Download Correlations and Counting Statistics of an Atom Laser

Diss. ETH No. 16731
Correlations and Counting Statistics
of an Atom Laser
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY
ZURICH
for the degree of
Doctor of Natural Sciences
presented by
ANTON W. ÖTTL
Dipl-Phys, University of Freiburg, Germany
born 12.03.1974
citizen of Germany
accepted on the recommendation of
Prof. Dr. Tilman Esslinger, examiner
Prof. Dr. Vahid Sandoghdar, co-examiner
2006
For the loveliest discovery
during my PhD
LISA
Zusammenfassung
Die Experimente, die im Rahmen dieser Doktorarbeit entwickelt und durchgeführt wurden,
realiseren kontinuierliche Atomlaser von unerreichter Stabilität und messen deren Intensitätsfluktuationen mit Hilfe von zeitaufgelöster Einzelatomdetektion durch quantisierte
Lichtfelder in einem optischen Resonator. Unter Verwendung der experimentellen Technik
von Teilchenkorrelationen, welche auf Hanbury Brown und Twiss zurückgeht, untersuchen
wir die Kohärenzeigenschaften des Atomlasers. In dem wir die Korrelationsfuntkionen höherer Ordnung und sogar die komplette Zählstatistik aufnehmen, zeigen wir die Kohärenz
des Atomlasers im Sinne der Definition von Glaubers Quantentheorie der Kohärenz. Wir finden keine überschüssigen Korrelationen in den Korrelationsfunktionen zweiter und dritter
Ordnung und bestätigen, dass der Atomlaser durch einen kohärenten Zustand, welcher einer Poisson verteilten Teilchenstatistik gehorcht, beschrieben wird. Die Resultate für den
Atomlaser stehen im Kontext zu den Messungen an pseudothermischen Atomstrahlen, welche Anhäufelung zeigen und eine Bose - Einstein Verteilung der Teilchen aufweisen.
Ebenfalls beschrieben wird der konzeptionell neuartige Apparat, in dem wir die experimentelle Vereinigung von einem optischen Resonator höchster Güte mit quantenentarteten atomaren Gasen erreichen. Die Verwendung eines in sich geschlossenen, austauschbaren “Wissenschaftsmoduls” gewährt geräumigen Zugang zum Bose - Einstein Kondensate
und dem Atomlaser für unterschiedliche Proben und Analysemethoden. Wir produzieren
87 Rb Kondensate von 2 · 106 Atomen und erzeugen extrem stabile Atomlaser mittels Radiofrequenzauskopplung. Der Atomlaser wird in die Mode des optischen Fabry - Pérot Resonators gerichtet, welcher sich 36 mm unterhalb des Kondensates befindet und über ein utrahochvakuumtaugliches Vibrationsisolierungssystem in dem Wissenschaftsmodul integriert
ist. Der Resonator mit einer Finesse von 3 · 105 arbeitet im Bereich starker Kopplung zwischen Atom und Lichtfeld und ermöglicht dadurch den Einzelnachweis von Atomen aus einer quantenentarteten Quelle. Die hohe Detektionseffizienz von ca. 25% für diese Atome
zeichnet ihn als empfindliche und dadurch minimalinvasive, zeitaufgelöste Messmethode
für ultrakalte atomare Gase aus. Das Leistungsvermögung wird beschrieben und charakterisiert durch Einzelatomnachweise für thermische und quantenentartete Atomgase.
Dieser experimentelle Aufbau ermöglicht uns kohärente Atomoptik auf Einzelteilchenniveaus zu studieren und dadurch das neuartige Forschungsgebiet der Quantenatomoptik
weiterzuentwickeln.
i
Abstract
The experiments developed and performed within the scope of this thesis realize continuous
atom lasers of superior stability and measure their intensity fluctuations by time resolved
single atom counting using cavity QED methods. By employing particle correlation measurements of the Hanbury Brown - Twiss technique we investigate the coherence properties
of the atom laser. We proof the coherence of atom lasers in the sense of Glauber’s definition
of the quantum theory of coherence by measuring higher order correlations and even the
full counting statistics. We find the absence of any correlations in the second and third order correlation function and confirm that the atom laser represents a coherent state with
a Poissonian atom number distribution. These findings are contrasted with measurements
on pseudo-thermal atomic beams that exhibit bunching and a Bose - Einstein distribution of
particles.
The conceptually novel apparatus in which we achieve the experimental integration of
an ultrahigh finesse optical cavity with quantum degenerate atomic gases is also presented
and characterized. It grants large scale spatial access to the Bose - Einstein condensate
and the atom laser for divers samples and probes via a modular and exchangeable “science
platform”. We produce 87 Rb condensates of 2 · 106 atoms and generate ultrastable continuous atom lasers by radio frequency output coupling. The atom laser is directed into the
mode of an optional Fabry - Pérot cavity which is situated 36 mm below the condensate.
It is mounted on the science platform by means of an ultrahigh vacuum compatible vibration isolation system. The cavity of finesse 3 · 105 works in the strong coupling regime of
cavity QED and serves as a quantum optical detector for single atoms from the quantum
degenerate source. The high detection efficiency (∼ 25%) for quantum degenerate atoms
distinguishes the cavity as a sensitive, time resolved and weakly invasive probe for ultracold
atomic clouds. The performance of the setup is presented and characterized by single atom
detection measurements for thermal and quantum degenerate atomic gases.
This system enables us to study coherent atom optics on a single particle level and to
further develop the new field of quantum atom optics.
iii
Contents
1 Introduction
1
2 Basic Theoretical Framework
7
2.1
Bose - Einstein Condensation in Harmonic Traps. . . . . . . . . . .
8
Ideal Bose Gas • Ground State Properties of the weakly Interacting Bose Gas •
Magnetic QUIC Trap
2.2
Atom Lasers . . . . . . . . . . . . . . . . . . . . . . . . .
18
Concept • The Output Coupling Process • Beam Propagation • Coherence Properties
2.3
Cavity Quantum Electrodynamics . . . . . . . . . . . . . . . . .
28
Resonator Basics • Atom Field Interaction • Single Atom Detection
3 Experimental Apparatus
3.1
Vacuum System . . . . . . . . . . . . . . . . . . . . . . . .
37
39
Main Chamber • MOT Chamber • Installation
3.2
Magnetic Field Configuration . . . . . . . . . . . . . . . . . .
43
Magnetic Transport • QUIC Trap • Magnetic Shielding • Auxiliary Coils
3.3
Science Platform and Cavity Setup . . . . . . . . . . . . . . . .
49
Cavity Design • Vibration Isolation System • Science Platform Layout
4 Characterization of the System
4.1
Experimental Procedure . . . . . . . . . . . . . . . . . . . .
55
56
Bose - Einstein Condensation • Atom Laser Output Coupling • Cavity Lock
4.2
Single Atom Detection Performance. . . . . . . . . . . . . . . .
61
Signal Analysis • Characteristics of Single Atom Events • Detector Qualities •
Detection Efficiency • Atom Laser Beam Profile • Guiding the Atom Laser
4.3
Investigation of Ultracold Atomic Gases . . . . . . . . . . . . . .
72
Thermal Clouds • Quantum Degenerate Gases • Phase Transition
v
CONTENTS
5 Correlations and Counting Statistics of an Atom Laser
5.1
5.2
77
Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
Background . . . . . . . . . . . . . . . . . . . . . . . . .
78
79
First Order Coherence • Hanbury Brown - Twiss and the Invention of Bunching •
Glauber and the Quantum Theory of Coherence • Counting Statistics
5.3
5.4
Experimental Methods and Techniques . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Cavity QED Detection of Interfering Matter Waves
6.1
6.2
6.3
105
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 106
Quantum Mechanical Measurement Process . . . . . . . . . . . . 106
Buildup of Matter Wave Interference . . . . . . . . . . . . . . . 110
7 Conclusion
113
A Appendix
115
A.1
A.2
A.3
A.4
vi
94
96
Breit - Rabi Formula . . .
Physical Properties of 87 Rb
D2-Line Energy Levels . .
Physical Constants. . . .
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115
116
117
118
Bibliography
119
Credits
133
Publications
135
Curriculum Vitae
137
1 Introduction
The research fields of Bose - Einstein condensation (BEC) [1] in dilute atomic gases and cavity
quantum electrodynamics (QED) with single atoms [2] both push forward the understanding, engineering, and harnessing of quantum mechanical states. A Bose - Einstein condensate is a collective quantum state of a large atom sample and provides maximum control
over external degrees of freedom. Optical cavity QED in the strong coupling regime allows
probing and manipulation of single atoms with the quantized electromagnetic field in the
cavity mode.
A Bose - Einstein condensate is a fascinating demonstration of the quantum character of
matter where indistinguishable, weakly interacting particles populate the motional ground
state and establish a macroscopic wave function. Its experimental realization [3, 4] in
1995 sparked an ongoing vivid experimental and theoretical research on this novel quantum phase. Initial experiments highlighted its phase coherence [5], superfluidity [6, 7] and
demonstrated the production of atom lasers [8, 9, 10, 11]. Current investigations explore inter alia quantum phase transitions [12, 13], tunable atomic interactions [14, 15] and particle
correlations [16, 17, 18].
Similarly, the way to cavity QED in the optical domain was paved by first experiments in
the 1990s reaching the strong coupling regime and demonstrating vacuum Rabi splitting of
the coupled atom cavity system [19]. In the strong coupling regime of cavity QED the atom
field interaction dominates over the dissipative losses of the quantum system. This system
was used to demonstrate single atom detection in a thermal atomic beam [20]. Recent experimental progress was made in the observation of the motional dynamics [21, 22] as well
as the trapping [23, 24] and cooling [25, 26] of single atoms within the cavity mode. This
provides an avenue towards implementation of technologies and concepts for quantum information processing, such as nonclassical light sources [27, 28] and quantum state transfer
[29].
The experimental combination of quantum degenerate gases with high finesse optical
cavities offers fascinating prospects [30, 31, 32] and develops the emerging field of quantum
atom optics, where both matter and light fields are quantized. The first experiments detecting single atoms from a coherent matter wave field with an ultrahigh finesse optical cavity
have been performed in the scope of this thesis [17, 33]. A different technique with the
potential of single atom detection in quantum degenerate samples has been demonstrated
1
1. INTRODUCTION
with metastable Helium atoms [34, 18, 35]. However, cavity QED detection of single atoms
is potentially nondestructive on the atomic quantum state and could be used to perform
atom interferometry with squeezed states and precision measurements at the Heisenberg
limit [36]. In addition, the single atom detection method offers an unprecedented sensitive
and weakly invasive probe to investigate physical processes in ultracold atomic clouds in situ
and time resolved. On the other hand, Bose - Einstein condensates and atom lasers provide
dense and coherent atomic sources with precisely controlled external degrees of freedom
for exploring and exploiting cavity mediated atom photon interactions. The integration of
a high finesse optical cavity in a Bose - Einstein condensation system, despite being a central
goal for atom chips [37, 38], has only recently been achieved with the apparatus described
here [39]. The experimental difficulties in merging these two experimental research fields
arise mainly from adverse vacuum requisites and sophisticated topological requirements on
both of these state-of-the art technologies. For example, limited spatial access prevented
the inclusion of a high finesse optical cavity in conventional Bose - Einstein condensation
setups.
Atom lasers represent a prime example of the dualism in the quantum description of nature. The characteristic feature of bosons is their tendency to clump together and multiply occupy the same phase space volume, thereby amplifying a single quantum mechanical
mode to a macroscopic and observable level. This phenomenon is based on the stimulated
enhancement factor (n + 1) and is the underlying principle of laser operation. The formal
analogy between light quanta and massive bosons, evident in the formalism of second quantization, suggests the straight forward extension of Glauber’s quantum theory of coherence
[40, 41] from optical to matter wave fields. Atom lasers, as their optical counterparts, feature
ultimate directivity and brightness, monochromaticity and phase coherence [42]. However,
all these features could in principle be achieved with a thermal beam upon sufficient filtering. To tell wether or not one has a genuine atom laser, requires to measure at least its
second order coherence. This was already pointed out by Kleppner after the first experimental realization of an atom laser [43] and performed within the scope of this thesis [17].
For optical lasers the experimental proof of their coherence was demonstrated by Arecchi
[44, 45] shortly after the invention of this extraordinary light source [46].
The quantum mechanical state of a laser is fully characterized by the notion of coherent states as introduced by Glauber. The coherent state comes as close as possible to an
ideal classical wave, having an uncertainty of both the conjugate variables phase and amplitude at the quantum limit. Glauber’s theory formulates higher order correlation functions
that clearly display the characteristic features of quantum radiation in terms of coherence.
A high order coherent state is defined by the factorization of its higher order correlation
functions and thus by the absence of any Hanbury Brown - Twiss correlations. Equivalently,
the particle statistics of a coherent state obey a Poissonian distribution. Therefore the intensity of a laser is utterly stable and only limited by the shot noise contribution, whereas
any thermal beam will exhibit excess fluctuations.
2
One aspect of these fluctuations, the comparison of the variance to the mean value, is
grasped in the second order correlation function. This quantity was first measured in the
innovative experiments by Hanbury Brown and Twiss, where they observed correlations of
intensity fluctuations in two coherent beams of light [47, 48], the so-called bunching effect.
Although initially measured with analog signals, in principle it represents the joint probability to detect two particles at distinct positions in space or time. The fact that the Hanbury Brown and Twiss method investigates two-particle correlation or interference effects
contrasts any previous interferometric techniques. Their revolutionary measuring technique, intended and successfully applied to measure stellar diameters [49], found its way
into many diverse scientific disciplines [50, 51], where the correlations between individual
particles reveal insight into the quantum state of a system or the particle statistics governing its behavior. The two-particle correlation measurements can be exploited in the temporal or spatial domain to shed light on the bandwidth or the physical size of the source of
particle origin. The latter is of interest in astronomy [52] and high energy physics [53]. For
modern quantum optics the Hanbury Brown - Twiss method is a cornerstone and key technology for investigating the photon emission process and identifying nonclassical photon
states, which form the essence of most quantum information schemes.
Correlations between massive particles in atomic physics have first been measured by
Yasuda and Shimizu [54]. They observed the analogous Hanbury Brown - Twiss bunching
effect for identical bosons in a thermal atomic beam. Using metastable neon atoms and a
segmented multi channel plate they detected temporal correlations of two successive particle detection events. Using the Hanbury Brown - Twiss correlation technique the underlying quantum statistics of fermions were investigated in mesoscopic systems [55, 56] and
in a beam of free electrons [57]. These experiments demonstrated the existence of anticorrelations or antibunching for fermions as a consequence of Pauli’s exclusion principle. For
quantum degenerate systems the decrease of the three-body recombination rate by the factor 3! as compared to a thermal cloud was predicted by Kagan [58] and it has been observed
in the group of Cornell and Wieman [59]. Recently, in a similar experiment to the one by
Shimizu [54], Hanbury Brown - Twiss correlations of atomic matter waves were measured in
all three spatial dimensions by the use of metastable helium atoms and a spatially resolving
fast multichannel plate detector [18]. Furthermore, the researchers in the group of Aspect
observed the absence of the two-particle correlations for the case of a quantum degenerate source. These studies, carried out independently and published simultaneously are in
agreement with our findings [17] and highlight different aspects.
Although all the above experiments employ some kind of beam splitter and make use of
several detectors to correlate particle detections at different space-time points, neither is
a necessity to perform a Hanbury Brown - Twiss type experiment. Having a fast detector
with individual particle sensitivity the temporal second order correlation function can be
recorded with a single detector. This was already pointed out by Purcell [60] in his paper
supporting the initially contended Hanbury Brown - Twiss experiment. In our experiment
3
1. INTRODUCTION
we make use of this fact. The dead time of our single atom detector is short compared to
the relevant time scales. We are able to record the detection time of each particle explicitly
and find the temporal correlations by analyzing all differential times. In fact this represents
a major improvement compared to conventional Hanbury Brown - Twiss experiments that
rely on start-stop signals due to limitations set by the dead time of the detectors. Start-stop
events only correlate exclusive neighboring particles and are therefore just an approximation to the second order correlation function which is a nonexclusive quantity. This poses
the constraint on the maximum count rate that the probability of having more than two
particles within the relevant time scale is negligible. The consequence are long integration
times, that for instance exceeded 70 hours in the experiment by Shimizu [54]
Moreover, employing a single atom detector with a high quantum efficiency permits the
determination of correlations beyond the second order correlation function. The knowledge of all higher order correlation functions is equivalent to the full counting statistics or
particle distributions within a beam. We reported [17] the first experiment that records the
full counting statistics of massive particles, which is a significant quantity in mesoscopic
physics [61, 62, 63]. There however, no single particle detectors are at the experimenter’s
disposal yet, so only some features of the particle distribution can be inferred from the measurement of certain statistical moments [64, 65]. The knowledge of all moments fully characterizes the statistical distribution functions that usually fall into two distinct categories:
sub- and super-Poissonian. The prefix refers to smaller and larger variance, in comparison
to the delimiting case of a Poisson distribution which marks the total randomness of a process. So both cases provide excess information about the arrival times of particles and are
related to the effects of antibunching and bunching in the second order correlation function, respectively.
In the present work the first experiments detecting single atoms from a coherent matter
wave field with an ultrahigh finesse optical cavity have been performed [17, 33]. The detection process of our single atom detector is based on cavity QED methods and relies on
the fully quantized interaction of the electromagnetic field in the cavity with a single twolevel atom as modeled by Jaynes and Cummings [66]. The partially open quantum system
of an atom coupling to the cavity mode has inherent dissipation channels through spontaneous scattering and cavity field decay. These sources of decoherence establish a quantum
measurement of the presence of a single atom inside the cavity mode with the consequence
of the breakdown of probe light transmission through the optical cavity. The implication of
the quantum measurement is the collapse of the longitudinally vastly extended atomic matter wave function of an atom from the atom laser. In an experiment in analogy to Young’s
double slit we observe the buildup of an interference pattern from single particle detection
events. The fact that we reduce the atom flux to such low levels that on average only single
atoms are at a time in the “which way” interferometer, represents a prime example of the
wave-particle dualism in quantum mechanics and is at the heart of quantum atom optics.
4
The structure of the thesis is as follows:
• In Chapter 2 the theoretical background for the three main ingredients of the experiment, Bose - Einstein condensation, atom lasers and cavity QED, is developed.
• Chapter 3 explains design considerations and the technical realization of the experimental apparatus, with the main focus on the novel vacuum system, the magnetic
transport and the ultrahigh finesse optical cavity design.
• Single atom detection measurements for thermal atomic beams and atom lasers are
presented in Chapter 4 to characterize and benchmark the performance of our apparatus.
• The main part of the thesis is Chapter 5, which gives an account of our measurements
of the second and third order correlation functions and counting statistics of an atom
laser compared to pseudo-thermal atomic beams, thereby disclosing the true nature
of an atom laser as a coherent state.
• Chapter 6 deals with the investigation of the quantum measurement process realized
with the cavity QED system and the localization of the longitudinally extended matter
wave function.
5
2 Basic Theoretical Framework
“Felix die Kirsche zu Manni, Manni Banane, ich Birne - Tor.” - Horst Hrubesch
Here, the basic theoretical principles relevant for describing and understanding the physical phenomena presented in this thesis are developed. The framework is based on three
pillars of modern quantum optics over which we cast a brief overview. First, the general formalism to describe many aspects of Bose - Einstein condensation in dilute atomic gases and
the ground state properties of the macroscopic wavefunction are presented. Secondly, we
describe the mechanism to create coherent and continuous atom lasers and illustrate some
of their characteristic features. Lastly, the ideas of single atom detection in the strong coupling regime of cavity QED along with the full quantum mechanical treatment are mapped
out briefly.
7
2. BASIC THEORETICAL FRAMEWORK
2.1. Bose - Einstein Condensation in Harmonic Traps
The idea that a macroscopic number of indistinguishable, noninteracting, integer spin particles (bosons) can, and actually will occupy the same quantum state below a certain critical
temperature was put forward by S.N. Bose [67] and extended by A. Einstein [68] to massive
particles in the 1920’s. Over many years the evolution of the theoretical description of such
a Bose - Einstein condensate (BEC) was connected with other macroscopic quantum phenomena like superfluidity of 4 He and superconductivity [69, 70, 71, 72, 73], where however
the strong interactions dominate and deplete the condensate fraction. Only a combination
of experimental techniques developed in the 1970’s and 1980’s like laser cooling and evaporative cooling applied to dilute alkali atomic vapors succeeded in 1995 to realize a pure BEC
of weakly interacting particles [3]. That breakthrough opened the door to investigate this
novel quantum state of matter both experimentally and theoretically, which is still an ongoing and active research field in numerous labs worldwide. The theory of Bose - Einstein
condensation is discussed comprehensively in current text books [1, 74] and review articles
[75, 76, 77].
2.1.1. Ideal Bose Gas
The phenomenon of Bose - Einstein condensation, i.e., the macroscopic population of the
ground state by integer spin particles can be derived in terms of quantum statistics [78].
The quantum statistical behavior of indistinguishable integer spin particles by the name of
bosons is governed by Bose - Einstein statistics. In the framework of the grand canonical
ensemble, where the temperature T and the chemical potential µ are the natural variables,
the distribution function can be found. The mean occupation number n̄ of a single-particle
state with energy ²i is then given by its degeneracy gi times
n̄ (²i ) =
1
e β(²i −µ) − 1
,
(2.1)
where β = 1/kB T and kB denotes Boltzmann’s constant. The chemical potential µ(N , T ) is
set and fixed by the normalization condition for the total number N of massive particles
N=
X
n̄ (²i ) ,
(2.2)
which is in contrast to photons, where the total number is not a constant quantity. From
these formulas it is already evident that the chemical potential µ is bounded above by the
ground state energy ²0 , since n̄ has to be positive, and that the occupation number of the
ground state is special, since it diverges when µ → ²0 . This defines the critical temperature
Tc of the phase transition. To correctly account for the total number of particles, the number
of particles N0 being in the ground state is treated separately from the number of thermal
atoms Nth . For the latter, the sum in equation (2.2) can be replaced by an integral under
8
2.1. BOSE - EINSTEIN CONDENSATION IN HARMONIC TRAPS
the condition that kB T À ∆², where ∆² is the spacing of the discrete energy levels. In this
semiclassical approximation the discrete spectrum is treated as a continuum, where the
density of states ρ (²) contains the details of the level structure
Z
N = N0 + Nth = N0 +
ρ (²) n̄ (²) d ² .
(2.3)
In order to calculate the transition temperature the topology and the dimensionality of the
system have to be taken into account. Generally, the external trapping potential V (r) for
ultracold atoms of mass m is well described by a three dimensional harmonic potential
´
1 ³
V (r) = m ω2x x 2 + ω2y y 2 + ω2z z 2
2
−→
ρ (²) =
²2
2ħ3 ω̄3
(2.4)
with eigenfrequencies ωx , ω y , ωz along orthogonal dimensions x, y, z , which yields a density
of states being quadratic in energy: ρ (²) ∼ ²2 . Planck’s constant is denoted by ħ = h /2π and
we introduced the geometric mean ω̄ = (ωx ω y ωz )1/3 of the oscillator frequencies.
In the thermodynamic limit for large atom numbers the zero point energy ²0 = ħ(ωx +
ω y + ωz )/2 of the trap can be neglected. For evaluating the integral in equation (2.3) it is
most useful to introduce the general Bose function (or polylogarithm)
Z
g α (z ) = Γ(α)
−1
0
∞
dx
∞
X
x α−1
=
z k k −α ,
z −1 e x − 1 k=1
(2.5)
with Γ(α) being the Gamma function. The result for the three dimensional harmonic potential is
µ
¶
Nth (T, µ) =
kB T 3
g 3 (z ) .
ħω̄
(2.6)
For temperatures above the critical temperature Tc , when Nth = N is valid, this serves as
the normalization condition on the chemical potential, which is expressed in terms of the
fugacity z = e βµ . The phase transition to a Bose - Einstein condensate occurs when µ → 0,
so the critical temperature Tc is obtained by evaluating equation (2.6) for z = 1 and found to
be
µ
¶
Tc =
ħω̄ N 1/3
≈ 0.94 ħω̄ N 1/3 ,
k B ζ(3)
(2.7)
where the Bose integral in equation (2.3) simplifies to the Riemann zeta function ζ(α) =
P∞ −α
k . Besides the interaction between particles, there are corrections to the critical
k=1
temperature due to the finite particle number and the anisotropy of the potential [1]. For
temperatures below Tc , when µ = 0, the number of particles in the ground state increases
for lower temperatures and can be found from equations (2.6) and (2.7). The condensate
fraction is then given by
µ ¶3
N0
T
= 1−
.
N
Tc
(2.8)
9
2. BASIC THEORETICAL FRAMEWORK
An important, since experimentally accessible quantity, is the density and the momentum
distribution of the ultracold atomic gas. Similarly, they are best treated for the condensed
and thermal fraction separately as
n (r) = n 0 (r) + n th (r)
and
n (p) = n 0 (p) + n th (p) ,
(2.9)
respectively. In the absence of interactions the density of the condensate is simply given by
the ground state wavefunction φ0 (r) of the anisotropic harmonic trap and the momentum
distribution by its Fourier transform φ(p), which is therefore also anisotropic
N
n 0 (r) = N |φ0 (r)|2 =
2 2
2 2
2 2
e −x /a x e −y /a y e −z /a z
π3/2 a x a y a z
2 2
2 2
2 2
N
n 0 (p) = N |φ0 (p)|2 =
e −p x /c x e −p y /c y e −p z /c z
π3/2 c x c y c z
p
(2.10)
(2.11)
p
with a i = ħ/mωi and c i = ħmωi being the typical length and momentum scales. Therefore the spatial extension of the ground state wavefunction is on the order of the harmonic
oscillator lengths a i .
The density and momentum distribution of the thermal cloud can be calculated in the local density approximation by considering the particle distribution function in phase space,
which is represented by the Wigner function
W (r , p) =
1
1
(2πħ)3
e β(²(r,p)−µ) − 1
(2.12)
and integrating W (r , p) over momenta and positions, respectively. The semiclassical local
density approximation is valid for de Broglie wavelengths small compared to the size of the
cloud or equally the variation of the trapping potential V (r), which can therefore be assumed to be locally homogeneous. The local energy is then given by ²(r , p) = p2 /2m +V (r).
In the classical limit for T À Tc the effects of quantum statistics, i.e., the “–1” in the
denominator of the Planck formula (2.1) can be neglected. The densities may then be calculated using the classical Boltzmann statistics, resulting in
n th (r) =
n th (p ) =
N
2
π3/2 σx σ y σz
N
(2πmk B
2
2
e −(x /σx ) e −( y /σ y ) e −(z /σz )
T )3/2
2
e −(p /2mkB T ) .
(2.13)
(2.14)
q
The radial extension of a cloud at temperature T is given by σi = 2kB T /mω2i . This shows
the isotropy in momentum distribution for thermal atoms, only depending on their temperature.
However, when T ≈ Tc the shape of the cloud deviates form a Gaussian and is more appropriately described by a Bose function, but remains isotropic in momentum. Evaluating
10
2.1. BOSE - EINSTEIN CONDENSATION IN HARMONIC TRAPS
nth(r)
nth(p)
y
py
pz
z
FIGURE 2.1.: Illustration of the Bose functions for a thermal cloud at the phase transition (µ =
0). The anisotropic density (left) and isotropic momentum (right) distributions exhibit both
a sharper peak than a Gaussian. The calculation was done with typical parameters of our
experiment (ωz = 4 ω y ). The plot range is 150 a z and 50 c z for the density and momentum
distribution, respectively.
R
W (r , p) d 3 p and
R
W (r , p) d 3 r yields
3
n th (r) = λ−
dB g 3/2(z (r))
(2.15)
n th (p ) = (λdB m ω̄)−3 g 3/2(z (p )) ,
(2.16)
2
where z (r) = e β(µ−V (r)) and z (p ) = e β(µ−p /2m ) . The characteristic behavior of the Bose function g 3/2 is illustrated in Figure 2.1. The thermal de Broglie wavelength λdB for an ideal gas
of massive particles at thermal equilibrium is defined as
s
λdB =
2πħ2
mk B T
.
(2.17)
Compared to a distribution of distinguishable particles, the density of a Bose gas is increased
by g 3/2(z )/z . From equation (2.15) the critical phase space density D = nλ3dB can be extracted
using the peak density for µ → 0
D = n th (0)λ3dB = g 3/2(1) ' 2.6212 ,
(2.18)
which coincides with the criterion of Bose - Einstein condensation in a uniform Bose gas.
This criterion has a simple physical interpretation: a macroscopic population of the ground
state happens when there is more than one particle per cubic thermal de Broglie wavelength
and so the particle waves necessarily overlap. However, while the condensation in the uniform case only takes place in momentum space for particles with p = 0, the condensation in
a harmonic trap also occurs in position space as can be seen from equation (2.10).
Experimentally, the discrimination of a Bose - Einstein condensate from the thermal gas
is usually manifested in their specific peak density and momentum distribution, which is
measured in time of flight technique [79, 75, 80, 81] as shown in Figure 2.5.
11
2. BASIC THEORETICAL FRAMEWORK
2.1.2. Ground State Properties of the weakly Interacting Bose Gas
In contrast to the thermal cloud, the condensate part with its increased density can only
be well described when taking into account interactions between the particles. Because the
condensate part occupies the ground state, the contribution of thermal energy becomes
negligible compared to the interaction energy. Therefore we have to shift from an ideal gas
description towards a real gas, where the particle interactions are governed by the interatomic potential U (r 0 − r). The reduction to binary collisions is usually justified in dilute
atomic gases, where the mean particle distance d = n −1/3 is much larger than the range of
the interatomic potential r e ¿ d .
