Download Investigation of an Atom-Ion Quantum Hybrid System

Investigation of an Atom-Ion
Quantum Hybrid System
Lothar Ratschbacher
St Catherine’s College
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
April 2013
This dissertation is my own work and contains nothing which is the outcome of work
done in collaboration with others, except as specified in the text and/or acknowledgments.
It has not been submitted in whole or in part for the award of a degree at the University of
Cambridge or any other University. This thesis does not exceed the prescribed word limit
of 60’000 words.
To Natalie, Karin, Jutta and Leo
Abstract
Investigation of an Atom-Ion Quantum Hybrid System
In this thesis I report on experiments that study the chemistry and spin evolution of
single trapped Yb+ ions immersed in an ultracold cloud of
87 Rb
atoms. The investiga-
tions are performed in a hybrid setup that overlaps the two species in the same physical
location by combining a linear Paul trap for ions and an apparatus to produce and magnetically/optically trap atomic quantum gases. With both systems well isolated from the
environment, we study in detail the effects of binary inelastic collisions to identify the
fundamental interaction processes between atoms and ions at low temperatures.
Our experiments on reactive collisions in Yb+ +87 Rb make use of the beneficial properties of the ion-neutral hybrid system to investigate chemical processes at the most elementary level. We use state-selected single particles, precise control of the interaction,
and detailed analysis of the reaction products to characterize the rate, branching ratio and
energy release of various inelastic reaction channels. In particular, the charge exchange
reaction rate
174 Yb+ +87 Rb→Yb+87 Rb+
is shown to have a very strong dependence on
the internal electronic quantum state of the ion. We use this effect to tune the exchange
reaction rate of a single ion over several orders of magnitude by exerting control over its
internal quantum states. Interestingly, a high sensitivity of the charge exchange process on
the hyperfine state of the atoms is also detected, which highlights the importance of the
hyperfine interaction in atom-ion collisions.
The investigation of the spin evolution in the hybrid quantum system represents another important part of this work. We perform measurements on the spin dynamics and
decoherence of two different kinds of ion spin qubits during interaction with the neutral
atomic cloud environment. For this purpose we implement quantum state preparation,
manipulation and detection techniques for the hyperfine and Zeeman spin systems of the
isotopes
171 Yb+
and
174 Yb+ ,
respectively. We observe spin coherence times T1 and T2
on the timescale of the Langevin collision rate and identify the spin-exchange interaction
and an unexpectedly strong spin-nonconserving coupling mechanism as the sources of spin
decoherence in our system. We further detect a strong ion-heating mechanism for atomic
clouds prepared in the upper hyperfine manifold that we attribute to the release of energy
due to
87 Rb
hyperfine spin relaxation. The insights into inelastic atom-ion interaction and
the experimental techniques described in this thesis should help to realize proposed quantum simulation and quantum information experiments in this apparatus and the steadily
increasing number of atom-ion hybrid systems worldwide.
Contents
1 Introduction
9
2 Theory of Ion-Atom interaction
15
2.1
Introduction to Binary Atomic Interactions . . . . . . . . . . . . . . . . . .
15
2.2
Atom-Ion Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
The Polarization Potential, V (R → ∞) ∝ −1/R4 . . . . . . . . . . .
18
Ab Initio Potential Energy Curves V (R) . . . . . . . . . . . . . . . .
18
Atom-Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3.1
Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.3.2
Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2.1
2.2.2
2.3
3 Theory of Ions and Atoms in a Paul Trap
3.1
33
Radio Frequency Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.1.1
Pseudo Potential Approximation . . . . . . . . . . . . . . . . . . . .
34
3.1.2
The Linear Paul Trap . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.1.3
Linear Coulomb Crystals . . . . . . . . . . . . . . . . . . . . . . . .
37
3.1.4
Trap Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.1.5
Micromotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.2
Micromotion Heating in Ion-Atom Collisions . . . . . . . . . . . . . . . . .
42
3.3
Effect of the Paul Trap on the Atoms
46
. . . . . . . . . . . . . . . . . . . . .
4 Experimental Setup and Methods
4.1
4.2
Preparation of Ultracold
87 Rb
49
Atoms . . . . . . . . . . . . . . . . . . . . . .
49
4.1.1
Magneto-Optical-Trap and Molasse Cooling . . . . . . . . . . . . . .
50
4.1.2
Magnetic Transport and Evaporative Cooling . . . . . . . . . . . . .
52
4.1.3
Tunability of the Atomic Bath . . . . . . . . . . . . . . . . . . . . .
53
87 Rb
4.1.4
Absorption Imaging of
Atoms . . . . . . . . . . . . . . . . . . .
56
4.1.5
Density Measurement of the Neutral Bath . . . . . . . . . . . . . . .
59
Trapping of
4.2.1
Yb+
Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Loading of Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
174 Yb+
4.2.2
Doppler Cooling and Fluorescence Detection in
. . . . . . .
65
4.2.3
Experimental Determination of the Ion Electronic State Occupation
67
171 Yb+
4.2.4
Doppler Cooling of
. . . . . . . . . . . . . . . . . . . . . . .
69
4.2.5
Optical Layout of the Experiment . . . . . . . . . . . . . . . . . . .
71
7
Yb+ Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Experimental Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.2.6
4.3
5 Cold Chemistry with Single Particles
5.1
Chemistry in the
81
. . . . . . . . . . . . . . . . . .
82
Measurement Sequence . . . . . . . . . . . . . . . . . . . . . . . . .
82
Inelastic Collisions in Electronically Excited States . . . . . . . . . . . . . .
88
5.2.1
Collisional Quenching . . . . . . . . . . . . . . . . . . . . . . . . . .
90
5.2.2
Inelastic Collision during Fluorescence Detection . . . . . . . . . . .
91
5.2.3
Error Estimation for the Survival Method . . . . . . . . . . . . . . .
93
5.2.4
Error Estimation for the Fluorescence Method . . . . . . . . . . . .
94
5.1.1
5.2
Yb+
+ Rb Ground State
6 Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms 99
6.1
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1.1
6.1.2
6.2
Zeeman Qubit in
174 Y b+
Hyperfine Qubit in
171 Yb+
. . . . . . . . . . . . . . . . . . . . . . . . 101
. . . . . . . . . . . . . . . . . . . . . . . 108
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2.1
Spin-Exchange and Spin-Relaxation in
171 Yb+
174 Yb+
. . . . . . . . . . . . 115
6.2.2
Hyperfine Spin-Relaxation in
6.2.3
Spin Relaxation Heating . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2.4
Spin Coherence of the
171 Yb+
. . . . . . . . . . . . . . . . . . 119
Qubit . . . . . . . . . . . . . . . . . . 123
7 Outlook
127
Appendix A
A.1 Spins of
133
174 Yb+ , 171 Yb+ , 87 Rb
in a Magnetic Field . . . . . . . . . . . . . . 133
A.2 Rate Equation Model for Spin Dynamics in
171 Yb+
. . . . . . . . . . . . . . 135
A.3 Axial Trap Frequencies of Small Ion Crystals . . . . . . . . . . . . . . . . . 136
Chapter 1
Introduction
“The whole is greater than the sum of its parts” [1]. The motivation for combining two
very successful experimental platforms, single trapped atomic ions and ultracold neutral
atoms to a single atom-ion hybrid system is rooted in the wealth of experimental possibilities and physical phenomena that has been predicted for this cold matter composite. For
its realization, the hybrid system can resort to the precise manipulation techniques for the
internal and external degrees of freedom that have been developed for both species over
the last few decades.
For ultracold neutral atoms, the realization of Bose-Einstein-Condensates (BEC)1 of dilute
alkali vapors [2, 3] in 1995 and the observation of the first fermi-degenerate quantum gas [4]
in 1999 have been key moments. The possibility to access and tune the microscopic parameters of macroscopic quantum objects has made dilute quantum gases valuable tools for
simulating and testing quantum many-body models of condensed matter theory. Prominent
examples of this process are the observations of superfluid flow [5], vortices [6] and matter
wave interference [7] in BECs. The investigations of the superfluid to Mott-insulator quantum phase transition [8] and the BEC-BCS crossover [9] in strongly correlated bosonic and
fermionic atomic gases, respectively, are remarkable successes and the field is now advancing towards the local probing of quantum phases [10]. Registers of ultracold neutral atoms
have been envisioned for large scale quantum information processing and the demonstration
of collisional entangling gates [11] and the recent focus on single-particle addressing [12, 10]
are important steps in this direction. At the same time, the excellent coherence properties
of ultracold neutral atoms have enabled applications in metrology, ranging from gravitational [13] and electromagnetic field [14] sensors to high precision atomic clocks [15, 16].
Single trapped ions have seen an equally impressive evolution in their state manipulation
techniques and now arguable constitute the best-controlled single particle quantum systems2 . The long trapping times, excellent coherence properties, and exquisite control over
the internal and external quantum states have put single trapped ions at the forefront of
ultrahigh resolution spectroscopy [17, 18] and quantum information science. With the fidelity of fundamental quantum logic gates [19, 20] routinely exceeding the thresholds for
1
2
Nobel prize in Physics in 2001.
Nobel prize in Physics in 2012.
9
Chapter 1. Introduction
efficient quantum error correction [21], the biggest remaining challenge for universal quantum computation [22] in ion traps is the scaling of current architectures to system sizes
beyond a few ions [23].
By harnessing the mutual interaction between these two quantum systems, an integrated
ion-atom hybrid setup has the potential to enhance the capabilities of the two individual
constituent quantum systems and at the same time give rise to new physical phenomena.
Theoretical predictions have ranged from the sympathetic cooling [24] of the ion due to momentum transferring elastic collisions, to the modifications of the local density distribution
of bosonic gases, caused by the presence of ion impurities [25, 26, 27]. Assuming accurate
control of the atom-ion interaction, single trapped ions inside a degenerate quantum gas
have further been suggested as powerful local probes for characterizing the properties of
strongly correlated many-body quantum systems [28, 29]. Advanced future applications
could also include the use of state dependent ion-atom interactions for entangling gates in
ion-atom quantum processors [30, 31] and to control the tunneling of atoms through a
Josephson junction [32].
Experimentally, ultracold quantum gases and trapped ions have only been combined very
recently [33, 34]. Indeed, the apparatus, employed for the experiments in this thesis, has
realized the first insertion of a single ion into a BEC in 2009 [33]. On immersion into
the degenerate gas of Rb atoms, the sympathetic cooling of single trapped Yb+ ions has
been observed [35] in combination with significant atom losses from the neutral cloud. The
results have also provided a first reality check for theoretical proposals that require ions
and the atoms to interact at ultralow energies. The steady state energies of the ion during
elastic collisions have been found to be far in excess of the atomic bath temperature and
instead limited by collisional heating effects due to uncompensated electric fields in the
ion trap [36, 37]. Recent theoretical studies suggest that immersion cooled ions in Paul
traps are fundamentally prevented from entering the s-wave collision regime [38] for most
ion-atom combinations, except for the ones with the largest ion-atom mass ratios [39].
Current ion-atom hybrid systems, including the growing number of experiments that combine single trapped ions with ultracold atoms in magneto-optical-traps [40, 41, 42, 43], are
therefore limited to the multiple partial wave regime, where collisional dynamics is well
described by semiclassical theory [44]. In this regime the elastic scattering properties are
indeed completely independent of the short range potential and can be understood from
the asymptotic behaviour of the long range atom-ion interaction potential.
In addition to elastic scattering colliding ions and atoms can also undergo inelastic events
that change their internal electronic state or spin state. Charge exchange [45, 40, 46, 35,
42, 41, 47] and molecule formation [48, 25, 49, 41, 50], which belong to the first kind of
inelastic events, represent simple prototypes of chemical reactions. The study of neutral-ion
10
reactions [51, 52] and molecules [53, 54] in ion traps offers the same richness of chemical
interactions as the chemistry between neutral atoms, but at the same time has the key
advantage that charged particles can be manipulated and analyzed for extended periods of
time in the trap. In recent years the study of chemical reactions at low temperature has
attracted considerable interest [55, 56, 42, 41, 57, 58, 50, 47] as the excellent level of control
in cold and ultracold matter experiments has the potential to enhance the understanding
and control over chemical reactions [59] at the most fundamental level.
In this thesis I present experiments that investigate in detail the dynamics of the internal
states of the pioneering hybrid quantum system of Yb+ ions and neutral Rb atoms. We
develop experimental techniques to obtain control over the quantum states of ions and
atoms in the complex setting of a hybrid experiment and apply our system to the study of
cold chemistry and the dynamics of spin-bath decoherence.
Our experiments on reactive collisions in Yb+ +87 Rb parallel similar efforts in a number of
other hybrid combinations [41, 42, 58] that control the motional degrees of freedom of both
reactants. We add to this the exact manipulation of the internal quantum state and perform studies with quantum state-selected single particles, precise control of the interaction,
and detailed analysis of the reaction products to characterize the rate, branching ratio and
energy release of various inelastic reaction channels [47]. In particular, the charge exchange
reaction rate 174 Yb+ +87 Rb→Yb+87 Rb+ is shown to have a very strong dependence on the
electronic quantum state of the ion and the hyperfine state of the neutral atoms. We show
that cold reactive processes can be steered by exerting control over the internal quantum
states of the ion and the atoms.
At the same time, the immersion of a single ion with a spin degree of freedom into the spin
polarized neutral atomic cloud also realizes the coveted concept of an isolated, controllable
single qubit that couples to a well-defined and adjustable environment. The dynamics of
a spin-1/2 particle interacting with its surroundings is a problem that has been intensely
studied in theory [60, 61]. It represents a model system for the understanding of the quantum to classical transition [62] and plays an important role in impurity effects in solid state
systems [63, 64].
Experimentally, a high level of control over the spin impurity, the tunability of the properties of the environment and a good understanding of the interaction are the key tools
required to address these questions [65, 66, 67]. In our hybrid system the spin-polarized
neutral cloud constitutes such a widely tunable bath environment which has both continuous (motional) and discrete (spin) degrees of freedom. The well-defined atomic nature of
both the impurity and the bath offers the possibility of a precise microscopic description
of their interaction. The localized spin impurities are implemented using one of two complementary ion spins: the ideally isolated two-level Zeeman qubit in the electronic ground
11
Chapter 1. Introduction
state of
174 Yb+
and the clock-transition hyperfine qubit in
171 Yb+
[68]. Our experiments
are the first study of the decoherence and spin dynamics in this all-atomic spin-bath system
formed by the combination of ions and atoms in a hybrid setup.
The thesis is structured as follows
I begin in Chapter 2 by presenting the theoretical framework that exists for the description of atom-ion interactions at low collision energies and explain how it can be applied
to the experimental conditions of our Yb+ +
87 Rb
hybrid system. Starting with a short
motivation of the binary interaction Hamiltonian, the atom-ion interaction potential and
its approximations at long (the polarization potential) and at short distance (the ab initio
Born-Oppenheimer potentials) will be introduced. Based on this model the predictions
for the elastic and the various inelastic processes in the classical, semiclassical or quantum
mechanical approximation will then be reviewed in detail.
Chapter 3 discusses the physics of both ions and atoms in the linear Paul trap, which is at
the centre of our hybrid experiment. In the following Sections the aspects most important
for ion trap chemistry, the stability of the trap for different particle masses and the mass
sensitive oscillation modes of ion crystal, are discussed in some detail. Moreover, we present
both an intuitive picture and quantitative estimates for the crucial problem of micromotion
induced radio frequency heating of ions in neutral buffer gas experiments, that is limiting
the collision energies in current hybrid systems.
In Chapter 4 the experimental apparatus for producing ultracold
trapped
Yb+
87 Rb
clouds and single
ions will be reviewed. We will give an overview of the pre-existing experi-
mental setup [69, 70] and describe the additions which have been made for the experiments
in this thesis. We explain the techniques for preparing and controlling the ultracold bath
of neutral atoms and the basic operation of the ion trap.
Chapter 5 contains the experimental description and the results of the chemistry experiments in the ion-atom hybrid system. The techniques for state selective preparation and
analysis, including in-trap mass spectroscopy are explained in the first part of the chapter. This is followed by a detailed presentation of the the various electronic state changing
inelastic processes in the Yb+ + Rb hybrid system. We find a strong dependence of the
charge exchange process on both the electronic state of the ion and the hyperfine state of
the atoms and demonstrate how control over the internal state of single ions and neutral
atoms can be employed to tune cold reaction processes.
Our investigations of the spin interaction of a single ion qubit with its ultracold atomic
bath are reported in Chapter 6. First, we introduce the methods we have developed for the
preparation, manipulation and detection of the two different ion spin qubits in this experiment. We then study the spin dynamics and decoherence of the single spin qubit coupled
12
to its environment of spin polarized neutral atoms and observe spin coherence times T1
and T2 on the timescale of the Langevin collision rate. Spin-exchange interaction and an
unexpectedly strong spin-nonconserving relaxation mechanism are identified as the sources
of spin decoherence. The strong spin relaxation in our system is further shown to give rise
to significant motional heating for certain atomic states of our hybrid system.
In Chapter 7, I conclude this report with an outlook onto future investigations using the
atom-ion hybrid setup, including some ideas for the experimental route towards the holy
grail [71, 72] of atom-ion interactions: s-wave scattering.
Publications in the context of this thesis
Decoherence of a single ion qubit immersed in a spin-polarized atomic
bath
Ratschbacher, L.; Sias, C.; Carcagni, L.; Silver, J.; Zipkes, C.; Köhl, M.
accepted for Physical Review Letters (2013)
Controlling chemical reactions of a single particle
Ratschbacher, L.; Zipkes, C.; Sias, C.; Köhl, M.
Nature Physics, 8, 649 (2012)
Kinetics of a single trapped ion in an ultracold buffer gas
Zipkes, C.; Ratschbacher, L.; Sias, C.; Köhl, M.
New Journal of Physics, 8, 053020 (2011)
Cold heteronuclear atom-ion collisions
Zipkes, C.; Palzer, S.; Ratschbacher, L.; Sias, C.; Köhl, M.
Physical Review Letters , 105, 133201 (2010)
13
Chapter 1. Introduction
14
Chapter 2
Theory of Ion-Atom interaction
A detailed understanding of the fundamental interactions of atoms and ions in the experimental regime of our cold matter apparatus is the essential foundation on which further
investigations and applications can build. In this chapter I will present the theoretical
framework that exists for the description of atom-ion interactions and apply it to the experimental conditions of our Yb+ +87 Rb hybrid system.
I start by motivating the simplified quantum mechanical Hamiltonian of binary atomic interaction that generally serves as the starting point for the treatment of the collisional phenomena and molecular dimer physics. Next, the polarization potential V (r → ∞) ∝ −1/r4
and the numerical results of Yb+ +Rb ab initio potential curves will be introduced as the
approximations for the atom-ion interaction potential at large and small atom-ion internuclear distances. Based on this model, we explore the theoretical predictions for the elastic
and the various inelastic processes in the classical, semiclassical and quantum regime of
Yb+ + Rb collisions.
2.1
Introduction to Binary Atomic Interactions
In the context of atomic physics, the complete quantum mechanical description of the
interaction of two atoms in field-free space is a many-body Schrödinger equation involving
the two nuclei and all electrons of the system. Its Hamiltonian consists of the kinetic energy
and the mutual interactions of all particles as well as the interactions arising from the
electron spin and nuclear spin couplings. For all but the lightest atoms, the computational
requirements for calculations with the complete many-body Hamiltonian are prohibitively
high. In order to arrive from this first principle model to the simplified single particle
Hamiltonian, that is used in the field of molecular chemistry and ultracold collisions, a
number of approximations are performed, that we will shortly review based on [73].
The first method to reduce the complexity is to approximate the effect of core electrons by
introducing effective core potentials for both nuclei and consider only the valence electrons
explicitly.
15
Chapter 2. Theory of Ion-Atom interaction
In the case of the Rb+Yb+ interaction such a Schrödinger equation would look like
~
~
~
~
HΨ = −
∇2e1 −
∇2e2 −
∇2Rb −
∇2
2me
2me
2mRb
2mY b Y b
e2
2e2
e2
2e2
−
−
−
−
4πǫ0 |r e1 − RRb | 4πǫ0 |r e1 − RY b | 4πǫ0 |r e2 − RRb | 4πǫ0 |r e2 − RY b |
e2
2e2
+
Ψ = EΨ.
(2.1)
+
4πǫ0 |r e1 − r e2 | 4πǫ0 |RRb − RY b |
Atom (r) =
Here, we have assumed the very crude effective core potentials UECP
Ion (r) =
UECP
2e2
4πǫ0 |r| ,
e
4πǫ0 |r| )
and
that correspond to a maximal shielding of the nuclei charges by the
core electrons. We do not consider the spin degree of freedom at this point.
Another important concept is the Born-Oppenheimer approximation. The basic idea behind it is that the electrons, due to their smaller masses, move faster than the nuclei and
therefore are able to immediately adapt to the slower nuclear motion. By making use of
the different time scales the problem can be split into two parts1 . First the solutions for
the electronic motion are calculated while holding the position of the nuclei fixed
~
~
e2
e2
2e2
HΨe = −
∇2e1 −
∇2e2 +
−
−
2me
2me
4πǫ0 |r e1 − r e2 | 4πǫ0 |r e1 − RRb | 4πǫ0 |r e1 − RY b |
2
e
2e2
−
−
Ψe = W Ψe .
