
iT
0
v̂ = Ĵ(1,1) , ..., Ĵ(k,l) , Ŝ(1,1) , ..., Ŝ(k,l )
(A.11)
which is readily rewritten as the direct sum of phase space vectors of the sub-systems
for atoms (A) and light (L)
v̂ = v̂A ⊕ v̂L .
= uA ⊕ uL + ŵA ⊕ ŵL
(A.12)
(A.13)
Analogous to the unsegmented case, we can write the displacement vector v̂ as the
sum of a vector of mean values u and a vector carrying the quantum and classical
fluctuations ŵ, see Eq. (A.5). The covariance matrix γ is completely defined via ŵ
as in the unsegmented case (A.6).
A.3. Unified description of physical processes
We will describe a variety of physical processes by their effect on v̂ and γ in the
Heisenberg picture. Some processes, such as light-matter interaction, can be described by Hamiltonian evolution, while others, such as photo-detection, cannot. To
handle both cases, we calculate the dynamics by difference equations which describe
small but finite changes between time-steps. This is a well established method which
is applied, for instance, in a Monte-Carlo wave-function approach by Mølmer et al.
[197] or, as here, in the context of covariance matrices described in Ref. [189]. It
allows modeling of coherent light-atom interactions, losses and decoherence, measurement and transport processes in a consistent way. The phase space vector and
covariance matrix are updated in finite time-steps τ as
v̂(t + τ ) = Fτ (v̂(t)) .
(A.14)
166
A. Appendix I: Gaussian state picture for light and atoms
After we introduce the main coherent and incoherent effects, we will specify limits
for the time-step in Sec. A.3.5.
The discretization of the continuous light-atom dynamics is one reason for the segmentation of light into longitudinal segments of length L = cτ . We note that in most
experiments the atomic sample is much shorter than the coherence time of the light.
That means, in each channel only one longitudinal light segment will overlap with
the ensemble at most times. In addition, the effect of several longitudinal atomic
segments is, from the light’s perspective, sequential: Ŝ(n,1) interacts with Ĵ(n,1) then
with Ĵ(n,2) , and so forth. During the time (m − 1) τ = t < mτ , Ŝ(n,m) interacts with
all atomic segments Ĵ(n,.) in the n-th channel.
A.3.1. Coherent effects
For effects described by a Hamiltonian Ĥ which is linear in the elements of the
phase-space vector v̂, the phase space-vector evolves (to lowest order in τ ) as,
i
iτ h
v̂(t + τ ) = v̂(t) −
v̂, Ĥ ≡ Tτ v̂(t) .
(A.15)
~
The last equality, which expresses the change in v̂ in terms of a matrix Tτ , is possible
because of the linearity of the Hamiltonian and the linearization of the atomic and
light variables (A.1). The covariance matrix evolves as
γ(t + τ ) = Tτ γ(t)TTτ .
(A.16)
Atom-light interaction
The most important unitary evolution of the system is due the dipole interaction of
light and atoms. This we will use to gather information about the atomic spin state.
At the same time the light influences the atomic variables.
For a homogeneous system of light and atoms, off-resonant interaction gives rise to
an effective Hamiltonian [165, 39]. For the F = 1 pseudo-spin system defined in
Chap. 2, it has the form
o
~n
Ĥeff =
G1 Ŝz Jˆz + G2 Ŝx Jˆx + Ŝy Jˆy
,
(A.17)
τ
where coupling constants G1 and G2 are proportional to the vector and tensor part of
the atomic polarizability. In (A.17) we neglect terms describing state independent AC
Stark shifts, which leave both the probe polarization and the spin state unchanged.
Using Eq. (A.15), we can calculate the evolution of the displacement vector v under
Ĥeff to first order in time, i.e., for τ being shorter than any time scale in the interaction process. We find separate transformation matrices acting on the mean value
u and the fluctuating term ŵ
v̂(t + τ ) = u(t + τ ) + ŵ(t + τ )
QS O3
QS RJ
=
u(t) +
ŵ(t) ,
O3 QJ
RS QJ
(A.18)
(A.19)
A.3. Unified description of physical processes
167
where O3 is the 3 × 3 zero matrix. The transformation matrices Q and R depend
on the mean values u(t).




1
−G1 Sz G2 Sy
1
−G1 Jz G2 Jy
1
−G2 Sx  , QJ =  G1 Jz
1
−G2 Jx 
QS =  G1 Sz
−G2 Sy G2 Sx
1
−G2 Jy G2 Jx
1
(A.20)