Furthermore, all collisions in Bose - Einstein condensates, where the temperature is bepr
low the critical temperature, are low momentum collisions (p → 0) satisfying ħe ¿ 1. This
implies that the scattering amplitude becomes independent of energy and scattering angle known as s-wave collisions. In addition, only elastic scattering processes that preserve
the atomic internal state are considered. In the low energy limit the scattering amplitude
is given by its asymptotic value, the s-wave scattering length a . In the first order Born
approximation the exact form of the interatomic potential is not relevant and it can be apR
proximated by an effective potential with U0 = Ueff (r) d 3 r, where
U0 =
4πħ2 a
m
(2.19)
represents the interaction energy between two particles. In order for these approximations
to be valid, the so-called diluteness condition
n|a|3 ¿ 1
(2.20)
for the so-called gas parameter needs to be satisfied. It can be violated using a so-called
Feshbach resonance, where the scattering length can be tuned to any desired values. To
yield downright stable Bose - Einstein condensates it has to be repulsive with a > 0. The
elastic s-wave scattering length a for alkali atoms is typically two orders of magnitude larger
their physical size given by the Bohr radius (a 0 = 0.5 Å). In a thermal gas, the total cross
section σ of two identical bosons is then σ = 8πa 2 , which is a factor two larger than σ for
distinguishable particles.
In a microscopic theory the quantum mechanical state of the Bose gas is fully described
by the many-body Hamiltonian Ĥ = Ĥkin + Ĥpot + Ĥint in terms of the Bose field operators
Ψ̂. Its time evolution is then determined by the Heisenberg relation
∂
Ψ̂(r , t ) = [Ψ̂(r , t ), Ĥ ] .
(2.21)
∂t
P
The field operator can be expanded as Ψ̂(r) = i φi âi , where the âi (â i† ) are the annihilation (creation) operators of a particle in state φi , obeying the Bose commutation relations.
iħ
12
2.1. BOSE - EINSTEIN CONDENSATION IN HARMONIC TRAPS
However, in the Bogoliubov theory, which provides a good description for the macroscopic
phenomena associated with Bose - Einstein condensation, the ground state component is
separated and treated as a classical field
Ψ̂(r) = Ψ0 (r) + δΨ̂(r) ,
(2.22)
thereby neglecting the quantum mechanical commutation relations. The ground state parp
ticle creation/annihilation operators can then be replaced by â 0(†) → N0 and the wavefuncp
tion of the condensate is expressed as Ψ0 = N0 φ0 . This ansatz is appropriate when N0 À 1
and yields the prominent Gross - Pitaevskii equation in its time dependent form
µ 2 2
¶
∂
ħ ∇
2
i ħ Ψ0 (r , t ) = −
+ V (r , t ) +U0 |Ψ0 (r , t )| Ψ0 (r , t )
∂t
2m
(2.23)
when the quantum fluctuation term δΨ̂ and thermal depletion are completely neglected.
The condensate wavefunction Ψ0 (r) plays the role of an order parameter and its time evolution can be derived to be Ψ0 (r , t ) = Ψ0 (r)e −i µt /ħ , where the chemical potential µ = ∂E /∂N
is the energy per particle. This quantities is now nonzero in a real Bose - Einstein condensate, due to the interaction energy between the particles. The stationary form of the Gross Pitaevskii equation is thus given by
µ
¶
ħ2 ∇2
2
+ V (r) +U0 |Ψ0 (r)| Ψ0 (r) = µΨ0 (r) ,
−
2m
(2.24)
where the external potential V (r) is usually time independent. The order parameter is norR
malized to the total number of particles N0 = |Ψ0 (r)|2 d 3 r and gives the density of the
gas n 0 (r) = |Ψ0 (r)|2 . The Gross - Pitaevskii equation is a nonlinear Schrödinger equation
where the nonlinear term, being proportional to the particle density, describes the meanfield potential produced by the other bosons in the condensate. It is therefore only valid
when Ψ0 (r) varies only weakly on a length scale compared to the scattering length a . The
eigenvalue of the condensate is given by the chemical potential µ. For a solution Ψ0 of the
Gross - Pitaevskii equation at T = 0, the many-body wavefunction, for a system of N bosons
in the ground state, in its symmetrized form ignoring particle correlations can be written as
Φ0 (r1 , r2 , . . . , rN ) =
N
Y
1
p Ψ0 (ri ) .
i =1 N
(2.25)
This illustrates the fact that a Bose - Einstein condensate, although consisting of a large
number of particles, is essentially a single wave. However, the solutions of the Gross Pitaevskii equation (2.24), due to its nonlinear character, can in general only be obtained
by numerical integration.
An important exception arises in the Thomas - Fermi approximation, when the kinetic
energy term in the Gross - Pitaevskii equation is neglected compared the mean-field term.
13
2. BASIC THEORETICAL FRAMEWORK
p
This is the case when N0 a /ā À 1, where ā = ħ/m ω̄. Although the fraction a /ā is generally on the order of 10−3 the condition is usually well fulfilled for
p typical atom numbers
4
(N0 > 10 ). In this limit an analytical solution is given by ΨTF (r) = n TF (r), where
n TF (r) = [µTF − V (r)]/U0
(2.26)
for µTF > V (r) and zero elsewhere [82]. Therefore, in the case of a harmonic trapping potential, the density profile has the shape of an inverted parabola as shown in Figure 2.2. The
exact ground state wavefunction can only be found by variational methods and differs only
at the sharp boundary slightly from the Thomas - Fermi profile.
|Ψ0|2/100
E
V(r)
|ΨTF|2
μ
RTF
r
0
FIGURE 2.2.: The harmonic trapping potential V (r ) and resulting density distribution
|ΨTF (r )|2 of the condensate wavefunction in the Thomas - Fermi limit, in comparison to the
ground state wavefunction |Ψ0 (r )|2 in the absence of interactions, scaled down by a factor
100.
The peak density is given by n TF (0) = µTF /U0 and the chemical potential follows from the
normalization condition (2.2)
µ
¶
ħω̄ 15N a 2/5
µTF =
.
(2.27)
2
ā
The physical content of this approximation is that the energy to add a particle at any point
in the condensate is the same everywhere. The Thomas - Fermi result for the total energy
per particle is 5/7 of µTF . The boundary of the wavefunction when µTF = V (r) is given by
the Thomas - Fermi radii
s
Ri =
2µTF
mω2
i
14
µ
= ā
15N a
ā
¶1/5
ω̄
.
ωi
(2.28)
2.1. BOSE - EINSTEIN CONDENSATION IN HARMONIC TRAPS
The density profile is anisotropic, as is the momentum distribution, which can be found by
taking the squared Fourier transform of the Thomas - Fermi wavefunction ΨTF (r) [82]
µ
15
J 2 (p̃ )
n TF (p) =
N R̄ 3
3
16ħ
p̃ 2
¶2
(2.29)
,
q
where J 2 is the second order Bessel function, p̃ =
(p x R x )2 + (p y R y )2 + (p z R z )2 /ħ and R̄ =
(R x R y R z )1/3 . The width of the momentum distribution n TF (p i ) scales as ∼ R i−1 and for large
samples would approach a δ-function as in the uniform case.
n(x)
−10
n(v)
0
x [ μm ]
10
−0.5
0
vx [ mm/s ]
0.5
FIGURE 2.3.: Results of a variational method (operator split-step FFT) showing the exact density (left) and momentum (right) distribution of the condensate wavefunction. The numerical simulation was performed for typical parameters of our experiment, having a 87 Rb condensate of 2 · 106 atoms and a trap frequency ω = 2π · 38 Hz.
Going to higher order approximations, the quantum fluctuations absent in the mean-field
approach can be taken into account. The Bogoliubov transformation [74] allows the diagonalization of the Hamiltonian Ĥ including first order quantum fluctuations (see Eq. 2.22).
This microscopic approach yields for instance the excitation spectrum of a homogenous gas
p
at zero temperature ²(p ) = 4g nm p 2 + p 4 /2m . The dispersion relation is phonon-like for
p ¿ mc and particle-like for p À mc , which defines a natural length scale, the healing
length
1
ξ= p
.
(2.30)
8πna
It plays the role of a coherence length and sets the minimal length scale over which the
condensate wavefunction can be perturbed. Therefore the sharp boundary of the Thomas Fermi profile is smeared out on the size of the healing length ξ.
The microscopic Bogoliubovptheory gives corrections to quantities derived by the meanfield theory on the order of na 3 , which is typically a few percent for common condensates. The most interesting consequence is the prediction of the quantum depletion
15
2. BASIC THEORETICAL FRAMEWORK
³
´
p
≈ 1.5 · na 3 of a condensate originating from the interactions between its constituting
particles. It is the reason for the large condensate depletion in strongly interacting quantum fluids like liquid helium.
2.1.3. Magnetic QUIC Trap
The method of choice, besides optical dipole traps, to realize a confinement for ultracold
neutral atoms are magnetic traps. According to the fundamental Earnshaw theorem no
local maximum of the electromagnetic field can exist in free space. Therefore magnetic
traps create a local minimum and thus apply exclusively to atoms in “low field seeking”
Zeeman states with a positive magnetic moment µm = g F mF µB > 0, being composed of the
Landé factor g F and the magnetic spin quantum number mF , times Bohr’s magneton µB .
All magnetic traps in cold atom experiments require a finite offset field to avoid Majorana
spin flip losses at their center where the magnetic field would be zero [83, 84, 4]. In this
region the atomic spins lack a quantization axis and they will undergo spin flip transition
into untrapped or antitrapped Zeeman states. Quite generally these types of magnetic traps
and in particular our QUIC trap (quadrupole Ioffe configuration) [85] can be described as
Ioffe - Pritchard traps [86] exhibiting an axial symmetry with an offset field B 0 along the
symmetry axis. An implementation of the QUIC being the simplest and most stable of the
Ioffe - Pritchard type traps is shown in Figure 2.4.
A small coil (Ioffe coil) is added in series to a quadrupole coil pair to lift the magnetic zero
and add a curvature to the resulting magnetic field. The magnetic potential for the atoms
with a magnetic moment µm is then well approximated by
00
V (ρ, y ) ' µm (B ρ00 ρ 2 + B y00 y 2 )/2
with
B ρ00
B 02 B y
=
−
,
B0
2
(2.31)
where our symmetry axis (Ioffe axis) is along the y direction. The
q trap frequency in a certain
µ
p
axis is then simply related to the field curvature through ω = mm B 00 and scales as ω ∼ I
with the electrical current. Usually Bose - Einstein condensates in Ioffe - Pritchard traps
have a cigar shaped form with aspect ratios ranging from 2–100.
To investigate the ultracold atomic gas, the trapping potential is switched off abruptly
and the cloud is allowed to expand in a free time of flight before resonant laser light casts
the shadow image on a CCD camera. Absorption imaging is the standard technique to detect
the cold atomic cloud and to identify a Bose - Einstein condensate by its high peak density
and its anisotropic expansion reminiscent of the trap geometry. A typical image showing the
bimodal distribution of a partly condensed cloud with inverted aspect ratio and an isotropic
thermal background is depicted in Figure 2.5.
16
2.1. BOSE - EINSTEIN CONDENSATION IN HARMONIC TRAPS
z
B
y
FIGURE 2.4.: Schematic sketch of the arrangement of electromagnetic coils forming the magnetic QUIC trap. The small lateral Ioffe coil lifts the magnetic zero of the quadrupole field
and adds curvature. The symmetry axis of the BEC is along the y-direction.
y
-z
100 μm
FIGURE 2.5.: Absorption image of a partly condensed cloud after a time of flight of 22 ms.
The Bose - Einstein condensate is clearly distinguished by the anisotropic expansion and
increased peak density compared to the surrounding thermal cloud.
17
2. BASIC THEORETICAL FRAMEWORK
2.2. Atom Lasers
The idea of an atom laser as a coherent matter wave and strategies on how to
convert random de Broglie waves into such a coherent state, was proposed by
a number of different authors [87, 88, 89, 90, 91, 92] even before the experimental realization of Bose Einstein condensation. But as soon as the creation
of single macroscopic wavefunctions provided ultimate control over matter
waves in a trapped state it was just a question about finding a suitable extraction method to create an atom laser beam. After the first demonstration
of a coherence preserving pulsed output coupler [8], the extraction methods
were extended and refined to give mode locked [9], well collimated [10] and
continuous [11] atom lasers. Theoretical models describing the characteristics of atom lasers range from rate equations and mean-field approaches to
fully quantum field theory usually under the Born - Markov approximation.
A review about these methods can be found in reference [93]. Of foremost
interest are the coherence properties of atom lasers, where a major contribution was provided with the measurements within this work [17].
The concept of an atom laser is in close analogy to its optical counterpart
and they share the same distinct features. An atom laser can quite generally
be defined as a device that emits an extremely bright, highly directional and
monochromatic beam of atoms rather than photons [94]. Ultimately, however, an atom laser is defined by its high order coherence, which is elaborated
more deeply in Chapter 5.
The quantum optical description of lasers can be transferred from photons and light fields to incorporate atoms and matter wave fields because of
their formal equivalence in quantum field theory. Atom optics is the field
which exploits the analogy between Schrödinger waves and electromagnetic
waves. The quantum mechanical treatment of the laser is indispensable [40]
although, quite ironically, it requires its output beam to be well approximated
by a classical wave of fixed intensity and phase. Consequently, the single
mode output field must be highly Bose degenerate to limit quantum fluctuations and approach a classical field.
The key mechanism to establish a vastly populated quantum mechanical
mode relies on the mode selectivity of a cavity and the stimulated bosonic
enhancement factor N0 + 1 which amplifies the population of the selected
mode. Therefore a fermionic atom laser will be the counterpart of a light
sabre and remain science fiction. The ingredients that make up an optical
18
100 μm
2.2.1. Concept
2.2. ATOM LASERS
laser can all be recovered as the elements to realize an atom laser as illustrated in Figure 2.6.
Although the fundamental principles of matter and light lasers are alike, their technical
realizations naturally have to differ. For instance, number conservation allows the atomic
system to be prepared in the absolute ground state of the cavity. The natural population inversion of the thermal spectrum can be transformed into a macroscopic ground state population by means of stimulated scattering events during the processes of evaporative cooling
towards Bose - Einstein condensation. The trapping potential serving as the cavity for the
matter wave has to be rendered partially penetrable to establish an output coupling of the
atom laser beam while preserving its coherence properties.
gain medium
laser beam
cavity
output
coupler
pump
evaporative
cooling
trap
BEC
thermal
atoms
atom laser
output
coupler
FIGURE 2.6.: Schematic diagram illustrating the key ingredients and similarities of an optical
laser and its matter wave counterpart, the atom laser.
Of course the analogy between optical and atom lasers is not exhaustive. Atoms are particles of matter rather than gauge bosons such as photons. That means the matter field can
not be directly measured but only bilinear combinations of the atom field are observables.
The virtue of that is the fact that atoms will not be annihilated by the detection process in
contrast to photon detection. However, an atom laser of massive particles will be affected
strongly by gravity and can only propagate undisturbed under very good vacuum conditions.
But it are these exceptional features that distinguish an atom laser as unique scientific
tool for novel applications and research with atom optics. Atom laser experiences the interactions between its constituting atoms. These interactions render an atom laser highly
nonlinear and for instance four-wave mixing has been demonstrated in Bose - Einstein con-
19
2. BASIC THEORETICAL FRAMEWORK
densates [95]. Also, atomic beams ill not propagate at a fixed speed (of light) and can be focused much more tightly due to their correspondingly shorter de Broglie wavelengths [96].
Therefore much smaller structures can be resolved and created in coherent atom lithography and microscopy. Further prospects of atom lasers include pushing the limits of resolution of the interferometric methods employed for precision sensors and metrology beyond
the standard quantum limit towards the Heisenberg limit [36].
2.2.2. The Output Coupling Process
An output coupler for an atom laser is a mechanism that extracts atoms from the confined
quantum state to a propagating mode while preserving or even extending (in the temporal
domain) its coherence properties. Quite generally, a coherent process couples the trapped
spin state of atoms in the Bose - Einstein condensate into an untrapped state that can escape
the trap. The principle is shown in Figure 2.7 where the atoms in the condensate state Ψ
are degenerate at the chemical potential and the wavefunction of the freed state Φ are the
energy eigenfunctions of the linear gravitational potential plus the “hump” from mean-field
repulsion of the remaining Bose - Einstein condensate.
Different output coupling mechanisms based on coherently induced spin flips [8, 10, 11]
and other methods [9, 97] to couple the condensate wavefunction to free states have been
demonstrated. In general, the output coupling process can be described quantum mechanically through a set of coupled nonlinear Schrödinger equations in the rotating wave approximation
µ 2 2
¶
£
∂
ħ ∇
2
2¤
iħ Ψ = −
+ V (r) − mg z +U0 |Ψ| + |Φ| Ψ + ħΩe −i ∆t Φ
∂t
2m
µ 2 2
¶
£
∂
ħ ∇
2
2¤
iħ Φ = −
− mg z +U0 |Ψ| + |Φ| + ħ∆ Φ + ħΩe i ∆t Ψ ,
∂t
2m
(2.32)
(2.33)
where the trapped state Ψ is held by the potential V (r) under the influence of gravity (g is
the gravitational acceleration) and the mean-field potential exerted by the sum of the densities of both states [98, 99, 100]. The untrapped state Φ only experiences the linear gravitational potential (in −z direction) as well as the combined mean-field term. The nonlinearity
constant U0 is well approximated to be the same for both states because the interstate scattering lengths are equal within a few percent [101]. The coupling term between the states
is proportional to the Rabi frequency Ω. For radio frequency output coupling the Rabi frequency is given by the magnetic dipole matrix element µ between the states Ψ → Φ and the
magnetic field Brf of the radio frequency
Ω = µ · Brf .
(2.34)
The detuning ∆ in eq (2.33) is the energy difference of the radio frequency photon to the
potential energy of the trapped versus the untrapped state, which will be taken up by the
atoms.
20
2.2. ATOM LASERS
E
VΨ(z)
mF = -1
ħωrf
mF = 0
Φ(z)
VΦ(z)
-z
FIGURE 2.7.: Diagram of the coherent atom laser output coupling mechanism. Trapped degenerate atoms (in state Ψ) that experience the overall trapping potential V (r ) plus their
mean-field are coupled via rf photons to the untrapped states Φ being the eigenfunctions
Ai of the linear gravitational potential. The mean-field of the remaining BEC produces the
minor hump.
For the case of strong output coupling when the Rabi frequency dominates the intrinsic energy of the system (Ω À µ), the resulting behavior of the coupled equations (2.32)
(2.33) exhibits synchronous Rabi cycling between the trapped and the untrapped state. The
Rabi oscillation is faster than the “reaction time” of the untrapped atoms and they are cycled back before they can leave the trap. Therefore, in this mode only pulsed operation
(π/2−pulse)
p is possible as demonstrated in [8, 102]. However, the effective Rabi frequency
Ωeff (r) = Ω2 + ∆2 (r) is spatially dependent due to the inhomogeneous detuning across the
trap.
The case of weak output coupling (Ω ¿ ωz ) is of foremost interest especially for the creation of continuous atom lasers. The valid assumptions of an undepleted trapped state and
a very dilute output state allow perturbative solutions which describe the key aspects of the
output field behavior The output state eigenfunctions usually form a continuum of states
ΦE labeled by their potential energy E to which the output coupling process can couple
the trapped state Ψ. However, the output field is dominated by a spectral filter function
which depends on the duration of the coupling time τ through Heisenberg’s uncertainty
21
2. BASIC THEORETICAL FRAMEWORK
groundstate
Ψ0
E
ħωrf
ΦE
∆E
continuum
FIGURE 2.8.: The temporary output coupling process of duration τ results in a
Fourier limited spectral width of ∆E =
1/τ in the outgoing energy eigenfunctions.
relation ∆E = ħ/τ. Initially the weak output coupler produces an output field that is
a copy of the original trapped state but for
longer times (τ À 1/ω) the spectral filter
narrows and approaches a Dirac δ-function.
Eventually the output state wavefunction
becomes proportional to the energy eigenfunction of the linear gravitational potential. However, despite the spectral narrowing a steady-state laser operation should
only be established after an q
initial switching
effect on a time scale τ = 2R z /g ≈ 2ms,
when the first atoms left the condensate
and the system can be modeled by a Markov
process.
An elegant realization of an output coupler is the radio frequency (rf) or microwave
(mw) output coupling process which allows the production of continuous atom laser beams.
It is the prime choice to coherently extract atoms from a magnetic trap. An alternative
method would be the use of a stimulated two-photon Raman transition being off resonant
from the excited state. Whereas Raman output coupling can provide a momentum kick of
twice the recoil velocity (ħk /m ≈ 6mm/s on the D2-line, see Appendix A), the transferred
momentum of the radio frequency photon is negligible due to the low rf energy and the
large atomic mass. The freed atoms are solely accelerated downwards under the influence
of gravity.
A diagram of the level structure of the hyperfine ground states including the magnetic
sub-levels of 87 Rb is shown in Figure 2.9. The trappable magnetic states (“low field seekers”)
are marked by (N) whereas the magnetic field insensitive (to first order) Zeeman states are
labeled by (Ä). The antitrapped high field seeking states are not specially marked. The
hyperfine splitting of the ground state in 87 Rb is an extremely well know frequency [103]
and given by ∆E hfs = h · 6.834682GHz. The Zeeman splitting of the magnetic sub-levels of
both ground states is given by ∆E Z = g F µB ∆mF B where the Landé g-factor for F = 1 and
F = 2 is very close to −1/2 and +1/2, respectively. In the linear Zeeman regime, which is
valid for the weak magnetic fields on the order of a few Gauss we usually apply, this yields
a splitting of ∼ 700kHz/Gauss. The exact energy dependence of the states on the magnetic
field can be calculated using the Breit - Rabi formula (see Appendix A).
Different output channels respecting the selection rules (∆mF = ±1) for magnetic dipole
transitions are possible from one of the stretched states |F = 1, mF = −1〉 and |F = 2, mF = 2〉
where magnetically trappable Bose - Einstein condensates can be produced in. However, rf
22
2.2. ATOM LASERS
mw
6.8 GHz
F=2
rf
F=1
-2
-1
0
+1
+2
( mF )
FIGURE 2.9.: Zeeman splitting (not to scale) of the two hyperfine ground states of 87 Rb for
a weak magnetic field. Trappable low field seeking states are indicated by (N) whereas untrapped states are marked by (Ä). Two possible output coupling channels for an atom laser
from a Bose - Einstein condensate in the |F = 1, mF = −1〉 state are drawn by the microwave
and radio frequency photon, respectively.
output coupling from the |F = 2, mF = 2〉 state involves additional dynamics from the intermediate level |F = 2, mF = 1〉. Therefore we usually prepare our Bose - Einstein condensate
in the |F = 1, mF = −1〉 state which allows well defined direct output coupling. For instance,
output coupling into the untrapped state |F = 1, mF = 0〉 is possible with a single radio frequency photon ( ∼ 1MHz). On the other hand, this transition also introduces a weak mutual coupling between all states in the F=1 Zeeman manifold due to its symmetric splitting.
Therefore we mostly apply microwave output coupling into the |F = 2, mF = 0〉 state which
constitutes a true two-level system (see Figure 2.9).
The wavefunctions of the outgoing freed state are the energy eigenfunctions of the linear gravitational potential, neglecting the small contribution of the mean-field potential.
Assuming a hard wall boundary far away, e.g., the bottom flange of the vacuum apparatus
or absorbing boundary conditions, the eigenfunctions of a linear potential are found [98] to
be Airy functions Ai(ζ) of the
qdimensionless parameter ζ = (z − E /mg )/l which is scaled by
the natural length unit l = 3 ħ2 /2m 2 g . The Airy function, being the eigenfunction of the
atom laser is shown in Figure 2.10 in comparison to the Thomas - Fermi wavefunction of the
trapped Bose - Einstein condensate in a dressed state picture whiteout the energy of the rf
photon (see Figure 2.7). The local wavelength of the atom laser constantly decreases corresponding to the acceleration in the gravitational field. Its wavelength after a propagation of
3.6 mm is only 5 nm.
23
2. BASIC THEORETICAL FRAMEWORK
|〈Ψ|Φ〉|2
ΨTF(z)
0
0
Φ0(z)
-15
-10
-5
0
5
z [ μm ]
10
-15 -10
-5 0 5
∆z [ μm ]
10 15
15
FIGURE 2.10.: (left) The spatial overlap in a dressed states picture of the trapped condensate
wavefunction with the real part of the Airy function for a centrally output coupled wave.
(right) Variation of the Franck - Condon factor with the detuning of the output coupling
resonance.
The transition probability, according to Fermi’s Golden Rule, depends on the overlap between the bound and free states. The total output coupling rate for a Rabi frequency Ω is
then
Γ ∝ Ω2 |〈Ψ0 (r)|ΦE (r)〉|2 .
(2.35)
The Franck - Condon factor also determines the spectral width for which output coupling
from the Bose - Einstein condensate is possible. The overlap integral is nonzero only for
radio frequency detunings ∆ in the interval of
|∆| <
g p
2mµ ,
ħωz
(2.36)
assuming a Thomas - Fermi profile of the trapped groundstate including gravity as shown in
Figure 2.10 b.
For continuous radio frequency output coupling with a well defined energy E exactly the
wavefunction ΦE having its classical turning point at ζ = 0 is selected out of the continuum
of states by the resonance condition ħωrf = V−1 (z res ) − V0 (z res ). Quantum mechanically, the
overlap of the Airy function ΦE with the groundstate wavefunction Ψ0 and is only significant in a narrow region (≈ l ) around the classical turning point due to its fast oscillatory
behavior. Therefore, in a classical picture, an atom laser can be regarded to emanate from
the resonant region z res that fulfills energy conservation
ħωrf = µm B (z res ) ,
(2.37)
because the different potentials only differ in the contribution from the magnetic trap.
Having this picture in mind it is important to consider all energy contributions. In the
vertical direction one has to include the effect of gravity which shifts the overall potential
24
2.2. ATOM LASERS
for massive particles downwards. For atoms with a magnetic moment µm = g F mF µB the
global potential can be expressed as
V (ρ 0 , y ) = 1/2µm |B (ρ, y )| − mg z ,
(2.38)
gravity
where the symmetry is retained through the coordinate transform z 0 = z − z sag in comparison to equation (2.31). So the minimum of the potential is shifted downwards by z sag = g /ω2z
with respect to the center of the magnetic field, where g denotes the gravitational acceleration. This is illustrated to scale in Figure 2.11. In our case for a very relaxed trap the sag
is around 200 µm, with the consequence that areas of equal magnetic field resemble two
dimensional planes that intersect the condensate at different vertical positions. This fact
permits the output coupling from a very well defined position, i.e., horizontal plane within
the Bose - Einstein condensate when output coupling weakly and continuously with a single
radio frequency.
z
z
0
zsag
x
BEC
y
FIGURE 2.11.: Equilibrium position and size of the Bose - Einstein condensate in the
anisotropic magnetic trap including the gravitational sag (to scale). Therefore the regions
where the resonance condition for atom laser is fulfilled, which are the areas of equal magnetic field strength, intersect the BEC as almost horizontal planes.
2.2.3. Beam Propagation
Along the axial direction the wavefunction of the atom laser is given by an Airy function.
Its propagation direction is governed solely by gravity with a slight influence of magnetic
field gradients due to the weak magnetic susceptibility of the atoms in second order Zeeman
effect (see Appendix A.1).
In the radial directions however, the propagation properties of an atom laser in terms
of divergence, brightness and beam profile are mainly determined by the influence of the
source, i.e., the Bose - Einstein condensate. Transversely, the atom laser wavefunction will
be a copy of the initial ground state wavefunction of the BEC. Being coupled to plane waves it
will expand quantum mechanically as a free wavepacket governed by the momentum distribution, i.e., the Fourier transform of the trapped wavefunction. That means a small source
size corresponds to a faster expansion and larger divergence.
25
n(x)
m
s
2. BASIC THEORETICAL FRAMEWORK
n(y)
n0
T
=
2.
5
n0
phase
0
x [ μm ]
-75
m
s
n0
0
y [ μm ]
n0
75
n(y)
T
=
n(x)
75
86
-75
phase
FIGURE 2.12.: Beam profiles from numerically solving the Gross - Pitaevskii equation of the
atom laser after different propagation times. The upper series shows the atom laser density
(blue) and its phase (green) along the fast (left) and slow (right) axis in comparison to the
Thomas - Fermi profile in the trap (gray). The lower series shows the atom laser when it
enters the cavity after a propagation time of 86 ms. Its divergence is due to the initial momentum spread (see Figure 2.3) and the mean-field repulsion of the remaining condensate.
26
2.2. ATOM LASERS
Apart form this Heisenberg limited expansion the source exerts a repulsive interaction
on the freed atoms on their way through the BEC. The mean-field potential on top of the
linear gravitational potential acts as a negative lens for the atom laser beam and increases
its divergence with increasing curvature and density of the BEC.
The radial expansion of the atom laser wavefunction typical for our experiment over
86 ms is shown in Figure 2.12. It demonstrates the increased spreading for smaller source
sizes and the appearance of quantum interferences at its wings due to the side lobes of the
ground state Fourier transform as shown in Figure 2.3. These higher components in the momentum distribution are present even in the exact solution and due to the non-Gaussian
shape of the condensate wavefunction.