(2.2)
4πǫ0 |r e2 − RRb | 4πǫ0 |r e2 − RY b |
For each fixed separation R = |RRb − RY b | of the two nuclei2 Equation 2.2 is solved by sets
of electronic states with energy values W (|R|) and distinguished by their corresponding
quantum numbers. These instantaneous electronic-state dependent energies W (|R|) act
as potential energy curves for the motion of the nuclei and give rise to the Schrödinger
equation
effective core
HΨn =
z }| {
electronic
z }| { ~
2e2
~
2
2
+ W (|R|) Ψn
∇ −
∇ +
−
2mRb Rb 2mY b Y b
4πǫ0 |R|
|
{z
}
=
ẼΨn .
(2.3)
effective potential
The result of the Born-Oppenheimer approximation is a description of the interaction
of two multi-electron atoms as a problem of two structureless point particles interacting
1
Formally one arrives at the individual Equations 2.2 and 2.3 by making a product ansatz of electronic and nuclear wave functions for the total wave function in Equation 2.1 Ψ(re1 , re2 , RY b , RRb ) =
Ψe (re1 , re2 , RY b , RRb )Ψn (RY b , RRb ), then neglect terms which are smaller by the order of mmReb and mmYeb
and finally consider the separation of timescales of the electron and nuclei motion.
2
A change of coordinates, for example by r → r + RY b makes it evident that Equation 2.2 indeed only
depends on the difference |RRb − RY b | and not on RRb , RY b individually.
16
2.2. Atom-Ion Potentials
through an electronic-state dependent effective potential of radial dependence V (|R|) =
UECP (|R|) + W (|R|).
The introduction of centre-of-mass and relative coordinates Rcm =
RRb − RY b allows to separate the wave functions Ψn (RRb , RY b )
mRb RRb +mY b RY b
, R
mRb +mY b
iK·R
cm
· ΨR (R).
= e
=
It
follows that the energy eigenvalues of the system are the combination of the free particle
energy spectrum
~2 K 2
2mcm
for the centre-of-mass motion and the eigenvalues described by the
effective one-particle Schrödinger equation
HΨR =
~ 2
−
∇ + V (|R|) ΨR
2µ R
=
EΨR
(2.4)
for the relative motion of the nuclei. Equation 2.4 is the Schrödinger equation for a single
mRb mY b
mRb +mY b
in a spherically symmetric potential. By decom∞
P
(R)
Pl (cos(θ)) ulR
posing the wave functions into the radial and angular parts ΨR (R) =
particle with reduced mass µ =
l=0
the problem of binary atomic interaction is finally described by the one-dimensional radial
partial-wave equation
2 2
~ ∂
~2 l(l + 1)
−
+
+ V (R) ul (R) = E · ul (R),
2µ ∂R2
2µR2
(2.5)
that, in addition to the adiabatic Born-Oppenheimer potential energy curves V (R) specific
to the electronic state of the atomic system, also includes the repulsive centrifugal term
~2 l(l+1)
,
2µR2
which depends on the rotational state as specified by its azimuthal quantum number
l.
2.2
Atom-Ion Potentials
In the Born-Oppenheimer approximation the ion-atom interaction problem has been reduced to a set of one-dimensional second-order linear differential equations that can be
solved for the scenarios of interest, once its effective potential energy terms are known.
The central challenge therefore lies in the accurate determination of the atom-ion interaction potentials.
For convenience the conversions of the various units commonly used by atomic physicist
and chemists for energy and length scales shall be mentioned here.
1eV ⇔ 8065.73 cm−1 ⇔ 11604.9 K ⇔ 0.0367 Eh ,
a0 = 0.529 · 10−10 m,
(2.6)
where the Bohr radius a0 and the Hartree energy Eh are the units of length and energy in
atomic units.
17
Chapter 2. Theory of Ion-Atom interaction
The notation that we use in the following Section for the characterization of heteronuclear
molecular states is the molecular term symbol
2S+1
(+/−)
ΛΣ
,
(2.7)
where S is the total spin quantum number, Λ is the projection of the orbital angular
momentum along the internuclear axis, Σ is the projection of the total angular momentum
along the internuclear axis and (+/−) is the reflection symmetry along an arbitrary plane
containing the internuclear axis.
2.2.1
The Polarization Potential, V (R → ∞) ∝ −1/R4
At large internuclear separations the dominant interaction between an ion and a neutral
atom is the attractive polarization interaction that can be understood as follows. The
charge of the ion generates the inhomogeneous electric field E(R) =
Q
R̂
4πǫ0 R2
at the location
R of the atom. Due to the static polarizability α of the atom, the field induces the electric
dipole moment p = α · E that itself interacts with the field according to F = p · ∇E =
αE · ∇E. The asymptotic R → ∞ interaction between an ion and an atom is therefore
described by the attractive polarization potential
Uind (R) =
Z∞
R
F (R′ )dR′ = −
αQ2
α|E(R)|2
=−
≡ −C4 /(2R4 ).
2
8πǫ0 R4
(2.8)
The polarization potential is independent of the electronic state of the ion, but varies for
different electronic states of the atom due to the state-dependent static polarizability α. For
the electronic ground state 2 S1/2 of 87 Rb it has the value α0 = h · 0.0794(16)Hz/(V/cm)2 [74].
2.2.2
Ab Initio Potential Energy Curves V (R)
At shorter atom-ion separations the electronic wave functions of the atom and the ion are
significantly modified and the interaction potentials become strongly state dependent, until
at distances on the order of several Bohr radii the interaction turns repulsive. In this regime
the molecular potential energy curves for different electronic states can be estimated by
numerical ab initio calculations. One method to assess the accuracy of such computational
models is to compare their predictions of the individual atomic spectroscopic properties
with experimental results. This procedure shows that Equation 2.2 is in fact a bad starting
point for the calculation of adiabatic potentials, as it would approximately produce the
atomic spectra of He+ and H, instead of Yb+ and Rb for well separated atoms. However,
even the most advanced ab initio quantum chemical calculation are currently not precise
18
2.3. Atom-Ion Collisions
enough to predict the collisional properties of atoms and ions in the quantum regime and
the energy of weakly bound molecular states.
In Figure 2.1 the potential energy curves of the few lowest electronic excitations of the
(Yb + 87 Rb)+ system are shown from two recent calculations [75, 76]. The direct comparison of the potential energy curves from the two independent quantum chemical calculations
reveals differences at the level of ≈ 1000 cm−1 between the two sets of data. We interpret
this as an estimate for the accuracy of the current state of the art quantum chemical calcu-
lations. The discrepancies between the calculated potential energies at large internuclear
separations and the well-know atomic energy levels of the separated atom and ion are in
the range of ≈ 100 cm−1 .
It is evident from Figure 2.1 that the entrance channel of electronic ground state Yb+ ions
and
87 Rb
atoms, which we prepare in most of our experiments, is in fact not the absolute
electronic ground state of (Yb +
exchanged channel
Yb(6s2 1 S)
87 Rb)+
+
87 Rb+
system. The R → ∞ asymptote of the charge-
is located 16752 cm−1 (2.08 eV, kB · 24216 K)
lower in energy as a result of the difference in the ionization potentials of Yb and Rb.
All other molecular potential curves in Figure 2.1 are electronic excitations of the chargeexchanged combination Yb + 87 Rb+ . These states correspond asymptotically to excitations
of the neutral Yb atom, since the excitations of Rb+ amount to transitions in the extreme
ultraviolet due to the ion’s Kr-like noble gas shell structure. The lowest of the chargeexchanged excitations, which dissociates to the fine structure triplet Yb(6s6p, 3 P0J ) and
Rb+ , lies asymptotically just above the entrance channel but crosses it at shorter internuclear distances.
The Bohr-Oppenheimer potential shown have been obtained from scalar-relativist calculations and therefore do not account for spin-orbit coupling effects. In addition, the quantum
chemical calculations are limited to excited state potentials with energies below the level of
Yb+ (4f 13 )(6s2 ) + Rb(6s) (see Figure 5.3). This state, that represents the lowest electronic
excitation of the Yb+ ion cannot be efficiently accounted for by the theoretical techniques
used in the two studies due to its open f -shell. Its presence also prevents the accurate
computation of potential energy curves for any higher-lying excited states [75].
2.3
Atom-Ion Collisions
The potential energy curves of Figure 2.1 indicate, that even for low energy collisions of
Yb+ ions with
87 Rb
atoms, several different types of processes can occur. Our theoretical
description will focus on the description of binary collisions of a single ion and a single
atom (Table 2.1). Three-body collisions of one ion and two atoms, that have recently been
observed [77], are not included into our treatment as they do not play an observable role
19
Chapter 2. Theory of Ion-Atom interaction
40000
30000
Energy (cm-1)
Rb+ + Yb (6s6p 1 P)
Rb+ + Yb (6s6p 3 P)
20000
Rb(5s) + Yb+ (6s)
1
Π
1 +
Σ
10000
3
Π
3 +
Σ
Σ
3 +
Σ
1 +
Σ
1 +
0
Rb+ + Yb (6s2 1S)
)
5
10
15
20
25
30
R (a.u.)
Figure 2.1: Potential energy curves for the (Yb + 87 Rb)+ system. The molecular singlet
(1 Σ+ ) and triplet (3 Σ+ ) channels that asymptotically connect to the state of separate
Yb+ (6s)+87 Rb(5s) are not the electronic ground of the system. The triplet (3 Σ+ ) channel,
corresponding to the charge exchanged state Yb(6s2 1 S)+87 Rb+ , lies 16752 cm−1 (2.08 eV,
kB · 24216 K) below. The graph shows a comparison of the datasets from two different ab
initio calculations (lines [75]), (lines+points [76]) and yields an estimate for the precision
of current quantum chemical methods on the order of 1000 cm−1 . To the right of the
graph the (r → ∞) asymptotic energy levels as determined from atomic spectroscopy are
indicated. The internuclear separation on the horizontal axis is given in atomic units.
20
2.3. Atom-Ion Collisions
in the dynamics of the hybrid system at our experimental atomic densities.
Yb+
+
Rb
→
Yb+
+
Rb
Yb+ | ↑i
Yb+ | ↓i
+
+
Rb | ↓i
Rb | ↑i
→
→
Yb+ | ↓i
Yb+ | ↑i
+
+
Rb| ↑i
Rb| ↓i
)
spin exchange
Yb+ | ↑i
···
+
···
Rb | ↑i
···
→
···
Yb+ | ↓i
···
+
···
Rb| ↑i
···
)
spin relaxation
Yb+
Yb+
Yb+
+
+
+
Rb
Rb
Rb
Yb
+ Rb+ + γ
Yb
+
Rb+
(Yb Rb)+ + γ
→
→
→
elastic
radiative charge transfer
non-radiative charge transfer
molecule formation
Table 2.1: Possible elastic and inelastic processes during binary collisions of Yb+ ions and
Rb atoms.
2.3.1
Elastic Collisions
In the case of purely elastic scattering the internal states of the atom and the ion do not
change and the outgoing channel is per definition identical to the incoming channel. Elastic
collisions in the singlet or triplet potential of the incoming channel lead to the transfer of
momentum and kinetic energy in the laboratory frame and are therefore of importance for
sympathetic cooling.
The total elastic cross section at the collision energy E = ~2 k 2 /(2µ) is given by [78]
σel. =
∞
X
l=0
∞
4π X
(2l + 1)sin2 (ηl )
σl = 2
k
(2.9)
l=0
as the sum over all partial-wave contributions. The partial-wave phase shifts ηl are determined by the continuum solutions ul (R) of the partial-wave equation 2.5. The asymptotic
behavior of the solutions ul (R) at large distances is of the form
lπ
ul (R) ∼ sin kR −
+ ηl (k)
2
(2.10)
and defines the elastic scattering phase shifts ηl .
For the lowest partial-wave (l = 0, s-wave), the potential energy term in Equation 2.5
21
Chapter 2. Theory of Ion-Atom interaction
simply corresponds to the adiabatic atom-ion interaction potential. The potential energy
terms of higher partial waves (l ≥ 1) contain the centrifugal contribution
combination with the attractive asymptotic polarization V ≈ −C4
height3
~4 l2 (l + 1)2
Ecf (l) ≈
8µ2 C4
at radial positions
For collision energies E ≪ Ecf (l = 1) = E ∗ =
rcf (l) ≈
~4
,
2µ2 C4
/(2R4 )
s
~2 l(l+1)
,
2µR2
that in
form barriers of
2µC4
.
+ 1)
~2 l(l
(2.11)
only the l = 0 partial wave
contributes to the elastic scattering, as for higher partial waves the effect of the inner part
of the potential is suppressed by the centrifugal barrier (see Figure 2.2). In the s-wave limit
the scattering cross-section approaches the energy independent value σel. = 4πa2 , where a is
the zero temperature scattering length. The direct determination of a = limk→0
tan(η0 (k))
k
by
quantum mechanical calculations [79] using the previously presented ab initio potentials and
Equations 2.5 and 2.10, is not possible. Ab initio potential energy curves are not accurate
enough to calculate the scattering length from first principles. Without any knowledge
about the details of the short range potential, a statistical argument can be made, that the
median of the probabilityqdistribution4 of possible cross sections has the value ∼ 4πr∗2 ,
4
where r∗ = rCF (l = 1) = µC
is the characteristic length of the polarization potential [80].
~2
In the case of atom-ion collisions, the characteristic energy for the occurrence of the s-
wave threshold behavior is in the nano-Kelvin temperature range as a result of the long
range of the 1/r4 potential5 . The value for collisions of Yb+ + Rb is E ∗ = kB · 44 nK.
Whereas these temperatures can, in principle, be achieved for the ultracold buffer gases,
the significantly higher energies of the ion motion in current experiments (see Section 3.2)
result in the contribution of many partial waves in atom-ion collisions.
Semiclassical Approximation
In the many-partial wave regime the energy-dependent cross section for elastic collisions
can be estimated using a semiclassical approach [44]. To this end the evaluation of the
partial wave phase shifts in Equation 2.9 is split into two different parts and independent
approximations are applied in each case.
For large values of the angular momentum quantum numbers l the partial wave phases are
assumed to be insensitive to inner part of the potential due to the centrifugal barrier. In
3
The result is readily obtained by finding the stationary points of the potential energy term Epot (R) ≈
−C4 /(2R4 ) + ~2 l(l + 1)/(2µR2 ) in Equation 2.5.
4
The actual value of the scattering length a can still take any value depending on the exact details of the
short-range potential. In fact the probabilities for positive or negative scattering lengths are equal in the
case of the polarization potential [80].
5
The characteristic energies for the s-wave threshold behavior in neutral-neutral collisions are typically in
the ∼ 100 µK range due to the much shorter range of the ∼ 1/r6 van der Waals potential
22
2.3. Atom-Ion Collisions
Figure 2.2: Centrifugal barriers for higher order partial waves. The red bar at 750 a0
indicates the energy distribution and spatial extend of the wavefunction for an ion-atom
collision with an average kinetic energy on the order of kB · 50 mK.
this case the phase shifts are determined by the long range potential and can be calculated
from the semiclassical approximation for phase shifts [44]
µ
ηl ≈ − 2
~
Z
∞
R0
6
V (R)
,
dR p
2
k − (l + 1/2)2 /R2
(2.12)
where R0 is the classical outer turning point. The evaluation of this equation for the
V (R) = −C4 /(2R4 ) potential yields
7
ηl ≈
πµ2 C4 E
.
4~4 l3
(2.13)
On the other hand, partial waves with small values of l, get affected by the inner part of
the potential and their phase shifts ηl quickly become large as l decreases. As we are taking
the sum over many partial waves with rapidly varying phase shifts ηl the terms sin2 (ηl ) in
the elastic cross section (Equation 2.9) can be approximated by the average value 1/2. If
we split the evaluation of the elastic cross section into two parts, such that Equation 2.13
and sin(ηl ) ≈ ηl apply for l > lsplit and that sin2 (ηl ) ≈ 1/2 is a good approximation for
6
The formula is derived by applying the WKB approximation to spherically symmetric potentials and
includes several further approximations
(see [81]
p
√ and references therein).
√
R
7
The integral formula 1/(R3 R2 − A)dR = R2 − A/(2A · R2 ) − arctan( A/(R2 − A))/(2A3/2 ) and
the approximation for the classical turning point R02 ≈ l(l + 1)/k2 ≈ (l + 1/2)2 /k2 are used during this
evaluation.
23
Chapter 2. Theory of Ion-Atom interaction
l < lsplit , then the cross section of Equation 2.9 reads
σel. (E) =
≈

4π 
k2

lsplit −1
X
(2l + 1)sin2 (ηl ) +
l=0
∞
X
l=lsplit
lsplit −1
4π  X
1
(2l + 1) +

k2
2
l=0
Z∞
lsplit
2π 2
≈ 2 lsplit
(1 + ηl2split ).
k

(2l + 1)sin2 (ηl )


dl 2l ηl2 
(2.14)
It has been confirmed by quantum numerical calculations [44] that the approximations for
both regimes are well satisfied, if lsplit is chosen such that ηlsplit = π/4 and as long as
lsplit > 50. In this case we obtain from Equation 2.13
lsplit =
µ2 C 4
E
~2
1/3
(2.15)
and finally
σel. (E) = π
µ C42
~2
1/3 π2
1+
16
E −1/3 .
(2.16)
The cross section shows a E −1/3 dependency on the collision energy, which implies that
the elastic collision rate of an ion in a neutral gas of density na
γel. = na σel. (E) v ∼ E 1/6
(2.17)
scales as E 1/6 in the many-partial wave scattering regime.
In practice, measurements of collisional cross sections cannot be performed at single welldefined energies in our atom-ion system. The broad energy distribution8 that the ion
acquires during immersion in a neutral cloud means that the theoretical results need to
be energy averaged for comparison with the observed experimental rates. The approximations underlying the semiclassical calculation start to fail for collision energies such that
lsplit < 50. For Yb+ + Rb this corresponds to a lower limit of E ≈ kB ·11µK for the validity
of semiclassical result. In the region of collision energy between the s-wave (E ∗ = kB ·44nK)
and the semiclassical (E ≈ kB ·11µK) thresholds the contributions of only few partial waves
and the occurrence of shape resonances lead to a strong variation of the elastic cross section
with energy.
Quantum mechanical calculations based on the multi-channel treatment of atom-ion collisions, have shown that already in the semiclassical regime the effect of Feshbach resonances
8
This energy distribution is generally not thermal, as will be explained in Section 3.2.
24
2.3. Atom-Ion Collisions
should be observable in ion-atom collisions [38, 82]. Feshbach resonances, that have become
a valuable tool to control the interaction in ultracold quantum gases by tuning the magnetic
field [83], would indeed also lead to very interesting experimental possibility in ion-neutral
collisions. Theoretical studies, albeit for a different combination of ions and neutral atoms
(Ca+ + Na), have demonstrated that a significant number of Feshbach resonances are expected for comparatively low magnetic field strengths in the ∼ 10 G range. Their effect
at finite temperatures should be particularly pronounced for the inelastic charge exchange
collision rates.
2.3.2
Inelastic Collisions
Ion-atom encounters also lead to a number of inelastic processes, that change the internal
states of the colliding particles. Depending on the original state of the colliding pair, these
internal state changes can involve both the spin and the electronic degree of freedom of
the atom and the ion. A first estimate for the energy dependent cross section of various
inelastic events can be made by considering the fact that inelastic dynamics is dominated
by interactions at short internuclear distances. In the semiclassical model of collisions, the
inner part of the potential is only sampled, if the energy of the kinetic energy of the colliding
pair of particles exceeds the height of the centrifugal barrier. From Equation 2.11 it follows,
that for a given collision energy E, it is possible for partial waves with angular momentum
l < lmax (E) ≈
(8µ2 C4 E)1/4
~
to contribute to the inelastic cross section. Classically this means
that collisions with a maximal angular momentum ~lmax (E) and therefore with a critical
impact parameter
rcritical = lmax /k =
2C4
E
1/4
(2.18)
can give rise to inelastic events. The corresponding cross section and collision rate read
2
σclassic = πrcritical
=π
r
2C4
E
and
γclassic = na σclassic v = na · 2π
s
C4
= γLangevin .
µ
(2.19)
Note, that a simple summation of partial waves (with sin2 (ηl ) = 1/2) up to lmax in the
quantum mechanical cross section of Equation 2.5 would have resulted in a two times larger
value of the cross section. We will later use this result in the estimation of the spin-exchange
cross section.
The classical scattering rate of Equation 2.19 is independent of the collision energy and
only depends on the long range properties of the potential. It has been originally derived
by Langevin [84] in 1905 in the context of the mobility of ions in gases. A purely classical
solution of particle trajectories in the ∼ − Cr44 potential [85] shows that for impact param-
25
Chapter 2. Theory of Ion-Atom interaction
eters r < rcritical the colliding particles attract each other to r = 0 in an inward spiraling
motion, whereas for r > rcritical the particles merely deflect from each other.
The probability for a specific inelastic collision to occur once the ion and the atom approach
each other to close distances, depends, of course, on the details of the interaction potential.
It is therefore useful to introduce the relative inelastic efficiencies ǫi,j to express the rate,
at which particles in the state j undergo certain inelastic process i, as
γi,j = ǫi,j · γLangevin .