0
G2 Sz −G1 Sy
0
G2 Jz −G1 Jy
0
G1 Sx  , RJ =  G2 Jz
0
G1 Jx  (A.21)
RS =  G2 Sz
G2 Sy −G2 Sx
0
G2 Jy −G2 Jx
0
The dynamics of the classical components u are particularly simple and can be
calculated explicitly. In QND measurements of atomic spin degrees of freedom, we
can, without lack of generality, assume that the mean values for the light, uL , is
practically unchanged on its passage through the atomic cloud. We can calculate
the dynamics of the classical atomic components uA as
uA (t) = exp (QS t) uA (0) ,
(A.22)
where QS = (QS − I) /τ and I the identity matrix. We can substitute the Stokes
components Si defined before for a pulse of length τ in QS (cf. Eq. (A.20)) by Si =
τ ΦSi , where Φ is the photon flux and Si the normalized Stokes vector component of
the probe pulse.
The dynamics of the quantum variables follows directly from (A.19) and we get for
the update formula of the covariance matrix (A.16)
γ(t + τ ) =
QS RJ
RS QJ
γ(t)
QS RJ
RS QJ
T
.
(A.23)
A.3.2. Noise
In addition to Hamiltonian evolution, loss, transport, and decoherence of atoms
and/or photons can be described. These processes introduce extra noise into the
system. A fully general description of a noisy Gaussian process is the Gaussian
completely-positive map [198, 71], which acts on the covariance matrix as
γ 0 = MγMT + N ,
(A.24)
where the real matrix M transforms the phase space vector and the real symmetric
matrix N describes added noise. These must obey [198]
N + iΣ0 − iMΣMT ≥ 0 ,
(A.25)
where iΣpq ≡ [vp , vq ] and Σ0 , similarly defined, are commutation matrices before and
(k,l)
after the transformation and vp is an element of the phase space vector v̂, e.g, Sy .
Note that the commutation relations (A.8) and (A.9), which include the “classical”
components Jx and Sx can change due to loss and decoherence. For example, loss of
168
A. Appendix I: Gaussian state picture for light and atoms
a fraction of the photons implies a corresponding reduction in Sx , and consequently
in Σ0 . This places a lower limit on the noise introduced. Specifically,
N = |iΣ0 − iMΣMT | ,
(A.26)
where | · | indicates the matrix absolute value, is the minimal symmetric matrix to
satisfy (A.25).
Interestingly, there is a more physical picture for completely positive maps (A.24).
The map can be modeled as a unitary transformation between different modes of the
system, e.g., segments or a reservoir. In App. A.6 we derive Eq. (A.24) using n-port
unitary transformations and phase averaging over input modes.
Loss and decoherence from photon scattering
Inevitably, the coherent interaction of equation (A.17) will be accompanied by spontaneous emission of photons, producing also incoherent changes in the atomic state.
We use equation (A.24) to calculate the effect of loss and decoherence of atoms and
photons. Here, “loss” of atoms refers to the decay of atoms into meta-stable states
which do not interact with the light. Decay of atoms into the |↑i- and |↓i-states
can cause decoherence of the spin state. While loss is not present in the ideal spin1/2 system proposed by Kuzmich et al. [59], in alkali metal atoms both processes
are observed. For light there is no decoherence process since spontaneously emitted
photons scatter into all possible spatial modes and are counted as losses.
The covariance matrix transforms as
γ(t + τ ) = Mτ γ(t)MTτ + Nτ ,
(A.27)
where the decay is described by
Mτ = (1 − ητ )I2 ⊕ (1 − ε)I2 .
(A.28)
Here, ητ is the probability for an atom to scatter a photon and ε the probability for
the complementary process. Noise will have the form Nτ = Nτ,loss + Nτ,dec . With
Eq. (A.26) we get
Nτ,loss = ητ (1 − ητ )
~2
~2
NA I2 ⊕ ε(1 − ε) NL I2 .
4
4
(A.29)
and
Nτ,dec = ρητ
~2
NA I2 ⊕ O2 .
4
(A.30)
Here, I2 is the identity matrix in two dimensions, O2 is the zero matrix, and ρ is the
fraction of the scattered atoms which return to the system, assumed to be in a fully
mixed state. This last assumption is convenient [63] but not required. It introduces
the maximum possible amount of noise. For example, scattering in optical pumping
does not produce fully mixed states. For all simulations that follow in section A.4,
we assume to have exclusively atomic decoherence and no loss, i.e.,ρ = 1.
A.3. Unified description of physical processes
169
A.3.3. Transport processes
When atoms move from one segment to another, for example by diffusion, they
transport with them a fraction of the collective spin. The effect of the transport is
described by a linear transformation of the phase space vector
v̂(t + τ ) = Dτ v(t).
(A.31)
The covariance matrix transforms according to equation (A.24) with M = Dτ and
N = iΣ0 − iDτ ΣDTτ .
(A.32)
That is, the movement of atoms does not introduce extra noise beyond what is
required to maintain the commutation relations.
Other transport processes, as for example collisions, could be treated as well. They
can be coherent or incoherent in nature and are relevant for interacting systems like
quantum gases of bosons or fermions.
A.3.4. Projective Measurement
The next class of operations we can apply are measurements of atomic or light
variables. While a measurement will collapse the value of an observable in a way that
is fundamentally random, the resulting variances change in a way that is completely
predictable: the variance of the measured observable becomes zero, the variance of
the conjugate observable becomes large or infinite. The variances of other observables
may also be reduced if they are correlated with the measured observable.
A measurement can be described by a projection matrix P. For example, to measure
(n,i)
(n,i)
(n,i)
a polarization component of the (n, i) light segment, Ŝθ
≡ cos θŜy +sin θŜz
≡
(n,i) T
(n,i)
(n,i)
(pθ ) · v, the projector would be the outer product P = pθ ∧ pθ . In practical
situations, measurement of a light variable also implies that a light segment has
reached a detector and thus is removed from the problem, reducing the dimension of
the vector v. Upon measurement, the covariance matrix becomes
γ 0 = γ − γ(PγP)− γ T (n,i) .
(A.33)
where (...)− indicates the Moore-Penrose pseudoinverse and |...|(n,i) removes the column and row corresponding to the measured, and no longer existing, light segment
(n, i). Equation (A.33) is well known in mathematical statistics to compute the conditional covariance matrix of multivariate normal distributions [64]. A more detailed
introduction of Gaussian operations on Gaussian states can be found in [71, 68].
For the calculations in Sect. A.4, we consider a large-area detector, i.e., one which
does not distinguish between different channels (see Fig. A.1). Therefore, we define
the measured light variable to be
X (k,l)
(l)
Ŝθ =
Ŝθ
.
(A.34)
k
170
A. Appendix I: Gaussian state picture for light and atoms
A.3.5. Combining effects
Before we explain how to incorporate simultaneous effects, we want to give some
boundaries for the time-step τ . We can identify lower and upper limits as follows:
The close connection between τ and the length of a light segment leads to a lower
bound. A longitudinal light segment should contain a sufficiently large number of
photons in order to justify a continuous variable description (see Sect. A.2.2), i.e.,
(k,l0 )
NL
= Φ(k) τ 1 where Φ(k) is the flux in the k-th channel. The limit is then
τ min[Φ(k) ]−1 . At the same time, an upper limit comes from the requirement that
the difference equations Eq. (A.15) and Eqs. (A.27),(A.28) accurately model what is
in fact a continuous evolution. This limit is more difficult to specify a priori, but in
general it should be sufficient that the change of v and γ in one time step are small
relative to v and γ themselves. For example, to limit the decoherence introduced
per time step, we require ητ = NL σ(∆)/A 1, or τ A/(σ(∆)Φ), where A is
the interaction area, σ(∆) the off-resonant scattering cross-section and Φ the total
photon flux. In practice, an appropriate τ , for a given scenario, can be found by
calculating with decreasing values of τ until the results of the simulation converge.
In the appendix we compare upper and lower limits for the conditions used in the
simulations.
After we have introduced several different coherent and incoherent effects in the
above sections, a natural question arises: How to incorporate effects which are present
simultaneously? For example, coherent Hamiltonian evolution (A.16) and incoherent
photon scattering (A.27) could be modeled as
γ(t + τ ) = Mτ Tτ γ(t)TTτ Mτ + Nτ
(A.35)
γ(t + τ ) = Tτ [Mτ γ(t)Mτ + Nτ ] TTτ ,
(A.36)
or
depending on which effect is applied first. Naturally, if the effects are simultaneous,
any difference between these is not physical. Fortunately, the difference vanishes as
τ becomes small and, we conclude that different simultaneous effects can be modeled