2.2.4. Coherence Properties
The coherence of an atom laser strongly depends on the coherence quality of the source
and the coherence preservation of the output coupling process. Coherence is a somewhat
complex notion in physics and will be discussed in more detail in Section 5.2. Basically it
characterizes the ability to interfere (first order) and the noise (higher order) of quantum
systems. In general, there exists both spatial and temporal coherence. However they are
usually treated separately and independently.
In many experiments [5, 104] the spatial coherence of a Bose - Einstein condensate has
been proven even to higher order [59] and it can therefore truly be regarded as a single
wave with a well defined phase. Output coupling continuous atom lasers from different
regions within a BEC and demonstrating their ability to interfere [105] also proved the first
order spatial coherence of atom lasers as did the observation of quantum interferences in
the atom laser beam profile [106].
The temporal evolution of the condensate wavefunction is given by its internal energy
−i µt /ħ
e
with perturbations stemming from quantum phase diffusion. The experimental investigation of temporal coherence is more difficult than its spatial counterpart because it
requires the use of a time delay or a reference phase. Only the first order temporal coherence of atom lasers has been determined to be limited by the duration of output coupling
[42] up to 1.5 ms. In the scope of the work presented in this thesis it was possible to proof
the full temporal coherence of atom lasers up to several seconds by measuring higher order
temporal correlations.
27
2. BASIC THEORETICAL FRAMEWORK
2.3. Cavity Quantum Electrodynamics
The elementary system in cavity quantum electrodynamics (QED) is a two-level emitter
coupled to a single mode of an electromagnetic resonator [107]. It constitutes the most
fundamental instance of light-matter interaction and is a topic of active research in divers
fields [2]. Quantum mechanical two-level systems employed range from neutral atoms, ions,
atomic Rydberg states and molecules to artificial structures like quantum dots and cooper
pair boxes. They can be coupled to confined optical and microwave photons in different
types of cavities like Fabry - Pérot resonators, nano-fabricated photonic crystal defects,
whispering gallery modes and microwave circuit structures.
In the early days of cavity QED it was shown that the quantized electromagnetic field and
the correspondingly modified density of states drastically influences the radiative decay
time of the excited state of a two-level system. The emission can be enhanced by a number (∼ Q · λ3 /V ) known as the Purcell factor. It is proportional to the quality factor Q of a
resonator times the ratio of cubed wavelength λ to the mode volume V .
However, the ultimate goal of all these attempts is to increase the coherent atom cavity
coupling to exceed the dissipation losses of spontaneous emission and cavity damping. In
this so called “strong coupling limit” of cavity QED the coherent exchange of energy between
the two-level system and the cavity mode is reversible. It provides the unique possibility to
deterministically control the quantum evolution of the system.
The following two paragraphs establish the discussion of cavity QED in the strong coupling regime with a single (two-level) atom coupled to an ultrahigh Q optical Fabry - Pérot
cavity, which is the system experimentally realized in this work.
2.3.1. Resonator Basics
A Fabry - Pérot type cavity is realized by placing two highly reflecting mirrors at a distance
L . The electromagnetic field along this axis is quantized by the condition nλ/2 = l for the
wavelength λ, where n = {1, 2, 3 . . .} labels the number of nodes of the standing wave. For
instance, in our cavity we have 456 nodes of the electromagnetic standing wave. By scanning either the length l of the cavity or the frequency ν of the probe laser the resonance
condition can be satisfied and the transmission be maximal. The frequency distance of consecutive cavity resonances is given by the free spectral range νFSR , being the inverse of the
round trip time
νFSR = c /2l .
(2.39)
The full width at half maximum (FWHM) ∆ν of each resonance depends on νFSR and the
finesse F of the cavity through
∆ν = νFSR /F ,
28
(2.40)
2.3. CAVITY QUANTUM ELECTRODYNAMICS
free spectral range
T
laser
∆ν
detector
n · λ/2
(n+1) · λ/2
FIGURE 2.13.: (left) Schematic drawing of the near planar Fabry - Pérot cavity and the electromagnetic standing wave field inside. (right) Spectrum of the resonator exhibiting narrow
Lorentzian cavity lines separated by the free spectral range.
where the finesse is a measure that is solely determined by the quality of the cavity mirrors.
Since we employ symmetric mirrors of ultrahigh reflectivity R with a transmission coefficient T and a loss factor L on the order of a few ppm satisfying 1 = R + T + L , the finesse can
be expressed as
p
F '
π R
π
≈
.
1−R
T +L
(2.41)
It will be dominated by the main leakage of photons out of the cavity, either through transmission or loss channels like scattering, absorption and diffraction. We employ a cavity of
finesse F ≈ 350.000 being limited by scattering losses of about L ≈ 7 · 10−6 compared to a
transmission coefficient of T ≈ 2 · 10−6 . The finesse furthermore determines the number of
reflections (F /π) and the 1/e -lifetime of a photon inside the cavity
τc =
F l
· = (2π∆ν)−1 .
π c
(2.42)
Experimentally, measuring the cavity ring down time τc or the cavity linewidth ∆ν are
trusted strategies to determine the finesse of a cavity. However, the length of the cavity
has to be determined independently, for example by the mode spacing of higher transverse
modes or by simultaneously transmitting two different known wavelengths.
Classically, the complete power transmission spectrum through the cavity shown in Figure 2.13 is given by frequency dependent Airy transmission function
I (ν) =
Tmax
1 + (2F /π)2 sin2 (πν/νFSR )
,
(2.43)
where the maximum normalized power transmission on resonance for lossy cavities can be
calculated considering the complex round trip gain [108]. For a high finesse it approximates
a Lorentzian line shape on resonance. Mirror losses reduce the maximum on-resonance
transmission to
T 2 (1 − L )
Tmax =
(2.44)
[1 − R (1 − L )]2
29
2. BASIC THEORETICAL FRAMEWORK
and therefore also increase the minimal reflection on resonance to a finite value. Only
matched mirrors satisfying the input impedance condition R in = R out (1 − L )2 would allow
perfect coupling of the laser intensity into and through the cavity.
Nevertheless, inside the resonator the intensity is magnified by the factor T /[1−R (1−L )]2 .
Comparing the circulating energy over the dissipated energy per cycle yields the so-called
quality factor Q . As the name already implies the Q quantifies the losses of the system
and can more conveniently be expression as the ratio of the resonance frequency ν0 to the
linewidth
Q = ν0 /∆ν .
(2.45)
Since we work at optical frequencies the quality factor can be very large. For a cavity
linewidth ∆ν ≈ 2 MHz we get to a Q on the order of 108 .
Photon confinement of the cavity mode in the radial direction is provided by the focussing
ability of curved mirrors surfaces with radii r i resulting in a stable resonator geometry
for 0 ≤ (1 − l /r 1 )(1 − l /r 2 ) ≤ 1. We employ a near planar cavity with symmetric mirrors
so the
cavityp
mode is well approximated to be cylindrical with a Gaussian waist given by
p
4
w = l λ/π · r /2l . In our case the radius of curvature r ≈ 77.5 mm of the mirrors is much
larger than the length of the cavity l ≈ 178 µm and the waist results to w ≈ 25.5 µm. For near
planar cavities different transverse modes (TEM) are not degenerate and their spacing is a
practical tool to determine the exact length of a cavity [109].
νl ,m,n = (q + 1)
c
1
+ (m + n + 1) arccos(1 − l /r ) .
2l
π
(2.46)
These modes can be identified by their transverse mode pattern as shown in Figure 2.15
when observing the transmitted light through the cavity with a camera.
(0,0)
(1,0)
(0,1)
(2,0)
(0,2)
(1,2)
(3,0)
(0,3)
FIGURE 2.14.: Series of pictures taken of different transverse modes excited in our cavity.
2.3.2. Atom Field Interaction
With ultrahigh finesse cavities it is possible to advance into the strong coupling regime of
cavity QED where the interaction between single quanta of both light and matter significantly alter the behavior of the system. We therefore require the full quantum mechanical
30
2.3. CAVITY QUANTUM ELECTRODYNAMICS
description of an atom inside the cavity mode while neglecting its motional degrees of freedom. This was developed by Jaynes and Cummings [66] and constitutes an important quantum mechanical system because it has physical significance and is analytically solvable.
The Jaynes - Cummings model consists of a
single two-level atom coupled to a quantized
ΩR
|e〉
|1〉
single mode field represented as a harmonic
|g〉
|0〉
oscillator. The coupling between atom and
field is characterized by a Rabi frequency ΩR .
atom
field
Loss of excitation in the oscillator appears as a
gain of excitation in the atom and vice versa.
¯
Considering a general two-level atom having a ground state ¯g 〉 and an excited state |e〉
separated by the energy ħωA = E e −E g . It is convenient to write the Hamiltonian of the bare
atom in its symmetric form as
1
(2.47)
Ĥ A = ħωA σz .
2
The Pauli spin matrix σz provides a positive one if it operates on an atom in the upper state
¯
|e〉 and a negative one if it operates on an atom in the lower state ¯g 〉.
The quantized field of a single mode is traditionally expressed in terms of the photon
creation (â † ) and annihilation (â ) operators. Their product gives the number operator and
the bare field Hamiltonian can be expressed as
ĤF = ħωC (â † â + 1/2) ,
(2.48)
in analogy to a harmonic oscillator with a set of eigenenergies E n = ħωC (n + 1/2) separated
by the photon energy ħωC . The operator for the electric field of the standing wave inside
the cavity is then found to be
Ê (r) = ²F E 0 (â + â † )χ(r) ,
(2.49)
where E 0 is the maximum electric field established by a single photon in the cavity mode, ²F
denotes the field polarization and χ(r) represents the normalized spatial mode function
χ(r) = cos(kx ) e −( y
2 +z 2 )/w 2
0
,
(2.50)
obeying the quantization condition that kl /π is an integer. Here, the x -direction is the
symmetry axis of the cavity and z denotes the vertical. By comparing the electromagnetic
energy density W = 12 (ε0 E 2 + B 2 /µ0 ) = ε0 E 2 , using E = B /c and c = (ε0 µ0 )−1/2 , with the
R
energy ħωC of a single photon contained to the cavity mode of volume VC = |χ(r)|2 d r =
w 02 l π/4, its maximum field can be simplified to
s
E0 =
ħωC
.
2ε0VC
(2.51)
31
2. BASIC THEORETICAL FRAMEWORK
Clearly, the field strength of a single photon depends on its spatial extent. It can be increased
by making the mode volume smaller which is not possible in free space but requires an
optical cavity.
Quite generally the first order atom field interaction Hamiltonian is given by the coupling of the atomic dipole to the electric field of a photon: Ĥ I = −dˆ · Ê . Introducing the
dipole moment d = 〈e|dˆ|g 〉 the dipole operator can be written in terms of the raising (σ̂+ )
¯
¯
and lowering (σ̂− ) operators, given by |e〉〈g ¯ and ¯g 〉〈e|, respectively and more conveniently
expressed through the Pauli spin matrices σ̂± = 12 (σ̂x ± σ̂ y ). Thus the slightly better looking
interaction Hamiltonian is
Ĥ I = ħg 0 (r)(σ̂+ + σ̂− )(â + â † ) ,
(2.52)
where we have introduced an important quantity playing the role of a Rabi frequency, the
atom field coupling strength g 0 which is given by
g 0 (r) = ²A ·²F
d E0
χ(r) ,
ħ
(2.53)
where the ²’s denote the polarizations of the atom and field, respectively and χ(r) characterizes the field distribution of the cavity mode. It is common to drop the polarization and
space dependence and label with g 0 the maximum atom field coupling strength for a given
mode volume VC .
The complete Jaynes - Cummings Hamiltonian is now given by the sum of the bare atomic
and field Hamiltonians plus the interaction term. Commonly, the zero point energy is just
an additive constant and disregarded. By applying the rotating wave approximation the
energy nonconserving terms σ̂+ â † and σ̂− â can be dropped and the Jaynes - Cummings
Hamiltonian consists of two cross terms describing absorption and emission of a photon by
the two-level system
HJC =
1
ħω A σ̂z + ħωC (â † â ) + ħg 0 (r)[σ̂+ â + â † σ̂− ] .
2
(2.54)
The Jaynes - Cummings model is well known in quantum mechanics because it is one of the
few systems for which analytical solutions can be derived.
¯
A canonical basis for the unperturbed Hamiltonian are the bare states |e, n〉 and ¯g , n + 1〉
labeling the state of the atom and indexing the number of photons in the closed cavity.
However, to describe the coupled system a coherent mixture of the bare states |n, ±, r 〉 =
¯
c i |e, n〉 + c j ¯g , n + 1〉 (called dressed states) is better suited. Since the coupling strength
g 0 (r) is in general space dependent, the position is included in the state. In this basis the
eigenenergies of the Jaynes - Cummings Hamiltonian result to
1
2
E ± (n ) = ħωC (n + 1/2) ± ħ ΩR (n, ∆, r) ,
32
(2.55)
2.3. CAVITY QUANTUM ELECTRODYNAMICS
+
=
z
|2,+〉
|3〉
|g,3〉, |e,2〉
|2,-〉
|1,+〉
|g,2〉, |e,1〉
|2〉
|1,-〉
|1〉
|e〉
ωA
ωc
|0〉
|g〉
ħωAσz
|0,+〉
|0,-〉
|g,1〉, |e,0〉
+
|g,0〉
ħωC(a†a)
=
bare
states
dressed
FIGURE 2.15.: Coupling a two-state atom with resonant photons contained in a cavity results
p
in the Jaynes - Cummings ladder of energy doublets split by the frequency 2ħ n + 1g 0 (r),
where the coupling strength g 0 is taken to be spatially dependent as χ(r ).
where the Vacuum - Rabi frequency Ω2R = Ω2 + ∆2 contains the resonant n-photon interp
action frequency Ω(n, r) = 2 n + 1 g 0 (r) and the detuning ∆ = ωC − ω A . The atom photon
interaction results in a splitting of the dressed states by E + − E − = ħΩR being dependent on
the spatially varying coupling strength and the detuning. For maximum coupling on resop
nance the spacings of the Jaynes - Cummings doublets are given by 2 n + 1 g 0 . The choice of
the zero level in the dressed states |0, ±〉 results from the historic wish to let an excited atom
interact with the vacuum field of an otherwise empty resonator (n = 0), known as Vacuum Rabi oscillations. The coherent exchange of energy between the atom and the cavity mode
leads to an oscillatory behavior of finding the atom in the ground or excited state with a
frequency ΩR /2.
Plotting the eigenenergies versus the detuning ∆ for maximum coupling exhibits the
33
2. BASIC THEORETICAL FRAMEWORK
|g,2〉
E
|e,1〉
|1,+〉
|1,-〉
|e,1〉
|g,2〉
|g,1〉
|0,+〉
|e,0〉
|e,0〉
|0,-〉
|g,1〉
|g,0〉
|g,0〉
0
Δ
FIGURE 2.16.: Avoided level crossing in the Jaynes - Cummings model for different laser detuning. On resonance the splitting is maximal and the eigenstates have rotated from the
bare states into the dressed states.
avoided crossing type behavior. The eigenstates rotate from the unperturbed bare states
to the dressed states and back. The mixing angle is given by tan θn = Ω/(ΩR − ∆). On resonance θn = π/4 and the dressed states result in
¯
1
|n, ±, r 〉 = p [|e, n〉 ± ¯g , n + 1〉] .
2
(2.56)
The individual properties of atom and field are completely amalgamated and define a new
system that is commonly know as the dressed atom or Jaynes - Cummings molecule.
2.3.3. Single Atom Detection
The Jaynes - Cummings model considered so far is quite academical in a sense that it applies
only to a perfectly closed system which could not be observed. However, all real systems
are lossy and interact with their environment which leads to decoherence. Furthermore, in
order to perform a quantum mechanical measurement we require a coupling to an external
system. The loss mechanisms present in the coupled atom cavity system are the field decay
rate κ = ∆ν/2 of the cavity field due to nonideal mirrors and the dipole decay rate γ = Γ/2 of
the excited atomic state leading to spontaneously scattered photons out of the cavity mode.
A coherent pump rate, in form of a laser field supplies the cavity mode with “fresh” photons
to compensate the losses.
34
2.3. CAVITY QUANTUM ELECTRODYNAMICS
For a complete description of an open quantum system the influence of the environment
has to be taken into account in form of a reservoir. If the interaction with the reservoir
is weak it can be treated perturbatively, know as the Born approximation. Under the assumption that the excitations and correlations in the reservoir die out quickly, the system
is independent of its past evolution and can be regarded as having no memory in the socalled Markov approximation. The temporal evolution of the open quantum system can be
determined by the quantum master equation for its density matrix ρ
ρ̇ =
i
i h
ρ, Ĥ + L relax ,
ħ
(2.57)
where the operator Lrelax is of the Lindblad form and contains all modes of dissipation
[110]. Here, Ĥ represents the total Hamiltonian of the system including a pump term that
replenishes dissipated photons in the cavity mode
Ĥ = ħ∆ A σ̂+ σ̂− + ħ∆C â † â + ²(â + â † ) + ħg 0 (â σ̂+ + â † σ̂− ) .
(2.58)
The pump term is proportional to the incident field and to the mirror transmittivity with
p
² = κ n . Furthermore, the two detunings between the laser and the atomic transition frequency and between the laser and the cavity resonance have been introduced through
∆ A = ωL − ω A
and
(2.59)
∆C = ωL − ωC .
The Lindblad operator Lrelax contains two dissipation channels: an atom spontaneously
emits a photon out of the cavity mode, which this is described by the dissipation operator
p
p
γ σ̂− , or a photon is lost through the cavity mirrors, having a dissipation operator κ â .
The relaxation operator thus reads
L relax = γ(2σ̂− ρ σ̂+ − σ̂+ σ̂− ρ − ρ σ̂+ σ̂− ) + κ(2âρ â † − â † âρ − ρ â † â ) .
(2.60)
In general, the stationary solutions of the master equation have to be calculated numerically, for instance with quantum Monte - Carlo methods [111]. However, for weak atomic excitation analytical solutions of two interesting observables have been derived [112, 113, 114].
The expectation value of the intracavity photon number and the mean atomic excitation are
given by
〈â † â〉 = ²2
∆2A + γ2
|Λ|2
and
〈σ̂+ σ̂〉 = ²2
g 02
|Λ|2
,
(2.61)
respectively with
Λ = γκ + g 02 − ∆ A ∆C − i (γ∆C + κ∆ A ) .
(2.62)
The single atom detection principle relies on the significant change of transmissions of
the resonant probe laser by the presence of a single atom in the resonator. Therefore, the
35
2. BASIC THEORETICAL FRAMEWORK
T
g0
η
κ
z
detector
y
γ
x
-g0
0
g0
Δ
FIGURE 2.17.: (left) An open cavity containing a single atom resulting in a coupling strength
g 0 . The dissipation channels are given by the spontaneous atomic scattering rate Γ = 2γ and
the loss rate of photons through the cavity mirrors (2κ). (right) Splitting of the transmission
p
spectrum by 2 N g 0 for a cavity containing one and two atoms, respectively. The significant
reduction of on-resonance transmission is shown compared to the empty cavity resonance.
normalized cavity transmission is an essential observable and for the resonance condition
∆ A = ∆C it can be written as
"
4κ(γ − i ∆)
T =
(2∆ + i (γ + κ))2 − 4g 02 N + (γ − κ)2
#2
(2.63)
,
p
where N is the number of atoms in the resonator [19]. The spectrum is split by 2 N g 0 and
the on resonance transmission is diminished by the factor (1 + g 02 N /κγ)−2 which becomes
significant even for single atoms when g 02 > κγ.
However, single atom detection could in principle be attained with bad cavities provided
the integration time is long enough. Ultimately, the strong coupling in cavity QED is the
regime when an atom exchanges a photon coherently many times with the cavity mode
before it is lost due to decoherence through one of the dissipation channels. We can characterize the atom cavity system by the dimensionless parameters of the critical atom number
N0 and the critical photon number n 0 which are given by
N0 =
2γκ
g 02
and
n0 =
γ2
2g 02
,
(2.64)
respectively. These are the number of quanta required to significantly alter the atom cavity
response. In the strong coupling regime g 0 À {γ, κ} the critical numbers are both less than
one, which serves as the definition of strong coupling {n 0 , N0 } ¿ 1.
36
3 Experimental Apparatus
“Der Worte sind genug gewechselt,
Lasst mich auch endlich Taten sehn.” - Faust, Johann Wolfgang von Goethe
Our apparatus overcomes the experimental challenges of integrating an ultrahigh finesse
optical cavity into a Bose - Einstein condensation machine with a conceptually novel design
as illustrated in Figure 3.1. It provides spacious access to the condensate for divers samples
and probes which are modularly integrable on our science platform. This is rendered feasible by means of a nested vacuum chamber design, a high vacuum (HV) enclosure inside
the ultrahigh vacuum (UHV) main chamber and a short in vacuo magnetic transport. Two
distinct pressure regions are required since the two common stages towards Bose - Einstein
condensation, a magneto-optical trap (MOT) for laser cooling and trapping a large number
of atoms and evaporative cooling have conflicting requirements on their vacuum environment.
We utilize a short magnetic transport [115, 116] to convey the cloud of cold 87 Rb atoms
from the MOT to the main chamber, where we perform evaporative cooling to quantum
degeneracy. From the Bose - Einstein condensate we output couple a continuous atom laser
and direct it into the cavity mode. The ultrahigh finesse optical cavity is integrated on
the so-called “science platform” and rests on top of an UHV compatible vibration isolation
system which is vital for its stable operation. The cavity is located 36 mm below the BEC
and enables us to detect single atoms from a quantum degenerate source. In the following,
the modular experimental building blocks of our hybrid BEC and cavity QED apparatus are
presented in more detail.
This chapter has been published at large in [39]: A. Öttl, S. Ritter, M. Köhl, and T. Esslinger,
Hybrid apparatus for Bose-Einstein condensation and cavity quantum electrodynamics: Single atom
detection in quantum degenerate gases, Rev. Sci. Instr. 77 (2006), 063118.
37
3. EXPERIMENTAL APPARATUS
BEC production rig
magnetic
pump &
dispensers
transport
MOT
BEC
10-9 mbar
transport
atom
laser
magnetic
cavity
10-11 mbar
to pumps
vibration
isolation
system
science platform
FIGURE 3.1.: Schematic sketch of the experimental setup illustrating the nested vacuum
chambers, the short magnetic transport, and the “science platform” bearing the ultrahigh
finesse optical cavity on top of the vibration isolation system. The atomic cloud captured
in the magneto-optical trap (MOT) is transferred through a differential pumping tube into
the ultrahigh vacuum region and evaporatively cooled towards quantum degeneracy. We
output couple a continuous atom laser from the BEC and direct it to the cavity mode where
single atoms are detected.
38
3.1. VACUUM SYSTEM
3.1. Vacuum System
The vacuum system presented here is out of the ordinary. It consists of two nested steel
chambers, where the higher pressure (high vacuum - HV) MOT chamber is situated inside
the lower pressure (ultrahigh vacuum - UHV) main tank. The HV region houses the alkali
dispenser source. Both vacuum regions are pumped separately and a differential pumping
tube maintains a pressure ratio of 102 . The setup grants multiple optical access for laser
cooling as well as for observation and manipulation of the resulting Bose - Einstein condensate.
3.1.1. Main Chamber
The objective of the vacuum system shown in Figure 3.2 is to attain an UHV environment at
10−11 mbar for efficient evaporative cooling of the laser precooled atomic cloud. Centerpiece
of our vacuum system is the custom-welded, cylindrical main tank of nonmagnetic steel
(type 1.4436). It has a diameter of 20 cm and features multiple access (see Figure 3.3) in
form of optical grade viewports and electrical feedthroughs with standard CF sealing. The
viewports are antireflection coated on both sides.
Two custom-made CF 200 cluster flanges cap the main chamber from above and below.
The top flange (called “BEC production rig”) features optical and electrical access (see Figure 3.3) since most of the electromagnetic coil configuration is mounted on this flange and
placed inside the UHV. In addition a liquid nitrogen compatible feedthrough is supplied for
cooling the magnet coils and resistive temperature sensors (PT 100) are used to monitor
their temperature. The bottom flange (called science platform) serves as an exchangeable
mount for the inclusion of samples and probes into our system. Besides viewports and electrical feedthroughs (see Figure 3.3) to connect to electromagnetic coils, PT100 sensors, and
the piezo element of the optical cavity design, it includes a cold finger and a 300 l/s nonevaporable getter (NEG) vacuum pump. The core vacuum pumping is performed by a titanium
sublimation pump and a 150 l/s ion getter pump. A right angle valve is included in this
pumping section for rough pumping the system.
The HV part of the system (see Figure 3.2) connects to the MOT chamber which protrudes
into the UHV main chamber and serves as a repository for rubidium atoms. It can be shut off
with a gate valve between the MOT chamber and the rubidium dispenser source [117, 118].
The HV region is pumped by an ion getter pump (75 l/s) whose pumping speed can be derated by a rotatable disk inside the tube reducing its conductance. This serves to control
the rubidium vapor pressure which is monitored with a wide range pressure gauge. Also a
right angle valve is included for rough pumping purposes. Our rubidium repository consists
of seven alkali metal dispensers fixed in star shape to the tips of an eight pin molybdenum
electrical feedthrough where the center pin serves as the common ground. Beforehand, the
conductors were bent by 90 ◦ so that the dispensers aim towards the MOT chamber. Dis-
39
3. EXPERIMENTAL APPARATUS
HV
valve
gauge
valve
r
dispense
source
uc
BEC prod
tion rig
k
main tan
latform
science p
ation
Ti-sublim
valve
UHV
p
ion pum
150 l/s
FIGURE 3.2.: Overview of the complete vacuum system showing the pumping sections for the two nested vacuum regions, high vacuum
(HV) and ultrahigh vacuum (UHV), respectively. The overall length is close to 2 m. The main tank offers multiple optical and electrical
access and is sealed off by two CF 200 cluster flanges called “BEC production rig” and “science platform”.
40
ion pump 75 l/s
3.1. VACUUM SYSTEM
penser operation may be viewed through a viewport mounted from above. The dispensers
can easily be exchanged without breaking the ultrahigh vacuum in the main chamber by
closing the gate valve between the MOT chamber and the dispenser source.
3.1.2. MOT Chamber
The MOT chamber as part of the high vacuum region is situated inside the ultrahigh vacuum
main tank as shown in Figure 3.3. However the fact that both pressure regions are well in the
molecular flow regime allows for relatively simple sealing techniques. The purpose of the
inner chamber is to contain a higher vapor pressure of 87 Rb atoms for an efficient loading
of the MOT.
Our MOT chamber was milled out of a single block of nonmagnetic steel (type 1.4436). This
material was chosen to reduce eddy currents produced by fast switching of the magnetic
fields. Bores of 35 mm diameter give optical access for the six pairwise counterpropagating laser beams forming the magneto-optical trap. These bores are sealed off by standard
optical grade laser windows (BK 7) with double-sided antireflection coating and clamped
to the MOT chamber by stainless steel brackets. At the metal glass interface we use thin
(0.2 mm) Teflon rings to protect the windows. Additionally we took precautions in the form
of ceramic screens to prevent coating of the windows by the titanium sublimation pump.
An additional bore provides the connection of the MOT chamber to the HV pumping section and the dispenser source. This connection is sealed against the UHV main tank with
a tight fit stainless steel bushing inside the CF 40 socket. The bushing is tightened to the
MOT chamber thereby pressing its circular knife edge into a custom-made annealed copper
gasket. A screen to prevent a direct line of sight from the hot dispensers to the center of the
MOT is included in the laser cut gasket.
The MOT chamber is sandwich mounted between the two magnet coil brackets for the
magnetic transport (see Section 3.2) and simultaneously functions as a spacer for the magnet coil assembly. The whole structure is suspended from the top flange by four M8 thread
bars and represents our BEC production rig. A differential pumping tube interfaces the
MOT chamber with the main tank. It serves for conveying the cloud of cold atoms with
the magnetic transport from the MOT into the UHV main chamber. The aluminum differential pumping tube is mounted with a press fit in the MOT chamber and can be exchanged.
It has an inner diameter of 6 mm over a length of 45 mm and can maintain a differential
pressure of 102 -103 depending on the actual pumping speed in the UHV main chamber. Its
conductance for rubidium at room temperature is about 0.3 l/s.
3.1.3. Installation
All components of the system were electropolished (the custom-welded parts were pickled
afore), cleaned, and air baked at 200 °C before assembly [119]. Additionally, all critical in
41
M8
thread
bar
MOT axis
feedthroughs (liquid)
(electrical)
BEC + Cavity
axis
3. EXPERIMENTAL APPARATUS
BEC production
rig
main tank
(UHV)
differential
pumping
tube
MOT
chamber
(HV)
BEC
axis
●
♦
to pump s
ser
& dispen
gasket,
screen
& bushing
NEG pump
MOT
axis
cavity
axis
windows
(teflon-sealed)
science platform
FIGURE 3.3.: Section through the UHV system illustrating the realization of the nested chambers design and revealing the details and objectives of the divers optical axes. The position
of the BEC and cavity are marked by (•) and (¨), respectively. The high vacuum MOT chamber is suspended from the BEC production rig and sealed by a tight fit bushing against the
UHV main tank. The science platform provides space for additional components such as the
ultrahigh finesse optical cavity. (Note: For clarity in the illustration the magnet coil configuration (see Figure 3.4) and the optical cavity assembly (see Figure 3.8) are omitted in this
figure.)