(2.20)
In this model, the rate of inelastic processes is assumed to be independent of energy and
to be limited by the Langevin rate. For Yb+ + Rb the numerical value of Langevin rate
constant is γLangevin /na = 2.115 · 10−15 m3 /s, which for typical atom densities on the order
of n ∼ 1018 m−3 results in Langevin collision rates on the order of 103 s−1 .
Spin-Exchange Collisions
Spin-exchange collisions are encounters, where electronic spin is transferred from one atom
onto another and the spin orientations of both particles are altered after the event. In the
case of a Rb atom and a Yb+ ion in their respective electronic ground states this simply
corresponds to the exchange of spin orientation between the two valence electrons9
Rb| ↑ia + Y b+ | ↓ii → Rb| ↓ia + Y b+ | ↑ii
Rb| ↓ia + Y b+ | ↑ii → Rb| ↑ia + Y b+ | ↓ii ,
(2.21)
where | ↑i ≡ |S = 1/2, ms = 1/2i and | ↓i ≡ |S = 1/2, ms = −1/2i for both the atom and
the ion.
The physical origin of the effect lies in the fact that particles in symmetric electronic
spin states interact differently with each other compared to those in antisymmetric states.
For a given spin symmetry, the Pauli principle imposes the opposite symmetry for the
spatial component, as the total wave function of the two electron system needs to be
antisymmetric. The Coulomb interaction between the electrons lifts the degeneracy of the
spatial configurations by the exchange energy and results in two potential energy curves.
√
In Figure 2.3 the singlet potential Vs (R) for the spin state 1 Σ+ : 1/ 2(| ↑a ↓i i − | ↓a ↑i i) and
√
the triplet potential Vt (R) for the spin states 3 Σ+ : | ↓a ↓i i, | ↑a ↑i i, 1/ 2(| ↑a ↓i i − | ↓a ↑i i)
are highlighted for the incoming channel of electronic ground state Yb+ and Rb. These
curves can be written as a single interaction potential in terms of the spin operators S a
9
In ultracold atoms coherent spin-exchange interactions have been intensely studied in spinor condensates [86] and used for entanglement generation [87]. For atoms and ions, coherent interaction is not yet
possible, since in the current energy regime many partial waves contribute to collisions.
26
2.3. Atom-Ion Collisions
and S i of the individual electron spins and the total spin S ≡ S a + S i :

V (R)
s
V (R) =
V (R)
t
for 1 Σ+ (S = 0)
for 3 Σ+ (S = 1)
3
1 1
1
1 1
+ 2 (S 2 − S 2a − S 2i ) + Vs (R)
− 2 (S 2 − S 2a − S 2i )
4 ~ 2
4 ~ 2
1
1
3
(2.22)
= Vt (R) + Vs (R) + 2 Vt (R) − Vs (R) S a · S i ,
4
4
~
= Vt (R)
where we have used S 2 ≡ (S a +S i )2 = S 2a +S 2i +2S a ·S i and S|S, ms i ≡ ~2 S(S +1)|S, ms i.
In general, colliding atom-ion pairs start in superpositions of the singlet and triplet
state. During collisions the phases of the singlet and triplet contribution of the wave
function evolve according to their potentials, and as a result the spin directions of the
atoms and ions will change. The total spin of the system S = S a + S i , however, commutes
with the potential in Equation 2.22 and m = ma + mi remains a good quantum number
throughout the interaction. The possible changes of the atom and ion electron spin are
limited to ∆ma = −∆mi = ±1. These results can be applied to isotopes with nuclear spin
by expanding the atom’s and ions’s hyperfine states in their respective electronic spin basis
|F, mF i =
X
M =mI +mS
F
Cm
|S, mS i|I, mI i,
I mS
(2.23)
F
where the terms Cm
are the Clebsch-Gordan coefficients. The spin-exchange interaction
I mS
only couples the electronic and not the nuclear spins and can therefore only give rise to the
transitions within the individual hyperfine structures with ∆F = −1, 0, 1 and ∆mF = ±1.
However, atoms and ions colliding in fully stretched states |Fa = ±(Ia + 1/2), mF,a =
Fa i + |Fi = ±(Ii + 1/2), mF,i = Fi i, only project into the electronic spin triplet state and
cannot undergo spin exchange collisions.
So far, we have neglected the role of the kinetic energy in the inelastic spin-exchange
collisions. Indeed, if the asymptotic energy splitting between different internal states is
much smaller than the collision energy of the particles, the partial wave solution for the
spin-exchange cross section can be expressed in terms of elastic scattering partial wave
phase shifts ηls and ηlt in the singlet and triplet potential [88]
σSE =
∞
π X
(2l + 1) sin2 (ηls − ηlt )
k2
(2.24)
l=0
The singlet and triplet potential only differ for small internuclear distance inside the centrifugal barrier. Thus for l > lsplit the phase shifts are equal ηls = ηlt and for l < lsplit we
can make again the approximation sin2 (ηls − ηlt ) ≈ 1/2. By summing up the Equation 2.24
27
Chapter 2. Theory of Ion-Atom interaction
and considering the factor of 2 of Equation 2.19, we obtain the semi-classical spin-exchange
cross section as
σSE =
1
σLangevin .
4
(2.25)
If the collision energy gets small enough to approach the Zeeman or Hyperfine splittings,
the degenerate internal state solution for the total-spin exchange cross section is not a
good approximation any more. In this case, the spin-exchange rates between the nonresonant internal states become asymmetric, favoring the exothermic over the endothermic
spin transitions. As we will see in Chapter 3, the typical ion-atom collision energies in our
hybrid system indeed fall into this intermediate range, where the energetic suppression of
endothermic spin transitions plays an important role.
Spin-Relaxation
We have shown, that stretched polarization states, such as
174 Yb+ |1/2, 1/2i
+
87 Rb|2, 2i,
are immune to spin-exchange collisions. The spins of atoms in these states are still affected
by spin-relaxation, if the spherical symmetry of the interaction potential that underlies
the conservation of the electron spin is broken due to the presence of other interaction
mechanisms10 . One such interaction, that will always be present, is the magnetic dipoledipole coupling of the two valence electronic spins [89]
V (R) =
µ0 µ2B gs2
µ0
(µ
·
µ
−
3(µ
·
R̂)(µ
·
R̂))
=
(S i · S a − 3(S i · R̂)(S a · R̂)),
i
a
4πR3 i a
4πR3
(2.26)
where R = Ri − Ra , R̂ =
R
R,
and µ0 , µB and gs are the vacuum permeability, the Bohr
magneton and the Landé factor, respectively. At small internuclear separations this interaction lifts the degeneracy of the three 3 Σ+ states with spin S = 1 and spin projections
Ω = 0, ±1 and leads to the spin relaxation [90].
As there are, to our best knowledge, no theoretical or experimental studies, that consider
spin-relaxation phenomena in ion-atom collisions, we base much of this analysis on what is
known about the analogous neutral system of two colliding alkali metal atoms.
In ultracold electronic ground state collisions of light alkali atom the spin-spin interaction
is indeed the leading contribution to the relaxation of spin stretched states [90]. In binary
collisions of heavier alkali-metal atoms, such as cesium, a mechanism based on spin-orbit
coupling becomes the dominant cause for spin-relaxation.
Spin-orbit coupling, the interaction between the spin of a particle and its motion, plays
10
In the most general case only the total angular moment, which consist of the spin degree of freedom,
the electronic orbital angular momentum, and the orbital angular momentum of the collision, will be a
conserved quantity of the problem.
28
2.3. Atom-Ion Collisions
an important role in many physical systems. The best known example of this interaction
is the fine-structure splitting of the electronic energy levels of atoms and molecules. In
the moving frame of an electron, the electric field of the nucleus is seen as a magnetic
field that couples to the spin degree of freedom. The same principle is at work for mobile
electrons and carriers in condensed matter systems, where it gives rise to the interesting
physics of the spin-hall effect [91] and topological insulators. The recent implementation of
effective spin-momentum coupling in ultracold atomic systems has opened the opportunity
to study these effects using the excellent control techniques in the toolbox of quantum
optics [92, 93, 94].
Crucially, the relativistic spin-orbit coupling mechanism also breaks the electronic spin
conservation and gives rise to spin relaxation phenomena. For two atoms collide in their
2S
1/2
electronic ground states, the four molecular states 3 Σ+ and 1 Σ+ that asymptotically
correspond to this atomic state, do not have any electronic orbital momentum. For this
reason no first order spin-orbit coupling L · S = 0 is present in these collisions. At short
internuclear distances, however, the three 3 Σ+ states with spin S = 1 and spin projections
Ω = 0, ±1 are split by a second order spin-orbit coupling, that is mediated by couplings
to distant electronic states with non vanishing electronic orbital angular momentum [90].
The effect has been shown to increase with atomic mass and for cesium atoms the contribution of the second order spin-orbit coupling exceeds that of the spin-spin interaction of
Equation 2.26.
In the context of ultracold atomic collisions, spin-relaxation can have a significant impact on the stability of ultracold gases. The spin-relaxation rate during collisions of
87 Rb
atoms in the |F = 2, mF = 2ia is sufficiently small compared to the elastic collision rate
to allow for efficient evaporative cooling of atoms in our magnetic trap. For
133 Cs
atoms
in the magnetically trapable |4, 4ia of |3, −3ia states, on the other hand, the binary spin-
relaxation collisions dominate the inelastic loss behavior. The unfavourable ratio of inelastic
spin-relaxation and elastic collision rates γelastic /γSR ≈ 100 has made it difficult to reach
quantum degeneracy for the magnetically trapable states of
133 Cs
[95]. Spin relaxation
effects of similar magnitude have also been found for optically pumped alkali metal atoms
in vapor cells at room temperature [96]. The comparison of different alkali metals confirms
the increase of spin relaxation effects with the mass of the colliding particles.
In conclusion, the analogy with the neutral two-valence electron system of alkali metal
collisions could suggest that also for atom-ion collisions the ordering γelastic ≈ γSE ≫ γSR
of elastic, spin exchange and spin relaxation rates should be conserved in a wide range of
regimes. The experimental results in Chapter 6 will, however, show that this is strongly
violated in Yb+ + Rb collisions.
29
a
30000
Energy (cm-1)
Chapter 2. Theory of Ion-Atom interaction
20000
b 30000
Energy (cm-1)
Σ+
1
10000
3 +
Σ
Σ+
1
20000
10000
1 +
0
5
Σ
0
10
15 20
R (a.u.)
25
30
5
10
25
15 20
R (a.u.)
30
Figure 2.3: Molecular states primarily involved in spin-exchange a and radiative association
b. Potential energy curve data from [75].
b
a
Yb++Rb
Yb +Rb+
2+
Yb
2+
Yb
+
Yb+*+Rb
+
Yb+ +Rb
Rb
Rb
2+
Yb
2+
Yb
+
Rb
+
Rb
Figure 2.4: Pictorial representation of the charge exchange reaction a and quenching from
an excited state b. Shown are the filled core electronic shells and the relevant valence
electrons.
Reactive collisions
Inelastic collisions that lead to transitions between electronic states in the ion and/or atom
can be classified into two categories. If the inelastic event entails a chemical change of
the collision partners, it constitutes a genuine reactive collision. On the other hand, if the
collision only involves the (de-)excitation of the electronic state of the ion and/or the atom,
we refer to it as an inelastic quenching event. In Figure 2.4 examples for each of the two
processes are illustrated.
There are in general three different pathways that can lead to a transition between chemically distinct states. Non-radiative charge transfer (NRCT) is mediated by non-adiabatic
couplings between molecular states. In radiative charge transfer (RCT) the transition to
a continuum state of a lower lying potential energy curve is induced by the emission of a
photon. Finally, radiative association describes the transition to a bound state of a lower
lying potential by emission of a photon. In the later case, YbRb+ molecular ions in vibra-
30
2.3. Atom-Ion Collisions
tionally excited levels are formed (see Figure 2.3b and 2.5).
For the entrance channel (Rb(5s) + Yb+ (6s)) the only energetically lower lying molecular
potential energy curve corresponds to the charge-exchanged (Rb+ +Yb (6s2 )1 S) state. The
dominant chemical reaction pathways are radiative charge transfer and radiative association between the 1 Σ+ excited state and the 1 Σ+ ground states. Radiative processes from
the 3 Σ+ state are forbidden by symmetry and non-adiabatic coupling between (Rb(5s) +
Yb+ (6s)) 1 Σ+ and (Rb+ +Yb (6s2 )1 S) 1 Σ+ is negligible due to the large energy gap. Nonradiative charge transfer is also not possible via the higher lying (Rb+ + Yb (6s6p 3 P))
channel. Its potential exhibits a crossing with the entrance channel potential for both the
singlet and the triplet potential, but due to the asymptotic energy gap of > 500 cm−1 the
channel is energetically closed for cold collisions.
a
b
Figure 2.5: Radiative charge-exchange and radiative association. a The overall radiative
cross section depends on the collision energy as ∼ E −1/2 and therefore leads to an energy
independent reaction rate. The rich structure in the cross section is caused by centrifugal
barrier tunneling (shape) resonances of the many partial waves that are contributing. b
Frequency resolved radiative cross section. Free-free (ff) transition to continuum states
above the dissociation limit of the (Rb+ +Yb (6s2 )1 S) potential correspond to radiative
charge-exchange. Free-bound (fb) transitions lead to the formation of (RbYb)+ molecular
ions with large vibrational quantum numbers. Figure adapted from [76].
In a recent theoretical study the charge exchange dynamics of Yb+ + Rb system has been
investigated in a parameter regime relevant to current experiments [76]. The ab initio
31
Chapter 2. Theory of Ion-Atom interaction
calculations reproduce the experimentally determined, energy-independent, reaction rate
in the electronic ground state of ≈ 10−5 γLangevin [35, 97] (see Section 5.1). In addition,
the theory predicts a relative strength of the radiative association and the radiative charge
exchange pathway as 70 : 30. In Figure 2.5 the frequency resolved radiative cross section is
plotted for an assumed collision energy of 0.01 cm−1 (≈ 14 mK). The data shows that all
charge exchange products should have kinetic energies of less than 500 cm−1 and that all
radiative associations will lead to the formation of molecules with large vibrational quantum numbers. We will compare these theoretical predictions with our experimental results
on reactive collisions in Chapter 5.
32
Chapter 3
Theory of Ions and Atoms in a Paul
Trap
In order to investigate the various interactions between neutral atoms and charged ions
without disturbance, it is necessary to isolate them from undesired effects due to the environment. The idea of studying the coupling between the two species at very low temperatures therefore relies on techniques to suspend, confine and cool both neutral and charged
atoms and to overlap them in the same physical location. The development of these techniques has undergone tremendous progress since the inception of charged particle traps by
Paul and Dehmelt in the 50s [98] and the pioneering works on laser cooling and neutral
particle traps in the 80s [99].
In this Chapter I will give a short outline of the basic principles of the radio frequency
trap that is at the heart of our cold-matter hybrid experiment. Since the general physics
of ions in Paul traps is covered extensively in the literature [100, 101, 102], I will focus in
the following on the aspects most relevant to ion-atom hybrid traps.
3.1
Radio Frequency Traps
Ion traps are devices that confine charged particles in effectively conservative potentials
over sustained periods of time. There are two main types of traps for charged particles,
that due to their net charge, interact strongly with static electric and magnetic fields. In
Penning traps charged particles are trapped by a combination of static electric and strong
magnetic fields. In contrast, in Paul traps, such as the one implemented in our experiment,
the confinement results from the time averaged action of an inhomogeneous radio frequency
electric field and from static electric potentials.
Quantum optic experiments routinely employ both Penning and Paul traps to trap charged
particles at low energies for high precision metrology and quantum information processing
purposes. In cold matter quantum hybrid experiments, so far, only Paul traps have been
implemented. The main disadvantages of the use of Penning traps for atom-ion hybrid
systems are the high magnetic fields, that strongly couple to the neutral atoms, and the
fact that the localization of ions is more difficult to achieve [103, 104].
33
Chapter 3. Theory of Ions and Atoms in a Paul Trap
3.1.1
Pseudo Potential Approximation
A fundamental understanding of the confining effect, that ions experience in a Paul trap,
can be obtained in the so called pseudo potential approximation [36].
The classical equation of motion of a particle with charge Q and mass m in an electric
field E(r, t) is given by
r̈ =
Q
E(r, t).
m
(3.1)
It is evident from ∇ · E = 0 that charges cannot be confined in vacuum in all three
dimensions only by electrostatic fields1 . However, the application of non-uniform timevarying electric fields can give rise to restoring forces in all three dimensions, under the
right conditions.
We start by considering a sinusoidally oscillating electric field of frequency Ω and spatial
dependency E0 (r)
E(r, t) = E0 (r) · sin(Ωt).
(3.2)
In a spatially uniform field, the mean force over one oscillation period averages to zero and
returns the particle to its original position and motional state. For an inhomogeneous field,
on the other hand, there remains an averaged force acting on the particle [101].
If the length scale
Q
mΩ2
· E(r) is small enough (see Section 3.1.4), the trajectory of the
particle is well described by decomposing it into a fast oscillatory excursion δr and a
slowly drifting mean position r D
r(t) = r D (t) + δr(t).
(3.3)
The equation of motion 3.1 now reads
Q
(E 0 (r D + δr)) sin(Ωt)
m
!
∂E 0 Q
sin(Ωt),
E 0 (r D ) + δr
=
m
∂r r=rD
r̈ D + δr̈ =
(3.4)
(3.5)
where we have Taylor-expanded the electric field around
r D assuming δr ≪ r D . By making
0
and integrating Equation 3.5,
the approximations δr̈ ≫ r̈ D and E 0 (r D ) ≫ δr ∂E
∂r r=r D
we arrive at the solution for the fast oscillating motion
δr = −
1
QE 0 (r D )
sin(Ωt).
mΩ2
(3.6)
The discovery of this fact predates the formulation of Maxwell’s equation and is known as Earnshaw’s
theorem.
34
3.1. Radio Frequency Traps
Inserting this term into Equation 3.5 and averaging over one field oscillation period 2π/Ω
yields the expression for the drift motion of the particle
hr̈ D + δr̈i = r̈ D
QE 0 (r D )
Q ∂E 0 1
Q2 E 20 (r D )
=−
=
−∇
,
m ∂r r=rD 2mΩ2
m
4mΩ2
(3.7)
where we have introduced the conservative pseudo potential
U (r D ) =
Q2 E 20 (r D )
.
4mΩ2
(3.8)
The time averaged force on a charged particle in an inhomogeneous oscillating electric field
is also known as the ponderomotive force and is independent of the sign of the particles
charge. The slow motion of the particle in the pseudo potential is called secular motion,
whereas the fast superimposed oscillation at frequency Ω (Equation 3.6) is termed micromotion.
2
VDC,0
~VRF,0
Figure 3.1: Schematic of the electrode configuration in a linear Paul trap. The radio
frequency voltages are applied between the pairs of diagonally opposed linear electrodes
and give rise to radial trapping of the charged particle. DC voltages are applied to the
electrodes on either end of the quadrupole waveguide to provide confinement in the axial
direction.
3.1.2
The Linear Paul Trap
The Paul trap used in our experiments is a linear quadrupole trap, that provides radio
frequency confinement in the radial direction and electrostatic trapping along the axial
direction. It consists of four parallel rod electrodes surrounding the trap axis and one
electrode on either end of the trap axis.
A sinusoidally oscillating potential in the radio frequency domain is applied between
the two pairs of diagonally opposed electrodes and equal static potentials are put on both
35
Chapter 3. Theory of Ions and Atoms in a Paul Trap
end-cap electrodes (see Fig. 3.1). The electric potential at time t created by four hyperbolic
electrode surfaces with infinite extension is the ideal quadrupole
VRF (x, y, t) = VRF,0 (
x2 − y 2
) cos(Ωt),
2R02
(3.9)
where VRF,0 and Ω are the amplitude and frequency of the electrode voltages and 2R0 is the
distance between diagonally opposed electrodes. In general, the need for optical access and
Figure 3.2: Numerical simulations of the electrostatic potential generated by our ion trap
geometry a, in comparison to the potential of an ideal quadrupole with identical electrode
spacing b.
manufacturing restrictions require more realistic geometries, which also generate higher
order multipoles. The reduced quadrupole contributions of the potential can be taken
into account by replacing VRF,0 with an effective geometry dependent voltage amplitude
κRF · VRF,0 . Figure 3.2 shows the simulated radial electric potential inside our ion trap in
comparison to that of an ideal quadrupole trap with identical electrode spacing during an
antinode of the radio frequency voltage.
According to Equation 3.8 the oscillating electric quadrupole of Equation 6.6 in the centre
of the trap gives rise to the radial harmonic confinement
U (r D ) =
2
Q2 κ2RF VRF,0
4mR04 Ω2
(x2 + y 2 )
(3.10)
for a charged particle.
In order to confine the particle along the axial direction, the two end-cap electrodes are
held at identical static voltages. The electrostatic potential they generate at the centre of
36
3.1. Radio Frequency Traps
the trap is given by
VDC (x, y, z) =
κDC VDC,0 2 1 2
1
[z − (x + y 2 )] = C · [z 2 − (x2 + y 2 )],
2
2
2
Z0
(3.11)
where κDC is a geometry-dependent factor and Z0 is the distance to the end caps. This
static electric quadrupole potential traps the ion in the axial direction, but it also leads to
the weakening of the confinement in the radial direction.