if the time-step is sufficiently short.
A.4. Instructive examples
Now, we give three examples how the model can be applied in the context of atomic
spin squeezing. Here, spin squeezing is studied because it serves as a benchmark
for more sophisticated applications like atom-light entanglement, atomic quantum
memory and other quantum information tasks. They could as well be analyzed with
the same technique.
For all simulations we consider a cold ensemble of rubidium 87 atoms in an optical
dipole trap. The set of used parameters can be found in the appendix. It is well
known that for large detunings from resonance we can reduce the dipole interaction
Hamiltonian (A.17) to
A.4. Instructive examples
171
~g (k,l)
Ĥeff ŝ(k,j) , ĵ(k,l) =
Ŝz(k,j) Jˆz(k,l) .
τ
(A.37)
The coupling constant g (k,l) is proportional to the vector polarizability α(1) and
defined in the appendix. Note, the Hamiltonian does not explicitly depend on τ
because the Stokes operators are proportional to the flux of photons multiplied by
τ . As initial states we assume a pulse of horizontally polarized light, i.e., Sx =
NL ~/2 and a coherent superposition of the Zeeman substates |F = 1, mF = −1i and
|F = 1, mF = 1i, i.e., Jx = NA ~/2.
The effect of the Hamiltonian (A.37) on the light is a rotation of Ŝx about the
z-axis by an amount proportional to Jˆz . At the same time, Jˆz is not altered in
this process. A projective measurement of Ŝy provides information about, and thus
reduces the uncertainty of, Jˆz . If both input states are minimum uncertainty states,
spin squeezing is obtained. To monitor the evolution of this process we evaluate
2Var ( Jˆz ) /Jx which is also known as the spin squeezing parameter [48]. For squeezed
states it will become less than unity. The smaller the spin squeezing parameter the
higher the degree of spin squeezing. There are other criteria, for instance by Wineland
et al. [49] derived in the context of metrology. Regardless which of the definitions is
applied, we obtain the same qualitative results.
To make the comparison between different experimental situations clearer, we normalize the timescale. We can define a time when the rotation of the light polarization
due to the atom-light interaction (A.37) exceeds the shot noise of the photons, i.e.,
when the signal-to-noise ratio becomes one. We want this time to be characteristic
for the system as a whole. Therefore, we define
t0 ≡
4
1
.
4
2
~ G NA Φ
(A.38)
We assume an atom-light interface with a number of atoms, NA , homogeneously
distributed over an effective interaction area A. Here, G is an effective interaction
strength and Φ the photon-flux. Details on the derivation are given in the appendix.
A.4.1. Detector time-resolution
As a first example, we study the influence of the detector time-resolution on the
amount of spin squeezing and show the importance of correct modeling of pulsed
experiments even for pulses much shorter than the detector time resolution.
We define an ideal detector as one capable of detecting individual light segments.
The covariance matrix would be updated in accordance to (A.33) each time a light
segment hits the detector. In contrast, we say a detector has no time-resolution if it
detects all segments at the same time. Mathematically, the measured variable is the
sum of all n light segments
n
X
Ŝy(1) =
Ŝy(1,l) ,
(A.39)
l=1
172
A. Appendix I: Gaussian state picture for light and atoms
2var(Jz)/Jx
1
c)
0.8
b)
0.6
a)
0.4
0.2
0
2
4
Figure A.2.: The spin squeezing achieved by using an ideal
detector (a) and a detector
with no temporal resolution
(b) are shown. The parameters are given in appendix A.
Two zero-dimensional calculations are given for comparison. In curve (c) the noise due
d)
to decoherence and loss was
8
6
10 added after and in (d) before
Pulse Duration [units of t0] the interaction.
and we apply the transformation (A.33) to the whole covariance matrix. (For simplicity we assume only one atomic segment. Nonetheless, we keep the transverse
index to avoid confusion.) The projector has the form
P = O2 ⊕
1
Un ⊗ I2 .
n
(A.40)
Where Un is the unit matrix of rank n.
In Fig. A.2 we show the results for both having (a) perfect and (b) no temporal
resolution. In the case of no temporal resolution, the achievable spin squeezing is
reduced at longer timescales. This is understood if we compare the information
carried by different longitudinal light segments. Early light segments interact with
the initial atomic state and later ones with a noisier version of it. If the detector is
lacking temporal resolution all this different information is mixed.
Now we compare the results to calculations which neglect all dynamics during the
pulse duration, e.g., in [192]. We call this type of model “zero-dimensional” because it
treats the light-atom interaction as a point-like event in time. Therefore, we assume
that the light pulse is not partitioned into longitudinal segments. Curve (c) and (d)
in Fig. A.2 show the results if the noise is added after (cf. Eq. A.35) and before (cf.
Eq. A.36) the interaction, respectively. It becomes obvious that even in the case of
a pulsed experiment it is important to model light as a stream of sufficiently short
segments.
The difference for the two extreme cases of adding the noise after (curve c) and before
(curve d) interaction and measurement (IM) can be understood in the following way.
If the noise is added before interaction and measurement it can actually be detected
and squeezed at the same time, which leads to a much larger squeezing. For the
opposite case of addition of the noise and the end, the process has no time to “learn”
about the presents of the introduced noise and we see less spin squeezing.
A.4. Instructive examples
173
A.4.2. Spatial inhomogeneities
In many experiments inhomogeneities in light or atomic distributions are present.
We give two examples for typical situations that can arise. As the simplest test
model we assume an atomic ensemble which consists of two equally sized transverse
segments.
For all the following calculations we assume an ideal detector. Furthermore, the total
number of atoms NA , the photon flux Φ, the total interaction cross section A, and
all other parameters are fixed and stated in the appendix. The figure of merit is the
variance of Jˆz for the complete atomic ensemble
!
X (i)
var Jˆz = var
Jˆz
.
(A.41)
i
To verify the validity of the segmentation model, we consider first a homogeneous
atom distribution either as a single or as two transverse segments. In both cases we
observe the same results in the presence of atom-light interaction, loss and decoherence and transport processes, independent of the segmentation. They produce the
solid line (a) in Fig. A.2.
The first example reflects the situation we would find for inhomogeneous light fields
interacting with homogeneously distributed atoms. We model this with two channels
of equal interaction cross-section A/2. We assume light is only present in one of the
channels. The result is plotted as the dotted curve in Fig. A.3. The overall spin
squeezing is reduced. To explain this, we can evaluate both channels independently.
One channel contains all the photons and the maximal obtainable amount of squeezing will be the same as for the homogeneous distribution (solid line in Fig. A.2). This
reflects a very important property in atomic spin squeezing. The achievable amount
of squeezing does not depend on the intensity of light (as long as it is not zero) but
rather on the optical depth of the atomic ensemble. For the second channel, without
light, we expect no change in the atomic state. If we combine these two results, we
get exactly the dotted curve shown in Fig. A.3.
The second example is the inverse situation. The light beam has a larger cross-section
than the atomic ensemble. We model this case by assuming all NA atoms are in one
of the channels and light is homogeneously distributed over both. The result, plotted
as the solid line in Fig. A.3, seems surprising. We see exactly the same dynamics as
for the homogeneous case. In this situation two effects are compensating each other.
The optical depth of the atoms is twice as large as in the previous examples and
leads to larger spin squeezing. On the other hand, the light which does not interact
with the atoms is also detected, and contributes noise but no additional information
about the atoms.
The two examples give some intuition about the influence of inhomogeneities. It is
now straightforward to apply it to more interesting and complicated experimental
cases.
174
A. Appendix I: Gaussian state picture for light and atoms
2var(Jz)/Jx
1
0.8
0.6
0.4
0
2
4
8
6
10
Pulse Duration [units of t0]
Figure A.3.: Two exemplary