42
3.2. MAGNETIC FIELD CONFIGURATION
vacuo materials such as Stycast 2850 FT and Kapton used for the magnet coil brackets (see
Section 3.2), Viton A and Wolfmet utilized for the vibration isolation stack (see Section 3.3)
as well as plastic [Teflon, Vespel] and ceramic [Macor, Shapal] parts were externally outgassed by vacuum baking them at 200 °C.
The bakeout [120] of the fully assembled system was performed at 120 °C which is the
maximum temperature rating of the piezotube used in our optical cavity assembly. The ultimate attainable pressure in the UHV system is 3 · 10−11 mbar. It is measured directly inside
the main chamber in close proximity of the magnetic trap for Bose - Einstein condensation.
In the HV part we maintain a pressure in the range of 10−9 mbar.
3.2. Magnetic Field Configuration
A magnetic transport [115, 116] is a reliable and controlled way to transfer the cold atomic
cloud from the MOT to a region of considerably lower background pressure for evaporative
cooling. Only an in vacuo magnet coil arrangement in conjunction with nested vacuum domains allows for a short transport design and grants spacious access volume inside the main
chamber. However, care must be taken to meet the UHV requirements with the materials
chosen for the magnet coil structure. Besides spatial and optical accessibilities the requirement on the magnetic trap is mainly magnetic field stability to enable stable atom laser
output coupling. Therefore we employ a magnetic trap in the quadrupole Ioffe configuration [85] (QUIC) because its simplicity allows for a compact design and ensures an easy and
stable operation at very low power consumption (∼ 2 W). A magnetic shielding enclosure
and additional in vacuo coils for manipulating atoms in connection with the cavity round off
the magnetic configuration of the system.
3.2.1. Magnetic Transport
The magnetic transport design consists of two partially overlapping electromagnetic coil
pairs (called “MOT coils” and “transfer coils”) producing quadrupole potentials and the final
QUIC trap coils [Figure 3.5(a)]. The overall potential minimum can be moved over a distance
of 82 mm so that the cold atoms in a low field seeking Zeeman state are conveyed from
the position of the MOT directly into the final magnetic QUIC trap. The transfer coil pair
provides sufficient overlap between the two to achieve a smooth transfer of the magnetic
potential without significant heating of the cold atomic cloud.
The magnet coils were wound from rectangular copper wire (3 × 1 and 1 × 1 mm2 ) for optimal filling fraction. We choose Kapton film isolated wire which is temperature durable and
suitable in the ultrahigh vacuum environment. The coils were integrated in two mirrorinverted, custom-made copper brackets and encapsulated with Stycast 2850 FT, a thermally
conductive epoxy. The brackets are slotted in order to suppress eddy currents from switch-
43
3. EXPERIMENTAL APPARATUS
towards liquid nitrogen feedthrough
MOT coil
vibration isolation
system (VIS)
gradient coil
cavity coils (5)
offset coil
extra windings
Ioffe coil
Ioffe frame
sapphire spacers
quadrupole coil
gradient coil
BEC
cavity
M8 thread bar
differential
pumping tube
transfer coil
window
(teflon sealed)
transport bracket
MOT chamber
tight fit bushing
gasket & screen
ceramic spacer
cooling circuit
FIGURE 3.4.: Section through the complete assembly inside the main vacuum chamber. It illustrates the arrangement of magnet coils,
the inner chamber, and the cavity with respect to each other. Functional units of the magnet coil configuration are the two transport
brackets that sandwich the inner chamber and the laterally mounted Ioffe frame (elements between dashed lines). These parts,
including the top gradient coil, are fixed to each other and mounted from the top flange. The optical cavity on top of the vibration
isolation system, the surrounding coils, and the bottom gradient coil are mounted on the science platform.
44
3.2. MAGNETIC FIELD CONFIGURATION
ing the magnetic field. The magnet coil assembly was fixed in a sandwich structure around
the MOT chamber and suspended from the top flange by M8 thread bars (see Figure 3.4). The
complete assembly including the QUIC trap represents the BEC production rig.
A cooling system to remove the heat dissipated by the electromagnetic coils is supplied
in form of a copper pipe with 4 mm inner diameter. It is soldered in a loop around each
coil bracket and connected to the liquid nitrogen feedthrough. A temperature stabilized
recirculating chiller permanently pumps pure ethanol cooled to −90°C through the system.
Thereby we maintain a maximum operating temperature below 0 °C. This in turn lowers the
power consumption. The surface temperature of the coils is monitored with PT 100 sensors
and interlocked to the power supplies.
The geometry and arrangement of the magnetic transport coils [Figure 3.5(a)] are dominated mainly by constraints set by the size of the MOT chamber, the required length of
the differential pumping tube, and the optical access to the MOT, BEC, and cavity. For instance, the square shape of the MOT coils best achieves a large overlap with the transfer
coils while granting optical access to the cavity axis. However, the aspect ratio A /R of the
coil separation (2 A ) to the coil radius (R ) could be tuned to a balanced tradeoff between a
maximally strong ( A /R = 0.5) and a maximally long ( A /R = 0.87) linear gradient region [86].
Anti-Helmholtz configuration is advantageous for tight confinement and deep trap depths
whereas long linear gradients yield large handover regions between two coil pairs. Furthermore, the power consumption of a coil pair for a given field gradient can be minimized by
choosing a well matched ratio of axial to radial windings.
In order to find an optimum current sequence for the magnetic transport [115] we calculate the magnetic field of the coil configuration analytically and discretize it along the
transport axis on a 100 µm grid. The currents needed to transfer the magnetic minimum
smoothly from the MOT to the QUIC are then computed numerically in accordance with
several constraints. Limited by a maximum available electrical current we optimized the
magnetic field gradients and trap depths especially during the handover. Furthermore we
tried to minimize deformations of the trapping potential. The resulting spatial sequence of
currents per coil is converted into a temporal sequence including an acceleration and deceleration phase by taking into account the limited bandwidth of the current control servo
[Figure 3.5(b)].
The magnetic transport sequence initiates with a fast (400 µs) ramp to 20 A in the MOT
coils after magneto-optical trapping and optical pumping the cold atoms into a lowfield
seeking state. The ramp needs to be fast with respect to the expansion of the cloud but
adiabatic on the spin degree of freedom. It is followed by a slow (100 ms) compression of
the atomic cloud to the maximum field gradients. Increasing the current in the transfer
coils pulls the atoms towards their center and by decreasing the MOT coil current the zero
of the potential is handed over [Figure 3.5(b)]. The field of the QUIC is aiding at this point
to maintain a constant aspect ratio. The magnetic transport finishes by ramping down the
current through the transfer coils in favor of the QUIC coils. The atomic cloud is conveyed
45
3. EXPERIMENTAL APPARATUS
(a)
M
OT
co
ils
Transfer coils
ls
IC coi
QU
current [ A ]
(b)
180
160
140
120
100
80
60
40
20
0
MOT
Transfer
QUIC
0
0.2
0.4
0.6
time [ s ]
0.8
1
FIGURE 3.5.: (a) Top view of the arrangement of coils for the magnetic transport. The line
denotes the trajectory of the atomic cloud from the MOT (filled circle) into the QUIC trap.
(b) Temporal sequence of currents through the different coils to realize the compression of
the cold atomic cloud (negative times) and the magnetic transport.
through the differential pumping tube directly into the magnetic QUIC trap which stays on
for the subsequent evaporative cooling stage.
The trajectory of the magnetic transport [Figure 3.5(a)] is slightly bent such that atoms in
the final magnetic trap position have no direct line of sight into the higher pressure MOT
chamber. This suppresses background gas collisions which would shorten the lifetime of the
Bose - Einstein condensate. The bend is achieved by laterally offsetting the center of the
QUIC trap by 3 mm from the differential pumping tube.
The MOT and transfer coils are powered by a general purpose interface bus (GPIB) controllable 5 kW dc power supplies. However, since their internal current control bandwidth
is too slow to sample the time-current sequence for the MOT coils we externally feedback
control it by a closed-loop servo. It is implemented with a current transducer and a MOSFET
bench. The fast initial ramp to 20 A is additionally supported by current from four large
46
3.2. MAGNETIC FIELD CONFIGURATION
capacitors (1 mF) charged to 60 V. The electromagnetic properties of the coils with resulting maximum currents and field gradients are listed in Table 3.1. The magnetic transport
is performed over a period of 1 s. We maintain a minimum trap depth of ∼ 70 G equivalent
to about 2 mK. The total power required is approximately 2 kW which corresponds to an
average power consumption of ∼ 34 W at a duty cycle of 1/60.
Resistance
[ mΩ ]
Inductance
[ µH ]
Maximum current
[A]
Maximum field gradient [ G/cm ]
MOT Transfer QUIC
200
50
300
1000
70
450
115
170
15
310
290
320
TABLE 3.1.: Electromagnetic Properties of the Magnet Coils
3.2.2. QUIC Trap
The magnetic QUIC trap consists of three coils connected in series. This is advantageous to
diminish relative current fluctuations and therefore magnetic field fluctuations. Two coils
(called “quadrupole coils”) produce a quadrupole field and one smaller coil (called “Ioffe
coil”), mounted orthogonally between the quadrupole coils lifts the magnetic zero to a finite value and adds a curvature to the resulting potential [85]. Having a nonzero magnetic
minimum is crucial when evaporatively cooling atoms towards quantum degeneracy in order to circumvent losses due to Majorana spin flips.
The geometry of the QUIC trap potential is approximately cylindrically symmetric with
respect to the Ioffe coil axis. Along this direction the curvature and therefore the confinement is weaker than in the radial directions. In our case this results in cigar shaped Bose Einstein condensates with an aspect ratio of 5:1. The exact position and dimension of the
Ioffe coil are very critical to yield the desired magnetic bias field B 0 which should be on the
order of a few Gauss. A low bias field is preferential because the trap frequencies scale as
p
B 0 / B 0 , where B 0 is the magnetic field gradient and high trap frequencies permit faster and
more efficient evaporative cooling.
The construction of the Ioffe coil is done in the same way as for the transport coils. It is
integrated in a slotted copper frame and potted with Stycast. The Ioffe frame is mounted
laterally between the transport coil brackets which hold the quadrupole coils. Additionally,
the Ioffe frame serves as a spacer for the two transport brackets. The mechanical contact
is accomplished with sapphire sheets in order to prevent eddy currents by simultaneously
maintaining good thermal conductivity (see Figure 3.4). The large mass of the complete
magnet coil structure functions as a thermal low pass filter which contributes to the good
temperature stability.
47
3. EXPERIMENTAL APPARATUS
In the Ioffe frame we have integrated additional coils on the same axis as the Ioffe coil to
be able to manipulate the final trap geometry inside the vacuum system after bakeout. Two
few-winding coils are employed to fine-tune the value of the magnetic field minimum B 0 .
One larger coil (called “offset coil”) a little further away from the trap center can be used
to change the aspect ratio of the trap and make it approximately spherical. Furthermore,
the Ioffe frame features a conical bore which allows us to image the BEC through the center
of the Ioffe coil. The electrical connections of the coils forming the magnetic QUIC trap are
realized outside the vacuum. We have included a 1.4 MHz low pass filter in parallel to the
Ioffe coil to avoid any radio frequency (rf) pickup because of its low inductance of 4 µH. The
QUIC trap is operated with a 150 W power supply specifically tuned to our inductive load.
The average power consumption of the magnetic trap is maximally 60 W but can be as low
as 2 W when operated at 3 A.
3.2.3. Magnetic Shielding
We clad the main vacuum chamber in a mu-metal shielding depicted in Figure 3.6 to minimize the influence of residual external magnetic field fluctuations on the cold atoms. A
magnetically quiet environment is essential for stable continuous wave (cw) operation of
the atom laser. Mu-metal is a magnetically soft nickel alloy with a very high magnetic permeability µ ∼ 105 which attenuates magnetic fields inside a cohesive enclosure. The screening effect depends very much on the completeness of the mu-metal box. Magnetic field
lines penetrate an opening roughly as far as its diameter. Therefore we have attached a stub
around the pumping tube of the main vacuum tank to attain a better aspect ratio at the
position of the BEC. The design of the mu-metal hull was aided by computer simulations of
the electromagnetic field. The mu-metal was machined and cured as recommended by the
manufacturer. After demagnetization we have measured a dc magnetic extinction ratio of
∼ 40 in the vertical and ∼ 100 in the horizontal direction at the position of the BEC.
3.2.4. Auxiliary Coils
Since the mu-metal shielding prevents any manipulation of the atoms with external magnetic fields, we have arranged supplementary magnet coils inside the mu-metal enclosure.
All extra coils were potted with Stycast either in a slotted copper or Shapal frame for good
thermal conductivity and mechanical sturdiness.
Two large coils (called “gradient coils”) are included in the main vacuum chamber to compensate the gravitational force for the weakest magnetic sublevel (30.5 G/cm) with 22 A.
Their total resistance and inductance is about 0.2 Ω and 0.9 mH, respectively. The gradient
coils were mounted inside the vacuum chamber on the transport bracket (see Figure 3.4)
and on the science platform around the cavity (see Figure 3.8), respectively. With the latter
we should be able to reach the widest Feshbach resonance of 87 Rb (∼ 1008 G) [121] at the
48
3.3. SCIENCE PLATFORM AND CAVITY SETUP
μ-metal hull
FIGURE 3.6.: Photograph of the preassembled mu-metal hull before it is mounted around the
main vacuum tank. It consists of seven large and several small individual pieces.
position of the cavity.
Around the cavity we have placed two pairs of tiny coils (4 Ω, 0.4 mH) along and perpendicular to the cavity axis (see Figure 3.4). They can be used to create magnetic field gradients
of about 200 G/cm (with 1 A) for tomography experiments. In combination with a fifth tiny
coil (1 Ω, 0.1 mH) mounted above a magnetic trap at the position of the cavity can be formed.
These five small coils (called “cavity coils”) were wound from 0.04 mm2 Kapton isolated copper wire on Shapal frames to be penetrable by radio frequency.
In addition to the magnet coils inside the vacuum tank we have wound three mutually
orthogonal pairs of large extra coils around the main tank. However, they are still within the
mu-metal hull and serve to produce homogeneous magnetic fields, e.g., for optical pumping.
3.3. Science Platform and Cavity Setup
We have designed this apparatus with attention to versatile access for samples and probes
to the BEC. Therefore we have implemented two independent sections of complementary
functionality, i.e., the BEC production rig (see Sections 3.1 and 3.2) and the science platform.
The latter is a modular, interchangeable flange, which in the current configuration supports
our single atom detector in form of the ultrahigh finesse optical cavity.
49
3. EXPERIMENTAL APPARATUS
The design of the cavity was guided by the need for stability, compactness, and ultrahigh
vacuum compatibility. It rests on top of a passive vibration isolation stack which can be
positioned on the science platform.
3.3.1. Cavity Design
The Fabry - Pérot optical cavity is formed by two dielectric Bragg mirrors of ultrahigh reflectivity and ultralow scattering losses . The reflection band is 40 nm wide and centered
around 780 nm. We have determined an ultimate quality factor Q = 1.6 · 108 after bakeout
from the linewidth of the cavity (∆ν = 2.4MHz). The initial Q immediately after cleaning
the mirrors was higher by about a factor of 2. The cylindrical mirrors (3 mm diameter, 4 mm
length) having a radius of curvature of 77.5 mm are separated by 178 µm which results in a
Gaussian mode waist of w0 = 25.5µm. We precisely measured the length of the near planar
cavity by simultaneously transmitting two different known wavelengths (see Section 4.1.3)
and determined a free spectral range of νFSR = 0.84 THz from which we derive a finesse of
F = 3.5 · 105 .
Each mirror was bonded with superglue into a specifically fabricated ceramic (Shapal)
ring structure. It positions and fixes the mirror inside the piezoceramic tube [113]. A piezo
is required to fine tune the length of the cavity (∼ 0.5V/nm) and as the actuator for the
cavity lock (see Section 4.1.3). The 7 mm long piezotube has inner and outer diameters of
5.35 mm and 6.35 mm, respectively. It is equipped with nonmagnetic wraparound electrodes
(silver) which allows the inner electrode to be contacted from the outside. Additionally, the
piezotube features four radial holes of 1 mm diameter for lateral access of atoms and lasers
perpendicular to the cavity axis.
The cavity assembly is mounted by a specifically designed compact fixture (called
“clamp”) making use of mechanical joints [Figure 3.7(a)]. It was manufactured by spark erosion from titanium in order to be nonmagnetic while having good elastic properties. Further design considerations aimed at high mechanical eigenfrequencies to avoid resonances
within the bandwidth of the cavity lock (∼ 40kHz), that means a small size and high stiffness
are favorable. We estimate the lowest eigenfrequency of our fixture with a simple mechanical fixed-hinged beam model [122] to be ∼ 50 kHz.
Our design of the cavity mount consists of the t-shaped clamp and a baseplate with integrated bearings to which the clamp is tightened with a plate nut. It converts the downward
force onto the cavity assembly and firmly holds it together. Moreover it provides the piezo
with a load. A hole of 1.2 mm diameter in the baseplate and plate nut grants optical access
to the cavity from below. This cavity setup is highly modular and easily interchangeable
because it freely rests on the vibration isolation stack [Figure 3.7(b)].
50
viton
baseplate
-120
-80
-40
0
0
200
400
800
(d)
1 cm
frequency [ Hz ]
600
transfer function [ dB ]
(b)
science
(c)
keel
platform
support
vibration
isolation
stack
FIGURE 3.7.: Elements of the optical cavity implementation. (a) Plane cut through the assembled cavity design, where the red arrows
indicate optical access. (b) Photograph of the cavity setup. The electrical leads for the piezotube are pinched in a slotted Viton piece
to efficiently decouple the cavity from the environment. (c) The cavity assembly resting on top of the vibration isolation stack which
is positioned on the science platform. (d) Modeled frequency response of our vibration isolation stack.
scale 1:1
joint
platenut
& screw
piezo tube
clamp
holder
mirror
(a)
3.3. SCIENCE PLATFORM AND CAVITY SETUP
51
3. EXPERIMENTAL APPARATUS
3.3.2. Vibration Isolation System
The aforementioned baseplate simultaneously acts as the top mass of our vibration isolation
stack [123] which consists of five layers of massive plates (Wolfmet) with rubber dampers
(Viton A) in between [Figure 3.7(c)]. Viton has good vibration damping properties and is
suitable for an ultrahigh vacuum environment. The 5 mm diameter Viton pieces rest in
hexagonal grooves that are radially arranged in 120 ◦ graduations. Consecutive layers are
rotated by 60 ◦ to prevent a direct “line of sound”. Hexagonal shaped grooves best avoid
squeezing and creeping of the rubber and provide good lateral stability. Position, angle,
and tilt reproducibility of this structure are excellent because of the frustum shaped bottom
mass with keel. It centers the stack in an inverted, truncated conelike support and assures
mechanical stability by lowering the center of mass below the support points. The complete
stack has a central 10 mm bore for vertical optical access to the cavity.
Its damping properties can be modeled by regarding the structure as a system of coupled masses and springs [124] and calculating its frequency dependent transfer function
[Figure 3.7(d)]. For attenuation at low frequencies large masses and small spring constants
are favorable [125, 126]. Therefore we have fabricated the plates from a heavy metal alloy
(Wolfmet) and employed short (10 mm) Viton pieces. Our vibration isolation stack works
well for acoustic frequencies above 200 Hz.
Additional precautions to counter low frequency excitations such as building vibrations
include setting up the experiment on a damped rigid optical table in a basement laboratory
having its own independent foundation and choosing a position with little floor vibration
within this laboratory. The quality of the vibration isolation system is such that we could
easily operate the cavity in the vicinity of a turbo-molecular pump. Furthermore the vibration isolation stack kept the cavity in place when the whole optical table accidentally
dropped by about 2 cm as we tried to tilt it.
3.3.3. Science Platform Layout
The self-contained, interchangeable science platform flange was prepared to support and
align the complete cavity mount. Its layout provides manual positioning ability of the cavity
mount by ± 2 mm along and perpendicular to the cavity axis, respectively. This is rendered
feasible by an octagonal support (nonmagnetic steel) of the vibration isolation stack which
can be deterministically moved and fixed in a larger octagonal millout on the flange. The
second objective of the support is to erect the arrangement of cavity coils with the gradient
coil (see Section 3.2.4) to be positioned around the cavity without direct contact. The coil
assembly is mounted on two nonmagnetic steel sustainers which are fixed to the vibration
isolation support. In order to remove the dissipated heat by the electromagnetic coils, we
have connected the copper bracket of the gradient coil to a power feedthrough serving as
a heat bridge, i.e., cold finger. Outside the vacuum the 19 mm diameter copper conductor
52
3.3. SCIENCE PLATFORM AND CAVITY SETUP
can be connected to the cooling circuit and cooled to −90°C. The copper rod serves as a the
main drain for the heat because of the low thermal conductivity of the steel sustainers and
support. The mounting of the independent BEC production rig and science platform within
the main vacuum chamber has to be noncontact but within a fraction of a millimeter. This
results in a final position of the optical ultrahigh finesse cavity being 36.4 mm below the
BEC. The orientation of the cavity axis is at 90 ◦ with the symmetry axis of the magnetic trap
(Ioffe axis).
cavity
magnetic coil
structure
sustainer
cold finger
NEG
pump
VIS
MOT
viewport
support
electrical
feedthrough
FIGURE 3.8.: Photograph of the mounted science platform. The support bears the vibration
isolation system (VIS) and the magnetic coil structure which surrounds the optical cavity.
53
4 Characterization of the System
“Durch diese hohle Gasse muß er kommen.” - Wilhelm Tell, Friedrich Schiller
The features of the unique apparatus developed within the scope of this thesis are explored and characterized. For the first time quantum degenerate gases and cavity QED are
brought together experimentally. The apparatus proves to be a very robust and reliable BEC
machine for delivering high atom numbers 87 Rb Bose - Einstein condensates (∼ 2 · 106 ) with
the experimental procedure described below. We can produce extremely stable atom lasers
from a very well defined spatial region within the condensate over long timescales (∼ tens
of seconds) thanks to the magnetic shielding in combination with superior mechanical and
thermal stability. The precisely controlled atom laser output coupling is characterized in
this chapter.
Furthermore, we present the locking scheme of the ultrahigh finesse optical cavity and
describe the best parameters for single atom detection. The cavity functions as a linear
single atom detector of atom fluxes spanning three orders of magnitude. We explain how
we aim the atom laser into the cavity mode and show measurements of the atom laser beam
profile. The threshold behavior of the atom laser is observed and we utilize the atom laser
as a bright and reproducible source of cold atoms for investigations of cavity QED effects.
The high detection efficiency of degenerate atoms (∼ 25%) enables us to inspect ultracold
atomic clouds noninvasively and therefore time resolved.
Parts of this chapter have been published in [39]: A. Öttl, S. Ritter, M. Köhl, and T. Esslinger,
Hybrid apparatus for Bose-Einstein condensation and cavity quantum electrodynamics: Single atom
detection in quantum degenerate gases, Rev. Sci. Instr. 77 (2006), 063118.
55
4. CHARACTERIZATION OF THE SYSTEM
4.1. Experimental Procedure
We operate the experiment periodically with a cycle time of 60 s. During each cycle we
produce a new BEC from which we output couple an atom laser. It is directed to the high
finesse optical cavity situated 36.4 mm below the BEC where single atoms are detected. The
cavity is probed by a resonant laser and its length is actively stabilized by an off resonant
laser with respect to the atomic transition. The experimental sequence is fully computer
controlled by a C++ program. Digital and analog channels interface the computer with the
elements of the experimental setup. The experiment is distributed on two self contained
optical tables, one for the laser system and one for the vacuum apparatus. They are linked
by optical fibers.
4.1.1. Bose - Einstein Condensation
We form a Bose - Einstein condensate of 87 Rb in dilute atomic vapor from a dispenser loaded
magneto-optical trap by means of rf-induced evaporative cooling [79, 116]. During the first
20 s of each cycle we load the magneto-optical trap with atoms from the pulsed alkali dispenser source [118]. The dispensers are operated at ∼ 7A with a temporal offset of −3s to
the actual MOT phase. We work on the D2 line of 87 Rb (5 2 S1/2 → 5 2 P3/2 ) at a wavelength
¯
λ = 780nm. For the cooling transition on the hyperfine ground state |F = 2〉 ↔ ¯F 0 = 3〉 a
laser power of 17 mW is employed in each of the six 34 mm diameter MOT beams. For optimum collection efficiency we choose a detuning of 3Γ, where Γ = 2 π· 6 MHz is the linewidth
of the cooling transition. In order to be frequency tuneable the laser is offset locked [127]
¯
from the |F = 2〉 → ¯F 0 = 2〉 transition by about 250 MHz and subsequently amplified with
a tapered amplifier. An additional laser (called “repumper”) to avoid atomic losses into
¯
the |F = 1〉 dark state is directly locked to the |F = 1〉 → ¯F 0 = 2〉 transition and delivers a
power of 1 mW in each MOT beam. All our lasers are home-built external cavity diode laser
[128] locked by Doppler-free rf-spectroscopy technique [129] to atomic transitions. For the
magneto-optical trap we apply a magnetic field gradient of 10 G/cm by applying a current
of 3.5 A to the MOT coils. Due to the mu-metal shielding no earth field compensation is required. We collect about 2 · 109 atoms with the magneto-optical trap before we switch off the
magnetic field and sub-Doppler cool the atoms in a 10 ms optical molasses phase.
Before magnetically transporting the cold atomic cloud we optically pump the atoms into
the low field seeking |F = 1, mF = −1〉 hyperfine state. Optical pumping is performed over
2 ms at a homogeneous magnetic field of ∼ 4 G. All light fields are off when the transport sequence starts with adiabatically compressing the cloud [Figure 3.5(b)]. The magnetic transport conveys the atoms within 1 s through the differential pumping tube over a distance of
82 mm directly into the magnetic QUIC trap. We estimate a transport efficiency of > 90% by
transferring the atoms back into the MOT and measuring their fluorescence. The losses are
mainly due to background collisions and depend on the pressure in the MOT chamber.
56
4.1. EXPERIMENTAL PROCEDURE
We operate the magnetic QUIC trap initially with a maximum current of 15 A. This yields
the highest trap frequencies of ωx = ωz = 2π· 135 Hz and ω y = 2π· 28 Hz with a bias field B 0 of
4.7 G and a field gradient B 0 of ∼ 300G/cm. Here ω y and ωx denote the trapping frequencies
along and perpendicular to the Ioffe axis, respectively and ωz is in the vertical direction.
Over a period of 23 s we perform rf-induced evaporative cooling with an exponential frequency ramp and a radio frequency power of 24 dBm. The radio frequency is radiated by
a coil which consists of ten turns of Kapton clad copper wire (1 mm2 ) encircling an area of
3 cm2 . It is mounted 2 cm away from the center of the trap and is oriented at 90 ◦ with respect
to the Ioffe axis. This results in a B rf of about 30 mG at the position of the cold atoms.
(a)
(b)
(c)
100 μm
FIGURE 4.1.: Absorption images of cold atom clouds, (a) thermal cloud at a temperature T
above the critical temperature Tc , (b) bimodal distribution for T < Tc , and (c) “pure” Bose Einstein condensate at T ¿ Tc , taken after 30 ms time of flight with a detuning of 2Γ to avoid
saturation.
Before reaching the critical phase space density for Bose - Einstein condensation we relax
the trap to the final parameters of ωx = 2π· 38.5 Hz, ω y = 2π· 7.3 Hz and ωz = 2π· 29.1 Hz with
B 0 = 1.2G and B 0 = 60G/cm by powering the QUIC trap with 3 A. The initial trap symmetry is
lifted by the large gravitational sag of about 290 µm. It is given by z sag = −g /ω2z , where g is
Earth’s gravitational acceleration. Furthermore, the long axis of the BEC is inclined by about
20◦ with respect to the horizontal plane. The opening of the trap is performed adiabatically
(ω̇/ω ¿ ω) over a period of 1 s. During this time a rf shield limits the trap depth to prevent
heating of the cold atomic cloud. In the weak trap we further cool the atoms evaporatively
over 5 s and achieve pure Bose - Einstein condensates of up to 3 · 106 atoms. The density in
the weak trap is considerably lower so the losses due to inelastic collisions are reduced. We
have measured a 1/e -lifetime for condensates of about 30 s. The typical size of the Bose Einstein condensate is 12 × 15 × 60µm3 (Thomas - Fermi radii) with a chemical potential µ
of about 1 kHz. Resonant absorption imaging of the cold atoms after a free expansion time
of 30 ms allows us to extract the number of atoms in the cloud and its temperature. We fit
the resulting density distribution with the sum of a Gaussian and a Thomas - Fermi profile.
The spatial resolution of our imaging system ( f /10) is limited to 9 µm by the diameter of the
windows. We employ a charge-coupled device (CCD) camera with an according pixel size.
57
4. CHARACTERIZATION OF THE SYSTEM
4.1.2. Atom Laser Output Coupling
An atom laser is a coherent atomic beam extracted from a Bose - Einstein condensate. The
trapped condensate, being in a quantum degenerate state, serves as the source for the freely
propagating atom laser. A steady-state output coupling process establishes a coupling between the ground state of the trap and the energy eigenfunctions of the linear gravitational
potential and produces a continuous wave (cw) atom laser. The resulting cw atom laser [11],
in contrast to optical lasers, consists of interacting massive particles propagating downwards in the gravitational field. But alike optical laser it is a matter wave in a coherent state
as defined by Glauber in the quantum theory for optical lasers [40] and exhibits higher order
coherence [17].