The total conservative potential for the secular motion of a single ion U (x, y)+QVDC (x, y, z)
results in the harmonic trap frequencies
ωz =
3.1.3
s
2QκDC VDC,0
,
mZ02
ωx,y
QκRF VRF,0
= √
2mR02 Ω
s
1−
κDC VDC,0 mR02 Ω2
.
2
Qκ2RF VRF,0
Z02
(3.12)
Linear Coulomb Crystals
Ion Equilibrium Positions
When multiple motionally cold ions are located in a single trapping potential, the ions
arrange into stable ion crystals to minimize their potential energy [105]. If the radial
frequency of the trap is sufficiently large compared to its axial frequency, the ions line up
in a linear chain. The potential energy of such a string of N ions consists of the overall
confining potential of the trap and the mutual Coulomb repulsion between ions and is given
by [106, 107]
V =
X1
i
2
QCzi2 +
1
e2 X
,
8πǫ0
|zi − zj |
(3.13)
i,j;i6=j
where zi denotes the position of the ith ion and C (Equation 3.11) describes the strength
of the axial confining potential, that traps a single ion of mass m with frequency ωz =
p
QC/m.
The minimization constraint
∂V =0
∂zi zi =z (0)
(3.14)
i
leads to the set of equations
ui −
i−1
X
j=1
N
X
1
1
+
= 0,
2
(ui − uj )
(ui − uj )2
(0)
for the equilibrium positions zi
l3
i = 1, ..., N,
(3.15)
j=i+1
= ui · l, where we have introduced the length scale
= Q/(4πǫ0 a0 ) to obtain the normalized equilibrium ion positions ui .
In general the solution to Equation 3.15 has to be found numerically, but for binary
37
Chapter 3. Theory of Ions and Atoms in a Paul Trap
crystals the analytic expression
z1,2 = l · u1,2 =
p
3
Q/(4πǫ0 a0 ) · (±
r
3
s
1
Q2
)=±3
4
16πǫ0 mωz2
(3.16)
exists. The equation is useful, for example, to obtain a calibration of the scale of fluorescence
images, once the axial trap frequency ωz has been measured of an ion of mass m.
The equilibrium positions of ions in an axial crystal are independent of their mass and
therefore do not convey information about the chemical composition of any non-fluorescing
ions. In order to identify dark ions in an ion crystal we make use of the mass dependence
of the ion crystal’s motional frequencies.
Axial Eigenmodes
At low energies the ions oscillate with small amplitudes around their respective equilibrium
positions. Their motion, which is strongly coupled by the Coulomb interaction, can be
described in terms of eigenmodes of the entire ion string2 .
The Langrangian for small oscillations of a string of N ions, with N-1 ions of mass m and
a single impurity ion of mass mn at position n is given by the expression [106, 107]
N
N
m X 2 mn 2 1 X ∂ 2 V qi qj ,
q̇ −
q̇i +
L=
2
2 n 2
∂zi ∂zj qi =0
(3.17)
i,j=1
i=1;i6=n
where q(t)i is the excursion of ion i around its equilibrium position. To obtain the displacement eigenvectors and axial eigenfrequencies of the linear crystal the equation is numerically
diagonalized. The results for the lowest eigenmode of ion crystals with up to 4 ions are
summarized in the Appendix A.3 The largest relative changes of eigenmode frequencies due
to a single impurity ion are obtained for ion crystals of two ions, for which the analytic
expression
ω
com
breath
q
p
= ω · (1 + µ) ∓ 1 − µ + µ2
can be derived for the centre-of-mass and the breathing-mode frequencies, where µ =
(3.18)
m
mn
is
the ion mass ratio and ω is the axial trap frequency of a single ion of mass m. In Section 5.1.1
we will describe the experimental implementation of an in-trap mass spectrometry method,
which is based on the mass dependence of the COM eigenmode frequency of binary ion
crystals.
2
This mechanism is of great importance for ion trap quantum information tasks. In combination with the
photon-recoil induced coupling of the internal and motional degrees of freedom of ions, the strong Coulomb
interaction forms the basis for multi-ion entanglement gates [19] and precision spectroscopy schemes [17].
38
3.1. Radio Frequency Traps
3.1.4
Trap Stability
In order to study the range of trap parameters, for which the pseudopotential approximation
is valid and stable ion trajectories are possible in our linear Paul trap [102, 108], we start
with the equation of motion 3.1 for an ion in the total electric field
E(x, y, z, t) = −κRF VRF,0
κDC VDC,0
xx̂ − y ŷ
cos(Ωt) −
(2z ẑ − xx̂ − y ŷ)
2
R0
Z02
(3.19)
and rewrite it in the form
∂ 2 ri
+ [ai + 2qi cos(2τ )]ri = 0,
∂τ 2
1
τ = Ωt,
2
(3.20)
where
4QκDC VDC,0
1
,
ax = ay = − az = a = −
2
mZ02 Ω2
2QκRF VRF,0
,
qz = 0.
qx = −qy = q =
mR02 Ω2
(3.21)
(3.22)
This equation is known as the Mathieu equation and its stable solutions in terms of the
dimensionless stability parameters a and q are well understood [108, 102]. Figure 3.3 shows
the region of stable solutions for the linear Paul trap in green. Practically all ion traps
operate in the regime a, q 2 ≪ 1, for which the solutions of the Mathieu equation 3.20 to
first order in a and q 2 are [102, 109]
with
qi
ri (t) = r1i cos(ωi t + Φi ) 1 + cos(Ωt) .
2
(3.23)
r
1
1
ωi = Ω ai + qi2 .
2
2
(3.24)
The slow secular harmonic oscillations along the three trap axis at frequencies ωi are
identical to the expressions of Equation 3.12, that were found in the pseudo potential
approximation. The amplitudes r1i and phases Φi of the ion motion are determined by the
starting conditions. In the radial directions, micromotion oscillations at the frequency Ω
of the quadrupole field are superimposed onto the secular motion.
The limits of the stability diagram for small a, q 2 in Figure 3.3 can be understood from
Equation 3.24. For a > 0 the voltage on the end caps has the wrong polarity with respect
to the charge of the ion and attracts the ion towards the end cap electrodes. The low
boundary is approximately given by a ≈ − 21 q 2 ⇒ ωr = 0, where the radial RF confinement
can just counterbalances the deconfinement due to the axial electrostatic quadrupole. The
39
Chapter 3. Theory of Ions and Atoms in a Paul Trap
Table 3.1: Trap parameters of our ion trap under typical operating conditions. R0 is the
distance between the ion and the radio frequency electrode, Z0 the distance between the ion
and the DC end cap, and Ω/(2π) the radio frequency drive frequency. The radial (axial)
trap frequencies ωr /2π (ωz /2π) and the stability parameters a0 , q0 are specified for a Yb+
ion with mass m0 = 174 amu.
R0
0.466 µm
Z0
4.3 mm
Ω/(2π)
42.88 MHz
ωr /(2π)
150 kHz
ωz /(2π)
55 kHz
a0
−3.3 ·
10−6
q0
1.02405 · 10−2
dotted curve crossing the stability region at about a ≈ − 31 12 q 2 (⇒ ωr ≈ ωz for a single ion)
represents an estimate of the lower boundary for a binary ion crystal to still crystallize along
the axial direction. For large values of q 2 , Equation 3.28 is no longer a good approximation
for the trajectories of the ion. The breakdown of the pseudo potential approximation on
the right boundary of the a-q stability region is highlighted by the fact, that the radial
confinement frequency would approach the drive frequency ωr ≈ 21 Ω for a ≈ 1 − 12 q 2 .
According to the definitions of a and q in Equation 3.22, stable trapping conditions in a
Figure 3.3: The stability diagram for the linear Paul trap in various levels of detail ac. The green shaded area marks regions in a-q space that give rise to stable solutions
of the Mathieu’s equation. For our typical ion trap operating conditions, the stability
parameters of singly charged positive ions of different masses fall onto the red line defined
by Equation 3.25.
specific ion trap depend on a range of physical quantities. The geometric constants R0 , Z0 ,
κDC , κRF and Ω are fixed by the design of the ion trap and its resonant radio-frequency
voltage generating circuitry. The operating parameters VRF,0 , VDC,0 can be used to set
certain stability parameters a0 , q0 for an ion of mass m0 . The a, q values for ions with
40
3.1. Radio Frequency Traps
masses mn then fall onto the line
an =
a0
qn
q0
with
qn = q0
m0
.
mn
(3.25)
2
ωz
For typical operating conditions of our trap, we have determined the values a0 = −2 Ω
2,
q
1 2
√
2
ωr − 2 ωz
from ion trap frequency measurements with an 174 Yb+ ion (see Table 3.1
q0 = 8
Ω2
and Section 5.1.1). The mass relation is shown as the red line in Figure 3.3 with a selection
of relevant ion masses. Our ion trap is expected to be stable for all singly charged ions
with masses in the range 2 amu < m < 2500 amu and to support axial crystals of two ions
of equal mass for m < 900 amu. The mass limit for the axial alignment of a binary crystal
consisting of one Y b+ ion m0 = 174 and a heavier ion of mass m is m < 1300 amu.
Modifications to the stability diagram due to higher order multipole fields have not been
considered here, as they do not affect the stability limits of large ion masses (i.e. a, q 2 ≪
1) [110].
3.1.5
Micromotion
In addition to the slow oscillations at the secular trap frequencies of Equation 3.24, the
ion trajectories along the radial trap axes (Equation 3.28) also exhibit a fast oscillating
micromotion at the frequency of the radio frequency driving field. This residual motion is
proportional to the local radio frequency electric field and hence depends on the position of
the ion in the quadrupole trapping field. If the ion trajectories are centered on the zero of
the quadrupole the amplitude r1i q2i of the intrinsic micromotion is reduced, as the secular
motion r1i decreases during laser cooling. In this case, the total kinetic energy of the ion
averaged over the period of one secular oscillation
EKin
1 2 2
1 2 2
qi2
1 2 1 2
2
= m ṙi = mr1i ωi + qi Ω = mr1i ωi 1 + 2
2
4
8
4
qi + 2ai
(3.26)
consists of a secular motion term and a contribution of similar size from the intrinsic
micromotion.
There are several mechanisms that can increase in the size of micromotion relative to
this ideal case. If a static electric background field radially displaces the minimum of the
time averaged potential ion trap potential from the centre of the radio frequency quadrupole
by
QE of f set
= −∇Upseudo ri =r0i
41
⇒
r0i =
QEof f set
,
mωi2
(3.27)
Chapter 3. Theory of Ions and Atoms in a Paul Trap
then additional excess micromotion is superimposed on the ion motion according to
qi
ri (t) = (r0i + r1i cos(ωi t + Φi )) 1 + cos(Ωt) .
2
(3.28)
Since the micromotion is a driven motion, its amplitude is set by the strength of the driving
force and cannot be cooled effectively. In order to minimize the amount of excess micromotion, the electrostatic background fields at the location of the ion need to be detected
and compensated with the application of the corresponding electrostatic potentials to the
trap electrodes [109].
Excess micromotion can additionally be caused by radio frequency fields pointing along the
trap axis. The misalignment of radio frequency electrodes as well as the pick-up of radio
frequency on the end-cap electrodes can both give rise to axial micromotion. A further
source of radial excess micromotion can be phase differences between the potentials applied to diagonally opposite radio frequency electrodes due to, for example, a mismatch in
the length of the connecting wires. For well compensated electrostatic background fields,
axial micromotion and out-of-phase radial micromotion remain the dominant micromotion contributions and should therefore be minimized through careful design and precise
construction of the ion trap.
For many experiments with single ions in spectroscopy and quantum information science
it is sufficient to reduce the amplitude of the excess micromotion uemm,i to below the
amplitude of the secular motion (ground state) to suppress its detrimental effects on line
shapes and gate fidelities. This condition corresponds to an excess micromotion kinetic
2
larger than the secular kinetic energy. Since both
energy that can be a factor of Ω
ω
motions of the ion are well decoupled, there is in general no significant heating effect of the
secular motion from the radio frequency driving field.
3.2
Micromotion Heating in Ion-Atom Collisions
The situation is significantly altered for ions in the presence of neutral atoms, where elastic
collisions strongly couple the motion of ions and neutral atoms. It has been realized early
on in buffer gas experiments, that micromotion in Paul traps can give rise to both heating
and cooling effects for an ion inside the neutral gas [36].
An intuitive insight into the effect of micromotion on the kinematics of buffer gas cooled
ions can be obtained by considering the motion of an ion and its collisional processes in
phase space. The explanation follows [111], to illustrate the problem for a simplified onedimensional geometry.
In the absence of micromotion, the harmonic secular motion of an ion corresponds to a
elliptical trajectory in phase space, where the enclosed area is proportional to the energy
42
3.2. Micromotion Heating in Ion-Atom Collisions
a
Harmonic Oscillator
p
p
x
x
mi>mn
p
mi=mn
Paul trap
b
p
x
x
π/2
π/2
Figure 3.4: Illustration of collisional energy transfer of a trapped particle in one dimension.
a In a harmonic potential without micromotion, collisions with neutral atoms at rest can
only lead to a decrease of ion energy. The energy loss is indeed fastest for equal masses of
the ion and the atoms. b A maximally rotated and stretched ellipse represents the phase
space distribution of an ion in a Paul trap at the exemplary phase ξ = π2 of the radio
frequency field. Depending on the size of the momentum transfer, collisions with neutral
atoms at rest can lead to both heating or cooling of the secular motion of the ion. (Figure
adapted from [111])
of the ion (Figure 3.4a). Following a head-on hard sphere collision with a neutral atom
at rest3 , the instantaneous position of the ion remains unchanged, but some of its initial
momentum c · m is transfered. The ion’s energy and momentum c′ · m after the collision
are given by
∆E
=
E
′ 2
c
ma m
,
− 1 = −4
c
(ma + m)2
(3.29)
where ma , m, c and c′ , are mass of the atom, the mass of the ion, and the instantaneous
velocity of the ion in the laboratory frame before and after the collision, respectively.
Due to the presence of micromotion, the distribution in phase space of the position and
the total momentum of an ion in a Paul trap is represented at each phase ξ = mod(Ωt, 2π)
of the radio frequency gradient field E0 cos(Ωt) by rotated and streched ellipses, as shown
in Figure 3.5. During one radio frequency period the phase space trajectory of the ion
motion samples the continuum of equal-area ellipses, that lie between the extremal cases of
ξ=
π
2
and ξ =
3π
2 ,
always returning to points on the same ellipse at times of identical radio
frequency phase ξ. Figure 3.4b shows how a random collision event at a radio frequency
phase ξ =
3
π
2
and at an ion position close to the turning point, instantaneously changes the
The kinetic energies associated with buffer gas are assumed to be much lower than the motional energy of
the ion.
43
Chapter 3. Theory of Ions and Atoms in a Paul Trap
3π/2
pharm
pmic
3π/2
xharm
p
x
xmic
π/2
π/2
macromotion
+
micromotion
=
combined motion
Figure 3.5: For any phase ξ of the radio frequency field the phase space distribution of
the instantaneous velocity of the ion in a Paul trap is given by a rotated and stretched
ellipse due to the presence of micromotion. The phase space trajectory of the ion (thick
black line in the panel on the right) returns to a given ellipse for times of identical radio
frequency phase and progresses clockwise around points on that ellipse within one period
of the harmonic oscillation. (Figure adapted from [111])
area of the ellipse and therefore the secular energy of the ion.
The phase-space picture demonstrates, that depending on the amount of transfered momentum (i.e. the mass ratio
m
ma ),
the phase of the micromotion and the phase of the secular
motion, collisions can lead both to an increase or a decrease of the secular ion energy. In
order to arrive at a more quantitative analysis for the three dimensional case, we retrace
the original treatment by Dehmelt [36].
The velocity of an ion in the laboratory frame after a purely elastic collision with a neutral
gas atoms at rest is
c′ =
m
ma
|c|Θ +
c,
ma + m
m + ma
(3.30)
where c is the velocity just before the collision and Θ is a unit vector that spans an angle
of Θ with c in the centre-of-mass frame. Assuming a collision time much shorter than
2π
Ω,
the velocity before c = u + v and after the collision c′ = u′ + v can be decomposed into
the secular velocities u, u′ before and after the collision and the field amplitude dependent
micromotion velocity v =
Q
mΩ
cos(Ωt).
Substitution, squaring and regrouping leads to the relation
u′2 − u2 = −2
2ma
ma m
(u2 + 2u · v + v 2 )(1 − cos Θ) +
(v 2 + u · v − |c|Θ · v). (3.31)
2
(m + ma )
m + ma
To describe the heating and cooling dynamics of the ions secular motion, the average secular
′2
2 per collision is the relevant quantity. Since the Langevin
energy change m
2 · u −u
44
3.2. Micromotion Heating in Ion-Atom Collisions
collision4 rate (Equation 2.19) is independent of the collision velocity, we can consider the
random phase of the driving field during collisions to be uniformly distributed. Averaging
over the radio frequency phase ξ yields the expression
h∆Eion iξ =
2
m ′2
mma
2ma 2 u − u2 ξ = −2
u + v 2 (1 − cos Θ) +
v (1 − cos Θ)
2
2
(m + ma )
m+a
(3.32)
for the secular energy change per collision, where we have made use of the fact hu · viξ = 0.
The average secular energy of the ion reaches a steady state (h∆Eion iξ = 0) for
hEion iξ ≡
m 2
ma m 2 u ξ=
v ξ.
2
m 2
(3.33)
For the illustrative example of a purely harmonically trapped ion in a homogeneous5
radio frequency field E(t) = E RF cos(Ωt), the quantity v 2 ξ is indeed independent of
2
u ξ and the average secular energy of the ion will be centered around the value
hEion iξ = m
ma Q2
E2
m 4m2 Ω2 RF
(3.34)
due to collisions with neutral atoms.
The ideal unbiased hyperbolic 3D Paul trap with ax = ay = az = 0 and qz = −2qx =
−2qy is another instructive case, for which a simple exact solution can be found. For the
collisionless motion of an ion in a Paul trap, Equation 3.26 described the size of the intrinsic
micromotion kinetic energy, relative to the secular kinetic energy, averaged over one secular
oscillation. If the trap frequency of the ion is larger than its collision rate, then the relation
2
2
v ξ,sec = q2q+2a u2 ξ,sec ≈ u2 ξ,sec remains a good approximation also in the presence of
neutral atoms. Inserting this expression into Equation 3.32 and considering the isotropic
angular distribution of Langevin scattering for the averaging over the scattering angle Θ,
we find that that the average relative change of secular energy
∆Eion
Eion
=2
ξ,sec,Θ
m
a
m
−1
ma m
(m + ma )2
(3.35)
strongly depends on ion atom mass ratio.
In the presence of intrinsic micromotion, Dehmelt’s model predicts exponential heating
due to collisions with the neutral buffer gas, if the ion is lighter than the atom (m < ma ).
4
Only Langevin collisions are relevant, since small angle scattering collisions do not transfer significant
amounts of kinetic energy.
5
This situation could arise experimentally, if recent experiments with ions confined in optical traps [112]
were combined with neutral atoms in the presence of radio frequency noise.
45
Chapter 3. Theory of Ions and Atoms in a Paul Trap
No average energy change occurs for equal masses m = ma , and exponential cooling is
expected, if the ion is heavier than the atom (m > ma ). The mass ratio
ma
m
=
mRB
mY b
=
1
2
in
our experiment is clearly favourable for fast collisional cooling. The cooling process in this
case would only be limited by the temperature of the ultracold buffer gas, which we have
neglected so far. Due to the large difference in the trapping frequencies of the ion and the
neutral atoms, this picture would result in a large ground state occupation probability of
p(n = 0) = 1 − e
− k~ωT
B
for the ion.
Unfortunately, excess micromotion in a Paul trap introduces a limit for the ion energy
that is many orders of magnitude higher in current ion traps . In analogy to Equation 3.34,
the radio frequency field amplitude E RF at the centre point of the ion’s secular motion will
give rise to a steady-state average ion energy
hEion iξ = ζ ·
ma
Q2
· m 2 2 E 2RF .
m
4m Ω
(3.36)
The exact value of the proportionality constant ζ, which will be on the order of 1, depends on the trap stability parameters a and q, the neutral-ion mass ratio, the cause of
the nonzero radio frequency field (i.e. uncompensated static offset fields, radio frequency
phase mismatches, axial micromotion, etc.) and on the potentially inhomogeneous density distribution of the neutral buffer gas. In recent years Monte-Carlo simulations have
become the method of choice for the calculation of heating and cooling dynamics due to
elastic collisions in Paul traps [113, 37, 43, 114].
Interestingly, these calculations have revealed significant deviations of the ion steady-state
energy distribution from the thermal Maxwell-Boltzmann distribution. The likelihood of
higher energies is significantly enhanced in the form of long range power-law tails of the
energy distribution and increases for larger neutral to ion masses [113, 37] Experimental
studies in our ion trap have confirmed the quadratic dependence of the ion energy on the
excess micromotion amplitude and detailed calculations have indicated average ion energies
in the (≈ kB · 10 mK) range, that are limited due to uncompensatable axial micromotion
in our trap [37, 69].