cases are compared to the
homogeneous situation where
light and atoms are evenly
distributed over space. For
more information see text.
A.4.3. Atomic motion
As a last application, we ask what happens when atoms can change places and
go from one segment to another. This is a relevant question for comparing different
experimental situations. Atoms in vapor cells, for instance, have approximately room
temperature. This corresponds to a root-mean-square (rms) velocity of hundreds of
meters per second as opposed to a few tens of millimeters per second for lasercooled atoms. Typical timescales, t0 , for the light-atom interaction are milliseconds
and microseconds, respectively. Hence, atoms in vapor cells have moved around
half a meter (effectively) whereas the atoms in the trap move less than one hundred
nanometers in a time t0 . This suggest that spatial inhomogeneities can be transferred
from the light to cold atoms. To find a more quantitative description, we consider
a two-channel atomic system, and introduce a mixing probability mτ per time-step
τ . The system is assumed to be in equilibrium, so that an equal fraction of atoms
move in each direction. From Sect. A.3.3, the transport matrix is
1 − mτ
mτ
Dτ =
⊗ I2 ,
(A.42)
mτ
1 − mτ
so that
~2
N = mτ (1 − mτ ) NA
4
1 −1
−1 1
⊗ I2 .
(A.43)
We note that while transport can increase the variances of individual spin components, the total spin components are unchanged. Interestingly, the same results for N and M can be found by considering a unitary transformation matrix
Uij exp[i(φi − φj )] which acts on the bosonic operators aj of different segments. The
mixing matrix is then Mij = |Uij |2 and the noise matrix N above follows from averaging over the φi . This suggests that the above model describes incoherent transport,
but could possibly be extended to include coherent transport such as diffraction of
light or the motion of condensed atoms.
A.4. Instructive examples
a)
175
b)
c)
0.01
0.01
0.1
0.4
0
2
4
0.1
0.4
1
1
6
Pulse Duration [units of t0]
8
10
10
0.01
0.1
0.4
8
10
10
1
8
10
10
Mixing rate r
[units of 1/t0]
Figure A.4.: Atomic motion is simulated with different mixing rates. a) shows the
degree of spin squeezing for the whole ensemble, b) for the illuminated and c) for the
un-illuminated segment. The discussion is given in the text.
As a concrete example we assume we have an ideal gas of atoms and derive the
mixing probability from kinetic gas theory. The number
√ of collisions per area and
time in an ideal gas is known to be R = N vrms / V 6π . Where, N is the number
of atoms in volume V , and their rms velocity is vrms . From this we can calculate the
0
rate at which individual atoms cross a surface of area A0 , r = RA
√ /N . Furthermore,
0
we assume that the atoms occupy a box of volume V = A A, where A is the
interaction cross section of a segment. The rate can therefore be written as
vrms
r=√
.
6πA
(A.44)
In the limit of small τ we can define a mixing probability as mτ = rτ .
vrms
τ .
mτ = √
6πA
(A.45)
p
Where vrms = 3kB T /m is the rms velocity of atoms at temperature T . This model
is not an exact treatment of the different physical situations we find in vapor cells and
atomic traps. Nevertheless, it suggests how atomic motion influences the formation
of spin squeezing.
In Fig. A.4 we plot the squeezing factor for different mixing probabilities. For
mτ /τ → 0 we have the same situation as in the dotted curve of Fig. A.3 For
increasing mixing probability we see that the squeezing is improved and the full
amount (compared to the homogeneous situation, i.e., solid curve of Fig. A.2) is
achieved again. In this limit, when the inverse mixing rate becomes the same order of
magnitude as t0 , sufficient atomic movement is present that all the atoms get enough
interaction to be uniformly squeezed. This suggest that approaches of matched
variables, e.g., by Kuzmich et al. [192] are more relevant for cold atoms than for hot
vapor cell experiments.
If we instead focus our attention only on the segment of atoms which is illuminated
we see in plot b) of Fig. A.4 that the spin squeezing for this segment reduces when
the mixing probability increases. One can interpret this as a decoherence mechanism
for the smaller segment [118].
176
A. Appendix I: Gaussian state picture for light and atoms
A.5. Conclusion
We have presented a model to compute the dynamics of interacting light and atomic
ensembles in the framework of Gaussian states. The model is based on covariance
matrices for the quantum components of collective angular momentum operators and
employs segmentation of the light and atom systems. The use of extrinsic variables,
as opposed to derived canonical variables [65], greatly facilitates partitioning and
inhomogeneity calculations. Furthermore, partitioning of the light beam, which was
not considered in previous treatments, allows us to see the effects of inhomogeneity
in the optical beam. This is expected to be important in many realistic situations.
As a general point, optimal interactions, for example the optimal probe strength
for spin squeezing, balance the desired coherent interaction against scattering noise
and generally do not happen in the low-scattering regime. Outside of this regime,
the interaction is essentially non-linear and power distributions must be considered.
When properly considered, an inhomogeneous distribution of light is mapped to the
atoms and vice-versa. The partitioning of the light also allows us to model thermal
motion and could be applied to multi-spatial-mode entanglement.
The difference between loss and decoherence in the atomic states was pointed out.
This becomes important when the atomic state is different from the spin 1/2 system.
Modeling of atomic motion addressed a fundamental question for existing atom light
interfaces in hot vapor cells and for new experiments with dipole trapped samples.
Employing this model, we have made the following observations: The dynamics of
spin squeezing requires time-dependent modeling, even when the atoms interact with
optical pulses which are shorter than the detection system can resolve. At the same
time, the detector time resolution has only a minor effect on the degree of spin
squeezing under realistic conditions. The effects of spatial inhomogeneities in the
light and atoms have non-equivalent effects on spin squeezing: concentration of the
light into a sub-region of the atoms produces equal squeezing of the sub-region, but
less squeezing of the entire ensemble, while concentration of the atoms into one subregion of the light gives equal squeezing of the spin ensemble for equal segments.
Finally, we observe that atomic motion between an illuminated region and a nonilluminated one tends to degrade squeezing of the illuminated region while increasing
squeezing of the entire atomic ensemble. This suggests that high-fidelity experiments
should use probe pulses which are either much shorter, or much longer, than the
time-scale of the atomic motion.
The model can straight-forwardly be adapted to more complicated experimental situations, for example a cold thermal cloud in a focused laser beam. Also, application
to multi-pass schemes as proposed by Takeuchi et al. [199] or Sherson et al. [200] is
possible.
A.6. Coherent and incoherent transport processes described by N-port unitaries177
A.6. Coherent and incoherent transport processes described by
N-port unitaries
Guassian positive maps as used in Appendix A to describe coherent Hamiltonian
evolution (cf. Eq. (A.16)) and incoherent evolution due to loss and decoherence (cf.
Eq. (A.24)) or atomic motion (cf. Eq. (A.31)) can be derived in a more physical
picture using N-port unitary transformations.
A.6.1. Transport processes
Redistribution of atoms among segments, diffraction of light among channels, and
scattering processes which add or remove atoms or photons can all be considered
as transport processes, possibly involving additional reservoir modes. Transport
processes can be modeled by considering a unitary transformation U which produces
N output modes ĉ0 from the N input modes ĉ as
ĉ0i =
N
X
Uij ĉj ,
(A.46)
j=1
where we have defined a phase space vector in terms of the bosonic operators for
atoms b and light a as
iT
h
0 0
.
(A.47)
ĉ = b(1,1) , ..., b(t,l) , a(1,1) , ..., a(t ,l )
We restrict our discussion to unitaries of the form UA ⊕ UL where UA acts on the
atomic modes and UL acts on the photonic modes. Note that the transport process
in itself does not change the polarization states, although more general transport
processes are easily constructed by adding single-species (+/ − or ↑ / ↓) rotations
before and/or after the transport.
The components v̂i of the phase space vector v̂ transforms as