In order to output couple atoms we locally change their internal spin state from
microwave antenna
the magnetically trapped |F = 1, mF = −1〉
into the untrapped |F = 2, mF = 0〉 hyperfine state. The spin flip is induced by
a coherent microwave field at the hyperfine splitting frequency of 87 Rb (∆E hfs /h =
6.8 GHz) [130]. This microwave output coupling scheme is equivalent to a two-level
system because of the Zeeman splitting in
the hyperfine niveaus (∼ 1MHz). Therefore it is superior to rf output coupling
CF 40
which mutually couples all states from a
FIGURE 4.2.: Helix antenna built for
Zeeman manifold [102]. The microwave sigatom laser output coupling at 6.8 GHz,
nal is produced by a global positioning sysmounted on a vacuum feedthrough.
tem (GPS) disciplined synthesizer. We use
a home-built resonant helix antenna with
14 dB gain (see Figure 4.2) placed inside the ultrahigh vacuum chamber to radiate the microwave field. The antenna is connected and impedance matched to a commercial microwave feedthrough.
The energy conservation for the microwave output coupling resonance
condition is only satisfied at regions of constant magnetic field where ∆E hfs −g F mF µB B (r) =
hνmw . Here νmw is the microwave frequency, B (r) the magnetic field of the trap at position
r and µB the Bohr magneton. The hyperfine Landé g factor g F and the magnetic spin
quantum number mF apply to the BEC state. The magnetic moment of the output coupled
atoms is zero to first order.
The resonant regions for output coupling are ellipsoidal shells with the geometry of the
magnetic trap, centered at the minimum of the magnetic potential. However, the center of
the actual harmonic trapping potential for massive particles is lowered by the gravitational
58
4.1. EXPERIMENTAL PROCEDURE
100 μm
gravity
atom laser
sag with respect to the magnetic field minimum. For our experimental conditions the
BEC
gravitational sag is ∼ 290 µm and therefore
the resonant output coupling shells intersect the Bose - Einstein condensate almost
as horizontal planes.
The Rabi frequency Ω of the microwave
output coupling process is given by
µ12 B mw /ħ, where µ12 is the magnetic
dipole matrix element between the two
coupled states and B mw the magnetic field
of the microwave radiation [131]. The
magnetic dipole transition has selection
rules ∆mF = ±1. In the weak output
coupling regime (Ω ¿ ωz ) an atom leaves
the condensate much faster than the Rabi
frequency and does not undergo Rabi
oscillations [132]. The atom laser output
coupling rate depends on the number of
atoms in the condensate NBEC and the
overlap | 〈ΨBEC |ΦE 〉 |2 between the BEC wave
function ΨBEC and the energy eigenfunction ΦE of the free atom laser [99, 93]. For
given atom number NBEC and microwave
frequency the output coupling rate is
proportional to Ω2 and therefore to the
incident microwave power [98].
Producing a coherent cw atom laser crucially depends on the temporal stability
of the resonance condition. We take experimental care to avoid any fluctuations
FIGURE 4.3.: Microwave output coupling
of a continuous atom laser. Resonant abor drifts of the magnetic resonance posisorption image with (left) and without
tion. A temperature controlled cooling cir(right) repumper to image the remaining
cuit for the large mass magnet coil structure
|F = 1〉 BEC. The atom laser propagation
and a GPS locked synthesizer permit excelis ∼ 2 mm.
lently reproducible conditions. The magnetic shielding enclosure together with the
hermetic steel vacuum chamber eliminate external electromagnetic field fluctuations (see
Sections 3.1.1 and 3.2.3). The only detectable noise source is the low noise current supply
powering the magnetic QUIC trap. We have measured a magnetic field stability of better
59
4. CHARACTERIZATION OF THE SYSTEM
p
than 5µG/ Hz (at 3 kHz) or 50 µG overall (bandwidth: 50 kHz), respectively. This enables
us to produce second order coherent atom lasers and output couple a cw atom laser over
the duration of the BEC lifetime. Due to the extremely low atom fluxes measurable with the
cavity detector we do not have to deplete the condensate significantly.
The atom laser freely propagates downwards for 86.1 ms before entering the high finesse
optical cavity where single atoms are detected. The cavity is placed 36.4 mm below the BEC
which results in a velocity of 0.84 m/s for the atoms traversing the cavity mode. This velocity
corresponds to a de Broglie wavelength of about 5 nm which could be useful for applications
in coherent atom lithography [133] or as an atom laser microscope [134, 135].
4.1.3. Cavity Lock
In order to engage the optical high finesse cavity as a single atom detector we have to stabilize its length to better than 0.5 · λ/F ≈ 1pm with respect to the wavelength of the probe
laser.
We choose a cavity locking scheme [136] that allows us to independently adjust the frequencies of the cavity resonance (ωc ) and of the probe laser (ωl ). Furthermore it enables
us to keep the cavity permanently locked even during atom detection since the action of a
single atom transit on the far-detuned stabilization laser is negligible and vice versa.
The cavity lock is realized with a far-detuned master laser at 830 nm and a resonant master laser at 780 nm referenced to a 87 Rb line. They are frequency stabilized by means of
Pound - Drever - Hall locks [137] to a transfer cavity having a free spectral rage νFSR of 1 GHz.
In order to be freely tunable the actual probe and stabilization slave lasers are phase locked
[138] with a frequency offset of 0–500 MHz to their respective master lasers. The length of
the science cavity is then actively controlled by a Pound - Drever - Hall lock on the stabilization slave laser with a bandwidth of 38 kHz. We create the necessary sidebands for the
lock with a home-built electro-optical modulator [139]. It works at 362 MHz to have the sidebands well off resonant with the cavity because its finesse for 830 nm is 3.8·104 and therefore
its linewidth 22 MHz.
We actively control the incident powers of the stabilization and probe laser on the cavity
to about 2 µW and 3 pW, respectively. In order to have a good spatial overlap, the two lasers
are guided through the same optical fiber. Their power ratio of 10−6 is realized with an
optical color filter. We can couple about 25% of the incident probe laser power into the
cavity TEM00 mode being limited by the nonoptimal impedance matching.
The atomic resonance (ωa ) we employ for single atom detection is the cycling transi¯
tion |F = 2〉 → ¯F 0 = 3〉 of the D2 line of 87 Rb. It yields a maximum atom field coupling rate
g 0 = d iso |Emax | = 2π · 10.4 MHz, were we have assumed an isotropic dipole
q matrix element
¡
¢
[130] diso and a maximum single photon electric field strength |Emax | = 4ħc / ²0 λw20 l according to our mode volume with a beam waist w0 = 25.5 µm and a cavity length l = 178 µm.
The atom field coupling rate g 0 is large compared to the dissipation losses being the cavity
60
4.2. SINGLE ATOM DETECTION PERFORMANCE
FM
PDH
transfer
cavity
780 nm
slave
830 nm
master
830 nm
slave
PDH
stabilization
probe
frequency
PDH
PLL
PLL
780 nm
master
cavity locking chain
Rb
science
cavity
FIGURE 4.4.: Schematic diagram of the cavity locking scheme employing a transfer cavity to
bridge the large frequency difference between the resonant probe laser (780 nm) and the
far off resonant stabilization laser (830 nm). Phase locks (PLL) to the master lasers provide
full and free tuning flexibility of the probe laser. The cavity locks are Pound - Drever - Hall
locks (PDH) and the lock to the rubidium reference is realized with a frequency modulation
(FM) lock.
field decay rate κ = 2π · ∆ν/2 and the dipole decay rate γ = Γ/2 where Γ = 2π · 6.1MHz is the
natural linewidth of the excited state. Furthermore the inverse atom transit time τ−1 is
orders of magnitude smaller than the coupling rate which means the atom is always in a
quasi steady state with the cavity field during the transit. The relevant parameters of our
experiment are thus (g 0 , γ, κ, τ−1 ) = 2π· (10.4, 3.0, 1.2, 3·10−3 ) MHz, which brings us into the
strong coupling regime of cavity QED defined by g 0 À (γ, κ, τ−1 ).
4.2. Single Atom Detection Performance
Single atom detection with an ultrahigh finesse optical cavity [20] can heuristically be
viewed as the refractive index of a single atom being sufficient to significantly shift the
cavity resonance. Consequently, the transmission of an initially resonant, weak probe laser
is measurably reduced. In quantum mechanical terms the coupling of a single atom with the
quantized electromagnetic field in the cavity mode dominates the dissipation losses (strong
coupling regime) which means the level splitting of the Jaynes - Cummings model [66, 140]
61
4. CHARACTERIZATION OF THE SYSTEM
can be resolved. On the other hand, the quantum mechanical detection process on the longitudinally delocalized atom within the atom laser beam projects and localizes them inside
the cavity mode [33].
We can efficiently study these cavity QED interactions of single atoms having an atom
laser as an unprecedented bright, controllable, reproducible, and well defined atom source.
Here we present experimental results that characterize the performance of our combined
BEC and ultrahigh finesse optical cavity system.
4.2.1. Signal Analysis
In order to identify single atom transits we record the transmission of a resonant weak probe
beam through the cavity with a single photon counting module (SPCM). A typical recording
showing single atom transits is presented in Figure 4.5. The light coming from the cavity
is filtered with a 780 nm bandpass and a 830 nm notch filter to block the stabilization laser.
Their combined relative optical density (OD) for 830 nm is 12. The SPCM is located inside a
black box and exhibits an overall photon dark count rate of ∼ 100s−1 .
cavity transmission
[ photons / 20 μs ]
120
100
80
60
40
20
0
1.645
1.650
1.655
time [ s ]
1.660
1.665
FIGURE 4.5.: Cavity detection recording of an atom laser. The atom flux is about four orders
of magnitude lower compared to Figure 4.3. Single atom transits are clearly identified by
their reduction of the shot noise limited empty cavity transmission.
The cumulative detection probability for intracavity probe photons taking into account
losses in the optical system and the quantum efficiency of the SPCM is about 7%. It is mainly
limited by the fact that we employ symmetric cavity mirrors with equal transmittivity (∼
2ppm) and by the scattering losses (∼ 7ppm) of the mirrors. In order to achieve a large
signal-to-noise ratio for single atom detection we usually work with an average intracavity
probe photon number of about 5 [33]. This level corresponds to an intensity of about 40
times the saturation intensity and yields a photon count rate of 2π·∆ν· 5 · 7% ≈ 5 photons/µs.
We integrate the signal from the SPCM over 20 µs with a temporal resolution between 1-4 µs
62
4.2. SINGLE ATOM DETECTION PERFORMANCE
and set the criterion for single atom detection events to a reduction of more than four times
the standard deviation (σ) of the shot noise limited empty cavity transmission. This reduces
false atom detection events to less than 0.5s−1 .
4.2.2. Characteristics of Single Atom Events
The coupling of a single atom with the cavity mode can be characterized by the magnitude
and duration of the resulting transmission dips. A recorded typical single atom transit is
shown in Figure 4.6(a). The 4σ threshold here corresponds to about 50% reduction in the
probe light transmission of about 70 photons/µs. We analyze detected events and show
histograms in Figures 4.6(b) and 4.6(c) for atom laser data taken in 184 iterations of the
experiment. The atom flux was set to ∼ 1 · 103 s −1 so the probability [17] for unresolved
multiatom events within the dead time of our detector (∼ 70 µs) is less than 0.3%.
The dead time of our detector is reflected in the distribution of coupling times, i.e., the
full width half maximum (FWHM) of the transmission dips [Figure 4.6(b)]. It is mainly determined by the radial size of the Gaussian cavity mode and the velocity of the atoms during
2
2
their transit. For a radial coupling strength g(r ) = g0 e −r /w0 with w0 = 25.5 µm and an initial
velocity of 84.1 cm/s we expect an average coupling time of 45 ± 12 µs [Figure 4.6(b), gray].
Taking the classical free fall velocity is justified since the induced momentum uncertainty
by projecting the longitudinally delocalized atom into the cavity mode is on the order of
10µs.
In the numerical simulation we take into account photon shot noise and the features of
our peak detect routine, namely the 20 µs sliding average. The effect of the dipole potential
on the transit time is negligible because the slight gain in velocity (< 2µs) is counteracted
by an effectively stronger and therefore longer coupling [Figure 4.6(d)]. The mean of the
measured coupling time distribution [Figure 4.6(b), red] is in accordance with the expected
value. However, the distribution deviates from the expected shape and exhibits an excess of
short and long transit times. We attribute the shorter transits to optical pumping of atoms
into the dark state |F = 1〉 because their number is intensity dependent on the probe light.
Longer transit times could be explained by diffraction of the atomic beam, scattering of
spontaneous photons or cavity cooling effects, if the cavity axis is slightly nonorthogonal
with respect to the atom laser (possibly 10−2 rad) and by unresolved multiatom events.
The magnitudes of the cavity transmission dips in Figure 4.6(c) reflect the different maximum coupling strengths for single atom transits. Depending on its radial position an atom
will experience a varying peak coupling strength according to the Gaussian profile of the
cavity mode. In the axial direction however, the light force is strong enough to channel the
atoms towards the intensity maxima of the standing wave [141]. Arbitrarily weak coupling
transits cannot be resolved due to the shot noise in the empty cavity transmission. We set
the single atom detection threshold to 4σ of the original transmission to achieve a large
signal-to-noise ratio.
63
4. CHARACTERIZATION OF THE SYSTEM
(a)
(b)
0
6
20
4
40
60
2
80
0
100
transmission reduction [ % ]
frequency [ x 1000 ]
67500
coupling time [ μs ]
20 30 40 50 60 70 80
30
(c)
(d) 30
40
40
50
50
60
60
70
70
80
80
90
90
0
2
4
6
8
frequency [ x 1000 ]
transmission reduction [ % ]
transmission reduction [ % ]
67300
time [ μs ]
67400
20 30 40 50 60 70 80
coupling time [ μs ]
FIGURE 4.6.: Characteristics of detected single atom events. (a) The transit of a single atom
significantly reduces the probe light transmission through the cavity. We integrate the signal with a 20 µs sliding average and set the detection threshold to 4σ of the photon shot
noise. (b) Distribution of measured coupling times (FWHM) (red) compared to the distribution of simulated events (gray). (c) Distribution of measured transmission reduction magnitudes. An evaluation with a 4σ threshold (red) is compared to a 2σ threshold [gray] revealing the discrimination of the events from the photon shot noise. (d) Dependency of the
transmission reduction on the coupling time due to the non-Gaussian shape of the dips.
64
4.2. SINGLE ATOM DETECTION PERFORMANCE
The resulting histogram of dip depths is displayed in Figure 4.6(c) [red] compared to data
for a lower threshold level of 2σ in Figure 4.6(c) [gray] unveiling the photon shot noise. The
weakest detectable single atom events correspond to peak atom field coupling strengths of
gmin
= 2π · 6.5 MHz. The strongest attainable coupling strengths for our cavity are gmax
=
0
0
2π · 10.4 MHz, which would be equivalent to a reduction of 80% in the cavity transmission
[33]. We do not observe a sharp cutoff in the histogram but rather an equal distribution
of transmission reductions from 50-80% with smeared out edges due to the comparatively
large photon shot noise at the minimum of the transmission dip. This is consistent with
numerical simulations for single atom events.
The dependence of remaining probe light transmission through the coupled atom cavity
system is nonlinear with the atom field coupling strengths [33]. Therefore the shape of the
transmission dips is not Gaussian as the cavity mode and we observe a dependency of the
magnitude in transmission reduction on the coupling time and vice versa, as illustrated in
Figure 4.6(d).
The knowledge about the signatures of single atom events could facilitate the discrimination of “true” single atom events from “false” shot noise events or unresolved multiatom events, but the broad distributions make it difficult to distinguish two weakly coupling atoms from a strongly coupling one. However, the observed characteristics of the
detected events are in good agreement with the theoretical predictions for single atom transits. Moreover, these characteristics remain valid even when reducing the atom flux to very
few single atom events.
4.2.3. Detector Qualities
Having a BEC and an atom laser as the source for atoms that couple with the cavity mode offers several advantages. For instance, it provides well reproducible starting conditions and
allows us to precisely control the flux of atoms over a wide range by varying the microwave
output coupling power. The attainable atom flux is orders of magnitude larger than in experiments employing a magneto-optical trap as the cold atom source.
We have confirmed that our single atom detector functions as a linear detector on the
atom flux over three orders of magnitude (see Figure 4.7). The measured atom count rate is
proportional to the output coupling microwave power (see Section 4.1.2). Saturation occurs
at a flux of about 5 · 103 atoms per second. At higher rates multiatom arrivals within the dead
time of our detector become dominant and single atom events cannot be resolved anymore.
At a very low atom flux the error bars become increasingly large due to atom shot noise,
i.e., the Poissonian distribution in the atom number determination. Additionally, a very
weak atom “dark count” rate without intentional output coupling may be present. It is
likely due to stray magnetic or optical fields and depends on the size of the Bose - Einstein
condensate. However, the dark count rate is still less than 5 atoms per second on average
for a BEC with 2 · 106 atoms, for instance.
65
4. CHARACTERIZATION OF THE SYSTEM
atom count rate [ s-1 ]
104
103
102
101
-60
-50
-40
-30
radiated microwave power [ dBm ]
-20
FIGURE 4.7.: The ultrahigh finesse optical cavity functions as a linear detector on the output
coupling rate, i.e., atom flux over three orders of magnitude. Saturation occurs at a count
rate of about 5 · 103 atoms per second.
4.2.4. Detection Efficiency
The single atom detection efficiency of the ultrahigh finesse optical cavity strongly depends
on the frequencies chosen [114] for the probe laser (ωl ) and the cavity resonance (ωc ) with
respect to the atomic transition (ωa ). Furthermore the effective coupling strength g0 and
therefore the detection probability are determined by the polarization of the probe light
with respect to the quantization axis of the atomic spin.
In our experimental configuration we have a residual vertical magnetic field at the position of the cavity of about 16 G which represents the quantization axis for the atoms. The
field originates from the magnetic QUIC trap which is on during the single atom detection in
the atom laser. We set the probe light to horizontal (within 10 ◦ ) polarization which yields a
four times higher atom count rate as vertically (within 10 ◦ ) polarized light. Only these two
distinct polarization settings are feasible since we experience a birefringence in the cavity
resonance of about twice its linewidth. The horizontal polarization of the probe light produces a higher atom field coupling rate because it drives σ+ and σ− transitions compared
to the fewer and weaker π transitions for vertically polarized light. The exact atom field
interactions are more complex because of the Zeeman splitting and the resulting optical
pumping dynamics inside the cavity.
However, for red-detuned probe light the atoms entering the cavity in the |F = 2, mF = 0〉
state will predominantly be pumped into the |F = 2, mF = −2〉 stretched state and undergo
cycling transitions driven by the σ− polarization component. Therefore this cycling tran-
66
4.2. SINGLE ATOM DETECTION PERFORMANCE
2.4
2.0
2.2
1.8
1.6
1.4
0
1.2
1.0
0.8
-1.0
atom count rate [ ms-1 ]
cavity detuning ∆c [ MHz ]
2.0
1.0
0.6
0.4
0.2
-2.0
0
10
20
30
40
50
probe laser detuning ∆l [ MHz ]
FIGURE 4.8.: Dependence of the single atom detection efficiency on the probe laser ∆l and
cavity ∆c detunings. The vertical dashed line represents the cycling transition which is
Zeeman shifted by 22 MHz from the zero field atomic transition. Best single atom detection
is performed with a probe laser red-detuned by about 3Γ from the cycling transition and a
cavity detuning of about ∆ν/2, corresponding to the maximum dipole potential created by
the probe laser. The second local detection maximum corresponds to a blue-detuned probe
laser. Therefore the dipole potential is repulsive and the atom count rate reduced.
sition will be the main contribution in the single atom detection process. The imbalance is
due to a redshift for the σ− component and a blueshift for the σ+ component of ∼ 22MHz
in the magnetic field of 16 G at the cavity.
The number of detected atoms critically depends on the atom - probe laser detuning
∆l = (ωa − ωl )/2π and probe laser - cavity detuning ∆c = (ωl − ωc )/2π as illustrated in Figure 4.8. Here ωa refers to the bare atomic transition without magnetic field. For most efficient single atom detection we work with an atom - probe laser detuning ∆l ≈ 30-40 MHz
and a probe laser - cavity detuning ∆c ≈ 0.5-1 MHz. By taking into account the 22 MHz Zee¯
man shift of the cycling transition |F = 2, mF = −2〉 ↔ ¯F 0 = 3, mF 0 = −3〉 (vertical dashed line
in Figure 4.8) the probe laser red-detuning for optimum single atom detection is about 3Γ.
This value corresponds to the maximum of the dipole potential created by the probe laser,
that means the dipole force channels the atoms in the axial direction towards the antinodes
of the standing wave [141] which are simultaneously the areas of the highest atom field coupling strength. In the radial direction the dipole force is too weak to significantly modify
the trajectory of the atoms within the cavity mode. Also the dipole potential created by the
stabilization laser is weak compared to the one created by the probe laser.
67
4. CHARACTERIZATION OF THE SYSTEM
The second set of parameters in Figure 4.8 where single atom transits are detected is
around ∆l ≈ 18 MHz and ∆c ≈ −1 MHz. However, the count rate is reduced considerably
because the probe light is blue-detuned from the cycling transition and therefore the dipole
potential is repulsive. In the other two quadrants spanned by the resonances of the cavity and the cycling transition of Figure 4.8 (dashed lines), atom transits result in increased
probe laser transmission versus the empty cavity transmission [24]. We do not use those
events for single atom detection because the efficiency is reduced by about a factor of 2 as
compared to evaluating dips. Additionally the peaks exhibit a substructure consisting of
single photon bursts which makes it more difficult to discriminate single consecutive atom
transits.
In order to determine the detection efficiency for single atoms from the Bose - Einstein
condensate we make use of the linear behavior of the atom flux on the microwave output
coupling power (see Figure 4.7). We output couple a significant number of atoms measurable by absorption imaging while still in the weak output coupling regime. This number is
compared to the number of atoms detected by the cavity with the corresponding factor of
the output coupling powers.
We have calibrated the atom number in absorption imaging with the atom number at the
critical temperature which is well known for our trap frequencies. For optimum settings of
the cavity and laser detunings we are able to detect (24 ± 5)% of the output coupled atoms
with the cavity detector. This number is mainly limited by the spatial overlap of the atom
laser beam with cavity mode (see Section 4.2.5).
4.2.5. Atom Laser Beam Profile
Obviously, in order to see single atoms with the cavity, the atom laser has to propagate
through the cavity mode. However, this was not self-evident because the “BEC production
rig” and the “science platform” are completely independent entities of the experimental
apparatus and the alignment has to be better than a few millirad without knowing the exact
position of the cavity mode. Furthermore, the second order Zeeman effect slightly bends
the trajectory of the atom laser in the |F = 2, mF = 0〉 state and modifies its final lateral position by hundreds of micrometers. Although we have aligned the cavity with respect to the
BEC position as accurately as possible with plummets during the assembly of the apparatus,
the atom laser did not innately hit the cavity mode. We correct these deviations by tilting the whole optical table on which the experiment rests employing its height adjustable
legs. The tilt is monitored with a dual-axis inclinometer having its axes aligned along and
perpendicular to the cavity axis. With this method we aim the atom laser directly into the
cavity mode [Figure 4.9(a)] and maximize the atom count rate.
Moreover, tilting the experimental setup deterministically enables us to deduce the diameter of the atom laser after a propagation of 36.4 mm. The active area of the cavity mode
is approximately 35 × 150 µm2 . The size in the radial direction is determined by the weak-
68
18
atom count rate [ ms-1 ]
(b)
Ioffe axis [ mrad ]
(a)
18
16
14
12
8
10
12
14
cavity axis [ mrad ]
8
10
12
14
cavity axis [ mrad ]
14
(d)
(c)
2
1
0
16
15
0μ
m
atom count rate [ ms-1 ]
6
1.8
1.6
1.4
1.2
1
2
1.0 0
atom count rate [ ms-1 ]
0.8
0.6
0.4
le
ofi
r
0.2
p
er
s
la
m
o
t
a
d e n si ty
20
Ioffe axis [ mrad ]
4.2. SINGLE ATOM DETECTION PERFORMANCE
6
35 μ
m
FIGURE 4.9.: (a) The detected atom count rate for a constant atom flux is shown with respect
to the inclination of the optical table along the two axes. The rectangle represents the active
area of the cavity mode and the ellipse is the reconstructed size (1/e -diameter) of the atom
laser at the position of the cavity. [(b) and (c)] Fit (red) to the measured data (black) by the
convolution of the active size of the cavity mode with a Gaussian beam profile along the Ioffe
(b) and cavity (c) axes. It is compared to the expected shape from numerical simulations of
the Gross - Pitaevskii equation (gray). (d) Visualization of the extracted two-dimensional
atom laser beam profile clipped by the active area of the cavity mode.
69
4. CHARACTERIZATION OF THE SYSTEM
est detectable atom transits corresponding to gmin
0 = 2π · 6.5 MHz. In the axial direction it is
given by the projection of the cavity length clipped by the curved mirrors.
A deconvolution of the measured angle dependent count rates with this active area, assuming a Gaussian atom laser beam profile, yields 1/e -diameters of 80 µm and 110 µm along
and perpendicular to the Ioffe axis, respectively [Figure 4.9(b) and Figure 4.9(c), red]. The
mapped atom laser, being output coupled from the center of a Bose - Einstein condensate
with 1 · 106 atoms, is slightly inverted compared to the trap geometry but almost round at
the cavity. Here, its divergence along the fast axis, i.e., cavity axis, is about 2 mrad and less
than 0.5 mrad along the Ioffe axis, which makes it the best collimated atom laser to date
[11, 10, 142].
The repulsive mean-field interaction from the remaining trapped BEC is considerable only
along the fast axis where it acts as a defocusing lens for the atom laser beam. This results
in an expansion about four times larger than expected from Heisenberg’s uncertainty principle. Along the weakly confining Ioffe axis the lensing effect is negligible and the size of
the atom laser is consistent with a free expansion of the initial ground state in the trap. The
atom laser size and therefore its divergence, especially along the fast axis, can be further
reduced by output coupling below the center plane of the BEC [142] and by smaller condensates (see Section 4.3.2).
We compare the measured atom laser profiles along its symmetry axes with numerical
simulations of the time evolution using the Gross - Pitaevskii equation. The resulting density distributions of the atom laser deviate slightly from a Gaussian shape [106], but the
measured convolutions with the cavity mode agree very well with the simulated curves
[Figure 4.9(b) and Figure 4.9(c), gray]. The overestimated width along the Ioffe axis can be
explained be the angle of the BEC axis with respect to the horizontal plane, reducing the
spatial width of the output coupling region along the Ioffe axis.
Along the cavity axis the slight deviation at the edges is probably due to pointing variations, i.e., transverse oscillations of the atom laser beam. Small collective oscillations in the
trap are translated into deflections of the atom laser beam over which we integrate with our
detection method. The collective oscillations, mainly center of mass dipole oscillations in
the trapped Bose - Einstein condensate, can be excited by radio frequency evaporation or
incautious relaxation of the magnetic trap.
However, we can exploit this effect to precisely determine the frequencies of excited collective oscillations in the trap by analyzing the Fourier spectrum of the atom count rate.
Such a spectrum is shown in Figure 4.10 exhibiting harmonics of the trap frequencies (dipole
oscillations) and mutual sidebands. The frequencies can be measured in situ with one and
the same experimental implementation of a Bose - Einstein condensate to high precision
(millihertz), Fourier limited by the duration of the atom laser recording.
70
4.2. SINGLE ATOM DETECTION PERFORMANCE
fourier amplitude [ arb. u. ]
5
4
7.2 Hz
3
29.1 Hz
2
2 x 38.5 Hz
1
0
0
10
20
30
40
50
frequency [ Hz ]
60
70
80
FIGURE 4.10.: The Fourier spectrum of the detected atom laser flux exhibiting the trapping
frequencies and their harmonics. A fast and precise tool to measure frequencies of collective
oscillation in the trap.
4.2.6. Guiding the Atom Laser
The reason for the single atom detection efficiency not being unity is mainly the mismatch
of the atom laser and cavity mode sizes [Figure 4.9(d)]. Their overlap is only about 50%
assuming a box given by the projected length of the cavity mode and a minimum peak atom
field coupling strength of gmin
= 2π · 6.5 MHz in the radial direction. Calculating the atom
0
trajectories taking into account the channeling effect of dipole potential we find a maximum
single atom detection efficiency of 80% and an averaged efficiency of about 50% within this
box. This is in good agreement that we detect about one quarter of the released atoms.
In order to increase the overlap and therefore the detectable number of atoms it is possible to funnel the atoms with a dipole potential created by a far red-detuned guiding laser
(850 nm, 15 mW, beam waists of 30 × 60µm2 ) into the cavity mode. By doing so we are able
to improve the single atom detection efficiency by about a factor of 2 to around 50%. This
number still differs from a perfect detection efficiency because the dipole potential formed
by the probe laser is simply not strong enough to perfectly localize the atoms in the axial
direction at the antinodes of the standing wave.
Although we are able to increase the single atom detection efficiency, employing the guiding laser involves some disadvantages. The scattering and heating rate in the dipole potential formed by the guiding laser can cause modifications of the atom arrival time statistics
which is undesirable for many experiments [17]. Furthermore, the guiding laser acts on both
thermal and quantum degenerate atoms and therefore diminishes a characteristic feature
of our detector, namely, the very sensitive discrimination of thermal and condensed atom
count rates (see Section 4.3).