3.3
Effect of the Paul Trap on the Atoms
We have seen in Section 2.2.1, that atoms due to their static polarizability α are suscepti2
. In the electric
ble to electric fields via the induced dipole potential Uind (r) = − α|E(r)|
2
quadrupole field of a Paul trap (Equation 3.19) the atoms experience the deconfining potential
Udec (r) = hUind (r)i =
2
2
α κRF VRF,0 2
mRb 2 2
ω r
r =−
2
4
2 dec
R0
46
(3.37)
3.3. Effect of the Paul Trap on the Atoms
in the radial direction, where we have neglected the contribution from the static axial
quadrupole of order
a2
q2
and we have averaged over one cycle of the radio frequency. The
comparison with Equation 3.10 shows that the ions radial trap frequency6 and atoms radial
deconfinement “frequency” are related according to
ωRb,dec
mY b
=√
ωY b
mRb
s
αΩ2
.
Q2
(3.38)
When turned on, the ion trap reduces
q the confinement frequency of the atoms in their
2
2
optical or magnetic trap by ωON = ωOF
F − ωRb,dec
For our trap parameters the deconfinement effect on the neutral atoms ωRb ≈ 0.0001 ·
ωY b ≈ 2π · 15 Hz is starting to limit the maximally attainable ion trap frequency. Further
increases of the ion confinement would impede experiments with atoms in our current
magnetic trap, that has radial trap frequencies ωON, |2,2i ≈ 2π · 30 Hz and ωON, |1,−1i ≈ 2π ·
10 Hz for 87 Rb atoms in their |F = 2, mF = 2i and their |F = 1, mF = −1i hyperfine states
respectively. In the optical dipole trap the characteristic neutral atom trap frequencies
are significantly higher (≈ 100 Hz) and ion trap frequencies in the MHz range would be
possible.
6
Here, we neglect the radial deconfinement of the ion due to the static axial quadrupole potential of order
a
.
q2
47
Chapter 3. Theory of Ions and Atoms in a Paul Trap
48
Chapter 4
Experimental Setup and Methods
The experiments in this thesis are performed in a cold matter hybrid trap that has allowed
for the first time the investigation of the interaction of a single trapped ion with a Bose
Einstein Condensate [33]. The realization of this atom-ion hybrid system relies on an apparatus that can simultaneous prepare single trapped Yb+ ions and clouds of ultracold
87 Rb
atoms and overlap them under well defined conditions in the same physical location. For
this purpose the tools and methods to generate both species need to be carefully joined in
a single experiment with a special focus on their interoperability.
In this chapter I introduce the experimental apparatus and the techniques that support
the basic functionality of the atom-ion hybrid system. The design and implementation of
the hybrid trap is described in much more detail in the PhD theses [70, 69]. I begin by
discussing the preparation of the neutral clouds and present the methods we use to manipulate and analyze the ultracold samples. Next, I will describe the trapping and laser
cooling of Yb+ ions in our setup. We will then review the laser system of the hybrid trap
and focus in particular on the modifications and additions to the original design for the
experiments in the Chapters 5 and 6.
4.1
Preparation of Ultracold
87
Rb Atoms
Since the first realizations of Bose-Einstein Condensates (BEC) in ultracold atomic gases
almost 20 years ago [2, 3] the experimental technologies have undergone a steady development, however, the basic route for reaching ultracold temperatures and ultimately quantum
degenerate have not changed much. In our experiments the critical steps, namely, laser
cooling of atoms in a Magneto Optical Trap (MOT), followed by evaporative cooling of
the atomic cloud under ultra-high-vaccum conditions, are implemented in the two-chamber
vacuum system shown in Figure 4.1. The MOT-chamber on the left of the picture is connected to the interaction chamber, where the ion trap is located, by a differential pumping
section. The laser cooled clouds of atoms are moved between both chambers by a magnetic
transport over a distance of 140 mm.
49
Chapter 4. Experimental Setup and Methods
z
x
y
Figure 4.1: Layout of the central part of the vacuum chamber system and the magnetic coils
in our ion-atom apparatus. Atoms are collected and laser cooled in the MOT-chamber, (enclosed by the pair of quadrupole coils shown in yellow) before they are magnetically trapped
and transported through the differential pumping section. The continuous displacement of
a step magnetic field minimum from the position of the MOT to the centre of the QUIC-trap
(formed by the magnetic fields of the red quadrupole coils and the Ioffe coil in magenta) is
aided by an overlapping pair of quadrupole coils in blue. The coil in pink, the pair of black
coils and a similar pair of coils in the vertical direction (not shown), provide homogenous
magnetic offset fields to align the postion of the magnetic trap with respect to the centre
of the ion trap.
4.1.1
Magneto-Optical-Trap and Molasse Cooling
Our experimental sequence starts with laser cooling of
87 Rb
atoms in a 3D MOT [99] from
thermal rubidium vapour. The rubidium pressure in this vapour cell section of vaccum
chamber is kept constant at ≈ 1.5 · 10−9 mbar by firing rubidium dispensers once every
experimental cycle. Six counterpropagating laser beams of 1.8 cm waist and a total power
of ≈ 300 mW intersect at the centre of a quadrupole magnetic field with an axial gradient of 10.6 G/cm. The cooling laser light is red detuned by three natural line widths
(3ΓD2 ≈ 18 MHz) from the 2 S1/2 , F = 2 → 2 P3/2 , F = 3 hyperfine transition of the
rubidium D2-line. Off-resonant excitation to 2 P 3/2 , F = 2 state leads to the leakage of
population into the 2 S1/2 , F = 1, from where it is returned into the cooling cycle with
laser light that is resonant with the repump transition 2 S1/2 , F = 1 → 2 P 3/2 , F = 2 (see
Figure 4.2).
After collecting approximately 109 atoms at the Doppler-cooling limit of T ≈
~ΓD2
2kB
=
140 µK over a 20 s interval, the MOT magnetic field is turned off. The atom tempera-
50
4.1. Preparation of Ultracold
87
Rb Atoms
ture is then further reduced during an optical molasse stage [115] in cooling light, that is
6 ΓD2 detuned from resonance. Subsequently, the atoms are optically pumped into the
low field seeking 2 S1/2 , F = 2, mF = 2 state with σ + polarized light1 resonant on the
780nm
F=2
52S1/2
87
6.8 GHz
F=1
Rb
Repumping
5 P3/2
F=3
267 MHz
F=2
F=1
F=0
opt.pump.
2
Imaging
trap.
F = 2 → 2 P 3/2 , F = 2 transition, before they are loaded into a magnetic quadrupole
Cooling
2S
1/2 ,
Figure 4.2: Hyperfine level structure of the 87 Rb D2-line. The various transitions relevant
for the manipulation of the neutral atoms are indicated.
87 Rb
Laser system system
Frequency and intensity controlled coherent laser light to address the various transitions
of the D2-line for trapping/cooling, optical pumping and resonant absorption imaging of
87 Rb
atoms (see Figure 4.2) is produced in a purpose-built optical setup [70, 69]. For
experimental convenience and in order to decouple the two tasks, the production of laser
light and its final alignment on the atoms are separated on two different optical tables.
Polarization maintaining single mode optical fibers are used to spatially filter the light and
deliver it from the optical table housing the lasers to the second optical table, which holds
the experiment’s vacuum chamber.
The D2-line transitions at 780 nm are conveniently addressed with semiconductor diode
lasers. In our setup, commercial Fabry-Perot laser diodes are operated in home-built
Littrow-type external cavity diode lasers (ECDL), which are based on a blueprint by the
Hänsch group [116]. We employ the same laser design [70] for the various Fabry-Perot laser
diodes2 , that generate the light for the manipulation of the Yb+ ions (see Section 4.2.6).
The laser system consists of three of these ECDL. The first Master laser is locked onto
the 2 S1/2 , F = 2 → 2 P 3/2 , F = 2 transition using Doppler-free frequency-modulation
1
2
A small magnetic offset field is applied to define a quantization axis for the atoms.
The collimation lenses and feedback diffraction gratings are adapted to the various wavelengths.
51
Chapter 4. Experimental Setup and Methods
spectroscopy [117] in a saturated
87 Rb
- vapour cell. A second ECDL is locked to this
reference laser with a positive frequency offset [118] that is tunable around ≈ 266 MHz
to address (2 S1/2 , F = 2 → 2 P 3/2 , F = 3) MOT cooling transition. Its light is passed
through a tapered amplifier, in order to increase its power to the levels required for the
MOT. The third laser provides the light for the 2 S1/2 , F = 1 → 2 P 3/2 , F = 2 repumping
process during the MOT and absorption imaging and is directly locked to an independent
spectroscopy vapour cell.
Tunable and fast switchable light for the absorption imaging of the atoms (Section 4.1.4)
on the 2 S1/2 , F = 2 → 2 P 3/2 , F = 3 transition is obtained by double-passing Master
laser light through an Acousto-Optical-Modulator (AOM), that is adjustable in frequency
around 133 MHz. In order to achieve short pulses at the original frequency of the Master
and Repumper laser for optical pumping and imaging, we send the light through two successive single-pass AOM of opposite order. Finally, a portion of the Master laser light is
coupled into an intermediate finesse transfer cavity. The length of this transfer cavity is
stabilized to the frequency of the laser and is in turn used as a reference for the frequency
locking of some of the infrared Yb+ lasers (see Section 4.2.6).
4.1.2
Magnetic Transport and Evaporative Cooling
After the free expansion of the atoms during the molasses phase and the subsequent optically spin pumping, the magnetic field of the MOT coils is ramped up to form a magnetic
quadrupole trap for the low field seeking states of
87 Rb.
In order to avoid heating and
losses from collisions with the room temperature background gas, the atoms are then magnetically transported within 1.4 s to the ultra-high-vacuum (∼ 2 · 10−11 ) of the interaction
chamber. The continuous displacement of the centre of the magnetic quadruple trap is
achieved by ramping the currents of three overlapping quadrupole pairs (see Figure 4.1).
Once the atoms arrive in the interaction chamber, the magnetic trap is converted into a
QUIC trap [119] configuration. This Ioffe-Pritchard type trap features a non-zero magnetic
field minimum in order to avoid non-adiabatic spin flip losses of the atoms. In this trap we
perform microwave induced evaporative cooling of the atoms by driving transitions from
the 2 S1/2 , F = 2, mF = 2 to the untrapped 2 S1/2 , F = 1, mF = 1 hyperfine states (see
Figure 4.4). The hottest atoms of the continuously rethermalizing cloud sample the highest
magnetic fields and fall out of the trap as their Zeeman shifted transition becomes resonant
with the gradually lowered frequency of the microwave radiation between 6.9 GHz and
6.8 GHz. After ≈ 30 s of evaporation we typically obtain ultracold clouds of up to ∼ 3 · 106
atoms at T =∼ 700 nK.
52
4.1. Preparation of Ultracold
87
Rb Atoms
Transport into the ion trap
At this point the atomic cloud is small enough to be transported through the φ = 600 µm
hole in the end cap of the linear Paul trap. The atoms are moved into the ion trap along its
axial direction by shifting the magnetic trap with additional currents in the axial coils (see
Figure 4.1). The precise alignment of the final atomic cloud position inside the ion trap
is achieved by the application of offset magnetic fields in all three directions. In this final
magnetic configuration the trapping frequencies for atoms in the 2 S1/2 F = 2, mF = 2 state
have been measured by exciting dipole oscillations as ωr = 2π · 28 Hz and ωaxial = 2π · 8 Hz
in the radial and axial directions, respectively.
4.1.3
Tunability of the Atomic Bath
All the steps described so far remain very much unchanged regardless of the details of
the experiments we perform with the ultracold atoms. The only parameters that are routinely changed are the duration of the molasses phase and the endpoint frequency of the
microwave evaporation. Increases in the free-expansion time after the MOT are used to
reproducibly decrease the number of atoms in the final ultracold cloud without changing
its temperature. The endpoint of the microwave evaporation, on the other hand, affects
both atom number and temperature, with longer evaporations leading to a higher overall
central density of the cloud in the magnetic trap.
In order to prepare the atomic bath in hyperfine spin states other than the initially generated 2 S1/2 F = 2, mF = 2 state, the atoms need to be transferred from the magnetic trap
into an optical dipole trap.
The Optical Dipole Trap
The optical dipole trap [120] is formed by two far-off-resonant Gaussian laser beams, that
cross at an angle of 90◦ at their mutual focus point in the very centre of the ion trap
(Figure 4.3). The laser light at 1064 nm is red detuned from both D-lines of
780 nm and 795 nm and provides a conservative attractive
potential3 .
87 Rb
at
The horizontal
laser beam is focused to an elliptical waist of dimension 160 µm × 64 µm. The beam
is confined stronger in the vertical direction to provide sufficient support for the atoms
against gravity, while keeping the trapping frequencies along the horizontal axis lower. In
this way, the density of neutral atoms in the centre of the trap is minimized and the cloud
size maximized for a given number of atoms in the optical trap. This condition is favourable
for our experiments, where the atoms act as a controlled bath for the ion, since it minimizes
the effects of cloud depletion and misalignment. The vertical beam has a circular focus
3
Heating due to spontaneous scattering of photons is negligible on the experimental timescale.
53
Chapter 4. Experimental Setup and Methods
with a waist of 110 µm.
The atoms are loaded into the optical trap by simultaneously increasing the power from
0 W to ∼ 3 W in both beams in a quadratic ramp of 1 s duration. By lowering the power
in the horizontal beam, the atoms are evaporatively cooled, as the hottest atoms leave the
trap in the vertical direction due to gravity. After approximately 5 s of optical evaporation,
we can obtain almost pure condensates of up to 150’000 atoms in the 2 S1/2 F = 2, mF = 2
state.
Bose Einstein condensates, however, are not very convenient as an atomic bath for the
controlled immersion of ions due to their high atomic density and atom-ion collision rate,
and their relatively small size. For this reason, all experiments reported in this thesis are
performed with non-degenerate atomic clouds. The number of
87 Rb
atoms in the thermal
clouds, their temperature and trap frequencies, as well as their exact vertical position4 ,
depend on the evaporation values used in a specific experimental sequence. Typical trap
frequencies for the atoms in the optical trap are ωx ≈ 2π · 50 Hz, ωy ≈ 2π · 50 Hz and
ωz ≈ 2π · 75 Hz. It is worth noting that the optical dipole trap will of course also have an
effect on the trapped ions. The extra potential due to the far-red detuned optical dipole
trap is, however, completely negligible with respect to the ions’ strong confinement in the
ion trap.
In order to fine adjust the alignment of the thermal cloud in all three spatial dimensions, the
horizontal and vertical dipole laser beams are each reflected off computer-controlled twoaxis piezo mirrors. For this purpose two standard optical mirror mounts have been fitted
with low-voltage piezo-stacks, which are powered by computer-interfaced piezo drivers5 .
The tip and tilt angles of the mirrors of α ≈ 3 µrad/V displace the laser foci by ∆d =
α · ∆V · f , where f denotes the focal length of the focusing lens. The actual displacements
of the trap have been determined experimentally from calibrated in-situ absorption images
of the atoms. The maximum computer controlled excursions of the trap are ∆dx = 48 µm,
∆dy = 34 µm and ∆dz = 154 µm. Figure 4.3 shows a scan of the position of the dipole
trap with respect to the location of the ion. The implementation of the piezo mirrors and
temperature stabilization (±0.1 K) of the cooling water for the ion trap and the magnetic
field coils have reduced the long term drifts of the relative position to a value smaller than
our measurement precision (∼ 1 µm).
The laser light for the dipole trap at 1064 nm is generated by a 500 mW YAG laser6 and
then amplified by a fiber amplifier7 . The combination yields a maximum optical power of
∼12 W out of a single polarization maintaining fiber. The light is subsequently distributed
4
due to the gravitational sag
Thorlabs MDT693A
6
Innolight Mephisto
7
Nufern
5
54
4.1. Preparation of Ultracold
87
Rb Atoms
z
y
x
Atom Number
150000
120000
90000
60000
30000
-15
-10
-5
0
5
10
15
Displacement y [μm]
Figure 4.3: Alignment of the crossed optical dipole trap with respect to the ion. a Illustration of the relative positions of the magnetic trap coils, the ion trap and the dipole
laser beams. b Loss of atoms from the neutral cloud due to elastic collisions with the ion
(interaction time 150 ms). The relative position of the neutral cloud and a single ion is
scanned by displacing the optical trap. Figure a adapted from [70].
into three different laser beams, two of which are used for the optical dipole trap. Each
beam is sent through a single-pass Acousto-Optical-Modulator and a mechanical shutter,
before it is coupled into a polarization-maintaining single mode fiber and delivered to the
experiment. The operating frequencies of the AOMs are detuned by several tens of MHz
in order to avoid interference effects, when the laser beams are spatially overlapped for the
formation of the crossed dipole trap8 .
The AOMs are used for the fast switching of the light and serve as the control elements
in the closed-cycle feedback loops that stabilize the dipole beam powers at the position of
the atoms to computer-controlled set values. The sensor inputs to the loops are provided
by photo-diodes that detect proportional fractions of the light fields, once its polarization
has been filtered after the optical fibers. The feedback algorithm is currently implemented
by an analog PID controller circuit.
Preparation of spin-polarized atomic clouds
The use of the spin-independent optical dipole trap allows the confinement of atoms in
any state of the
87 Rb
electronic ground state spin manifold, shown in Figure 4.4. Of
particular interest as atomic bath states are the states |F = 2, mF = −2i, |F = 2, mF = 2i,
|F = 1, mF = −1i and |F = 1, mF = 1i, that are protected against spin-exchanging atom-
atom collisions due to spin conservation or a hyperfine energy barrier9 . The population
8
The beams are also set to have different polarization, in order to avoid interference. The frequency detuning
helps to average out any residual effects.
9
Spin-exchange collisions that require changes from the F=1 to F=2 manifold are energetically completely
suppressed at the ultracold temperatures of the gas.
55
Chapter 4. Experimental Setup and Methods
transfers from the initial |F = 2, mF = 2i state are performed by driving adiabatic Landau-
Zener radio frequency and microwave transitions in the presence of a homogeneous magnetic
field of several Gauss. The duration and range of the linear frequency chirps of constant
amplitude are experimentally optimized to yield adiabatic passage efficiency of ∼ 100 %
and ∼ 60 % for the radiofrequency and the microwave transition, respectively. We measure
the decoherence limited transfer efficiencies by detecting the spin populations with a SternGerlach experiment. For this purpose a gradient magnetic field along the direction of the
ion trap axis is applied during the time-of-flight expansion to separate the individual spin
components spatially, before they are detected by resonant absorption imaging.
In order to increase the polarization purity of the states in the F = 1 manifold after a
microwave transfer, we selectively push out the remaining atoms in the F = 2 manifold with
a resonant light pulse on the 2 S1/2 |F = 2i - 2 P 3/2 |F = 3i transition. The combination of
radiofrequency and microwave population transfers with hyperfine selective push-out light
pulses enables the preparation of all four polarization bath states in the optical trap with
high fidelity.
87
Rb
g F = 1/2
F=2
6.8 GHz
g F = -1/2
F=1
mF
-2
-1
0
1
2
Figure 4.4: Preparation of spin polarized atomic clouds in the optical trap. Starting from
the |F = 2, mF = 2i state, the |F = 2, mF = −2i, |F = 1, mF = 1i and |F = 1, mF = −1i
spin states of the 87 Rb electronic ground state are populated with high fidelity by radio
frequency and microwave adiabatic passage.
4.1.4
Absorption Imaging of
87
Rb Atoms
The standard tool for determining the properties of ultracold quantum gases is optical
imaging. In its various forms optical imaging makes use of the absorptive or dispersive
effects of atoms on light to generate a projection of the instantaneous spatial distribution
of the atoms. In the case of absorption imaging, atoms are illuminated from one direction
with a collimated beam of resonant laser light. The atoms absorb and randomly reemit
part of that light and thereby cast a shadow that can be imaged onto a CCD camera.
56
4.1. Preparation of Ultracold
87
Rb Atoms
For reasons that will become evident in this section, the standard approach of low saturation imaging is not practical for our experimental conditions. The following theoretical
description of the high saturation absorption imaging is based [121] and [122].
The attenuation of the intensity of the probe beam, as it propagates through an atomic
cloud is given by
dI
= n3d (x)σ(x)I(x),
dx
(4.1)
where n3d (x) denotes the local density of the atoms and σ(x) = σ0 /(1 + s(x) + δ 2 ) is
the light scattering cross section. Here, s(x) = I(x)/Isat is the saturation parameter and
σ0 describes the resonant (detuning δ = 0) scattering cross section in the limit of very
low saturation parameters. In the two-level approximation their values Isat =
σ0 =
3λ20
2π
hcπΓ
3λ30
and
only depend on the transition line width Γ and the optical transition wave length
λ0 . To calculate these quantities for the 2 S1/2 |F = 2i - 2 P 3/2 |F = 3i transition used in
our setup, the polarization of the imaging light and the spin structures of the electronic
states need to be considered as well.