 σx 
 σx 
X
~
~
†
σ y ĉ0i =
v̂i0 =
ĉ0
Uij∗ Uik (ĉj )† σ y ĉk



2 i 
2
j,k
σz
σz




 σx 
 σx 
X
~X ∗
~
Uij Uij (ĉj )† σ y ĉj +
Uij∗ Uik (ĉj )† σ y ĉk
=




2
2
j
j6=k
σz
σz
X
=
Mij v̂j + ξˆi
(A.48)
j


 σx 
X
where Mij ≡ Uij∗ Uij and ξ̂ i ≡ ~2
Uij∗ Uik (ĉj )† σ y ĉk . The matrix M describes


j6=k
σz
the mixing of observables associated with different segments. In contrast, ξ̂ describes the influence of coherence between different segments. These coherences are
178
A. Appendix I: Gaussian state picture for light and atoms
not included in the phase-space description
and it can be easily verified that their
D E
expectation values will vanish, i.e., ξ̂ = 0. However, we will see they contribute
noise to the state.
f ≡ M⊗I2
In order to compute the covariance matrix from (A.6) we define a matrix M
which has the same dimension as γ. I2 is the identity matrix in two dimensions. If
we employ the symmetry of M and the fact that the mean values of ŵ and ξ̂ vanish,
we find
E D
E
D
E
D
f + ξ̂ ∧ ξ̂ .
f M
f+M
f ŵ ∧ ξ̂ + ξ̂ ∧ ŵ M
(A.49)
γ 0 = Mγ
The above description is appropriate to coherent transport processes such as diffraction of coherent light or movement of Bose-condensed atoms. For incoherent processes such as movement of thermal atoms, we can average over unitaries with different phases, i.e., we take Uijφ = Uij exp(iφj ) and average over {φ}.
D E
ξ̂ i
{φ}
=


+
 σx 
†
i(φl −φk )
∗
σ y âl e
=0
Uik Uil (âk )


k6=l
σz
{φ}
*
~ X
2
while
ξ̂ ∧ ξ̂
ij
{φ}
=
~2
4
=
≡


 σx 
∗
Uil (âk )† σ y âl
Uik


k6=l
σz


+
 σx 
X
†
∗
i(φl −φk +φn −φm )
σ y ân e
×
Ujm Ujn (âm )


m6=n
σz
{φ}




 σx 
 σx 
~2 X ∗
Uik Uil (âk )† σ y âl Ujl∗ Ujk (âl )† σ y â
(A.50)