71
4. CHARACTERIZATION OF THE SYSTEM
4.3. Investigation of Ultracold Atomic Gases
The combination of a Bose - Einstein condensate with an ultrahigh finesse optical cavity
enables us to detect single atoms from a quantum degenerate gas with very high sensitivity.
Therefore we can employ the cavity as a minimally invasive probe for cold atomic clouds.
This allows us to perform nondestructive measurements on the ensemble of cold atoms in
situ and time resolved. Assuming a constant, weak output coupling power, the atom count
rate of the cavity detector depends on the properties of the source via two factors. First of
all the number of output coupled atoms is proportional to the number of atoms fulfilling
the resonance condition, i.e., the one-dimensional density at the output coupling plane.
And secondly the atom count rate depends on the probability for an output coupled atom
to hit the cavity mode. Because of its finite active area the cavity functions as a filter in
momentum space.
4.3.1. Thermal Clouds
For a thermal cloud the density at the output coupling central plane is proportional to
Nth T −1/2 and the probability to hit the detector is proportional to 1/T assuming Gaussian
density and momentum distributions. Therefore the thermal atom count rate detected with
the cavity is proportional to Nth T −3/2 . This dependency is shown in Figure 4.11. At the critical temperature of Tc ≈ 180 nK for 107 atoms only about 0.6% of the output coupled thermal
atoms will fly through the cavity mode and can possibly be detected.
atom count rate [ ms-1 ]
8
6
atom laser threshold
(onset of BEC)
4
2
0
0
0.2
0.4
0.6
0.8
-3/2
-3/2
NthT [ 1 / (NcTc ) ]
1.0
FIGURE 4.11.: Investigation of atom count rates for thermal beams. The atom flux is proportional to Nth T −3/2 for temperatures above the critical temperature and exhibits a threshold
behavior when cooling across the phase transition. Just above Tc the density and momentum distributions of the thermal cloud are governed by a Bose distribution and obey a different scaling law than expected for a Gaussian distribution.
72
4.3. INVESTIGATION OF ULTRACOLD ATOMIC GASES
The onset of Bose - Einstein condensation can be clearly seen in the sharp increase of detected atoms [143, 35]. Close to the critical temperature, however, the detected atom flux
slightly deviates from the expected behavior because the approximated Gaussian distributions for density and momentum are not valid anymore near Tc . The thermal cloud is described by the more peaked Bose distribution which yields an increased atom detection rate
of about 30% near the critical temperature of 180 nK compared to the Gaussian distribution.
4.3.2. Quantum Degenerate Gases
atom count rate [ s-1 ]
For Bose - Einstein condensates the probability for an atom to hit the cavity mode and therefore the atom count rate detected with the cavity is independent of temperature. The number of resonant atoms participating in the output coupling process is proportional to the
density of the BEC and the area of the output coupling plane. This means the atom flux is
proportional to N04/5 because the Thomas - Fermi radius of a BEC scales as N01/5 .
104
N04/5
103
102
101
100
103
104
105
N0
106
107
FIGURE 4.12.: Investigation of detected atom count rates for pure quantum degenerate samples. The scaling with the atom number in a pure BEC exhibits three different regimes. The
expected N04/5 behavior is only valid for intermediate particle numbers. Very small and
very large condensates obey different scaling laws due to an increased Heisenberg limited
momentum spread and the mean-field repulsion of the remaining condensate, respectively.
However, this dependency is only true for Bose - Einstein condensates of intermediate
size and deviates for very small and very large condensates as shown in Figure 4.12. Output
coupling from small condensates accounts for a faster quantum mechanical expansion of
the initial ground state wave function in the atom laser. Therefore the overlap between the
transverse atom laser wave function and the cavity mode is reduced. Large condensates on
the other hand exhibit increased divergence because of the mean-field repulsion exerted on
the atom laser propagating through the BEC. The condensate acts as an imperfect diverging
73
4. CHARACTERIZATION OF THE SYSTEM
lens and displaces the maximum density outward [142, 106, 144]. This results in a weaker
scaling of the detected atom flux with the number of atoms in the BEC and possibly a decrease when the atom laser profile becomes more “donut-mode-like”. These three regimes
are displayed in Figure 4.12 for measured atom count rates versus the number of atoms in
the “pure” BEC. The exact position of the crossover between these regimes depends on the
active area of the single atom detector.
4.3.3. Phase Transition
Our single atom detector in form of the high finesse optical cavity is extremely sensitive
and selective to quantum degenerate atoms not only because of the increased density at the
output coupling region but also due to the filtering in transverse momentum space.
This means we can more accurately observe the onset of Bose - Einstein condensation as
compared to absorption imaging techniques. The exact determination of the critical temperature, in combination with the precisely measured trap frequencies (see Figure 4.10),
allows one in turn to calibrate the atom number obtained by the absorption images.
Furthermore we are able to survey the density distribution in the trap along the vertical direction by scanning the resonant plane for the output coupling process through the
trapped cloud of cold atoms as shown in Figure 4.13. For temperatures close to the critical temperature the density distribution of the thermal cloud already deviates from the
Gaussian shape and has to be described by the more peaked Bose distribution [Figure 4.13,
(134 nK)]. For temperatures slightly below Tc single atom detection with the cavity allows
us to observe and map very small condensates that are not visible in absorption images
[Figure 4.13, (128 nK) and (123 nK)]. This could be a valuable tool to study the temporal and
spatial evolution of the bosonic gas at the phase transition.
74
4.3. INVESTIGATION OF ULTRACOLD ATOMIC GASES
4
atom count rate [ ms-1 ]
123 nK
3
128 nK
2
1
134 nK
0
-40
-20
0
20
output coupling position [ μm ]
40
absorption
images
FIGURE 4.13.: Analysis of the density distribution of the trapped ultracold atom gas by output
coupling at different vertical positions relative to the center of the BEC and measuring the
resulting atom count rate with the cavity. The profiles for three different temperatures
around Tc are shown in comparison with the absorption images. The high sensitivity of the
cavity detector to quantum degenerate atoms allows for precise observation of the onset of
Bose - Einstein condensation and the deviations from a Gaussian profile (gray curve).
75
5 Correlations and Counting Statistics of
an Atom Laser
“. . . a widely believed fact.” - Fry, Futurama
We demonstrate time resolved counting of single atoms extracted from a weakly interacting Bose - Einstein condensate of 87 Rb atoms. The atoms are detected with an ultrahigh
finesse optical cavity and single atom transits are identified. An atom laser beam is formed
by continuously output coupling atoms from the Bose - Einstein condensate. We investigate the full counting statistics of this beam and measure its second order correlation function g (2) (τ) in a Hanbury Brown and Twiss type experiment. For the monoenergetic atom
laser we observe a constant correlation function g (2) (τ) = 1.00 ± 0.01 and an atom number
distribution obeying Poissonian statistics, whereas a pseudo-thermal atomic beam shows a
bunching behavior and a Bose distributed counting statistics.
Alternatively, the two-particle correlations are illustrated by evaluating the exclusive
probabilities of particle detection both for the conditioned and the unconditioned case. In
addition, we show the increased third order correlations (by a factor ∼ 3! = 6) of the pseudothermal compared to the uncorrelated atom laser.
The experimental results of this chapter were published in [17]: A. Öttl, S. Ritter, M. Köhl,
and T. Esslinger, Correlations and Counting Statistics of an Atom Laser, Phys. Rev. Lett. 95 (2005),
090404.
77
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
5.1. Introduction
Correlations between identical particles were first observed by Hanbury Brown and Twiss in
light beams [48]. Their idea was that intensity fluctuations and the resulting correlations
reveal information about the coherence and the quantum statistics of the probed system.
This principle has found applications in many fields of physics [51] such as astronomy [145],
high energy physics [53], atomic physics [54] and condensed matter physics [55, 56]. In
optics, the reduced intensity fluctuations of a laser have been observed by Arecchi [44] only
a few years after its invention, thereby disclosing the extraordinary properties of this light
source.
With the realization of Bose - Einstein condensation in dilute atomic gases a novel weakly
interacting quantum system is available. The interpretation of a Bose - Einstein condensate
representing a single, macroscopic wave function has been supported in numerous experiments highlighting its phase coherence [5, 146, 147, 105]. Correspondingly, atom lasers are
atomic beams which are coherently extracted from Bose - Einstein condensates [8, 9, 10, 11].
Their first order phase coherence has been observed both in space [5] and time [42]. However, only the second order coherence reveals whether atom lasers exhibit a truly laser-like
behavior. Here we present a measurement of the temporal second order correlation function g (2) (τ) of an atom laser in a Hanbury Brown and Twiss type experiment. Measuring all
atom arrival times explicitly with a single detector of high quantum efficiency enables us to
extract the full counting statistics of atomic beams.
The second order correlation function g (2) (τ) represents the conditional likelihood for
detecting a particle a time τ later than a previously detected particle and quantifies second
order coherence [40]. For a thermal source of bosons g (2) (τ) equals 2 for τ = 0 and decreases
to 1 on the time scale of the correlation time which is given by its energy spread. For a
coherent source g (2) (τ) = 1 holds for all times and therefore intensity fluctuations are reduced to the shot noise limit. Higher order coherence in quantum degenerate samples was
so far only studied in the spatial domain where atom atom interactions reveal the short distance correlations [59]. In an interferometric measurement g (2) (r ) has been determined for
elongated, phase fluctuating condensates [148], and recently spatial correlation effects in
expanding atom clouds were observed [16].
We demonstrate the detection of single atoms from a weakly interacting quantum gas by
employing an ultrahigh finesse optical cavity [20, 21] as illustrated in Figure 5.8. A different
technique with the potential of single atom detection in quantum degenerate samples has
been demonstrated with metastable Helium atoms [34]. Detecting the arrival times of all
atoms at the cavity explicitly gives access to the full counting statistics that reveals the atom
number distribution function and its statistical moments [62, 149]. Determining the full
counting statistics goes far beyond a measurement of the intensity correlation function only,
because it represents the full statistical information about the quantum state. Despite recent
78
5.2. BACKGROUND
progress, especially in mesoscopic electronic systems [64], the full counting statistics has
not been measured for massive particles before. For neutral atoms this quantity is of special
interest, since the strength of the interaction does not overwhelm the quantum statistics as
it is often the case for electrons.
5.2. Background
Before we present our results, an overview is given about coherence of first and higher order and its relation to correlation functions. These put into context the bunching effect
in two-particle correlations of thermal beams, first observed by Hanbury Brown and Twiss
and the quantum theory of coherence developed by Glauber. Lastly, the features of statistical distribution functions are described which contain the complete information about a
stochastic quantum system in form of the full counting statistics of particle distributions.
The textbook by Paul [150] provides a good introduction to photon statistics and coherence.
5.2.1. First Order Coherence
Conventional optical interferometers measure the correlation function 〈E ∗ (r1 , t 1 ) E (r2 , t 2 )〉,
which compares the complex values of the electromagnetic field at different points in spacetime. This quantity is called first order coherence or phase coherence and describes the
ability of fields to interfere and exhibit interferences fringes. In its normalized form
〈E ∗ (r1 , t 1 ) E (r2 , t 2 )〉
g (1) (r1 , t 1 ;r2 , t 2 ) = rD
¯
¯ E D¯
¯ E
¯E (r1 , t 1 )¯2 ¯E (r2 , t 2 )¯2
(5.1)
the magnitude determines the contrast of the interference fringes in terms of the visibility
V as introduced by Rayleigh
|g (1) | = V =
I max − I min
.
I max + I min
(5.2)
The visibility ranges from zero, for incoherent electric fields, to one, for coherent electric
fields. Anything in between is described as partially coherent.
The above defined field correlation function refers both to spatially and temporally separated points in space-time and is therefore suited to describe all single-particle interference phenomena, like Young’s double slit experiment or optical interferometers such as the
Michelson or Mach - Zehnder type. All these geometries have in common that the electric
field is split into two components, one part being spatially offset and/or time delayed with
respect to the other part, and finally recombined. The intensity of the resulting field is measured as a function of spatial variation or time delay and exhibits interference fringes as
long as the two quantum mechanical paths (amplitudes) are indistinguishable. For theses
experiments considered so far the famous quote of Dirac applies that “each photon interferes
79
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
only with itself”. However, this can not be generalized to higher order phenomena like the
Hanbury Brown - Twiss effect [151], since not photons interfere but quantum mechanical
amplitudes !
In practice, most optical interferometers are concerned with measuring the temporal coherence which allows insights to the spectral properties of the light and its emission process.
For stationary fields only the time delay τ = t 2 − t 1 enters g (1) and assuming plane waves the
spatial retardation can be included through τ = ξ/c , where ξ = x2 − x 1 . So the discussion
simplifies to the temporal case
g
(1)
­ ∗
®
E (t )E (t + τ)
(τ) = D¯
.
¯ E
¯E (t )¯2
(5.3)
The angle brackets 〈 〉 denote the ensemble or statistical average, which for nonstationary
states, such as pulses, is made up of many shots. For stationary states the statistical properties do not change with time, so the ensemble average is equivalent to a time average
∗
〈E (t ) E (t + τ)〉 =
1
Z
T
T 0
E ∗ ( t )E ( t + τ )d t .
(5.4)
This is equivalent to the autocorrelation function of the electric field and visualized in Figure 5.1. Since a convolution is most easily performed in Fourier space, where it reduces to a
multiplication it is obvious that the first order correlation function relates to the normalized
power spectrum of the light source via an inverse Fourier transform
Z ∞
1
(1)
−1
g (τ) = F [P (ν)] = p
P (ν)e i 2πντ d ν ,
(5.5)
2π −∞
which is a form of the Wiener - Khintchine theorem.
time
t+τ
t
FIGURE 5.1.: Schematic illustration of the overlap integral in equation (5.4) for two wavetrains incident on a detector. In the marked region it is ambiguous which of the wavepackets
is detected.
For classical fields, general properties of the correlation function g (1) can be derived [152].
It is symmetric g (1) (τ)∗ = g (1) (−τ) and maximum for no time delay, i.e., g (1) (0) = 1, dropping
off to zero over a timescale τc . This correlation time τc is defined as
Z
τc =
∞
|g (1) (τ)|2 d τ
−∞
and is approximately given by the reciprocal of the spectral bandwidth ∆ω.
80
(5.6)
5.2. BACKGROUND
The discussion of first order coherence can equally be applied to matter wave fields (see
Chapter 6) in the formalism developed by Glauber which is presented in setion 5.2.3. For
output coupling two atom lasers at two different planes within the Bose - Einstein condensate it is the phase coherence between the “slits” that determines the ability of these atom
lasers to interfere [105]. The g (1) (r1 , r2 ) of atomic clouds is basically the Wigner function in
the phase space representation of the many-particle system [80].
5.2.2. Hanbury Brown - Twiss and the Invention of Bunching
In the mid-fifties Robert Hanbury Brown, in collaboration with Richard Twiss who worked
out the theoretical details, invented a revolutionary new measuring technique (see Figure 5.2). It is based on measuring correlations between intensities and fluctuations thereof,
rather than correlations of the electromagnetic field. The intensity interferometer was
particularly developed to measure stellar diameters, where it overcomes major difficulties
faced with conventional Michelson interferometers, because the very delicate phase relation of two signals at different space-time points is discarded. However, the Hanbury Brown
- Twiss effect caused considerable uproar and disbelief in the physics community: how could
photons coming from opposite edges of a star exhibit correlations although being emitted
by independent sources ? The mystery was resolved by Purcell in a notable publication to
support the Hanbury Brown - Twiss experiment [60], where he gives an intuitive explanation of the origin and calculates the magnitude of the positive cross-correlations observed
by Hanbury Brown and Twiss in two coherent beams of thermal light, known nowadays as
“bunching”. Purcell praises the contended Hanbury Brown - Twiss method as “an instructive
illustration of the elementary principles of quantum mechanics” and in along with his discovery
of modified spontaneous emission [153], this can be regarded as the birth of quantum optics. The Hanbury Brown - Twiss technique found its way into many divers fields of physics
where correlations between particles reveal insight into fundamental processes [53, 50, 51]
and quantum statistics [54, 55, 56, 57, 17, 18].
The measuring principle of the Hanbury Brown - Twiss technique is based on interrogating and correlating the local intensities simultaneously at two different positions within a
beam. A coherent beam of thermal light will show intensity correlations in excess to completely random coincidences of incoherent beams. By varying the separation of the detectors, the coherence angle of a light beam could be mapped out and the size of the source
inferred. In the original version [47], the Hanbury Brown - Twiss experiment was carried
out in the radio frequency band with two antennas as square law detectors of the electromagnetic field from the same stellar object as illustrated in Figure 5.2.
They measured increased correlations in the fluctuations of the two distinct detectors
of the form 〈∆I 1 ∆I 2 〉 that vanish when the detectors are separated by more than a certain
length scale ξc . The AC term ∆I i is given by I i −〈I i 〉, where 〈I i 〉 is the time averaged intensity.
Considering classical electromagnetic waves, the correlations in the intensity fluctuations
81
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
star
τ
filter
antennas
filters
lamp
aperture
correlator
c
beam
splitter
c
FIGURE 5.2.: Schematic illustrations of the renowned Hanbury Brown - Twiss experiments.
(left) The original version in the radio frequency band, developed and successfully applied
to measure stellar diameters. The intensity correlations from two antennas illuminated by
the same rf field are recorded. (right) Table top experiment as a proof of principle in the
optical domain. Thermal light from a lamp is filtered spatially (aperture) and energetically
(color filter). The beam is divided by a half-silvered mirror and cast onto two symmetric
photon counters. The read out is done by an electronic coincidence unit. The spatial and
temporal second order correlation function can be measured by varying the lateral position
of a detector or the time delay.
can relatively easy be explained [154, 51]. Assuming two spherical waves of distant origin,
the total intensity at the detectors can be calculated. The correlation term 〈I 1 I 2 〉 depends
on the distance ξ between the detectors through a compact spatial function f (ξ) that depends on the size of the source 〈I 1 I 2 〉 = 〈I 1 〉 〈I 2 〉 + I 2 f (ξ). Because the average intensities are
independent of detector separation 〈I 1 〉 = 〈I 2 〉 = I there exist correlations
〈∆I 1 ∆I 2 〉 = 〈I 1 I 2 〉 − 〈I 1 〉 〈I 2 〉 ∝ I 2 f (ξ)
(5.7)
for ξ < ξc . For separations ξ > ξc the function f (ξ) tends to zero and consequently
〈∆I 1 ∆I 2 〉 = 0, as expected for completely uncorrelated signals, where 〈I 1 I 2 〉 factorizes.
The Hanbury Brown - Twiss effect can equally be observed in the temporal domain by
introducing a delay time τ between intensity measurements at the same position. However,
since the excess correlations measured by Hanbury Brown - Twiss are not normalized, it
became more practical to represent them in the more universal form of the second order
82
5.2. BACKGROUND
correlation function g (2) [155]
g (2) (τ) =
〈I (t ) I (t + τ)〉
〈I (t )〉
2
(5.8)
.
A typical behavior of the second order correlation function exhibiting the bunching effect
for thermal fields is illustrated in Figure 5.3. The original Hanbury Brown - Twiss correlations are proportional to g (2) − 1. For thermal light, the stochastic behavior of I (t ) can be
described through the first order correlation function g (1)
〈I (t ) I (t + τ)〉 = 〈I 〉2 (1 + |g (1) (τ)|2 )
(5.9)
and consequently by its spectral properties [60, 152]. This formula shows the relation between the first and second order coherence. Limiting values for g (2) (τ) of thermal, chaotic
light follow for instance from the Cauchy - Schwartz inequality 〈I 2 〉 ≥ 〈I 〉2 yielding
g (2) (0) ≥ 1
and
g (2) (τ À τc ) → 1 .
(5.10)
For spatially coherent thermal beams g (2) (0) = 2. This follows from eq. (5.9), or the expression 〈I 2 〉 = 2 〈I 〉2 when summing up all complex electric fields of infinitely many random
emitters (Wick’s Theorem), which is the case in thermal or chaotic radiation [152].
g(2)
2
1
τc , ξc
τ,ξ
FIGURE 5.3.: Characteristic second order correlation function g (2) for thermal radiation. The
distinct feature of bunching can be observed both in the temporal and spatial domain. The
width of the bunching is determined by the inverse bandwidth and the inverse momentum
spread, i.e., divergence angle, respectively.
These considerations for classical waves notwithstanding, the scepticism centered on the
Hanbury Brown - Twiss experiment in the optical regime [48, 156, 145] illustrated in Figure 5.2. It was intended as a proof of principle for their measuring technique and further
developed to be applicable in astronomy as demonstrated with measurement on Sirius [49].
83
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
In the optical however, unlike analog signals from rf antennas which were subsequently
multiplied, more or less digital clicks from photon counters were correlated. This caused
people to call for a revision of quantum mechanics, because then the controversy about the
photoelectric detection process and the quantization of the electromagnetic field was still
vivid. However, starting from a gedanken experiment using a single perfect detector, Purcell
in his paper supporting paper [60] showed that these correlations are a natural consequence
of the Gaussian fluctuations of the electric field. He calculated the excess coincidences over
random events to be proportional to the squared count rate and a reduction factor which
arises from the finite time resolution T of the photo counters
〈∆n 1 ∆n 2 〉 =
1
〈n〉2 τc /T .
2
(5.11)
This illustrates the fact that the Hanbury Brown - Twiss correlations diminish when integrated over long times compared to the coherence time τc . In fact τc /T was on the order
of 1/5000 in the original Hanbury Brown and Twiss experiment and only thanks to the high
count rate they succeeded in observing the bunching effect, in contrast to their contenders
[156].
a
source
b
1
detector
2
FIGURE 5.4.: A generic Hanbury Brown - Twiss setup. Independent sources emit particles
that are detected by independent detectors. However, when the particles are indistinguishable for the detectors, Bose - Einstein statistics comes into effect. This results in a positive
interference term, giving a factor two in the joint count rate as opposed to distinguishable
classical particles.
Although the Hanbury Brown - Twiss experiment can be understood completely in terms
of classical electromagnetism it is a fine verification of the laws of quantum mechanics for
identical bosons. What the experiment shows is that the bosonic nature of photons is already contained in the superposition principle obeyed by classical electromagnetic fields.
To derive a quantum mechanical interpretation of the Hanbury Brown - Twiss effect, let us
regard the generalized gedanken experiment [157] illustrated in Figure 5.4. Two bosonic
particles a and b , which do not necessarily have to be photons, are emitted by independent
sources and are detected by two independent detectors 1 and 2. The quantum mechanical amplitude of detecting particle a in detector i can be written as 〈a|i 〉. In the same
way, the amplitudes 〈b|i 〉 for the second particle are defined. For simplicity, we will assume
that the amplitudes of detecting one particle in either one of the detectors is equal, i.e.,
〈a|1〉 = 〈a|2〉 ≡ a and 〈b|1〉 = 〈b|2〉 ≡ b . Now, we ask for the probability of detecting exactly
84
5.2. BACKGROUND
one particle in each detector. This probability depends on whether or not the particles are in
principle distinguishable, e.g., due to their energies, momenta, internal degrees of freedom,
etc.
• For distinguishable particles, we just have the classical sum of probabilities. The total
probability is the probability of detecting a in 1 and b in 2 plus the reversed case:
P d = | 〈a|1〉 〈b|2〉 |2 + | 〈a|2〉 〈b|1〉 |2 = 2|a|2 |b|2
(5.12)
• In the case of indistinguishable bosonic particles we have constructive interference of
the two possible paths which means that in this case the probability amplitudes add:
P i = | 〈a|1〉 〈b|2〉 + 〈a|2〉 〈b|1〉 |2 = |2ab|2 = 4|a|2 |b|2 = 2P d
(5.13)
Consequently, for quantum mechanically indistinguishable particles it is twice as probable
to detect two of them simultaneously than for classically distinguishable particles. This
effect accounts for the increased correlations in chaotic light sources on short time scales
τ < τc and is called bunching, or the Hanbury Brown - Twiss effect. Clearly, this phenomenon is
not a property of the source(s) since the emission happens to be completely independent.
It is solely the indistinguishability of quantum mechanical amplitudes at each detector that
renders this effect possible.
Pauli’s exclusion principle prevents two identical fermions to be in the same quantum
mechanical state or likewise at the same position in space-time. Their two-particle wave
function is stated in its antisymmetric form and therefore the quantum mechanical amplitudes don’t sum up but are subtracted, leading to a complete absence of correlations. This
behavior, called antibunching has been observed in three remarkable experiments performing the Hanbury Brown - Twiss technique with confined [55, 56] and free [57] electrons.
The notion of identicalness can be further discussed in terms of coherence. Two particles
a, b arriving at a detector are quantum mechanically indistinguishable when they are detected within their coherence volume. This virtual volume spanned by the overlap of their
extending wavefunctions can be quantified by the spatial coherence length in the transverse
to the propagation direction and the longitudinal or temporal coherence.
The spatial coherence length ξc is given by the inverse spread of the wave vector and is a
measure of collimation of the beam, i.e., the planarity of its wavefronts
ξc =
2π
∆k
=
λ
,
θ
(5.14)
where λ is the wavelength of the light and θ = s /l the angular size of the source as seen from
the detector. The spatial coherence can be determined in the Hanbury Brown - Twiss technique by varying the separation of the detectors (see Figure 5.2), provided the integration
time is short compared to the coherence time. It is interesting to note, that the strategy to
85
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
use a beam splitter solely arose form the requirement to measure towards zero separation,
i.e., for completely overlapping detector images, which would otherwise not have been possible due to the physical size of the detectors. The use of a half-silvered mirror as a beam
splitter, in contrast to a polarizing beam splitter, is essential to ensure the identicalness of
photons in each path.
The transverse coherence length ξc is also the characteristic size of the speckle pattern
[158], which illustrates that the correlations are diminished for detectors having an active
diameter d > 2ξc . So, if either the source or the detector are spatially too extended, the
particles become in principle distinguishable and the Hanbury Brown - Twiss effect is lost.
The criterion to observe bunching is therefore the “single mode” behavior of a beam which
means that all particles of a beam are within a single phase space cell. In momentum space
this can be achieved when being in the far field limit or being diffraction limited in other
words. This was well granted for observation of stars, but poses a limitation on the experiments with cold atoms. There, the correlation length after a time of flight t is given by
ξc =
ħt
,
ms
(5.15)
where s is the size of the source and m the mass of the particles [18]. The bunching is then
reduced by ∼ ξc /d in one dimension, where d is the size of the detector.
The coherence time, as already stated in equation (5.9), is related to the bandwidth ∆ω of
the light or more generally to the energy spread of the particles
τc =
2π
∆ω
=
ħ
.
∆E
(5.16)
The temporal second order correlation function can be recorded when varying the delay
time between particle detections at the same position. When associated with the velocity v
of massive particles the temporal coherence can be translated into a longitudinal coherence
ξc = v τc . For a continuous beam, the correlation time, or equivalently the longitudinal
correlation length, depends only on the velocity width of the source and not on the source
size. Thus, the longitudinal and transverse directions are qualitatively different. For thermal
atomic gases the second order correlation function is given by [54]
g (2) (τ) = 1 + p
1
1 + (∆ωτ)2
with
∆ω =
mv s2
2ħ
,
(5.17)
where v s represents the velocity width of a thermal source with a Gaussian velocity distribution.
Since Hanbury Brown and Twiss were primarily interested in spatial coherence as a tool
to determine stellar diameters, the issue of temporal correlations was only later addressed
in separate experiments [159, 160].
Although a single detector would be sufficient in the time domain, the double detector
geometry proved useful to overcome the dead time of photon counters for time correlated
86
5.2. BACKGROUND
measurements. So the time dependence of the intensity correlation function could be determined by coincidence counting techniques down to zero time delay [45]. However, measuring start-stop signals with two consecutive events only gives an approximate second order
correlation function. The photon count rate has to be much less than the inverse coherence time, such that having a third event (of unrecorded time delays) within the relevant
correlation time is negligible.
τi
t
time
FIGURE 5.5.: Extracting g (2) (τ) from all possible detection time differences rather than simple
start-stop signals.
When recording single events, their temporal second order correlation function can be
equally expressed as the conditional likelihood P c (t |t + τ) to detect a particle at time t + τ
after detecting an initial one at time t
g (2) (τ) =
〈I (t ) I (t + τ)〉
〈I (t )〉
2
= P c (t |t + τ) .
(5.18)
The second order correlation function in that sense is a nonexclusive property, because it
does not matter if any other particles are detected within the time interval τ. The exact
g (2) (τ) is recovered by taking all possible time differences of recorded single particle detection events (see Figure 5.5) and plotting them as a histogram. If the overall recording
time T is comparable to the relevant time scales, a correction factor T /(T − τ) must be applied to take into account that longer time differences become less likely. A proper normalization (5.10) of the histogram is obtained by considering the total number of pairs, being
N (N − 1)/2 for N single particle events, per available time bins.
5.2.3. Glauber and the Quantum Theory of Coherence
Glauber’s theory deals with the quantum theory of electromagnetic fields, in particular with
the theory of coherence and photon correlations [40]. It provides the required analytical
tools, such as coherent states and P-representation, to describe light fields and the photon
detection process in a full quantum mechanical treatment and in terms of correlation functions. This became necessary to explain the photon correlation experiments established
by Hanbury Brown and Twiss and through the development of quantum light sources such
as lasers. Naturally, it is applicable to nonclassical quantum states of light. In the context of atom optics the quantum theory of coherence finds its natural extension to matter
wave fields and successfully incorporates all classical (thermal) and quantum (bosonic and
fermionic) states of de Broglie waves.