With the boundary conditions defined as the transmitted intensity IT and the initial10
intensity IT , the solution of Equation 4.1 reads [121]
s0
s0 − nσ0
1 + δ2
W
exp
,
T =
s0
1 + δ2
1 + δ2
where W is the Lambert function of the first kind, T =
IT
I0
(4.2)
is the relative transmission of
light through the atomic cloud, s0 = I0 /Isat is the saturation parameter of light before
R
passing the atoms11 and n = n3d (x)dx is the column density. Using the defining equation
X = W [Y ] ⇒ Y = XeX of the Lambert function, we can rewrite this equation to express
the column density along direction of the imaging
s0
1 + δ2
−ln(T ) +
(1 − T )
n=
σ0
1 + δ2
(4.3)
as a function of the measurable intensities of probe light and in the presence (IT ) and
absence of atoms (I0 ). The well-known Beer-Lambert laws n = −ln(T )/σ is in fact an
approximation to this solution for the case of small saturation parameters (s0 ≪ 1) and/or
small absorbance ((1 − T ) ≪ 1).
The large dynamic range of in-situ atomic densities and the limited choice of time-of-
flight expansion times tT OF in our quantum hybrid system make it challenging to obtain
10
11
The initial intensity equals the transmitted intensity in the case that no atoms are present.
Here, we have neglected the collection of re-emitted fluorescence photons by the imaging optics and
therefore underestimate the column density by a factor (1 − Ω) [121]. The fraction of the total solid angle
covered by the imaging optics is Ω = 0.005 for the numerical aperture NA=0.17 of our imaging system.
57
Chapter 4. Experimental Setup and Methods
satisfactory imaging conditions12 in the low saturation regime for all atomic cloud sizes.
Due to the shadow cast by the ion trap electrodes, the possible time-of-flight expansion
times for the horizontal imaging are restricted to the interval 0 − 7 ms, for imaging in
between the electrodes, and to times > 27 ms, for imaging below the electrodes.
The absolute measurement of atomic densities is critical for several of our atom-ion experiments and it is therefore important that saturation effects in the absorption imaging are
correctly accounted for in Equation 4.3.
Experimentally, the intensities IT and I0 are obtained from a series of three picture. The
first picture records the counts in each pixel (NAtoms (yp , zp )) of the CCD image, while the
atoms are present. The second picture (NBright ) is taken with an identical imaging light
pulse, but without atoms. In both shots, the imaging light, will inevitably contain intensity fringes as a result of the interference of light, scattering from imperfect surfaces. In
order to have similar illumination conditions in the two pictures it is essential to capture
them in quick succession to avoid changes in the fringe pattern. We reduce the time between subsequent images to 5 ms by implementing the fast-kinetics function of the camera.
In this mode the previous image is transferred to a shaded part of the CCD, before the
next image is taken and all stored images are jointly read out at the end of the sequence.
The third picture (NDark ) is recorded with neither laser light nor atoms. From these
three pictures we calculate the relative transmission of imaging light through the cloud as
T (yp , zp ) =
IT
I0
=
NAtoms −NDark
NBright −NDark .
In contrast to the Beer-Lamberts law, Equation 4.3 also requires s0 = I0 /Isat and
therefore the absolute amount of imaging light. In principle, the intensity I0 =
~ω0 M
ηCCD ηO tl A
·
(NBright − NDark ) = C1 · (NBright − NDark ) at the location of the atoms can be calculated
from the recorded pixel counts with the knowledge of the CCD’s quantum efficiency ηCCD
and pixel area A = d2p , the imaging optics’ transmission ηO and magnification M and
the imaging time tl . In practice, it is more convenient to determine the proportionality
constant C2 between the saturation parameter and the detected pixel counts s0 = I0 /Isat =
C1 (NBright − NDark )/Isat = C2 (NBright − NDark ) experimentally. To this end, a large
number of identically prepared atomic clouds are imaged with varying amounts of light.
The correct C2 is found as the value for which the column densities n, calculated from
Equation 4.3, are as similar as possible for all measured light intensities [122]. Typical
imaging parameters we use for magnetically trapped thermal clouds are tl = 40 µs, s0 ≈ 1
and tT OF = 27 ms.
12
Too little transmitted light leads to bad signal-to-noise ratios and to biases in the detection. Long imaging
times, accelerate the atoms and distort their absorption cross sections due to the Doppler-effect. Ideal
values for the relative transmission of light are typically T ≈ 0.3.
58
4.1. Preparation of Ultracold
4.1.5
87
Rb Atoms
Density Measurement of the Neutral Bath
In a harmonic trapping potential
1
V (r) = ma (ωx2 x2 + ωy2 y 2 + ωz2 z 2 )
2
(4.4)
the density distribution of an ideal Bose gas above the BEC transition temperature follows [123]
n(r) =
1
g3/2 (z(r)),
λ3dB
(4.5)
where λdB = (2π~2 /ma kB T )1/2 is the thermal deBroglie wavelength, z(r) = exp((µ −
V (r))/kB T ), µ is the chemical potential and T is the temperature of the gas. Compared
to a thermal gas of distinguishable particle, the density of the Bose gas is enhanced by the
P
factor g3/2 (z)/z, with the function gj (z) = i z i /ij being the polylogarithm. For the phase
space densities of thermal clouds used in our ion-atom experiments, the Bose enhancement
is negligible even in the centre of the cloud and the neutral density distribution is well
described by
n3d (r) = n3d (0) · e
V (r)
BT
−k
3/2
2 2 +ω 2 y 2 +ω 2 z 2
y
z
2kB T
Na ma ωx ωy ωz −ma ωx x
·e
=
(2πkB T )3/2
,
(4.6)
where Na is the number of atoms in a neutral cloud of temperature T . By integrating the
R
density along the imaging direction n(r) = n3d (r)dx the column density
2 2
2 2
Na ma ωy ωz −m ωy y2k+ωTz z
B
·e
n(y, z) =
(2πkB T )
(4.7)
of the cloud is obtained.
In-situ resonant absorption imaging of atoms in their trapping potential is hampered by
the high optical densities n ≫ 1 of ultracold clouds and the limited spatial resolution of
the imaging optics (≈2.5 µm). The solution two both limitations, is time-of-flight imaging,
where the atoms are abruptly release from the trap and expand, before they are imaged.
Once the trap is switched off, the atoms fly ballistically13 from their in-trap position r 0
with their original velocity v to reach the points r(t) = r 0 + p/ma · t after the time t.
As a consequence, the 3D density of the cloud evolves during the time-of-flight expansion
as [123]
x,y,z
3/2
− 2kmaT
Na ma ωx ωy ωz Y
1
B
q
)e
n3d (r, t) =
(
(2πkB T )3/2
2
2
1+ω t
i
x,y,z
P
i
ωi2 x2i
.
(4.8)
i
13
This neglects the effect of interactions during the expansion, which do not play a role due to the low
atomic densities in our neutral clouds.
59
Chapter 4. Experimental Setup and Methods
The self-similar expansion does q
not change the shape of the density distribution, but scales
its width in the direction ri by
1 + ωi2 t2 .
In conveniently reparameterized units, the column density of the time-of-flight cloud
n(yp , zp , t) = n0 · e
−
(yp −y0 )2
2
2wy
·e
−
(zp −z0 )2
2
2wz
(4.9)
serves as the fitting function for the experimental column density pixel maps (Equation 4.3),
obtained from the absorption imaging. The pixel fit parameters n0 , wy , wz relate to the
quantities of interest T and Na of the imaged atomic cloud through
k B Ti =
ma ωi2 wi2 A
n0
(1 + ωi2 t2 )M 2
and
Na =
πn0 wy wz A
,
M2
(4.10)
where M is the magnification of the imaging system, A = d2p is the area of a pixel, and the
lengths yp , zp , wy and wz are measured in pixels.
The central density of an atomic cloud, while it is held in the trap,
3/2
Na ma ωx ωy ωz
.
n3d (0) =
(2πkB T )3/2
(4.11)
follows from the Equation 4.6 and the fitted temperature and atom number.
When imaging identically prepared, large thermal clouds of neutral atoms, we typically
observe fluctuations in the detected atom number and temperature of 15% and 5% standard deviation, respectively. Theses fluctuations correspond mainly to real variations in
the properties of the prepared clouds due to imperfections in the repeatability of the experiment. For very small atomic samples, fitting errors due to image imperfects also start
to contribute significantly to the observed variations.
Considering the uncertainties of all quantities entering into Equation. 4.11, we estimate
that the absolute values of the central densities derived from absorption imaging have a
systematic uncertainty on the order of 40%. The main contributions to the error budget
result from the determination of the polarization-dependent optical cross section, the estimation of the saturation intensity, and the corrections to the expansion of neutral atoms
due to the deconfining effects of the ion trap.
4.2
Trapping of Yb+ Ions
The second ingredient in our quantum hybrid system are trapped Ytterbium ions. In recent years singly ionized Yb+ ions have become an increasingly popular choice for ion trap
experiments in quantum metrology and quantum information science due to their versatile
60
4.2. Trapping of Yb+ Ions
atomic level structure.
Being the member of the lanthanide (rare earth) series of chemical elements with atomic
number 70, singly ionized Ytterbium atoms have the electron ground state [Xe](4f 14 )(6s)S1/2 .
In its lowest energy state the ion has a single electron in an open s-shell, which gives it an
electronic level structure, similar to that of the heavier earth-alkali ions. In Figure 4.5 the
electronic level scheme, the most prominent optical transitions and the laser wavelengths
used in our experiment are summarized.
The electronic ground state is strongly coupled to the fine structure split (4f 14 )(6p)
1
τ[3/2]=37.7ns
2
2
P3/2
P1/2
3
[3/2]1/2
1.8%
98.2%
τp=8.12ns
935 nm
2
0.5%
99.5%
[5/2]1/2
τD5/2=7.2ms
D3/2
D5/2
2
τD3/2=51ms
369 nm
638nm
2
83%
τF≈years
F7/2
17%
411 nm
2
S1/2
Yb+
Figure 4.5: Energy level scheme of the electronic ground and excited states of Yb+ (drawn
to scale). The transitions driven by diode lasers in our Experiment are marked as straight
lines. Life times and decay branching ratios are taken from [124].
2P
1/2 and
The 2 S1/2
2P
369nm and 329nm, respectively.
-
of Γ369 =
3/2 states via dipole allowed transitions at
2P
1/2 transition with a natural linewidth
2π
τp
=
1
8.12ns
= 2π ·
19.60(5)MHz [125] serves as the cycling transition for Doppler cooling and fluorescence
detection (Section 4.2.2) in most Yb+ ion trapping experiments14 . Dipole allowed decay
from the 2 P 1/2 state leads to the leakage of population into the (4f 14 )(6d)
2D
3/2
state
with a branching ratio of 0.501(15)%. In order to close the fluorescence cycle, repump laser
light at 935 nm depopulates the metastable state with a radiative lifetime of τD3/2 = 51 ms
via the short lived 2 D[3/2]1/2 state and returns the ion into the ground state. A detailed
description of the relative state populations during the 4-level fluorescence cycle will be
given in Section 4.2.3.
14
In [126] the narrower 2 S1/2 - [3/2]1/2 transition was recently explored for Doppler cooling.
61
Chapter 4. Experimental Setup and Methods
The second state of the metastable fine structure doublet, the 2 D5/2 state has a lifetime of
7 ms [127] and couples to the lower lying (4f 13 )(6d)2 2 F7/2 state (83 %) and via a quadrupole
transition (17 %) to the 2 S1/2 ground state. The later state exhibts an extremely long natural lifetime of several years due to the fact that its radiative relaxation mechanism is
an electric octupole transition to the 2 S1/2 ground state. The (4f 13 )(6s)2 2 F7/2 state is
usually not populated during Doppler cooling, as neither the 2 P 3/2 , nor 2 D5/2 state form
part of the 4-level fluorescence cycles. However, collisions of electronically excited ions with
hot background gas atoms can lead to the occupation of the F-state. At background gas
pressures of ≈ 10−11 mbar these collisional transfers occur several times per hour. The
F-state is also efficiently populated when we drive the narrow 2 S1/2 - 2 D5/2 quadrupole
transition with laser light at 411 nm. In order to return ions from the F-state back into
the fluorescence cycle we apply a further repump laser at 638 nm in our experiment.
4.2.1
Loading of Ions
Whereas the fast and efficient loading of a deterministic number of ions is a beneficial feature for many trapped ion experiments, it is a critical requirement for the investigations
of our atom-ion system. In the hybrid trap, the reactive collisions of ions with ultracold
neutral atoms make ion reloading rates necessary, that can exceed those of pure ion trap
experiments15 by several orders of magnitude. In addition, the frequent reloading process
must not lead to any deterioration of the vacuum, in order to not affect the lifetime of the
ultracold quantum gas.
I will briefly summarize the ion loading procedure we use in our ion trap to satisfy these
stringent criteria. The oven design and the experimental sequence underlying this functionality are described in detail in [69].
Isotope Selective Loading
Ytterbium features seven stable isotopes, of which five (171, 172, 173, 174, 176) have a natural abundance of larger than 10% and their respective nuclear spins are (1/2, 0, 5/2, 0, 0).
In our experiment we primarily use
spin-free isotopes, and
171 Yb+ ,
174 Yb+ ,
since it is the most abuntant of the nuclear
which is of interest for our quantum information experi-
ments due to its hyperfine structure (see Section 6.1.2). The preparation of isotope-pure
ion crystals of a selected isotope is achieved by means of resonance enhanced two-photon
ionization [128]. For this purpose a beam of thermal atoms is generated by resistively heating a thin steel tube containing Yb metal of natural abundance with a short current pulse
of 90 A and 100 ms duration [69]. After being collimated by a small aperture, the beam
15
In pure ion trap experiments the need to replace ions usually arises from losses due to background gas
collisions.
62
4.2. Trapping of Yb+ Ions
crosses the centre of the ion trap, where it is intersected by two laser beams. The first laser
beam at 399 nm addresses the 1 S0 - 1 P1 transition of the neutral ytterbium atoms. The
laser crosses the beam at the Doppler-free angle of 90◦ , in order to resolve the isotope shifts
of the line, that are on the order of hundreds of MHz. The laser resonantly excites atoms
of a single isotope species to 1 P1 , from where they are ionized by a second photon that
exceeds the energy gap to the continuum. The ion cooling laser at a wavelength of 369 nm
naturally provides the light required for this second, non-resonant step of the process, so
that no further lasers are required.
Ion Number Preparation
The average number of ions that are simultaneously loaded into the trap in a single loading
attempt, depends on several experimental parameters. The most important parameters are
the size and length of the current pulse, that determine the temperature of the Yb in the
oven and therefore the amount and velocity distribution of neutral Yb in the atomic beam.
Since the atoms get ionized on passing the very centre of the trap, the energy of newly
formed ions is given by the thermal Maxwell-Boltzmann distribution of the initial atomic
beam
16 .
The kinetic energy of the ions (∼ kB · 1000 K) is significantly smaller than the
maximum radial trap depth of ∼ 0.250 eV (≈ kB · 3000 K), which explains why for strong
axial confinements, the loading efficiency saturates in our experiment. Since every ionized
atom will stay in the trap and will eventually be cool, the number of loaded ions is directly
proportional to the number of atoms in the neutral beam. The resonant light at 399 nm
is not a critical parameter of the loading process either, because the 1 S0 - 1 P1 transition
is saturated even for moderate laser powers (> 10 µW) due to the small laser focus size
(< 100 µm). We observe, however, a gain in the ion loading efficiency by increasing the
intensity of 369 nm cooling light beyond the ion’s Doppler cooling saturation value, which
results from the enhancement of the second step in the two-colour ionization process.
In order to increase the efficiency for preparing a desired number of ions N beyond the
probabilistic value of the Poissonian loading distribution, an automated feedback scheme
has been implemented for our experiment previously [69]. For this purpose the trap loading
parameters of the trap are adjusted to produce a number of ions per loading pulse that
on average exceeds N . Following a loading pulse the ions are Doppler cooled to form a
Coulomb crystal and the number of ions is automatically detected from a CCD camera
fluorescence image. If the ion count is found to be larger than N , the axial confinement of
the trap is reduced to a experimentally adjusted value for a short period of time to expel
16
The fact, that the energy of the second photon at 369 nm significantly exceeds the ionization threshold
corresponding to 394 nm does not lead to a relevant kinetic energy increase in the ion. Its surplus energy
is almost exclusively taken up by the electron due to the large ion to electron mass ration.
63
Chapter 4. Experimental Setup and Methods
ions from the trap. The success of the ion dump is checked by another ion image and, if
necessary, a limited number of further attempts is made to reach N .
Electrostatic Offset Field Drifts
In practice, the capability of preparing fresh quasi-deterministic ion crystals in every experimental cycle (∼80 s) comes at the price of the creation of significant electrostatic offset
field at the location of the ion. These fields are the result of the formation of electric charges
on the ion trap electrodes due the deposition of neutral atoms from the oven beam. The
presence of cooling laser light at 369 nm (∼3.36 eV) is sufficient to extract photo electrons
from metallic Yb, that has a work function of only 2.6 eV [129]. The extraction of electrons
can lead to the charge-up of isolated patches on the blade electrodes.
The problem is evident in Figure 5.2 that shows the drift of the axial equilibrium position
of ions due to the creation of uncompensated static offset fields. The data in theses plots,
that contain 830 oven firing events over the course of 20 hours, can be used to quantify the
change of the electrostatic offset field per single oven firing.
In order to convert the pixel position into ion displacements the scale of the camera images
first needs to be established. According to Equation 3.16, the separation between the two
ions of a binary crystal in a harmonic trap of trap frequency ωz = 2π · 51.40(2) kHz is
d=24.81(1) µm. The comparison with the measured ion separation of 10.1(2) pixels on the
CCD camera yields a scale of 4.9(1) µm/pixel. We approximate the change of ion positions
by a linear drift with a fitted velocity of 0.14(1) pixel/h or ∆z = 0.016(1) µm per oven
firing event. The electrostatic field change is therefore estimated to be ∆E = ∆z · mωz2 /Q
= 0.0033(3) V/m per oven firing event for the specific loading conditions of that measurement.
Changes in the micromotion compensation voltages (Section 3.1.5) during times of intensive ion loading have shown that electrostatic fields of similar size are also generated in
the radial direction. As expected from the accumulation of localized charges, the loadinginduced offset fields always increase, pointing in the same spatial direction. We also observe
that during long intervals without continuous loading the electrostatic offset fields slowly
return to their pre-loading values. This indicates the existence of a mechanism that lets
accumulated charges diffuse over long timescales.
The detrimental effects due to loading induced contamination could be reduced in a future
setup by improving the collimation of the hot atomic beam as it emerges from the oven.
A more elegant solution to the challenge of efficient and clean ion loading, however, is the
idea of sourcing the ions from a cold cloud [130]. The method is particularly convenient for
atom-ion hybrid systems, that already have a vacuum system with two connected chambers. In this way, the contaminating part of ion trap loading process can take place in the
64
4.2. Trapping of Yb+ Ions
chamber that houses the magneto-optical-trap for the neutral atoms. After the loading of a
small MOT with Yb atoms, the cold cloud will have to be transported optically17 through
the differential pumping section. Once the cloud has been positioned in the centre of the
ion trap, the loading of ions by two-photon ionization should be both efficient and clean.
4.2.2
Doppler Cooling and Fluorescence Detection in
174
Yb+
The laser cooling of trapped atoms and ions in the regime Γ ≫ ω, where the natural
linewidth of the addressed atomic transition Γ is much larger than the mechanical oscillation
frequency of the trapped particle ω, is known as Doppler cooling. In this weak binding limit
the coherence of the atomic state over one oscillation period in the harmonic potential can
be neglected and the interaction of light with the atom can be treated as if occurring at an
instant of time. The steady-state rate of absorption and spontaneous emission of photons
by a two-level atom
R(v) = Γ · ρee (v) = Γ
1 I
2 Isat
1+
I
Isat
2
+ ( 2∆(v)
Γ )
(4.12)
depends on the light’s frequency detuning from the resonant atomic transition ∆(v) =
∆Laser + ∆Doppler (v) and on its intensity I. The saturation intensity Isat , specifies the
intensity light that is needed on resonance to maintain an excited state population ρee =
1/4. Importantly, the detuning from resonance includes a frequency shift that is caused
by the instanteous motion of the particle due to the doppler effect. As in the case of free
atoms18 , the preferential absorption of momentum-transfering photons by an atom with
velocities v · k ≈ ∆Laser can be used to modify the motional state of trapped ions.
For light that is red-detuned from the atomic resonance, it can be shown that the energy
distribution of an ion during Doppler cooling reaches a lower limit of
TD =
~Γ
2kB
for the optimal detuning
∆Laser = Γ/2.
(4.13)
The application of Doppler cooling to the 2 S1/2 - 2 P 1/2 transition in Yb+ results in Dopplerlimited ion temperatures of ≈ 470 µK. The energy states of the ion motion in its trap
potential are the discrete energy eigenstates E = ~ω(n + 1/2) of the quantum harmonic
oscillator. With the knowledge of the radial and axial trap frequencies (ωAxial ≈ 2π·55 kHz,
ωRadial ≈ 2π · 150 kHz), we obtain the average energy occupation numbers nAxial ≈ 180
17
Magnetic transport is not possible due to the lack of a magnetic moment in the ground state of neutral
Yb atoms.