 k
4
k6=l
σz
σz
*
X
Nij2
As a result for an incoherent transport, the covariance matrix changes as
f M
f + N2 ,
γ 0 = Mγ
which is equivalent to the Guassian completely positive map [198, 71].
(A.51)
B. Appendix II: Numeric simulation of
multi-level atoms
Contents
B.1. Scattering cross section for different hyperfine ground
states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
B.2. AC Stark shifts and the limit of the rotating wave approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 180
B.2.1. AC Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . 180
B.3. Numerical simulation techniques . . . . . . . . . . . . . . 183
180
B. Appendix II: Numeric simulation of multi-level atoms
This appendix summarizes the important technical details for numerical simulations
of the density matrix and the light shifts of the atoms in the dipole trap.
B.1. Scattering cross section for different hyperfine ground
states
The scattering cross section σ connects the microscopic description of the interaction
of two quantum particles, e.g., photons and atoms, to macroscopic observable effects
as the probability of absorption [17]. The size of the scattering cross-section has little
relation to the actual physical size of the atom. For a two level atom interacting with
monochromatic light the scattering cross-section is
λ2
σ(∆) = 3
2π
Γ2
4
Γ2
4
+ ∆2
.
(B.1)
For a multi-level atom as 87 Rb, where we have a number of excited state hyperfine
levels the total scattering cross section is the sum over all the possible transitions.
For atoms in F = 1 we find
λ2 Γ2 1
σ(∆) =
2π 32
"
4
Γ2
4
+ (∆ − ∆00 )2
+
5
Γ2
4
+ (∆ − ∆10 )2
+
7
Γ2
4
+ (∆ − ∆20 )2
#
,
(B.2)
where ∆F 0 = νF 0 − ν00 is the energy splitting between F 0 = 0 and the higher excited
hyperfine states.
B.2. AC Stark shifts and the limit of the rotating wave
approximation
B.2.1. AC Stark shift
Light shifts play an important role in this work, because we optically trapped atoms
to work with. Here, we want to derive some important expressions for the level shift
and the scattering rate of atoms in far-detuned laser fields. We don’t want to restrict
the treatment to the ground state light-shift but also discuss the shift of the excited
hyperfine levels, since this will be important for the experiments in Chapt. 6 and is
interesting in itself1 .
There are different possibilities to derive the light shift of an atomic transition that
is subject to a detuned laser field. One is a completely classical Lorentz oscillator
model assuming a dipole and a classical field (for details see [201, 202]). Another
applies second order time-independent perturbation theory [203].
1
Some times it is important to find a wavelength for the trapping laser at which the ground state
light shift is the same as the excited state one. These wavelengths are sometimes called magic.
B.2. AC Stark shifts and the limit of the rotating wave approximation
181
within RWA
Let us assume we have only a two level system and the semi-classical Hamiltonian
has the form
1
0
Ω
2
H=~ 1
.
(B.3)
2 Ω −∆
The combined atom-light system will have new eigenstates and eigen-energies, the
so-called dressed states [121]. By diagonalizing (B.3) we find the new eigen-energies
p
1
ω
e± =
(B.4)
∆ ± ∆2 + Ω2 .
2
In the limit of large detuning and weak excitation, |∆| Ω, the expression (B.4)
simplifies to
Ω2
Ω2
ω
e+ = − ∆ +
,ω
e− =
.
4∆
4∆
For all practical purposes, it is possible to achieve |∆| Ω by choosing an intensity
such that the Rabi frequency fulfills the assumption. Using the linearity of the light
shift, one can scale to the intensity of interest. We find that the initial energy levels,
of the bare atomic system, have changed by
∆e
ω=±
Ω2
.
4∆
(B.5)
Note that ∆ is negative for red detunings. Hence, for a red detuned laser the energy
of the ground state is lowered in the same way as the excited state energy is raised.
In a multilevel atom, like 87 Rb, the level structure is more complicated and involves
many ground and excited states. One way to calculate the light shift of any atomic
state is to write the atom-light Hamiltonian for the multitude of states and calculate the new eigenvalues. If we consider all characterized [204, 205, 206, 207, 208]
transitions from the 5S1/2 ground and from the 5P1/2 and 5P3/2 excited state we get
26 fine structure levels and 84 hyperfine levels. If we are interested in the light shift
of individual magnetic substates, the matrix can be very large (384 × 384 entries at
maximum) but feasible to diagonalize.
Alternatively, the light shift in multi-level atoms can be calculated with Eq. (B.5)
for the two level system and summed over all relevant transitions
∆e
ωi =
X Ω2if
f 6=i
4∆if
,
(B.6)
where the detuning is defined as ∆if = ωL − ωif , with i and f labeling initial and
final atomic energy levels.
beyond RWA
For very large detunings, when |∆| ω0 is no longer valid, i.e., the detuning gets
comparable to the atomic transition frequency, the RWA stops being applicable. It
182
B. Appendix II: Numeric simulation of multi-level atoms
12
w/o RWA
w/ RWA
Ac Stark shift [MHz]
10
8
6
4
2
ground state
0
excited state manifold
ï2
ï4
ï6
ï1
0
+1 ï2 ï1
0
+1 +2
0 ï1
0
+1 ï2 ï1
0
+1 +2 ï3 ï2 ï1
0
+1 +2 +3
Figure B.1.: AC Stark shift in dipole trap of magnetic substates in the ground and
the excited state manifold calculated without and with RWA. The light intensity corresponds to the value we expect in the focus of the dipole trap for 6 W of laser power.
is no longer true that the counter rotating terms disappear. Using the Lorentz model
for a classical oscillator, one can arrive at expression which has the same form as Eq.
(B.6) [202].
∆e
ωi =
X Ω2if
,
˜ if
4∆
(B.7)
f 6=i
where
˜ if ≡ −
∆
1
1
+
ωif − ωL ωif + ωL
−1
.
(B.8)
The first term is the detuning term we used in the expression before. The second is
the counter-rotating term which is usually neglected when the RWA is applied. The
error in the light shift we make in neglecting the counter-rotating term is
∆e
ωerr =
ωL − ωif
∆e
ωnoRWA − ∆e
ωRWA −
=
.
∆e
ωnoRWA
ωL + ωif
(B.9)
For red detuning, i.e., ωL < ωif , we under- and for blue detuning we overestimate
the light shifts. In practice, for the used dipole laser with λ = 1030 nm and the D1
and D2 transitions at 795 nm and 780 nm, the error is around 12 %. Therefore, we
will use Eq. (B.6) and the corrected detuning (B.8) to calculate the light-shift in the