87
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
Starting with the more customary optical theory, the real electric field E = E + +E − is split
into two complex conjugate terms, where E + contains only the positive frequencies ∼ e −i ωt
and vice versa. The E ± are physically equivalent and define the electric field operators Ê (±)
in the Heisenberg picture. For a single mode we therefore have
q
³
´
Ê (χ) = Ê (χ) + Ê (χ) = ħω/2ε0V â e −i χ + â † e i χ ,
+
−
(5.19)
in analogy to equation (2.49). The phase angle χ = ωt − kz − π/2 contains the explicit time
and position dependence. The â and â † represent the annihilation and creation operators
of a photon in the particular mode under consideration
a † |n〉 =
p
n + 1 |n + 1〉
and
a |n〉 =
p
n |n − 1〉 .
(5.20)
The normalized time dependent second order correlation function in its quantized version
then reads
〈â † (t 1 )â † (t 2 )â (t 2 )â (t 1 )〉
(2)
.
(5.21)
g (t 1 , t 2 ) = †
〈â (t 1 )â (t 1 )〉〈â † (t 2 )â (t 2 )〉
Here, the angle brackets 〈 〉 represent quantum mechanical expectation values. For a pure
¯
state ¯φ〉 the observable Ô is given by 〈φ|Ô|φ〉. For a system in a mixed state the average
P
value is given by the statistical mixture i P i 〈φi |Ô|φi 〉 or equivalently by the trace over the
density operator ρ̂
〈Ô〉 = Tr{ρ̂ Ô} .
(5.22)
In contrast to its classical form (eq. 5.1), the only limiting relation for the second order correlation function in its quantum form is that it is positive
0 ≤ g (2) (t 1 , t 2 ) ≤ ∞ .
(5.23)
This follows from the fact that the numerator in equation (5.21) is an expectation value of
Hermitian operators.
The g (2) (t 1 , t 2 ) represents the joint probability to detect two particles at times t 1 and t 2 ,
respectively. For stationary light beams, the second order correlation function will only
depend on the time difference τ = t 2 −t 1 between two particle detection events. It is because
the correlation is defined as the average of a normally ordered product of field operators,
that it expresses the correlation of pairs of particles. Normally ordered refers to the sorting
of all annihilation operators to the right of the creation operators and time sorting from
the “outside in”. The density correlation 〈â † (t 1 )â (t 1 )â † (t 2 )â (t 2 )〉 is equivalent to the second
order correlation 〈â † (t 1 )â † (t 2 )â (t 2 )â (t 1 )〉 except for the autocorrelation at t 1 = t 2 due to the
bosonic commutation relation [â (t 1 ), â † (t 2 )] = δ(t 1 − t 2 ).
The quantum treatment establishes the interpretation of g (2) as particle correlations
rather than intensity correlations. Therefore it can be equally applied to massive particle
by simply replacing the photon creation/annihilation operators â (†) with the atomic field
88
5.2. BACKGROUND
operators Ψ̂(†) . Moreover, Glauber’s theory applies for general correlations of order n and
he was constructing a state that allows all higher order correlation functions to factorize
according to
〈Ψ̂† (t 1 )Ψ̂† (t 2 ) . . . Ψ̂(t 2 )Ψ̂(t 1 )〉 = ψ∗ (t 1 )ψ∗ (t 2 ) . . . ψ(t 2 )ψ(t 1 ) ,
(5.24)
where ψ is a wave function. This is achieved with a coherent state. Full coherence is then
defined as g (n ) ≡ 1, which can only be fulfilled up to order O (N ) of the particle number N .
Because of the normal ordering N annihilation operators are applied before the N creation
operators. This is in stark contrast to a thermal state. With the gedanken experiment illustrated in Figure 5.4 it is straightforward to calculate the joint detection probabilities of
multiple bosonic particles on multiple detectors. Summing up all processes, it turns out that
the probability of simultaneously detecting n thermal bosons is enhanced by the factor n !
compared to random coincidences. This was confirmed in measurements of the three-body
recombination rates of thermal and Bose condensed atomic clouds [58, 59]. The same result is found when calculating the higher order moments for a thermal light field having a
Gaussian distribution of the electric field value
n
〈I (t )n )〉 = n ! 〈I (t )〉 .
(5.25)
Coherent states were introduced by Schrödinger in the treatment of the harmonic oscillator
in search for solutions to his equation that satisfy the correspondence principle. Glauber
coined their use in quantum optics and other bosonic quantum field theories [40, 41]. The
coherent state |α〉 is defined to be an eigenstate of the annihilation operator â
â|α〉 = α|α〉 ,
(5.26)
where α is just a complex number. This means the coherent state |α〉 remains unchanged
when a quantum is removed. Therefore it is clear that it can not contain a definite number
of photons. The representation of a coherent state in the basis of the number or Fock states
|n〉 reads
1
|α〉 = e − 2 |α|
2
∞ αn
X
p |n〉 .
n=0 n !
(5.27)
When a mode is in a coherent state the probability of finding n quanta is a Poisson distribution
P (n ) = | 〈n|α〉 |2 =
|α|2n −|α|2
e
,
n!
(5.28)
and the mean number of quanta in this distribution is found to be 〈n〉 = |α|2 .
Glauber showed that coherent states can be generated by a displacement of the harmonic
oscillator ground state |0〉, which is a coherent state itself with α = 0. He defined the displacement operator
†
∗
D (α) = e αâ −α â
(5.29)
89
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
that achieves just that |α〉 = D (α) |0〉. Since a coherent state is just a displaced version of
the the vacuum state |0〉 it has also the same properties in terms of uncertainty. The only
contribution to the quantum noise are vacuum fluctuations. The coherent state of a field is
the closest a quantum state can come to the ideal classical limit of simultaneously perfectly
well defined amplitude and phase. This is true for any other pair of conjugate variables and
illustrated in the phase space representation in Figure 5.6. The coherent states reduce the
uncertainty product to its minimum value ∆x · ∆p = ħ/2.
p
Δφ
Δp
ħ/2
α
Δx
x
FIGURE 5.6.: Phase space representation of a coherent state |α〉.
Glauber is also noted for the introduction of the P- and Q-function representation of the
density matrix. These functions are quasi probability distributions in phase space, related
to the Wigner function. They are not limited to positive values, like classical probability
distributions and apparently well suited to solve complex problems in quantum optics.
5.2.4. Counting Statistics
The full counting statistics contain the complete information about a mixed quantum state.
Therefore the knowledge of the counting statistics is more fundamental than any particular correlation function. In fact it is equivalent to determining all higher order correlation
functions [161]. However, it is experimentally much more difficult to record the full counting statistics, than the second order correlation function. The counting statistics measure
the probability p (n, T ) to detect n particles within a certain time window of width T . Of
course, this quantity is only of significance when the binning time is shorter then the correlation time (T < τc ), which can only be achieved when the integration and dead time of
the detector are also much shorter than the correlation time. For T À τc the contribution
90
5.2. BACKGROUND
of the correlations is negligibly small and the statistics will be totally random and therefore
always approach a Poissonian.
Since the inverse bandwidth of conventional thermal radiation is in general much shorter
than the resolution of photon counters, the invention of novel light sources [46] and the development of sophisticated experimental techniques [158] were required to record photon
counting statistics. Pseudo thermal, chaotic light with arbitrary correlation time can be
produced by scattering of laser light on rapidly and randomly moving objects like a ground
glass disc. Arecchi [44, 45] used this technique to contrast the photon correlations and statistical properties of such a pseudo-thermal light beam with that of a laser. In a series of
experiments he unveiled the extraordinary properties of laser light by measuring its temporal second order correlation function to be constant and the photon counting distribution
to be Poissonian.
P
A classical probability distribution is positive definite and normalized to unity ∞
0 p (n ) =
1, where p (n ) is taken to be discrete, here. By means of the probability distribution the
expectation values and all higher moments µk can be calculated
µk = 〈n k 〉 =
∞
X
n k p (n ) .
(5.30)
0
The moments provide a useful way to characterize a probability distribution function, which
can be expanded in an infinite series of all moments. However, besides these moments defined above there exists a multitude of alternative and better adapted parameters to express
the features of a probability distribution function, like standardized, central or factorial moments and cumulants.
The moments µk and cumulants κk can be derived from the characteristic function
P i λn
Θ(λ) = 〈e i λn 〉 = ∞
p (n ), which represents the Fourier transform of the probability dis0 e
tribution, through
µ
d
µk = −i
dλ
¶k
µ
Θ(λ)|λ=0
and
d
κk = −i
dλ
¶k
ln Θ(λ)|λ=0 ,
(5.31)
respectively. Further, the central moments mk are the moments µk of a distribution centered around its average value µ1 . The central moments or the cumulants define the features of a distribution on the basis of its standardized moments, where the first is simply
the mean κ1 = µ1 = 〈n〉 and describes the expectation value. The second is the variance
κ2 = Var(n ) = 〈(n − µ1 )2 〉 = σ2 and indicates how far from the expectation value the events
typically are, where σ is the standard deviation. The third standardized moment, called
skewness is a parameter to characterize the asymmetry of a distribution. It is defined as
p
m 3 /σ3 or alternatively as κ3 / κ2 3 . When the skewness is positive, the distribution has a
longer tail to the right, and vice versa. The (excess) kurtosis is the fourth standardized moment κ4 /κ22 = µ4 /σ4 − 3 and expresses the excess peakedness of a function with respect to a
91
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
Gaussian. A distribution with a positive kurtosis has a sharper peak and longer tails, while a
negative kurtosis distribution has a more rounded or flat-top peak.
As an example, two important probability distributions can particularly easy be grasped
by their cumulants κi . The normal distribution (Gaussian) has only two nonvanishing cumulants, the first being the mean and the second being the variance. A Poissonian distribution
is distinguished by the equality of all its cumulants κ1 = κ2 = κ3 = . . .. Therefore, in a Poisson distribution, which describes completely random processes, the standard deviation of a
p
p
measurement of N particles is N . The relative error therefore scales as ∝ 1/ N . This is
the shot noise inherent in quantized random processes.
However, for discrete distributions, especially with respect to particle correlation functions, it is more convenient to introduce the factorial moments ϕk =
〈n (n − 1)(n − 2) . . . (n − k + 1)〉. They are generated by the alternative characteristic function
Q (λ) = 〈(1 − λ)n 〉 in the following way
µ
¶
d k
Q (λ)|λ=0 .
ϕk = −
dλ
(5.32)
The factorial cumulants defined in a way analog to equation (5.31) are of less importance.
The interest in the factorial moments becomes obvious when considering the normally
ordered correlation functions. Re-sorting the operators â (†) to measure particle numbers
n = â † â and respecting the bosonic permutation relation [â, â † ] = 1 yields
〈(â † )k â k 〉 = 〈n (n − 1)(n − 2) . . . (n − k + 1)〉 .
(5.33)
The knowledge of the complete probability distribution function contains more information as the pure correlation functions. Only from all correlation functions could the counting statistics be reconstructed. However, evaluating the counting statistics p (n, T ) for all
different time windows T allows for instance the derivation of the second order correlation
function from the second factorial cumulants
µ
g
(2)
(T ) − 1 = c
2
¶
∂ 2
ϕ2 (T ) ,
∂T
(5.34)
where the factor c 2 contains the squared average count rate. It therefore depends only on
the second moment (variance) of a distribution, related to the mean for proper normalization.
The proportionality is due to the nonideal detection efficiency. A similar relation holds for
the other higher order particle correlation function [162, 163], which also depend only on
the cumulants of the same order. This can equally be seen by applying the bosonic commu2
tation relation to the normally ordered second order correlation function 〈â † â † â â〉 / 〈â † â〉 ,
so then
g (2) = 1 +
92
σ2 − 〈n〉
〈n〉2
,
(5.35)
5.2. BACKGROUND
since 〈n 2 〉 − 〈n〉2 = σ2 ≥ 0. It follows that the condition for the single mode second order
correlation function g (2) ≥ 1 − 1/ 〈n〉. So it can be nonclassical, i.e., below one for a mode
with very few particles. A number state, or a sub-Poissonian distribution will have σ2 < 〈n〉.
time
2
0.4
g(2)(τ)
1.5
0.3
1
0.2
0.5
0.1
0
τc
τ
0
P(N)
2
4
6
8
N
FIGURE 5.7.: Schematic illustration of the relation between the time series of particle detection events, the second order correlation function and the full counting statistics. Three
typical cases with equal mean values are shown for Bose - Einstein distributed thermal states
(red), a coherent state having a Poissonian distribution (blue) and a sub-Poissonian distribution exhibiting antibunching (green).
The effect of a beamsplitter or attenuator in the beam, which is equivalent to a reduced
detection efficiency, can be expressed in terms of the Fano factor f = σ2 / 〈n〉. It is one for a
coherent state and zero for a number state. A sub-Poissonian state has a Fano factor between
zero and one, and the Fano factor of a super-Poissonian or chaotic state is larger than one
and unlimited. It can be shown [164], by applying the rules of counting statistics, that the
output state has a Fano factor f o corresponding to
f o = 1 − ²(1 − f i ) ,
(5.36)
depending on the input Fano factor f i and the loss coefficient 0 ≤ ² ≤ 1. That means the
intensity fluctuations of either sub- or super-Poissonian beams will be diminished in heavily
attenuated beams (² → 0) and approach a completely random Poissonian state with f =
1. This might be the reason why we don’t observe the second and third order correlation
functions to go to two and six respectively, besides the possibly nonlinear mixing process.
Two statistical distribution functions of importance for particle counting experiments in
quantum optics are generally the Poisson distribution and the Bose - Einstein distribution.
Both are discrete probability distributions and their mean value µ = cT here depends on the
93
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
time window T and the average count rate c per time window. The particle count distribution for a coherent state is a Poissonian distribution
p (n ) =
µn −µ
e .
n!
(5.37)
The characteristic function is Q (λ) = e −λµ and all cumulants are given by the same value
p
n which is the variance. A Poisson distribution describes totally random processes. It’s
p
characteristic noise is the shot noise (∼ 1/ n ) which is the quantum limit for quantized
observables. A narrower distribution can only be achieved by squeezed states with the consequence of increased uncertainty in the conjugate variable. The Poisson distribution is
p
p
positive skew (κ3 = n ) and more peaked than a Gaussian (κ4 = n ). It is the limes of a binomial distribution for very large sample sizes. On the other hand, a chaotic state such as a
thermal bosonic beam is represented by a Bose - Einstein or geometric particle distribution
p (n ) =
µn
(1 + µ)n+1
.
(5.38)
The characteristic function is Q (λ) = (1 + λµ)−1 . One of its distinguishing features is the
monotonically decreasing character, being highest for n = 0. Therefore it becomes very
broad for large mean values. The Bose - Einstein distribution can be derived from Planck’s
law [152] and is in principle a discrete exponential distribution function. This is evident
when considering the electric field of a chaotic radiation, which is a Gaussian distribution
2
∼ e −E around the mean of zero. Therefore chaotic light is also called Gaussian light. Thermal light is a form of chaotic radiation, as is pseudo-thermal light produced by random scattering of a coherent beam [45]. Consequently, the intensity distribution of Gaussian light
has an exponential behavior ∼ e −I .
5.3. Experimental Methods and Techniques
Our new experimental design combines the techniques for the production of atomic Bose
- Einstein condensates with single atom detection by means of an ultrahigh finesse optical
cavity. The apparatus consists of an ultra high vacuum (UHV) chamber which incorporates
a separated enclosure with a higher background pressure. Here we collect 109 87 Rb atoms in
a vapor cell magneto-optical trap which is loaded from a pulsed dispenser source. After polarization gradient cooling and optical pumping into the |F = 1, mF = −1〉 hyperfine ground
state we magnetically transfer the atoms over a distance of 8 cm out of the enclosure into a
magnetic trap. All coils for the magnetic trapping fields are placed inside the UHV chamber
and are cooled to below 0 °C . In the magnetic trap we perform radio frequency induced
evaporative cooling of the atomic cloud and obtain almost pure Bose - Einstein condensates
with 1.5 · 106 atoms. After evaporation we relax the confinement of the atoms to the final
94
5.3. EXPERIMENTAL METHODS AND TECHNIQUES
BEC
85
90
95
100
80
60
40
20
0
90
91
92
80
60
40
20
0
93
cavity transmission [ photons / 20 μs ]
80
atom laser
36 mm
75
time [ ms ]
cavity
laser
photon counter
FIGURE 5.8.: Schematic of the experimental setup. A weak continuous atom laser beam is
extracted from a Bose - Einstein condensate. After dropping a distance of 36 mm the atoms
enter the ultrahigh finesse optical cavity and single atoms in the beam are detected. A typical signal recorded with the single photon counter after time averaging with 20 µs.(right)
trapping frequencies ω⊥ = 2π · 29 Hz and ωk = 2π · 7 Hz, perpendicular and parallel to the
symmetry axis of the magnetic trap, respectively.
For output coupling an atom laser beam we apply a weak continuous microwave field to
locally spin flip atoms inside the Bose - Einstein condensate into the |F = 2, mF = 0〉 state.
These atoms do not experience the magnetic trapping potential but are released from the
trap and form a well collimated beam which propagates downwards due to gravity [11].
The output coupling is performed near the center of the Bose condensate for a duration
of 500 ms during which we extract on the order of 3 · 103 atoms. After dropping a distance
of 36 mm the atoms enter the high finesse optical cavity as illustrated in Figure 5.8. Fine
tuning of the relative position between the atom laser beam and the cavity mode is obtained
by tilting the vacuum chamber. We maintain a magnetic field along the trajectory of the
atom laser, which at the position of the cavity is oriented vertically and has a strength of
approximately 15 G.
The cavity consists of two identical mirrors separated by 178 µm. Their radius of curvature is 77.5 mm resulting in a Gaussian TEM00 mode with a waist of w0 = 26 µm. The cou-
95
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
pling strength between a single Rb atom and the cavity field is g 0 = 2π · 10.4 MHz on the
F = 2 → F 0 = 3 transition of the D2 -line. The cavity has a finesse of 3 · 105 and the decay rate
of the cavity field is κ = 2π · 1.4 MHz. The spontaneous dipole decay rate of the rubidium
atom is γ = 2π · 6 MHz. Therefore we operate in the strong coupling regime of cavity QED.
The cavity mirrors are mounted inside a piezo tube which enables precise mechanical control over the length of the resonator [21]. Four radial holes in the piezo element allow atoms
to enter the cavity volume and also provide optical access perpendicular to the cavity axis.
The cavity resides on top of a vibration isolation mount which ensures excellent passive stability. The cavity resonance frequency is stabilized by means of a far detuned laser with a
wavelength of 831 nm using a Pound - Drever - Hall locking scheme.
The cavity is probed by a weak, near resonant laser beam, whose transmission is monitored by a single photon counting module. The presence of an atom inside the cavity results
in a drop of the transmission as shown Figure 5.8. The stabilization light is blocked from the
single photon counter by means of optical filters with an extinction of 120 dB. The probe
laser and the cavity are red-detuned from the atomic F = 2 → F 0 = 3 transition by 40 MHz
and 41 MHz, respectively. The polarization of the laser is aligned horizontally and the average intracavity photon number is 5. These settings are optimized to yield a maximum
detection efficiency for the released atoms which is 23%. This number is primarily limited
by the size of the atom laser beam which exceeds the cavity mode cross section. The atoms
enter the cavity with a velocity of 84 cm/s giving rise to an interaction time with the cavity mode of typically 40 µs, which determines the dead time of our detector. The dead time
of our detector is short compared to the time scale of the correlations, which allows us to
perform Hanbury Brown and Twiss type measurements with a single detector [165].
5.4. Results
We record the cavity transmission for the period of the atom laser operation and average the
photon counting data over 20 µs(see Figure 5.8). Using a peak detection routine we determine the arrival time of an atom in the cavity, requiring that the cavity transmission drops
below its background value by at least four times the standard deviation of the photon shot
noise. From the arrival times of all atoms we compute the second order correlation function
g (2) (τ) by generating a histogram of all time differences within a single trace and normalizing it by the mean atomic flux. Due to the finite duration of the measurement, long time
differences between atom pairs are less likely than short intervals and we account for this
by an additional normalization. We average these histograms over many repetitions of the
experiment to obtain g (2) (τ) with a high signal-to-noise ratio.
Figure 5.9 shows the measured second order correlation function of an atom laser beam.
The value of the correlation function is g (2) (τ) = 1.00 ± 0.01 which is expected for a coherent
source. The second order correlation function being equal to unity reveals the second order
96
5.4. RESULTS
coherence of the atom laser beam and is intimately related to the property that it can be
described by a single wave function. Residual deviations from unity could arise from technical imperfections. Magnetic field fluctuations either due to current noise in the magnetic
trapping coils or due to external fluctuations could imprint small intensity fluctuations onto
the atom laser beam. We employ a low noise current source and magnetic shielding to minimize these effects. In addition, we use a highly stable microwave source which is stabilized
to a GPS disciplined oscillator. A further contribution to a potential modification of the
second order correlation function could be due to the output coupling process itself. The
spatial correlation function of atoms output coupled from a weakly interacting condensate
has been studied theoretically in a situation neglecting gravity [100]. The modification from
a constant unity second order correlation function was on the order of one percent, which
is on the same order of magnitude as the uncertainty in our data.
Measuring the second order correlation function requires to detect the particles within
their coherence time and coherence volume [165]. The uncertainty of the detection time
of an atom must be smaller than the correlation time, because otherwise the correlations
vanish [54]. We estimate that the acquired time delays resulting from a possibly misaligned
detector are shorter than the dead time of our detector. It has been measured that the
coherence time of the atom laser is given by the duration of output coupling at least for
durations of 1.5 ms [42]. Moreover, for the total flux of 5.2 atoms per ms, the probability of
having two or more atoms arriving within the dead time of our detector is 5%. Therefore
multiatom arrivals do not significantly influence the data.
Trapped Bose - Einstein condensates have been demonstrated to be phase coherent and
to have a uniform spatial and temporal phase [146, 10, 105]. The atom laser beam has been
theoretically described by a single wave function [100, 166] and its spatial coherence was
observed [5]. Moreover, a full contrast interference pattern was observed between two atom
laser beams extracted from separate locations inside a condensate [105]. This indicates a
high degree of spatial overlap between the two propagating modes and a negligible distortion of the uniform spatial phase due to interactions with the remaining condensate. From
this we conclude that the atom laser leaves the condensate region with a well defined spatial
wavefront.
Many overlapping spatial modes at the detector wash out the correlations. In our experimental geometry this is the case when output coupling from a thermal source, since we can
not resolve a single diffraction limited spatial mode. Therefore we do not observe thermal
bunching of noncondensed atoms.
Determining the arrival times of all detected atoms explicitly allows us to extract the full
counting statistics of the atoms. We choose a time bin length of T = 1.5 ms in which we count
the number N of detected atoms and plot the probability distribution p (N ) in Figure 5.10.
The distribution is close to a Poissonian distribution
p (N ) = 〈n〉N e −〈n〉 /N !
(5.39)
97
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
2.0
g(2)(τ)
1.5
1.0
0.5
0
0
3
6
9
time interval τ [ ms ]
12
15
FIGURE 5.9.: Second order correlation function of an atom laser beam. The data are binned
with a time bin size of 50 µs. The average count number is 2 · 105 per bin. We have omitted
the first two data points since they are modified by the dead time of our detector.
probability p(N)
0.3
0.2
0.1
0
0
1
2
3 4 5 6 7 8
number of events N
9 10
FIGURE 5.10.: Probability distribution p (N ) of the atom number N detected within a time
interval of T = 1.5 ms. The (+) symbols show a Poissonian distribution for the same mean
value of 〈n〉 = 1.99 as the measured data. The errors indicate statistical errors.
98
5.4. RESULTS
with a mean of 〈n〉 = 1.99. For the measured distribution we have calculated the 2nd, 3rd
and 4th cumulant to be κ2 = 1.75, κ3 = 1.34 and κ4 = 0.69, respectively.
We realize a direct comparison with a pseudo-thermal beam of atoms by output coupling
a beam with thermal correlations from a Bose - Einstein condensate. This is in close analogy to changing the coherence properties of a laser beam by means of a rotating ground
glass disc [44]. Instead of applying a monochromatic microwave field for output coupling
we have used a broadband microwave field with inherent frequency and intensity noise. We
have employed a white noise generator in combination with quartz crystal band pass filters
which set the bandwidth of the noise. The filters operate at a frequency of a few MHz and
the noise signal is subsequently mixed to a fixed frequency signal at 6.8 GHz to match the
output coupling frequency. For atomic beams prepared in such a way we observe bunching
with a time constant set by the band pass filter as shown in Figure 5.11. To compare our data
with the theoretically expected correlation function we have measured the power spectra
of the band pass filters and calculated |g (1) (τ)|2 of the rf field before frequency mixing. In
Figure 5.11 we plot 1 + β|g (1) (τ)|2 . The normalization factor β = 0.83 accounts for the deviation of the experimental data from g (2) (0) = 2 due to imperfections in the frequency mixing
process.
For the pseudo-thermal beam we also calculate the counting statistics and find a significantly different behavior than for the atom laser case. For a filter with a spectral width
(FWHM) of 90 Hz we have chosen the time bin length of T = 1.5 ms, smaller than the correlation time. The atomic flux with a mean atom number 〈n〉 = 1.99 is equal to the case of the
atom laser. We compare the measured probability distribution to a Bose distribution
p (N ) = 〈n〉N /(1 + 〈n〉)1+N ,
(5.40)
which is expected for a thermal sample and find good agreement presented in Figure 5.12.
From the distribution we have extracted the 2nd, 3rd and 4th cumulant to be κ2 = 4.6, κ3 =
14.5 and κ4 = 50.6, respectively.
In comparison to the nonexclusive second order correlation function it is interesting to
evaluate the exclusive probability when the next particle is detected [162]. Two cases can be
distinguished. First, the probability P (t ) to detect a particle after a random initial time and
secondly the conditional probability P c (0|t ) to detect a particle after an initial particle was
detected. For an atom laser in a coherent state, both these cases should be totally random
and therefore decrease exponentially
P (t ) = ce −ct
and
P (0|t ) = ce −ct .
(5.41)
When normalized with the count rate c these exponential probabilities are unity for t = 0
as shown in Figure 5.13(a). The lines are fits with only the count rate c as a free parameter,
which agrees with the experimental values.
99
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
2.0
g(2)(τ)
1.5
1.0
0.5
0
0
3
6
9
time interval τ [ ms ]
12
15
FIGURE 5.11.: Second order correlation functions of pseudo-thermal atomic beams. The
square symbols correspond to a filter band width (FWHM) of 90 Hz, the triangles to a bandwidth of 410 Hz, and the circles to a bandwidth of 1870 Hz. The data are binned with a time
bin size of 50 µs in which the average count number is 8 · 104 . We have omitted the first
two data points since they are modified by the dead time of our detector. The lines are the
experimentally determined correlation functions of the broadband microwave fields.
probability p(N)
0.3
0.2
0.1
0
0
1
2
3 4 5 6 7
number of events N
8
9 10
FIGURE 5.12.: Probability distribution p (N ) of the atom number N within a time interval of
T = 1.5 ms for the 90 Hz bandwidth data. The (+) symbols indicate a Bose distribution with
the same mean value of 〈n〉 = 1.99. The errors indicate the statistical errors.
100
5.4. RESULTS
For a chaotic state, on the other hand, these exclusive probabilities decrease geometrically like ∼ t −2 and ∼ t −3 and are given by
P (t ) =
c
and
(1 + ct )2
P (0|t ) =
2c
(1 + c t )3
(5.42)
,
(a)
1
P(0|t)/N
P(t)/N
respectively. The different exponent in the unconditional and conditional probability simply stem from the Bose - Einstein distribution (eq. 5.38) with n = 1 in the first case and
n = 2 for the second. The fact that the conditional probability describes a two-particle phenomenon, in contrast to the unconditional one, is also manifested in the limiting value for
t = 0. The conditional probability normalized by the count rate c tends to two, as does the
second order correlation function, whereas P (t ) just goes to one as shown in Figure 5.38(b).
Here, the lines are also fits with only the count rate c as a free parameter, which agrees with
the experimental values.
0.1
0.01
0
(b)
1
0.1
0.5
1.0
1.5
2.0
time to first count t [ ms ]
2.5
0.01
0
0.5
1.0
1.5
2.0
time interval t [ ms ]
2.5
FIGURE 5.13.: Exclusive probabilities (normalized) when to detect the next atom. The data
for the atom laser is shown in blue, the data for the pseudo-thermal beam with bandwidth
90 Hz is red. (a) Unconditional probability to detect the next atom after t for a random
initial time. (b) Conditional probability to detect two atoms, one at 0 and the next one at t .
It is 2 for t → 0 due to bunching.
Investigating three particle correlations for thermal beams one expects to find increased
correlations by a factor 3! = 6. This effect was predicted [58] and observed [59] in the reduced
three body recombination for Bose - Einstein condensates compared to a thermal cloud.
However, this only measures equal time and equal position correlations. We investigate the
temporal third order correlation function g (3) (τ1 , τ2 ) which depends on two time differences
τ1 and τ2 for a triple. We plot the symmetric three particle correlation function given by
g (3) (τ1 , τ2 ) =
〈Ψ̂† (τ1 )Ψ̂† (0)Ψ̂† (τ2 )Ψ̂(τ2 )Ψ̂(0)Ψ̂(τ1 )〉
〈Ψ̂† Ψ̂〉
3
.