18
In Section 4.1 we have described the use of Doppler cooling as the method to decrease the temperature
of neutral atoms in the first step of the experimental preparation of ultracold rubidium clouds.
65
Chapter 4. Experimental Setup and Methods
and nRadial ≈ 65. The corresponding spatial extend of the ion wave function is given by
√
zi = z0i 2ni + 1,
where
z0i =
r
~
2mωi
(4.14)
is the size of the harmonic ground state wave function. The numbers for our current trap
are z0 Radial = 23 nm, z0 Axial = 14 nm, z Radial ≈ 160 nm and z Axial ≈ 430 nm.
Fluorescence Detection
The detection of ions in our trap and the readout of information about their internal states
relies on the collection of fluorescence photons that are scattered by the ion, while it is
driven on the 2 S1/2 - 2 P 1/2 Doppler cooling transition.
A simple estimate for the rate of photons from a cold ion can be made from the two state
approximation in Equation 4.12. The photon scattering rate is directly proportional to
the occupation probability of the excited state and takes the form of a power broadened
Lorentzian curve as a function of cooling laser intensity and detuning from resonance. For
typical Doppler cooling parameters
I
Isat
= 2 and ∆ = Γ369 /2 the ion scatters photons into
free-space at a rate of R = Γ · ρee ≈ 40 · 106 s−1 .
In order to obtain a more accurate model of the fluorescence behaviour of Yb+ , it is
necessary to take into account the ion’s multilevel nature, that can lead to the formation
of dark. Optical pumping into the electronic 2 D3/2 state due to the weak dipole allowed
decay from the excited 2 P 1/2 state is countered by the application of repump laser light at
935nm. The fact that the additional higher lying electronic 3 [3/2]1/2 state is used to close
the fluorescence cycle has the benefit that coherent population trapping effects between the
2S
1/2
and the 2 D3/2 cannot arise19 . However, the formation of coherent dark states due
to quantum interference effects is also possible, if a single laser field is driving a Zeemandegenerate transition with a high multiplicity in the ground state than in the excited state.
This is the case for the 2 D3/2 to 3 [3/2]1/2 repump transition in both
as well as for the 2 S1/2 (F = 1) to 2 P 1/2 (F = 0) cooling transition in
174 Yb+
171 Yb+
and
171 Yb+
[131].
The two common experimental methods for the destabilization of these dark states [132]
are the fast modulation of the polarization of the driving laser field or the application of
a static magnetic field to rotate the ground-state spin-polarization out of the dark state.
In our experiment we make use of the second method and apply static offset fields in the
range of several Gauss to achieve efficient ion fluorescence detection. As a result of the
complex dark state dynamics, the ion fluorescence rate is a function of the detunings, and
the polarization and propagation angles of both cooling and repump beams as well as the
19
The physics of Λ three level system has of course also many interesting properties that can for example
be used for electromagnetically induced transparency cooling of ions, etc..
66
4.2. Trapping of Yb+ Ions
magnetic field strength and direction.
4.2.3
Experimental Determination of the Ion Electronic State Occupation
For our experiments on the role of electronically excited ion states in the inelastic collisions
of Yb+ +Rb in Chapter 5 it is essential to accurately determine the average electronic
state population during fluorescence scattering of the ion. Traditionally, this is done by
trying to adapt the various laser and magnetic field parameters in a many-level/two-laser
optical-Bloch-equation model such that the model’s predicted fluorescence rates best fit the
experimental results over a large range of laser detunings.
Here, we develop an experimental method that allows us to directly determine the relative
state occupation for a single experimental condition. Most importantly the technique is
significantly more sensitive to repumping efficiency and therefore the relative time the ion
spends in the 2 D3/2 state.
We start by defining the average state populations during fluorescence cooling as
px = Tx /Tc ,
(4.15)
where we have introduced the average fluorescence cycle time
Tc =
X
Tx
(4.16)
x
as the average time between two spontaneous decays on the 2 P 1/2 - 2 D3/2 transition and
Tx as the average time spent in state x ǫ {S, P, D, [3/2]} during one fluorescence cycle.
The time TP = τP /BP D is set by the natural lifetime τP = 8.12 ns of the 2 P 1/2 and the
branching ration BP D = 0.005 of the decay from the P to the D state. The same statement
can be made for T[3/2] = τ[3/2] /B[3/2]S with τ[3/2] = 37.7 ns and B[3/2]S = 0.98. The times TS
and TD are determined by the dynamics of the cooling and repump transition and depend
on the laser parameters and the magnetic field. With these definitions the fluorescence
photon rate of a single ion is given by
R=
1
B P D · Tc
.
(4.17)
Since in Yb+ the relation TS & TP ≫ TD & T[3/2] is obeyed during typical fluorescence
detection conditions, the scattering rate is very insensitive to the repump parameters.
For this reason we determine TS and TD by using the pulsed sequence in Figure 4.6,
which permits the direct measurement of the times for a single set of laser parameters. The
sequence consist of alternating pulses of cooling (369 nm) and repump (935 nm) light with
67
Chapter 4. Experimental Setup and Methods
369nm
935nm
N369 Int xegrat5000
Repetitions
ed Count s
Count s per [ µs]
sum m ed over
5000 repet it ions
0
2000
1000
800
τ1
τ2
τ3
C1
C2
C3
τ4
TS+TP =8 μs
0
100
200
Tim e[ μs]
s]
C4
300
400
C4
0.5%
C1
99.5%
369 nm
C2
τT935D = 0.4[µs]
μs
b
600
400
1.8%
C3
200
0
τ3
0
d
τ2
τ1
1.5
Pulse Tim e 935nm
t935 [μs]light [ µs]
0.5
98.2%
τ4
1
2
935 nm
c
Figure 4.6: Determination of the electronic state populations during fluorescence detection.
Our measurement sequence alternates between the 369 nm and 935 nm light periods, in
order to directly observe, how long the ion will stay on average in the cooling and repump
part of the continous fluorescence cycle. The time TS + TP is observed as the exponential
decay constant of fluorescence after the 369 nm laser is pulsed on. The pulse length is set to
get complete depletion of the S-P system into the D-state. d For a given setting of intensity
and detuning of the 935-nm laser, the average number of photons N369 per 369-nm pulse
depends on the duration t935 of the preceding repump pulse. We vary the length of the
repump pulse t935 to determine the average time TD that is needed to repump the ion for
a given set of experimental conditions. During the 8 s measurement 16 different repump
times are repeated 5000 times each in order to gather sufficient statistics. The solid line is
a fit to the data with Equation 4.18.
the same detuning and intensity as the fluorescence steady-state for which the population
of electronic states shall be determined.
After the start of the 80 µs cooling pulse the ion scatters fluorescence on the 2 S1/2 to 2 P 1/2
until it is completely pumped into the 2 D3/2 dark state. The decay constant of the recorded
ion fluorescence curves directly measures the average time TS + TP the ion spends in the
S - P system. The efficiency of a repump light pulse, on the other hand, decides the total
number of fluorescence counts N369 during the following cooling pulse. Varying the length
t935 allows us to retrieve both TD and the photon detection efficiency ηDet from the fit
N369 = NDark +
ηDet
(1 − e−t935 /TD )
BP D
68
(4.18)
4.2. Trapping of Yb+ Ions
to the recorded photon counts. Here, NDark denotes the experimentally determined average
number of stray light dark counts during the 80 µs detection interval.
Experimentally, Acousto-Optical-Modulators are used to time accurately switch the cooling
and repump lights according to pulse sequences that are generated by an FPGA. The
population measurement for a single fluorescence condition is performed in a single 8 s
interval, by repeating a sequence of 16 different 935 nm repump durations, 5000 times in
order to gather statistics.
The experimental results for the measurement shown in Figure 4.6 are TD = 0.40(2)µs,
TS = 8.0(2)µs − TP = 6.4(2)µs and Tc = 8.4(2)µs. The value of TD = 0.40(2)µs represents
in fact the shortest repump time observed in our experiments and was obtained for several
mW of resonant 935 nm light. It is still significantly longer than the minimal theoretical
two-level value of TD ≈ T[3/2] ≈ 38.5 ns and is limited due to the formation of coherent
dark states.
According to Equation 4.15 the state populations in this case are pS = 0.76(1), pP =
0.19(1), pD = 0.047(2) and p[3/2] = 0.0044(1).
From the fit to the data in Figure 4.18 we also find ηDet = 2.1(1) × 10−3 for the probability
that an emitted fluorescence photon is detected by the photon counter. The value is in
good agreement with the simple estimate of the overall detection efficiency ηDet = 12 (1 −
√
1 − N A2 ) · ηT · ηP C ≈ 0.0019, where we have considered the design numerical aperature
N A ≈ 0.27 of the light collecting optics, the measured overall transmission ηT ≈ 0.75
of all optical components and the specified quantum efficiency ηP C ≈ 0.14 of the photon
multiplier tube at 369 nm.
4.2.4
Doppler Cooling of
171
Yb+
The Doppler cooling scheme for the nuclear spin 1/2 isotope
is very similar to one of
174 Yb+
171 Yb+ ,
shown in Figure 4.7,
and other ions with nuclear spin 0, that we have previously
presented in Section 4.2.2. In order to efficiently Doppler cool
171 Yb+
ions two additional
transitions need to be addressed to cover all hyperfine sublevels of the 2 S1/2 and 2 D3/2
states.
The main cycling transition is driven by 369 nm cooling light that is Doppler detuned
(∆369 = −Γ369 /2) from the 2 S1/2 , F = 1 → 2 P 1/2 , F = 0 resonance. From the excited
2P
1/2 ,
2D
3/2 ,
F = 0 state, the ion can only return to the 2 S1/2 , F = 1 state or decay into
F = 1 due to selection rules. As mentioned in Section 4.2.2, in
171 Yb+
ions the
photon scattering rate on the cooling transition is limited by the formation of coherent
dark states [131]. For this reason, the typical fluorescence rate we obtain for
171 Yb+
ions
in a magnetic field of several Gauss and with optimized polarization angles is only about
1/3 of the fluorescence rate of ions without nuclear spin like
69
174 Yb+ .
Chapter 4. Experimental Setup and Methods
During fluorescence scattering the cooling light also off-resonantly excites the 2.105 GHz
detuned 2 P 1/2 , F = 1 state, from where the ion can leak into the dark 2 S1/2 , F = 0
state. There are two possible ways to clear out the 2 S1/2 , F = 0 state. The use of the
2S
1/2 ,
F = 0 → 2 P 1/2 , F = 1 optical transition to repump the population requires laser
light that is 14.7 GHz blue detuned from the 369 nm cooling light. Alternatively, the dark
2S
1/2 ,
F = 0 state can be coupled to the bright 2 S1/2 , F = 1 state by driving the hyperfine
transition at 12.6 GHz with microwave radiation. We have tested both options and found
that the fluorescence counts during Doppler cooling are reduced by ∼30% for the microwave
in comparison to the laser repumping method. Due to its experimental convenience we,
nevertheless, currently use the microwave rather than the additional optical transition in
our setup.
To return the ion from the metastable D3/2 -states back to the 2 S1/2 , F = 1 ground state,
the 935 nm repump laser is tuned to the 2 D3/2 , F = 1 → 2 [3/2]1/2 , F = 0 transition. In
order to clear out population from the 2 D3/2 , F = 2 hyperfine state, we use an Electro-
Optic-Modulator (EOM) operating at 1.535 GHz to imprint sidebands onto the repump
laser. The light of the red detuned second-order sideband at 3.07 GHz is resonant with the
2D
3/2 ,
F = 2 → 2 [3/2]1/2 , F = 1 repump transition.
3
F=0
F=1
2
P1/2
2.105 GHz
D[3/2]1/2
2.2095(11) GHz
F=1
F=0
F=2
F=1
2
D3/2
0.86(2) GHz
2
S1/2
F=1
12.643 GHz
F=0
171 Yb+
Figure 4.7: Doppler cooling of
ions. Compared to the nuclear spin-0 isotopes, two
additional transitions need to be addressed to avoid optical pumping into dark hyperfine
states. Adapted from [68].
70
4.2. Trapping of Yb+ Ions
Ion Microwave Setup
Microwave radiation resonant with the 12.6-GHz hyperfine transitions is used for coherent
qubit manipulation on the clock transition, for Rabi π-flips on the |0, 0i → |1, −1i and
|0, 0i → |1, 1i transitions (see Section 6.1.2), as well as for repumping during Doppler
cooling. The coherent manipulation of hyperfine states requires a system that can generate
microwave pulses with high power, accurate timing, and well-defined phase relations.
The microwave source in our setup is an Agilent 5183A synthesizer that is locked to a
10-MHz rubidium atomic clock20 for long term stability. The system is not referenced to
any primary frequency standard, since our experiments are not concerned with absolute
frequency accuracy21 . The synthesizer’s continuous wave output at 12.24 GHz is frequency
mixed22 with the tunable radio frequency signal of a computer-interfaced DDS solution23 to
provide timing, amplitude and frequency-controlled microwave pulses around 12.64 GHz.
The signal is amplified24 up to 10 W and delivered to a SMA-waveguide adapter25 that
serves as a horn antenna. The microwave antenna is pointed at the ion trap through a
vacuum chamber window from a distance of about 120 mm (see Figure. 4.8).
4.2.5
Optical Layout of the Experiment
The simultaneous manipulation and detection of both atoms and ions in the same physical
location presents a formidable challenge in terms of optical access in our experiment. Figure 4.8 and Figure 4.9 show the layout of laser beams and optical detectors with respect
to the location of the ion trap and the surrounding vacuum chamber. In Figure 4.8 the
experiment has been cut open horizontally at the height of the ion trap and is now viewed
from the top. In Figure 4.9 the experiment is seen from the side of the camera CCD1. For
simplicity the six laser beams of the magneto optical trap have not been included into the
drawing. All other laser beams, optical detectors, and microwave antennas are aligned to
the interaction region in the centre of the ion trap.
Basic Ion Trapping
The laser light at 399 nm, that addresses the resonant transition in the neutral Yb atoms,
and the 369 nm Horiz. Cooling beam, that ionizes the excited atoms in the second step of
the two-photon ionization, are coupled out of the same optical fiber. The beams are located
20
Efratom LPRO
Indeed the comparison of two identical atomic clock-synthesizer combinations in our lab has shown a
stable frequency offset between both systems of several Hz at the 12 GHz level.
22
Minicircuits ZX05-153MH+
23
designed by Christoph Zipkes http://decads.scondaq.com/
24
Microwave Amplifiers Ltd AM51-12.4-12.8-40-40
25
Flann microwave 18094-SF40
21
71
Chapter 4. Experimental Setup and Methods
369nm Detection
935 nm
369 nm Horiz. Cooling
399 nm
Figure 4.8: Overview of the optical layout. In this rendering the experimental chamber has
been cut open horizontally at the height of the ion trap and is now viewed from the top.
The black components form part of the vacuum chamber, the µ-metal box that is used for
magnetic shielding is shown in grey and the arm holding the ion trap is displayed brown.
The 1064 nm Horiz. Dipole Trap and the 369 nm Pol. Pump. laser beams are both aligned
with the ion trap axis and enter the trap through the holes in its end caps. Details about
the individual beams and detection devices are given in the text.
in the horizontal plane where they form an angle of 60◦ with the ion trap axis and at the
same time intersect the beam of hot atoms from Yb oven at the Doppler-free angle of 90◦ .
Both beams are shaped into an elliptical focus that extends along the axial direction of the
ion trap in order to better illuminate long axial ion crystals, while minimizing the amount
of stray light from the blade electrodes of the ion trap. A second UV beam, 369 nm Vertical
Cooling, crosses the ion trap in the vertical direction from the bottom and is needed to
perform micromotion compensation measurements along the second radial trap axis. The
ion repump light at 935 nm and the second ion repumper at 638 nm again originate from
the same optical fiber and cross the ion trap axis at an angle of 60◦ .
We employ two devices to detect fluorescence photons at 369 nm wavelength from the ion
(CCD3 and PMT ). In order to record pictures of single ions and spatially resolved small
ion crystals we image the central region of our ion trap from the top with a home-built
multi-lens objective and the camera26 CCD3. The objective lens with a numerical aperature
26
Andor iXon DV-885
72
4.2. Trapping of Yb+ Ions
of 0.15 is designed from of-the-shell spherical lenses to enable diffraction limited imaging
at both 369 nm and 780 nm wavelength with magnifications M369 = 5.7 and M780 = 8.0,
respectively. A dichroic mirror is used to separate the UV light from light at 780 nm,
with the later casting a resonant absorption image of the neutral atom cloud onto a second
camera27 (CCD2 ).
In the horizontal plane a second home-built multi-lens objective [133] of numerical aperture
0.27 is used to collect ion fluorescence photons onto a photo multiplier tube28 (PMT ).
In order to reduce stray light counts, we spatially filter an intermediate image with a
small aperture and block nonresonant light with a narrow band pass filter29 at 370 nm.
The absolute detection efficiency of the system has been measured in Section 4.2.3 to
be ηDet = 0.0021(1). Individual photon detection events are time tagged30 with a time
resolution of 25 ps and made available to the control software for real time processing.
CCD3
CCD2
1064 nm Vertical. Dipole Trap
λ/4
Wave Plate
1064 nm Horiz.
Dipole Trap
369nm
Pol. Pump.
z
x
780nm Vertical Imaging
411nm
369nm Vertical Cooling
Figure 4.9: Overview of the optical layout. The experiment is seen from the side from the
position of the camera CCD1 (see Figure 4.8). Different laser beams and detection lights
are separated by the use of dichroic mirrors, polarizing beam splitters or by spatial displacement. The vertical absorption imaging with camera CCD2 is only used for alignment
purposes. Details about the individual beams and detection devices are given in the text.
27
Apogee Alta U1
Hamamatsu H7360-01
29
Semrock FF01-370/10-25
30
RoentDek TDC8HP
28
73
Chapter 4. Experimental Setup and Methods
Ion Zeeman Qubit Manipulation
The experiments on the dynamics of the
174 Yb+
Zeeman qubit (Section 6.1.1) additionally
require the 369 nm Polarization Pumping beam. The beam propagates along the ion trap
axis, which is during these measurements also the direction of the homogeneous magnetic
offset field. With the help of a computer-controlled motorized λ/4 waveplate the polarization of the beam can be set to σ + or σ − polarization in order to initialize the ion in its
upper or lower Zeeman state. The frequency of the UV light in this beam is the same as
in the two 369 nm Cooling beams.
The propagation direction and the linear polarization of the 411 nm light were chosen to
be mutually orthogonal with each other and the direction of the magnetic field in order to
only address electric quadrupole transitions with ∆m = ±2.
Ion Hyperfine Qubit Manipulation
For the preparation, manipulation and detection of the hyperfine qubit in
171 Yb+
we use a
further pair of laser beams with 369 nm light in the horizontal plane of the experiment. The
369 nm Detection light is applied during the hyperfine detection. Compared to the 369 nm
Cooling beams it has a significantly smaller waist to minimize the amount of stray light
background counts and operates closer to the resonance frequency of the ion in order to
maximize the number of resonantly scattered photons before off-resonant pumping occurs.
The function of the 369 nm Frequency Pumping light is to initialize the hyperfine qubit
through frequency selective optical pumping. The microwave radiation resonant with the
ground state hyperfine transitions at 12.6 GHz is emitted by a waveguide-SMA adapter
that is located at a distance of ≈120 mm from the ion.
Rb Manipulation
The optical dipole trap consists of a horizontal beam, that is aligned to the trap axis and
propagates through the holes in the end-caps of the ion trap, and a vertical beam that
crosses the ion trap from the top. The linear polarizations of both beams are orthogonal
and both lie in the horizontal plane of the experiment shown in Figure 4.8.
The spiral antenna for microwaves at 6.8 GHz is used during the microwave evaporation
in the QUIC trap as well as for Landau-Zener sweeps to transfer atoms in the dipole trap
from the F = 2 to the F = 1 manifold of the Rb ground state.
Resonant absorption imaging of the Rb atoms can be performed, both in the vertical and
horizontal direction. For the imaging in the horizontal direction resonant light at 780 nm
is launched from the tip of a bare single mode fiber, that is mounted in the centre of the
PMT objective. The image of the cloud is captured by the camera CCD1 that serves as
74
4.2. Trapping of Yb+ Ions
our main quantitative imaging, both in-situ and after time-of-flight expansion.
4.2.6
Yb+ Laser System
The cooling and manipulation techniques for Yb+ ions, described in the previous Sections,
require the generation of a significant number of intensity and frequency controlled coherent
light fields. The purpose-built laser system that supports the basic functionality of the ion
experiment, has been designed and built by the former PhD student Stefan Palzer [70].
Here, I will present an overview of the laser system and describe the most important
additions that we have made.
The laser system for Yb+ currently consists of the six external cavity diode lasers (ECDL),
two tapered amplifiers, one frequency-doubling cavity and two frequency transfer cavities.
A schematic drawing of the complete setup, also including the three ECDL and the tapered
amplifier of the Rb laser system described in Section 4.1.1, is shown in Figure 4.10. All
Yb+ lasers are intermediate line width lasers, that require frequency stabilization to (near-)
resonantly address the various optical transitions in the ion over long timescales. Several
different ways of frequency referencing diode lasers are currently implemented in our laser
system.