dipole trap.
In Fig. B.1 we plot the AC Stark shift for the ground and the excited hyperfine
ground state manifold without and with applying the RWA. This clacualtion indicates another important issue which is not immidiately obvious. The light shifts
for the different magnetic substates in the excited state manifold are not equal as
they are in the ground state. The reason is that the states in the excited state are
coupled to hyperfine state manifold which have also smaller J quantum numbers.
The coupling are therefore not equal to all of them.
B.3. Numerical simulation techniques
183
All considered transitions
Transition
5S 1
2
5S 1
2
5P 1
2
5P 3
2
5P 1
2
5P 3
2
5P 3
2
5P 1
2
5P 3
2
5P 1
2
5P 3
2
5P 3
2
5P 1
2
5P 3
2
5P 1
2
5P 3
2
5P 3
2
5S 1
2
5S 1
2
5S 1
2
5S 1
2
5S 1
2
5S 1
2
→ 5P 1
2
→ 5P 3
2
→ 6S 1
2
→ 6S 1
2
→ 4D 3
2
→ 4D 3
2
→ 4D 5
2
→ 7S 1
2
→ 7S 1
2
→ 5D 3
2
→ 5D 3
2
→ 5D 5
2
→ 8S 1
2
→ 8S 1
2
→ 6D 3
2
→ 6D 3
2
→ 6D 5
2
→ 6P 1
2
→ 6P 3
2
→ 7P 1
2
→ 7P 3
2
→ 8P 1
2
→ 8P 3
Wavelength
[nm]
|hn0 J 0 kDk nJi|
[10−29 Cm]
fnJ,n0 J 0
AnJ,n0 J 0
[MHz]
rnJ,n0 J 0
6
[10 s−1 mW−1/2 ]
794.9749
780.2374
1323.8743
1366.8700
1475.6393
1529.2558
1523.3608
728.1962
741.0173
762.0990
776.1531
775.9745
607.2394
616.1389
620.7985
630.0925
630.0026
421.672
420.298
359.259
358.807
335.178
334.968
3.5787
5.0497
3.4922
5.0980
6.6530
3.0013
9.0159
0.8088
1.1463
1.3701
0.6725
1.9788
0.4271
0.6020
1.0004
0.4731
1.4057
0.2823
0.4587
0.0975
0.1713
0.0500
0.0941
0.3404
0.6905
0.1946
0.2009
0.6338
0.0622
0.5615
0.0190
0.0187
0.0520
0.0061
0.0533
0.0064
0.0062
0.0341
0.0038
0.0331
0.0040
0.0106
0.00056
0.0017
0.00016
0.00056
35.926
37.783
7.408
14.343
9.707
1.775
10.675
2.388
4.551
2.989
0.671
3.937
1.149
2.183
2.948
0.630
3.712
1.498
3.994
0.289
0.895
0.094
0.332
3.8010
5.3633
3.7091
3.8287
7.0662
2.2541
6.7711
0.8591
0.8609
1.4552
0.5011
1.4862
0.4538
0.4521
1.0626
0.3553
1.0557
0.3000
0.4874
0.1036
0.1820
0.0532
0.1000
2
Table B.1.: All 17 transitions used to calculate light shifts of the 5S 21 , 5P 12 , and 5P 23 .
The reduced matrix elements |hn0 J 0 kDk nJi| are found in Ref. [206, 205, 204, 208, 207].
The oscillator strength fnJ,n0 J 0 and the spontaneous emission rate AnJ,n0 J 0 are given.
The last column shows the Rabi frequency divided by the square root of the intensity.
B.3. Numerical simulation techniques
Throughout the thesis we have gained a lot of insight into the physical processes
by simulating the evolution of the single atoms density matrix. As we explained
in Sect. 2.3.3 we use the master equation to evolve the density matrix under the
influence of external fields. The system is modeled with all magnetic substates of the
two hyperfine ground and the 4 hyperfine excited state. In total we have a density
184
B. Appendix II: Numeric simulation of multi-level atoms
matrix which has the dimension 24 × 24. Time-dependent fields are simulated by
dividing the evolution into short time steps, in which we assume the Hamiltonian
to be constant. The duration of the time step is chosen to much shorter than the
fastest timescale involved.
√
The jump operators Cse ≡ Γdrel, ge in Eq. (2.35) for the spontaneous emission are
calculated from the relative matrix element
0
p
F0
1 F
J F0 I
I+J+1+2F 0 −mF 0
0
0
drel, ge ≡ (−1)
[J ][F ][F ]
.
−mF 0 q mF
F J 1
(B.10)
Other dissipative effects, e.g., the influence of the laser linewidth are not included,
but are in general feasible. Despite the existence of the wonderful “Quantum Optics
Toolbox for Matlab” [209] we developed most of the routines ourselves to be able
to incorporate special requirements. Nevertheless, we use a set of routines from the
QO-toolbox which allow us to perform the pre- and post multiplication of the density
matrix by super-operators as required by Eq. (2.35).
Scattering rate number of scattered photons
From the evolution of the density matrix we can deduce interesting parameters from
a physical point of view. For instance, we can calculate the instantaneous scattering
rate in the light-atom interaction be evaluating the following expression
Rsc (t) = 2πΓ
X
ρjj (t) ,
(B.11)
j6=g
where Γ is the excited state decay rate and ρjj (t) is the population if an excited
state at time t and we sum over all excited states. From the scattering rate we can
calculate the number of scattered photons for each atom.
Z t
X
nsc =
Rsc (t)dt = ∆t
Rsc,k ,
(B.12)
0
k
where the second equality is the one we actually use since we evolve the density
matrix in discrete time steps ∆t in which we calculate the scattering rate Rsc,k at
time t = k∆t.
Mixed state fidelity
In Chapter 3 when simulating the optical pumping efficiencies, we use the mixedstate fidelity [210, 90]:
q
√
√ 2
F = Tr
ρ2 ρ1 ρ2
,
(B.13)
where ρ1/2 are density matrices describing arbitrary states. The above is the generalization of the pure state state overlap |hΨ1 | |Ψ2 i|2 probability [21].
Abbreviations
AOM
c.c.
CSS
ECDL
e.m.
h.c.
HWP
MOT
PM
PBS
PDL
PZT
QND
QWP
RWA
SGE
SNR
TOF
acouto optical modulator
complex conjugate
coherent spin state
extended cavity diode laser
electro-magnetic
Hermitian conjugate
half wave plate or λ/2 plate
magneto optical trap
polarization-maintaining
polarizing beam splitter
polarization-dependent losses
Piezoelectric Transducer
quantum non-demolition
quarter wave plate or λ/4 plate
rotating wave approximation
Stern-Gerlach experiment
signal-to-noise ratio, used in connection with optical QND measurements
time of flight
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List of Publications
Journal articles
High resolution magnetic vector-field imaging with cold atomic ensembles,
M. Koschorreck, M. Napolitano, B. Dubost, and M. W. Mitchell, to be submitted
Chapter 5
Quantum Nondemolition Measurement of Large-Spin Ensembles by Dynamical Decoupling, M. Koschorreck, M. Napolitano, B. Dubost, and M.W. Mitchell, Phys.
Rev. Lett. 105, 093602 (2010)
Chapter 7
Squeezed-light Optical Magnetometry, F. Wolfgramm, A. Cerè, F.A. Beduini, A. Predojević, M. Koschorreck, and M.W. Mitchell, Phys. Rev. Lett. 105, 053601
(2010)
content not part of thesis
Sub-Projection-Noise Sensitivity in Broadband Atomic Magnetometry, M. Koschorreck, M. Napolitano, B. Dubost, and M. W. Mitchell, Phys. Rev. Lett. 104, 093602
(2010)
Chapter 7
Unified description of inhomogeneities, dissipation, and transport in quantum lightatom interfaces, M. Koschorreck and M.W. Mitchell, J. Phys. B: At. Mol. Opt.