(5.43)
In Figure 5.14 the three particle bunching peak is shown for an atomic beam with a pseudothermal character, produced with the 90 Hz filter. The result is in good agreement with
101
5. CORRELATIONS AND COUNTING STATISTICS OF AN ATOM LASER
the expected shape. Clearly, the symmetry and the lower order correlation functions are
visible. The peak value is ∼ 5 < 6, which is probably due to the nonlinear microwave mixing
process or maybe due to the reduced Fano factor (eq. 5.36) for imperfect detection efficiency.
However, since the second order correlation function also does not reach the ideal value of
two, this is not so surprising.
In contrast, the atom laser g (3) depicted in Figure 5.15, shows the absence of any third order correlations. This is in agreement with a coherent state, where all correlation functions
factorize.
102
5.4. RESULTS
6
5
g(3)(τ1,τ2)
4
3
2
1
0
2
4
6 8 10
time interval τ [ m12
s]
1
14
16
18 20
20
4 0
8
]
16 12 erval τ2 [ ms
t
in
time
FIGURE 5.14.: Symmetric third order correlation function g (3) (τ1 , τ2 ) for a pseudo-thermal
atomic beam of 90 Hz bandwidth
g(3)(τ1,τ2)
2
1
0
2
4
6 8 10
time interval τ [ m12 14
s]
1
16
18 20
20
4 0
8
]
16 12 erval τ2 [ ms
t
time in
FIGURE 5.15.: Symmetric third order correlation function g (3) (τ1 , τ2 ) for an atom laser beam
103
6 Cavity QED Detection of Interfering
Matter Waves
“. . . has in it the heart of quantum mechanics.
In reality, it contains the only mystery.” - Lectures on Physics, Richard P. Feynman
We observe the buildup of a matter wave interference pattern from single atom detection
events in a double slit experiment. The interference arises from two overlapping atom laser
beams extracted from a Rubidium Bose - Einstein condensate. Our detector is an ultrahigh
finesse optical cavity which realizes the quantum measurement of the presence of an atom
and thereby projects delocalized atoms into a state with zero or one atom in the resonator.
The experiment highlights the granular nature of matter while simultaneously reveals its
wave properties, because the atom flux is so low that on average there is only one particle
at a time in the interferometer.
This chapter has been published as [33]: T. Bourdel, T. Donner, S. Ritter, A. Öttl, M. Köhl,
and T. Esslinger, Cavity QED detection of interfering matter waves, Phys. Rev. A. 73 (2006), 043602.
105
6. CAVITY QED DETECTION OF INTERFERING MATTER WAVES
6.1. Introduction
The prediction of the duality between particles and waves by de Broglie [167] is a cornerstone of quantum mechanics. Pioneering experiments have observed interferences of massive particles using electrons [168, 169], neutrons [170], atoms [171, 172, 173] and even large
molecules [174]. However, the simple picture that matter waves show interferences just like
classical waves neglects the granularity of matter. This analogy is valid only if the detector
is classical and integrates the signal in such a way that the result is a mean particle flux.
With quantum detectors that are sensitive to individual particles the discreteness of matter
has to be considered. The probability to detect a particle is proportional to the square amplitude of the wave function and interferences are visible only after the signal is averaged
over many particles.
In the regime of atom optics, single atom detection has been achieved for example by fluorescence [175], using a micro-channel plate detector for metastable atoms [176], and high
finesse optical cavities [20]. In these experiments the size of the de Broglie wave packet of
the particles was much smaller than the detector area. Therefore localization effects during the detection have been negligible. With the realization of Bose - Einstein condensation
in dilute gases particles with a wave function of macroscopic dimensions have become experimentally accessible. Only very recently the single atom detection capability has been
achieved together with quantum degenerate samples reaching the regime of quantum atom
optics [17, 18]. The quantum nature of the measurement opens perspectives for atom interferometry at and below the standard limit [36].
6.2. Quantum Mechanical Measurement Process
For atoms with a spatially extended wave function, such as in a Bose - Einstein condensate
or in an atom laser beam, a measurement projects the delocalized atom into a state localized
at the detector [177]. This quantum measurement requires dissipation in the detection process. For single atom detection we employ an ultrahigh finesse optical cavity in the strong
coupling regime of cavity quantum electrodynamics (QED) [178, 21, 179]. We study this open
quantum system including the two sources of dissipation, cavity losses and spontaneous
emission. In particular, we calculate the time needed for the localization of an atom in the
cavity measurement process. We then experimentally investigate atomic interferences on
the single atom level using our detector.
A schematic of our experimental setup is shown in Figure 6.1. We output couple two weak
atom laser beams from a Bose - Einstein condensate and their wave functions overlap and
interfere [105]. The flux is adjusted in such a way that there is on average only one atom at
a time in the interferometer. Using the ultrahigh finesse cavity we measure the atom flux in
the overlapping beams with single atom resolution. We observe the gradual appearance of
106
6.2. QUANTUM MECHANICAL MEASUREMENT PROCESS
a matter wave interference pattern as more and more detection events are accumulated.
Single atom detection in an optical cavity can be captured in a classical picture: an atom
changes the index of refraction in the cavity and thereby brings it out of resonance from the
probe laser frequency. In the absence of an atom, the probe beam is at resonance with the
cavity and its transmission is maximal. Experimentally we use a probe power corresponding to five photons on average in the cavity. The cavity transmission noise is found to be
dominated by photon shot noise and not by the quality of the lock. The presence of an atom
results in a drop of the cavity transmission as shown in Figure 6.2. We set the threshold for
an atom detection event to a drop in transmission of four times the standard deviation of
the photon shot noise in our 20 µsintegration time. Then the overall detection efficiency of
atoms extracted from a Bose - Einstein condensate is measured to be 0.23(8). The probability
of an artefact detection is lower than 2 · 10−4 in the measurement time of 0.5 s.
36 mm
BEC
100 μm
laser
cavity
photon
counter
FIGURE 6.1.: Schematic of the experimental setup. From two well defined regions in a Bose Einstein condensate (BEC), we output couple atoms to an untrapped state. The real parts of
the resulting atom laser wave functions are sketched on the right hand side. The absorption
image showing an interference pattern corresponds to a flux ∼ 106 times larger than the
one used for the actual single atom interference experiment. Monitoring the transmission
through an ultrahigh finesse optical cavity with a photon counter, single atom transits are
detected.
107
6. CAVITY QED DETECTION OF INTERFERING MATTER WAVES
cavity
probe
laser
(a)
single photon
counter
atom
photon counts per 20 μs
100
(b)
80
60
40
20
0
0
0.2
0.6
0.4
time [ ms ]
0.8
FIGURE 6.2.: (a) Cavity single atom detection principle. An atom detunes the ultrahigh finesse cavity from resonance and the cavity transmission consequently drops. (b) Photon
flux through the ultrahigh finesse optical cavity when an atom is detected. The photon
count rate is averaged over 20 µs. The detection threshold is set to be 4 times the standard
deviation of the photon shot noise (dashed line).
Our cavity has been described before (see Section 3.3.1) and we only recall here its main
figures of merit. Its length is 178 µm, the mode waist radius is w0 = 26 µm, and its finesse
is 3 · 105 . The maximum coupling strength between a single 87 Rb atom and the cavity field
g = 2π · 10.4 MHz is larger than the cavity field decay rate κ = 2π · 1.4 MHz and the atom
spontaneous dipole decay rate γ = 2π·6 MHz. The probe laser and the cavity are red-detuned
as compared to the atomic resonance such that a light force pulls the atoms to regions where
the coupling is large, therefore enhancing the detection efficiency.
To understand the actual detection process we study the dynamics of the atom cavity
quantum system taking into account dissipation. We first consider a classical atom entering a simplified square shaped cavity mode so that its coupling to the cavity field increases
suddenly to a constant value g . The cavity field is initially coherent with a few photons. We
use a two level approximation for the atom description and assume a 30 MHz red-detuning
of the probe laser compared to the atomic resonance. Our probe beam is linearly polarized orthogonally to the magnetic field direction. We therefore do not probe on a cycling
transition. The 30 MHz detuning is an effective value chosen to match the experimental
conditions. In the case of strong coupling the following dynamics occur. On a short time
scale given by 1/g , the atom cavity system exhibits coherent oscillations. It progressively
reaches an equilibrium state on a time scale given by 1/κ and 1/γ due to cavity loss and
atomic spontaneous emission. These are the two sources of dissipation. In the equilibrium
state, the mean photon number in the cavity is reduced and the cavity transmission drops.
To evaluate this drop quantitatively, we find the steady-state of the master equation for
the density matrix numerically [180, 181, 182, 110]. For our parameters, the transmission as
a function of the coupling strength g is plotted in Figure 6.3. For a maximally coupled atom
108
6.2. QUANTUM MECHANICAL MEASUREMENT PROCESS
g = 2π · 10.4MHz, the average intracavity photon number is found to be reduced from 5 to
0.9, and the number of detected photons is then reduced by the same ratio. Such a reduction corresponds well to the largest observed transmission drops with an example shown in
Figure 6.2. The detection threshold corresponds to a coupling of g = 2π · 6.5MHz. Experimentally, unlike in our model, an atom feels a position dependent coupling as it transverses
the mode profile. However the atom transit time through the cavity mode (40 µs) is long
compared to the cavity relaxation time scales 1/κ and 1/γ and the atom cavity system adiabatically follows a quasi equilibrium state. Therefore the experimental transmission drops
can be compared to the calculated ones.
1.0
(a)
0.8
0.6
0.4
(b)
0.8
coherence
transmission
1.0
0.6
0.4
0.2
0.2
0
0
0
2
4 6 8 10 12
coupling g0 [ MHz ]
0
0.1
0.2
0.3
time [ μs ]
0.4
0.5
FIGURE 6.3.: (a) Normalized transmission as a function of coupling strength. The solid line
corresponds to our probe strength of 5 photons in the cavity in the absence of an atom.
The dashed line is the weak probe limit. The dotted line corresponds to 10 photons in the
cavity. (b) Coherence between the states with one and no atom as a function of time. The
initial coherence is normalized to 1. Solid line: g = 2π· 10MHz. Dashed line: g = 2π· 6.5MHz.
Dotted line: g = 2π · 3MHz.
Specific to our experiment is that a spatially extended matter wave and not a classical
atom enters the cavity. Our system allows us to realize a quantum measurement of the presence of an atom. For our low atom flux, we can neglect the probability of having more than
one atom at a time in the cavity. The incoming continuous wave function is thus projected
into a state with one or zero atom in the cavity. This projection necessarily involves decoherence that is introduced by spontaneous scattering and cavity photon loss. The origin of
the decoherence can be understood as unread measurements in the environment [181, 177].
For example, if a spontaneously emitted photon is detected, there is necessarily an atom in
the cavity and the wave function is immediately projected. Similarly, the more different the
light field with an atom in the cavity is from the one of an empty cavity, the more different
is the scattered radiation out of the cavity, and the projection occurs correspondingly faster.
We now quantify the time needed for the projection to occur. For simplicity, rather than
a continuous wave function, we consider a coherent mixture of one and zero atom entering
109
6. CAVITY QED DETECTION OF INTERFERING MATTER WAVES
a square shaped cavity at a given time. We take the limit when the probability to have
one atom is low. The initial cavity field is the one of an empty cavity. Dissipation effects
are studied by computing the time evolution of the density matrix [182]. The degree of
projection of the initial state can be extracted from the off-diagonal terms between states
with one atom and no atom in the density matrix. More precisely, we define the coherence
as the square root of the sum of the square modulus of the off-diagonal terms mentioned
above. This quantity is maximal for a pure quantum state with equal probability to have an
atom or not. The coherence is zero for a statistical mixture.
In Figure 6.2, the evolution of the coherence is plotted. As expected, it decays to zero at
long times due to dissipation. The decay time increases as the coupling to the cavity is weakened. In the limit where the coupling vanishes, the coherence is preserved. The atomic wave
function then evolves as if there was no cavity. For g > 2π · 6.5MHz, the decoherence time is
found to be a fraction of a microsecond. This value is much lower than the 40 µstransit time
of an atom through the cavity and for all our detected atomic transits, the wave function
is thus well projected to a state with one atom, realizing the quantum projection theorem.
Our detection scheme realizes a quantum measurement of the presence of an atom in the
cavity. However during an atom transit some photons are spontaneously scattered and the
velocity of an atom is slightly modified.
6.3. Buildup of Matter Wave Interference
Using our cavity detector, we can observe matter wave interferences on the single atom
level. The starting point of the experiment is a quasi pure Bose - Einstein condensate with
1.5 · 106 Rubidium atoms in the hyperfine ground state |F = 1, mF = −1〉 [17]. The atoms are
magnetically trapped with frequencies ωk = 2π · 7 Hz axially and ω⊥ = 2π · 29 Hz radially.
A weak and continuous microwave field locally spin flips atoms from the Bose - Einstein
condensate into the untrapped |F = 2, mF = 0〉 state. This process is resonant for a section
of the condensate where the magnetic field is constant. Because the magnetic moment of
the spin flipped atoms vanishes they fall due to gravity and form a continuous atom laser
[11].
When we apply two microwave fields, we are able to output couple atom laser beams
from two well defined regions of the condensate [105]. The two distinct atom laser wave
functions overlap and interfere. At the entrance of the cavity, the atomic wave function ψ is
well described by the sum of two plane waves with the following time dependence
ψ(t ) ∝ exp(i ω1 t ) + exp(i ω2 t + φ)
(6.1)
∝ cos((ω2 − ω1 )t /2 + φ) ,
where ħω1 and ħω2 are the energies of the two laser beams and φ is a fixed phase difference. The radial dependence of the wave function is neglected. The probability to detect
110
6.3. BUILDUP OF MATTER WAVE INTERFERENCE
an atom is given by the square norm of the wave function which is modulated in time and
behaves like a cosine square. The modulation frequency of the interference signal is given
by the energy difference between the two atom lasers. Experimentally, it is determined by
the frequency difference of the two microwave fields and is chosen to be ∆ f = 10 Hz, which
corresponds to a distance of 5 nm between the two output coupling regions. The two microwave fields are generated such that the interference pattern is phase stable from one
experimental run to the other.
1
iterations of the experiment
191
time [ ms ]
100
200
300
400
500
atom counts per 5ms time bin
FIGURE 6.4.: Histograms of the atom detected in 5 ms intervals. (a) Single experimental run,
(b) sum of 4 runs, (c) sum of 16 runs, and (d) sum of 191 runs, where the line is a sinusoidal
fit. Please note the different scales.
The results of the experiment are presented in Figure 6.4. Each experimental run corresponds to output coupling from a different condensate and on average ∼ 6 atoms are detected in 0.5 s. After the detection of a few atoms, their arrival times appear to be random
(Figure 6.4 a). Nevertheless, after adding the results of several runs, an interference pattern
progressively appears (Figure 6.4 4b-d). The atom number fluctuation is found to be dominated by the atomic shot noise and the signal to noise ratio of the interference increases
as more data are included. A fit to the histogram leads to a contrast of 0.89(5). The slight
reduction of contrast can be attributed to an extremely low uncontrolled atomic flux corresponding to one atom every 3 or 4 runs observed in the absence of output coupling. Those
111
6. CAVITY QED DETECTION OF INTERFERING MATTER WAVES
atoms could be output coupled either by a stray microwave field or by scattered light from
the cavity mirrors.
We work with a flux of one detected atom per 83 ms, which is about the time an atom
needs to travel from the condensate region to the cavity. We are thus in a regime where the
atoms fall one by one in the interferometer. A single atom behaves both like a wave because
its time arrival probability shows an interference pattern and like a particle as single atoms
are detected. This can be similarly expressed by saying that each individual atom is released
from both slits simultaneously. Our experiment is an atomic counterpart of the Young’s
double slit experiment with individual photons.
In conclusion we detect matter wave interferences with an ultrahigh finesse optical cavity
detector which realizes a quantum measurement of the presence of an atom. It is explained
how dissipation plays a crucial role in the detection process and for the localization of the
atom inside the cavity. Using this detector, we are able to detect a high contrast atom interference pattern at the single atom level. The coupling of a matter wave to a cavity QED
system opens the route to the quantum control not only of the internal state of the atoms
but also of their positions [183]. Using the presented detection technique we can probe an
atomic gas with a good quantum efficiency and introduce only a minimum perturbation
through the measurement. This could facilitate nondestructive and time resolved studies of
the coherence of a quantum gas, for example during the formation of a Bose - Einstein condensate. Similar interference experiments between two distinct condensates would permit
investigations of their relative phase evolution [184, 185] or diffusion [186].
112
7
Conclusion
Within the scope of this thesis an apparatus was designed, constructed and successfully
tested that achieves the fusion of quantum degenerate gases with the single atom detection
ability of cavity QED in the strong coupling regime. The challenges to experimentally merge
these two fields were overcome by forging new paths for the Bose - Einstein condensation
setup and the ultrahigh finesse optical cavity design.
The novel system distinguishes itself by a very reliable and reproducible operation for
the production of large Bose - Einstein condensates and superior stability for the output
coupling of continuous atom lasers. It features flexibility for research on and applications of
atom lasers through the vast, free and accessible half-space below the BEC. In particular, we
have implemented a modular and interchangeable science platform, which, in the current
implementation, incorporates the very compact realization of an ultrahigh finesse optical
cavity design on top of a proper UHV compatible vibration isolation system.
With this experimental setup we are able to detect single atoms from a quantum degenerate source with high efficiency by aiming the atom laser into the cavity mode. We have
demonstrated the coherence of atom lasers by measuring a constant second and third order
correlation function in contrast to pseudo-thermal atomic beams, which show bunching behavior. Moreover, we have recorded the full counting statistics of atomic beams, confirming
a Poissonian distribution for the atom laser and a Bose - Einstein distribution for the chaotic
beam.
We detect matter wave interferences with an ultrahigh finesse optical cavity detector
which realizes a quantum measurement of the presence of an atom. We explain how dissipation plays a crucial role in the detection process and for the localization of the atom
inside the cavity. Using this detector, we are able to detect a high contrast atom interference pattern at the single atom level. This opens the potential to probe the coherence of
two distinct Bose - Einstein condensates, the evolution [184, 185] and diffusion [186] of their
relative phase and possibly the establishment of phase through the detection process [187].
Since the quantum measurement in the cavity projects the coherent beam onto a number
state, the atomic state leaving the cavity is highly squeezed. It would be interesting to exploit this prepared number squeezed state by performing a readout measurement with a
second cavity, which could be realized in a new implementation of the modular science
platform.
113
7. CONCLUSION
Our current system facilitates research in the field of cavity QED, because the atom laser
is a source of unprecedented brightness and provides very high and controllable reproduction rates of atoms coupling with the cavity mode. On the other hand, the ultrahigh finesse
cavity functions as a single atom detector in coherent matter waves which develops the
new field of quantum atom optics. It is an extremely sensitive tool to detect atomic beams
for high precision interferometry measurements and to investigate particle correlations.
Moreover, the capability of detecting single atoms and their coherence is especially useful
to probe cold atomic clouds in situ and time resolved in an essentially nondestructive manner. This allows the observation of kinetics and coherence growth in the process of Bose Einstein condensation and the mapping of the correlation length across a phase transition.
This behavior is scale invariant and governed by universal critical exponents which could
be measured with our setup.
Future prospects with the system include single molecule detection and the setup of a
heterodyne detection technique [136] for the presence of an atom inside the cavity mode,
which can potentially be nondestructive on the atomic quantum state. Furthermore, we
intend to transport the Bose - Einstein condensate into the ultrahigh finesse optical cavity
by loading it into the dipole potential of a moving optical standing wave and conveying
it downwards over the distance of 36 mm into the cavity mode. There, the arrangement
of coils around the cavity could be used to apply magnetic field gradients for tomography
experiments. The lateral optical access enables us to create a three dimensional optical
lattice inside the ultrahigh finesse optical cavity, which will open the route to study strongly
correlated systems with single atom resolution.
114
A Appendix
A.1. Breit - Rabi Formula
The energy shift of the hyperfine states of an atom with a single valence electron, i.e., alkali
atoms, in an external magnetic field B is described by the Hamiltonian
HB = g e µB S · B + g I µN I · B + 2C S · I ,
(A.1)
where S corresponds to the electron spin and I to the nuclear spin, and g e and g I denote
the g -factor of the electron and the nucleus, respectively. µB is the Bohr magneton and µN
the nuclear magneton. C = νhfs /(2 I + 1) is related to the frequency splitting νhfs between the
hyperfine states at zero magnetic field. Diagonalizing (A.1) yields the Breit - Rabi formula
[188]
E (B ) = −
±
hνhfs
2(2 I + 1)
hνhfs
2
+ µK g I B m F
s
1+
g e µB − g I µN 2
4mF g e µB − g I µN
B +(
B) ,
2I + 1
hνhfs
hνhfs
(A.2)
which can be used to calculate the energy shift of a hyperfine state in the F = I ± 1/2 manifold with the magnetic quantum number mF at arbitrary magnetic fields.
For not too high magnetic fields where F = J + I is a good quantum number, the hyperfine
Landé g factor is calculated by
gF = g J
F (F + 1) + J ( J + 1) − I ( I + 1)
2F (F + 1)
− gI
F (F + 1) + I ( I + 1) − J ( J + 1)
2F (F + 1)
.
(A.3)
115
APPENDIX
A.2. Physical Properties of 87Rb
Mass
Natural abundance
Nuclear spin I
Nuclear g I -factor
Landé g J -factor for D transitions
Hyperfine ground state splitting νhfs
Vacuum Wavelength λD 1 (52 S 1/2 → 52 P 1/2 )
Natural linewidth ΓD 1 (FWHM)
Vacuum Wavelength λD 2 (52 S 1/2 → 52 P 3/2 )
Natural linewidth ΓD 2 (FWHM)
Saturation intensity I sat F = 2 → F 0 = 3
(σ± -polarized light, D 2 -line)
Resonant cross section σ0 F = 2 → F 0 = 3
(σ± -polarized light, D 2 -line)
Saturation intensity I sat F = 2 → F 0 = 3
(isotropically polarized light, D 2 -line)
Resonant cross section σ0 F = 2 → F 0 = 3
(isotropically polarized light, D 2 -line)
Scattering length a S (singlet)
Scattering length a T (triplet)
van der Waals coefficient C 6
86.9091835(27)u
27.83(2)%
3/2
-0.0009951414(10)
2.00233113(20)
+6.83468261090429(9)GHz
794.9788509(8)nm
2π · 5.746(8)MHz
780.241209686(13)nm
2π · 6.065(9)MHz
1.669mW/cm2
2.907 · 10−9 cm2
3.576mW/cm2
1.356 · 10−9 cm2
90(1) a0
106(4) a 0
4691(23) a 06 α2 me c 2
TABLE A.1.: Properties of 87 Rb [130].
116
A.3. D2-Line Energy Levels
F=3
266.650(9) MHz
2
3
gF = 2/3
0
-1 (0.93 MHz/G)
-2
-3
F=2
2
1 gF = 2/3
0
-1 (0.93 MHz/G)
-2
F=1
1 gF = 2/3
0
-1 (0.93 MHz/G)
2
5 P3/2
1
156.947(7) MHz
72.218(4) MHz
F=0
0
780.241 209 686(13) nm
384.230 484 468 5(62) THz
1.589 049 439(58) eV
12 816.549 389 93(21) cm-1
5 2S1/2
F=2
12 gF = 1/2
0
-2-1 (0.70 MHz/G)
F=1
-1
0
6.834 682 610 904 29(9) GHz
gF = -1/2
1 (-0.70 MHz/G)
FIGURE A.1.: D2 line of 87 Rb. Energy level structure of the hyperfine and Zeeman sublevels
[130].
117
APPENDIX
A.4. Physical Constants
In this thesis, the following physical constants are used:
Speed of light
Planck constant
Fine-structure constant
Electric constant
Elementary charge
Electron mass
Atomic mass constant
Boltzmann constant
Bohr radius
Bohr magneton
Nuclear magneton
Electron g -factor
c
h
α
²0
e
me
u
kB
a0
µB
µB
µN
ge
299792458
6.6260693(11) · 10−34
7.297352568(24) · 10−3
8.854187817 · 10−12
1.60217653(14) · 10−19
9.1093826(16) · 10−31
1.66053886(28) · 10−27
1.3806505(24) · 10−23
0.5291772108(18) · 10−10
9.27400949(80) · 10−24
1.39962458(12) · 106
5.05078343(43) · 10−27
2.0023193043718(75)
TABLE A.2.: Physical constants.
118
m/s
Js
F/m
C
kg
kg
J/K
m
J/T
Hz/G
J/T
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131
Credits
Ich möchte diese Gelegenheit nutzen, um mich bei all denen zu bedanken, die zum erfolgreichen Abschluß meiner Doktorarbeit maßgeblich beigetragen haben.
Mein Dank gilt:
• zuvorderst und ganz herzlich Prof. Dr. Tilman Esslinger für die freundliche Aufnahme
in seine Forschungsgruppe und die verwantwortungsvolle Aufgabe, ein so spannendes
Experiment von Anfang an mitzugestalten.
• Dr. Michael Köhl, der jederzeit mit Rat und Tat zur Seite stand und der mit seinem Elan
und Wissen die Sache mächtig vorangetrieben hat.
• Stephan Ritter, dem Optikmeister, für die hervorragende Zusammenarbeit über die
Jahre hinweg und seine unermüdliche Diskussionsbereitschaft, nicht nur im Labor
sondern auch auf dem Balkon.
• Tobias Donner, Thomas Bourdel und Ferdinand Brennecke, mit denen es mir eine grosse Freude war zu experimentieren und die Physik der kalten Wolke heiß zu diskutieren.
• Thilo Stöferle und Henning Moritz, den “alten Hasen” vom Gitter-Experiment, von
deren Erfahrungsschatz und Hilfsbereitschaft wir viel profitieren konnten.
• Ken Günter, Bruno Zimmermann und Niels Strohmaier, den “jungen Wilden”, für das
tolle Klima, das sie mit ihrem Einsatz innerhalb der Gruppe schaffen.
• bei unseren Diplomanden Roger Gehr, Gabriel Puebla-Hellmann und Robert Jördens,
daß sie ab und zu auch meinen Rat einforderten. Ein besonderer Dank gebührt dem
“TEX-bert” für seine Hilfsbereitschaft und typographischen Anregungen.
• Alexander Frank für seine vielen tollen elektronischen Gerätschaften, ohne die wir die
Geheimnisse des Atomlasers niemals hätten ergründen können, und seine Neugier,
neue Ideen gleich in die Tat umzusetzen.
• Jean-Pierre Stucki, Paul Herrmann, Hans-Jürg Gübeli und Peter Brühwiler mit seinen
Mannen von der Werkstatt für die phantastische “Hardware”, die sie für unser Experiment geschaffen haben.
133
CREDITS
• Veronica Bürgisser für ihre fleißige Arbeit im Büro, die uns unglaublich viele Dinge
erleichtert hat, und für ihre Nachsicht bei meinen schlampigen Eintragungen.
• Christian Schori, Martin Schultze, Sebastian Slama, Tobias Gemperli, Patrick Maletinsky und allen ehemaligen Mitgliedern der Gruppe, mit denen zu arbeiten mir eine grosse Ehre war.
• meinen WG-Mitbewohnern und Liselotte für die gute Stimmung im Haus.
• meiner Familie für den wunderbaren Platz den man Daheim nennt, und im Andenken
an meine kürzlich verstorbene Oma.
• und am allermeisten danke ich meiner Lisa ganz lieb für ihre tolle, aufopferungsvolle
Unterstützung und ihre Geduld.
134
Publications
• A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Correlations and Counting Statistics of an Atom
Laser, Phys. Rev. Lett. 95 (2005), 090404
• T. Bourdel, T. Donner, S. Ritter, A. Öttl, M. Köhl, and T. Esslinger, Cavity QED detection of
interfering matter waves, Phys. Rev. A. 73 (2006), 043602
• A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Hybrid apparatus for Bose-Einstein condensa-
tion and cavity quantum electrodynamics: Single atom detection in quantum degenerate gases,
Rev. Sci. Instr. 77 (2006), 063118
• M. Köhl, A. Öttl, S. Ritter, T. Donner, T. Bourdel, and T. Esslinger, Observing the time
interval distribution of atoms in atomic beams, accept. for Appl. Phys. B
◦ S. Ritter, A. Öttl, T. Donner, T. Bourdel, M. Köhl, and T. Esslinger, Observing the Formation
of Long-Range Order during Bose-Einstein Condensation, subm. to Phys. Rev. Lett.
135
Curriculum Vitae
Personal Details
ANTON W. ÖTTL
Date of birth: 12. March, 1974
Place of birth: Salzburg, Austria
Citizenship: German
Rötelstrasse 63
8037 Zürich
Switzerland
phone: +41 44 63 32346
fax: +41 44 63 31254
[email protected]
Education and Degrees
03/2002–07/2006 Swiss Federal Institute of Technology (ETH) Zürich, Switzerland
PhD Research in Quantum Optics
09/2001–02/2002 Institute National Polytechnique de Grenoble (INPG), France
Internship
March 2001 Diplom
09/1997–08/1998 University of Toronto, Canada
Exchange Program Baden-Württemberg - Ontario
October 1996 Vordiplom
10/1994–03/2001 Albert-Ludwigs University in Freiburg, Germany
Studies in Physics (major), Mathematics and Chemistry
07/2003–09/2004 Civil Service
June 1993 Abitur
09/1984–07/1993 High School (Karlsgymnasium) Bad Reichenhall, Germany
Honors and Awards
Government of Canada Award 1999
Research Project at the University of Toronto (01–07/2000)
US Patent # 6,552,301 Burst Ultrafast Laser Machining Method
P. R. Herman, R. S. Marjoribanks, and A. Öttl
137