Phys 42, 195502 (2009)
Appendix A
Polarization-based light-atom quantum interface with an all-optical trap, M. Kubasik,
M. Koschorreck, M. Napolitano, S.R. de Echaniz, H. Crepaz, J. Eschner, E.S.
Polzik, M.W. Mitchell, Phys. Rev. A 79, 043815 (2009)
Chapter 4
Ultra low-noise differential ac-coupled photodetector for sensitive pulse detection applications, P.J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E.S. Polzik, J.H. Müller, Meas. Sci. Technol. 20, 055301 (2009)
Chapter 3
Hamiltonian design in atom-light interactions with rubidium ensembles: A quantuminformation toolbox, S.R. de Echaniz, M. Koschorreck, M. Napolitano, M. Kubasik,
M.W. Mitchell, Phys. Rev. A 77, 032316 (2008)
Chapter 2
Conditions for spin squeezing in a cold 87 Rb ensemble, S.R. de Echaniz, M.W.
Mitchell, M. Kubasik, M. Koschorreck, H. Crepaz, J. Eschner, E.S. Polzik, J.
Opt. B 7, S548 (2005)
overall relevant for thesis
Conference Contributions
Oral presentations
Spin Squeezing via Quantum Non-demolition Measurements in a Cold 87 Rb Atomic
Ensemble
M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood and M.W. Mitchell
CLEO/QELS 2010, San Jose, USA
Ultra-sensitive Spin Measurements below the Standard Quantum Limit
M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood and M.W. Mitchell
ICSSUR 2009, Olomouc, Czech Republic
Ultra-sensitive Faraday Rotation Measurements from an Atom-Light Quantum Interface
M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood and M.W. Mitchell
CLEO/EQEC 2009 Munich, Germany
Ultra-sensitive Faraday Rotation Measurements from an Atom-Light Quantum Interface
M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood and M.W. Mitchell
CLEO/IQEC 2009 Baltimore, USA
Quantum-Enhanced Measurements of Atomic Spin
A. Predojević, M. Koschorreck, M. Napolitano, F. Wolfgramm, B. Dubost, Y. de
Icaza, N. Behbood, A. Cerè, and M. W. Mitchell
CLEO/IQEC 2009 Baltimore, USA
State Tomography with Ultra-sensitive Faraday Rotation Measurements
M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood and M.W. Mitchell
QOIT meeting 2009 Madrid, Spain
Spin squeezing experiments in a cold ensemble of 87 Rb
M. Kubasik, M. Koschorreck, H. Crepaz, S.R. de Echaniz, E.S. Polzik, M.W.
Mitchell
CLEO/IQEC 2007 Munich, Germany
Conditions for Spin Squeezing in Laser-Cooled Rubidium
S.R. de Echaniz, M.W. Mitchell, M. Kubasik, M. Koschorreck, H. Crepaz, J. Eschner and E.S. Polzik
ICSSUR 2005 Besancon, France
Posters
Better-than-Heisenberg Scaling of Sensitivity with Light and Cold Atomic Ensemble
M. Napolitano, N. Behbood, A. Cerè, B. Dubost, M. Koschorreck and M. W.
Mitchell
CLEO/EQEC 2009 Munich, Germany
Towards spin squeezing in a cold Rb ensemble
M. Kubasik, M. Koschorreck, M. Napolitano, S.R. de Echaniz and M.W. Mitchell
QOIT meeting 2008 Barcelona, Spain
Next-generation atomic ensemble for quantum information
M.W. Mitchell, A. Cerè, M. Kubasik, M. Koschorreck, M. Napolitano, E.S. Polzik,
J. Eschner, A. Predojević and F. Wolfgramm
QCMC 2008 Calgary, Canada
Inhomogeneities in atom-light interfaces and spin squeezing dynamics
M. Koschorreck, M. Kubasik, S.R. de Echaniz and M.W. Mitchell
CLEO/IQEC 2007 Munich, Germany
Quantum Light Matter Interaction with Cold Atoms
M. Kubasik, S.R. de Echaniz, M. Koschorreck, E.S. Polzik and M.W. Mitchell
CQO9 2007 Rochester, USA
Conditions for Spin Squeezing in Laser-Cooled Rubidium
S.R. de Echaniz, M.W. Mitchell, M. Kubasik, M. Koschorreck, H. Crepaz, J.
Eschner and E.S. Polzik
CLEO/EQEC 2005 Munich, Germany
Acknowledgments
Here, I wish to express my deep appreciation for all those who have contributed to
the success of this work.
I thank Morgan W. Mitchell who directed this work in his own special way. I had the
pleasure to work with him as a great teacher and a never ending source of insights
into physics. It was a great experience to work under his supervision which extended
beyond physics and science.
I’d like to express special gratitude to the people who shared many hours in the
lab with me. First of all, I enjoyed the companionship of Marcin Kubasik on this
journey towards the ambitious goal of creating a low noise quantum state in cold
atomic ensembles in the heat of the Spanish sun. Mario Napolitano joined the
experiment a bit later but contributed a lot to the success of the work. I’m grateful
for his friendship and commitment to offer help whenever it is needed. Many thanks
to Brice Dubost for his never ending hunger for answers, which triggered many
interesting discussions. I’m thankful to Herbert Crepaz and Sebastian de Echaniz
who spent many hours in the lab and getting the experiment running in the first
place.
I owe special thanks to Rob Sewell who joined at the very end. Despite our short
overlap, his contribution to this manuscript and the number of papers as a ghostreader, is invaluable. In his hands, together with Naeimeh Behbood, the experiment
will produce many interesting results in the future.
Many thanks to the ’light-side’ of the quantum-light-atoms group. It was always a
pleasure to have you around to share ideas, cakes and cava.
Thanks to the people of the Niels Bohr Institute in Copenhagen, I had the pleasure
to work with, especially Eugene Polzik, Jörg-Helge Müller, Patrick Windpassinger,
Andrew Hilliard, Franziska Kaminsky and Rodolph Le Targat.
I also want to thank Jürgen Eschner who always had an open ear for my problems
and needs. It was a pleasure to discuss physics and to do things beyond science
with him. I thank all the people in ICFO giving technical support, especially Juli
Céspedes i Capdevila, Jose-Carlos Cifuentes, Ricardo Saiz, and Xavier Menino. They
were always helpful in solving all kinds of technical issues.
I would also like to thank all the people which crossed my way in Barcelona and
which became close friends. I’m glad to have you met and apologize not to list you
here. Fortunately, the list would be too long.
Thanks to my family, especially my parents, who supported me and my ideas invariably, and my brother who has been always an inspiration.
The biggest gratitude is reserved for my love, Katharina, who convinced me to come
to Spain. All this would have never happened without your loving support at all
times. You have brought me always back to the ’real’ world and showed me the
beauty of discovering new things. Thank you ...