Frequency locking of Lasers
The light at 399 nm for the resonant first step in the two-colour photo-ionization process
of Yb, is provided by an ECDL that is stabilized by frequency modulation spectroscopy to
an Ytterbium hollow cathode lamp. The Doppler-free saturation spectroscopy resolves the
hyperfine structure of the 1 S0 →1 P1 transition in neutral Yb, so that the laser can be locked
to a specific hyperfine transition for isotope selective loading. Part of the spectroscopystabilized 399 nm laser light now serves as a frequency reference onto which a blue transfer
cavity is actively stabilized. With the exception of the mirror coatings, the blue cavity is
identical in design to the infrared transfer cavity in our setup, which is described in detail
in [70]. The blue transfer cavity can be used to referenced lasers in the wavelength range
365 − 440 nm to the Yb-spectroscopy.
The cooling laser light at 369 nm is created in our setup by frequency-doubling laser light at
739 nm in a nonlinear LBO crystal inside a bow tie enhancement cavity. The infrared light
is provided by amplifying the light from an ECDL in two successive tapered amplification
stages. The seeding ECDL is locked with a tunable frequency offset to another diode laser,
which itself is Pound-Drever-Hall locked to the infrared transfer cavity. The Master and
Seed laser solution allows to continuously frequency tune the cooling laser light over a range
75
1064nm Horiz. Dipole
1064nm
Fiber Ampifer
Mirror
Beam splitter
1064nm Vert. Dipole
1064nm
Seed
Yb+ Repump
935nm and 638nm
Yb+ 411nm
Yb+ 369nm
Freq. Pump.
76
Chapter 4. Experimental Setup and Methods
Rb Repump
MOT
Blue Transfer
Cavity
Doubling Cavity
Yb+ 369nm
Pol. Pump. or
369nm Detection
Rb Img
Hori
Locking
Feedback
Mechanical
Shutter
Yb+ 369nm
Horiz. Cooling
and Yb 399nm
Yb+ 369nm
Vert. Cooling
Rb SigmaPump
Laser
Tapered
Amplifier
Fiber Coupler
Polarizing
Beam Spiltter
1064nm Lattice
Rb Img
Vert
Optical Isolator
AOM
Rb Cool
MOT
Offset
Lock
780nm TA
MOT
Rb Repump
Img
Rb Pi-Pump
WM
Lock
739nm
TA
Red Transfer
Cavity
Offset
Lock
Vapor Cell Lock
780nm Rb Repumper
780nm Rb
Repump
780nm Rb
Seed
WM
Lock
Yb Hollow
Cathode
Lock
Vapor Cell Lock
780nm Rb Master
780nm RB
Master
935nm Yb+
Repump
740nm Yb+
Master
740nm Yb+
Seed
399nm Yb
411nm Yb+
Seed
369nm Yb+
Diode
638nm Yb+
Repumper
Figure 4.10: Simplified schematic of the laser system of the atom-ion hybrid setup. Details are given in the text.
4.2. Trapping of Yb+ Ions
of several hundred MHz. The main output beam of the cavity passes through an AOM for
intensity control and fast switching and is then divided into the 369 nm Horiz. Cooling and
the 369 nm Vertical Cooling beam. A second output beam of the cavity, that results from
the reflection of UV light on the output facet of the nonlinear crystal, is sent through an
independent AOM and is used for either the 369 nm Detection or the 369 nm Polarization
Pumping beam, depending on the experimental setting.
The light for the repump transition at 935 nm is provide by an ECDL, that is frequency
locked to the infrared transfer cavity. A double-pass AOM is used to switch and frequency
tune the repump light over a range of 50 MHz.
In order to conveniently lock laser that do not require the ∼ MHz stability provided by the
transfer cavities, we have implemented a digital frequency lock based on our commercial
High Finesse WS7 wave meter. The wavemeter’s programming interface is used to make
the wave readings available to the ion control software for real-time feedback and control
purposes. The measurement shown in Figure 4.11 characterizes the short- and intermediateterm stability of the wavemeter for various wavelengths by logging the apparent frequency
drifts of the spectroscopy referenced ion lasers. In general, wavemeter drifts are on the
order of several MHz per hour with worse performance in the ultraviolet and for low light
intensities. Over timescales of days and weeks and the wavemeter drifts within the specified
absolute accuracy of 60 MHz.
Frequency [MHz]
We currently implement a simple digital integrator loop by feeding the analog-channel
3
0
-3
740nm Seed
3
0
-3
9
6
3
0
-3
-6
-9
740nm Master
935nm
399nm
15
12
9
6
3
0
-3
-6
-9
-12
0
1
2
3
4
5
6
7
8
9
Time [h]
Figure 4.11: Short- and intermediate-term stability of the commercial wavemeter for the
experimental conditions in our laboratory.
77
Chapter 4. Experimental Setup and Methods
voltage Vn+1 = Vn +α·(νSet −νW avemeter ) back to the piezo of the ECDL grating. The linear
frequency-voltage tuning slope is determined experimentally for each grating stabilized
diode laser. The small loop frequencies on the sub-Hz scale31 are sufficient for efficient
frequency feedback due to the intrinsic short term stability of the diode lasers and the
excellent linearity of the grating feedback (α(ν) = const.).
The wavemeter locking technique is used for the repumper laser at 638 nm. In order
to efficiently repump from all Zeeman states of the 2 F7/2 -state, that span, for example,
175 MHz at a magnetic field of 13.8 G, a triangular modulation voltage at 200 Hz is added
to the grating feedback voltage to continuously sweep the laser.
The stability of the wavemeter feedback is also sufficient for the locking of the 369 nm diode
laser that generates the light for the 369 nm Frequency Pumping beam. In the long term,
direct 369 nm diode lasers could completely replace the current frequency-doubling system
in our experiment, that requires a relatively high maintenance effort due to the rapid aging
of the inefficient 739 nm Fabry-Periot laser diodes. The Yb spectroscopy stabilized blue
transfer cavity could be used as a stable frequency reference in this case.
The direct generation of 411 nm light for the 2 S1/2 - 2 D5/2 transition is achieved with
an ECDL that employs a frequency selected blue-ray Fabry-Periot diode, which is heated
to 45 ◦ C. Since the laser is not intended for coherent manipulation of the 139 Hz narrow
electric quadrupole transition (see Section 6.1.1), its frequency is only referenced to the
blue transfer cavity. We have determined a instantaneous linewidth of 380 kHz for this
diode laser using a self heterodyne measurement [134].
Intensity and Pulse Control
The generation of light pulses with computer-controlled duration and intensity is implemented in the same way for the 369 nm Horiz. Cooling, 369 nm Vert. Cooling, 935 nm,
411 nm, 369 nm Detection, 369 nm Pol. Pump.32 and the 369 nm Freq. Pump.32 laser
beams. Each control loop includes an Acousto-Optical-Modulator as the control element,
a logarithmic photo diode as the sensor, and a FPGA-controlled unit33 which combines the
functionality of a digital PID and a DDS-sourced AOM driver.
The typical extinction rations of ∼ 104 − 105 , that we obtain, when switching UV laser
beams with AOMs are not sufficient for some of our experimental pulse sequences. Whereas
the ion should scatter on the order of 107 photons during detection and Doppler cooling, it
must not scatter a single photon during coherence time measurements that can extend over
several hundred milliseconds. In order to increase the extinction ration we synchronize the
31
The loop frequencies are in practice limited to the typical light integration times of the wavemeter.
Since the intensity is not critical we only use open loop control for these beams.
33
developed by Christoph Zipkes http://decads.scondaq.com/.
32
78
4.3. Experimental Cycle
Figure 4.12: Experimental cycle of the hybrid trap. During time needed for the production
of ultracold neutral cloud, a range of operations are performed on the ion to compensate for
long term drifts of laser frequencies and electrostatic offset fields. The controlled interaction
window between the ion and atoms is only ∼ 8 s out of a total experimental cycle time of
∼ 80 s. Figure adapted from [69].
open and closing of mechanical shutters in the beams with the AOM light pulse sequences.
4.3
Experimental Cycle
The production of an ultracold cloud of neutral atoms as described in Section 4.1 takes
about 65 s. After this time the atomic sample is available for several seconds for interactions with the ion until it is finally destructively analyzed by resonant absorption imaging.
The total experimental cycle time of the experiment is 80 s (Figure 4.12). On the ion side,
the time before and after the interaction is used for the preparation and the detection of
the ion state as well as for a number of compensation steps to keep the conditions of the
experiment constant from one experimental cycle to the next.
The low rates of data acquisition in the experiments on ion-atom interaction, that only
obtain a single binary outcome once every experimental cycle, make it necessary to accumulate data over long time scales. For this reason it is important to guarantee the
repeatability of the experiment with continuous measurements that compensate for drifts
in the micromotion [109], the frequency of the lasers and the relative positioning of the ion
and the atom cloud [69].
With only ∼ 1100 repetitions of the experiment available in 24 hours, the autonomous
operation of the experiment, including the automatic reloading of ion (Section 4.2.1)) is
essential for many of our experiments. The flexible experiment control software has been
79
Chapter 4. Experimental Setup and Methods
designed34 with this objective in mind from the very start35 .
34
35
by Christoph Zipkes http://decads.scondaq.com/,[69]
Our recent implementation of a Skype interface in the software has made it now possible for the experiment
to call, if a laser falls out of lock or a problem arises with one of the interlocks of the experiment.
80
Chapter 5
Cold Chemistry with Single Particles
In Chemistry the control over the structure of matter and its properties, and the study of the
underlying principles are traditionally performed under conditions dominated by the laws of
thermodynamics. Chemical reactions are analyzed and steered by the choice and amount
of agents and by varying macroscopic variables such as temperature and pressure. In
recent years the booming field of cold and ultracold, dilute matter experiments has handed
researchers new tools to investigate and control chemical reactions. At low temperatures
the influence of externally applied radiation or static electromagnetic fields on chemical
reactions is not attenuated by the averaging effect of large motional and internal state
distributions of the reactants [59]. One can imagine, that the ultimate degree of control
and understanding of a particular chemical reaction could be achieved, if all motional
and internal quantum states of the reacting particles were manipulated precisely and the
states of the final reaction products were analyzed in detail. In the context of neutral
atoms, photo association, for instance, has been employed to efficiently transfer colliding
atoms into electronically excited bound molecular states [135]. Using Feshbach resonances
ultracold atoms have also been associated into weakly bound molecules [83]. The interaction
between ions and neutral atoms as outlined in Chapter 2 is predicted to be subject to the
same range of experimental control methods.
In this Chapter we describe our first use of the Yb+ +Rb hybrid trap as a tool to
investigate cold chemical processes and inelastic collisions1 . To obtain a clean and wellcontrolled experimental system, we use state-selected single particles, precise control of the
interaction parameters and finally analyze the reaction outcomes in detail using in-trap
mass spectrometry. This allows us to study the large spectrum of electronic state changing
collisions that we have introduced in Section 2.3.2, including the various types of charge
exchange and collisional quenching. We demonstrate how control over the internal electronic state of a single ion and the hyperfine state of neutral atoms can be employed to
tune cold exchange reaction processes. Our measurements show a large sensitivity of the
charge-exchange reaction rates to the atomic hyperfine state, highlighting the influence of
the nuclear spin on atom-ion collisions.
1
The first results on ground state charge exchange collisions were published in [35] and also form part of C.
Zipkes thesis [69]. The main Sections of this Chapter are based on the publication [47].
81
Chapter 5. Cold Chemistry with Single Particles
5.1
Chemistry in the Yb+ + Rb Ground State
The theoretically considerations of Chapter 2 have given an idea of the complexity of
inelastic interactions in the heteronuclear Yb+ + Rb system.
We therefore start our investigation into cold chemistry with the simplest scenario and
considering collisions of ions with atoms in the electronic ground state. In the absence of
resonant light, collisions in the Yb+ (6s)+87 Rb(5s) state natural arise from inserting a Yb+
ion into a neutral bath of Rb atoms. In addition to sympathetic cooling [33] by elastic
scattering the immersed ion can also undergo collision induced transitions to the single
lower lying binary molecular state that corresponds asymptotically to Rb+ +Yb(6s2 )1 S
(see Section 2.3.2). The existence of a lower lying charge exchange channel is a feature
Yb+ +87 Rb(5s) shares with several other atom-ion combinations of current quantum hybrid
experiments [34, 42, 41]. It is the result of the generally higher ionization potential of earth
alkali metals used as the ion species, compared to alkali metals that make up the ultracold
neutral gas.
5.1.1
Measurement Sequence
Experimentally, we measure the inelastic loss rate γℓ at the single particle level by immersing single ions for a time t into the neutral atom cloud and determining the survival
probability [33, 35, 136, 65], Ps = exp (−γℓ t). The measurement sequence starts with the
confirmation of the presence of a single bright Yb+ ion in our trap (see Section 4.2.1).
Prior to the arrival of the neutral cloud, the ion is displaced from the interaction region
along the ion trap axis, in order to avoid atom-ion interactions during the transport of
the atoms. By ramping the end cap voltages, the ion is then moved into the centre of
the cloud within a few ms to start interactions. The interaction interval is terminated by
releasing the atoms from the magnetic or optical trap. We infer the central in-trap density
na ≡ n3d (0) of the neutral cloud at the end of the interaction from the time-of-flight ab-
sorption images according to Equation 4.11. Depletion and heating effects due to elastic
collisions play a minor role for the large, low-density clouds of magnetic trapped atoms.
For optically trapped atoms the changes in the neutral bath during the interaction interval
are taken into account by a simple model that linearly interpolates between the measured
cloud conditions before and after ion interaction. Following the time of flight imaging of
the neutral cloud the ion cooling laser light is turned on and the presence or absence of a
fluorescing Yb+ ion is detected.
In order to describe the measured loss rates we make use of the Langevin capture model
82
5.1. Chemistry in the Yb+ + Rb Ground State
|2,2ia
of Equation 2.20, that introduce the inelastic collision loss efficiency ǫS
of collisions of electronic ground (S) state
174 Yb+
ions with
87 Rb
for the case
atoms prepared in the
|2, 2ia hyperfine spin state of their electronic ground state manifold. We have shown that
|2,2ia
the measured loss rate with a value of ǫS
= 10−5±0.3 is indeed energy independent and
scales linear with the neutral atom density in [35]. The measurements in Chapter 6 will
show that the spin of the
174 Yb+
ion is in an almost completely mixed state and that col-
lisions happen three times more often in the triplet potential than in the singlet potential.
Similarly low inelastic rates have been found for other ion-atom combinations in hybrid
traps [34, 42, 58] except for resonant charge exchange [40, 43].
The permanent loss of fluorescence can come about in two different ways. On the one
hand, it can mean that the ion has undergone a chemical reaction, such that a chemically
distinct, dark ion, such as (RbYb)+ , Rb+ or Rb+
2 is left in the trap after the interaction.
Whereas ions (RbYb)+ and Rb+ are the result of radiative association and radiative charge
exchange, respectively, the Rb+
2 ion can be formed in the secondary process
(RbYb)+ + Rb → Yb + Rb+
2
(5.1)
from a previously associated (RbYb)+ ion2 . On the other hand the absence of fluorescence
can indicate the loss of the charged particle from the trap. If in an exothermic inelastic
collision the kinetic energy imparted on the charged particle exceeds its trap depth, then
the particle will be lost from the trap. From numerical simulations of our ion trap we
estimate the depth of our ion trap to be on the order of 250 meV. For Yb+ and Rb
colliding in their electronic ground state the only inelastic processes that lead to ion loss
are the various pathways of the charge exchange reaction, since the release of energy in
spin changing collisions is only on the order of µeV (see Chapter 6). To distinguish the
two events, ion loss and the presence of a dark ion, we perform inelastic fluorescence loss
experiments with binary ion crystals. The four possible outcomes after the interaction are
shown in Figure 5.1a. In the case of a single bright ion left in the trap, the position of the
ion in the trap clearly reveals the presence of a further dark ion.
Mass Spectrometry
In order to further determine the chemical composition of the dark ion we identify the ion
by its mass. For this purpose we perform in-trap mass spectrometry[137, 35] on the binary
ion crystal consisting of one bright ion and one dark ion of unknown mass.
The principle of this in-trap mass spectroscopy technique is based on probing the mass
2
For molecules in the absolute electronic ground state of the system, this is the only two-body decay channel,
as the dissociation of molecules by the process (RbYb)+ + Rb → Rb2 + Yb+ is energetically not allowed
(see Table 5.1).
83
Chapter 5. Cold Chemistry with Single Particles
a
b
Fluorescence counts [a.U.]
dependent eigenmode frequencies (see Section 3.1.3) by resonant excitation. In our experi-
40
30
20
10
0
(RbRbYb)
30
+
35
(RbYb)
+
40
+
Yb
45
+
Rb
50
55
Common mode excitation frequency [kHz]
Figure 5.1: Identification of the inelastic reaction outcome. a Fluorescence images of the
binary bright ion crystal before interaction (top) and images of the four possible reaction
outcomes (bottom). b Example light curves of mass spectroscopy scan for two different
dark ion mass, a dark Yb+ ion (black) and a charge exchange Rb+ ion (red). We do not
observe RbYb+ nor RbRbYb+ in our mass spectroscopy scans.
ment we perform the mass spectroscopy by monitoring the fluorescence light of the binary
crystal while exciting its centre-of-mass mode frequency through light pressure modulation.
To this aim a frequency tunable square wave is imprinted onto the intensity of the Doppler
detuned 369 nm Horiz. Cooling light, that illuminates the ion pair, by chopping the laser
beam with an Acousto Optical Modulator3 . In a typical measurement as seen in Figure 5.1
the frequency of the modulation is linearly swept over the region of possible centre-of-mass
frequencies during a scan time of several seconds. As the modulation of the light pressure
becomes resonant with the centre-of-mass resonance of the trap, the oscillation amplitude
of the collective ion mode becomes large and we observe a reduction in the number of
detected fluorescence counts. Numerical simulations show that for our experimental parameters the decrease in the photon rate is the result of two different effects. In addition
to the direct reduction of photon scattering rate of the ion due to the Doppler shift of the
cooling laser transition, the detected photon rate is also suppressed because the bright ion
spends less time in the ∼60 µm wide field of view of the photon counter. The large ion
oscillation amplitudes needed to achieve a detectable reduction in the fluorescence intensity
also explain why the eigenfrequencies of linear ion crystals beyond the COM mode cannot
be detected with this method and why the dip in the fluorescence rapidly becomes smaller
for the centre-of-mass mode of crystals with increasingly unequal ion masses. In both cases
3
The laser beam is aligned at 60◦ with respect to the trap axis and can therefore be used to excite both the
axial and the radial trap frequencies.
84
5.1. Chemistry in the Yb+ + Rb Ground State
the large amplitudes of the relative motion between ions, which are required for a fluorescence drop, exceed the validity region of the linear approximation in Equation 3.17 and the
motion becomes unharmonic.
Compared to the alternative method of electronic excitation, our optical excitation scheme
is self regulating so that ions can not be heated out of the trap by the mass spectrometry
measurement. By repeatedly sweeping a small frequency range we can detect the COM
mode frequency with a relative error of less than 10−3 , which is sufficient to resolve mass
differences of 1 amu at ion masses of ≈ 174 amu (see Figure 5.2).
Molecular Ions
The fluorescence image and mass spectroscopy analysis shows that the fluorescence loss in
the collisions of S-state 174 Yb+ ions with 87 Rb atoms in |2, 2ia is the result of ion loss from
the trap in 65% of all events, whereas the presence of Rb+ is detected in 35%. We do not
observe any ions with masses that would correspond to radiatively associated RbYb+ ions.
We have previously considered the 65:35 splitting as the branching ratio of non-radiative to
radiative charge exchange [35]. In light of a recent theoretical study [76] of charge exchange
in our system this interpretation is becoming increasingly unlikely. The study confirms that
radiative charge exchange indeed will lead to trapped Rb+ as the kinetic energies of the
Rb−1 ion (< 2/3 · 500 cm−1 ≈ 20 meV, see Figure 2.5) in this case are expected to be
considerably smaller than our trap depth of 250 meV. The simulated occurence of radiative
charge exchange exchange (30%) also agrees well with the observed 35% of trapped Rb−1
ions within experimental and theoretical uncertainties. The study, however, predicts the
formation of bound RbYb+ molecule for 70% of reactive events and negligible non-radiative
charge exchange, whereas we detect ion loss in 65% of all cases. The formation of molecular
ions has recently been observed for excited state collisions of Ca+ in Rb [41] and Ba+ in
Rb [50].
The analysis in Section 3.1.4 has shown that binary ion crystals are formed for a very
large range of ion mass and we can therefore exclude ion trap stability as a possible reason
for the absence of detected molecules. It has been suggested [42] that RbYb+ molecules
could be collisionally dissociated in a secondary collision with Rb to form Rb+
2 ions and
a neutral Yb atom (Equation 5.1).
of nucleon as
174 Yb+
87 Rb+
2
molecules incidentally have the same number
so that in order to distinguish the molecule from the Yb+ ions4 by
mass spectrometry, we have performed experiments with the isotope
measurement results in Figure 5.2 could indicate the creation of
4
176 Yb+ .
YbH+
Whereas the
molecules in two
This measurement predates the implementation of the 638 nm repump laser, which is now used to efficiently
avoid the creation of dark Yb+ in the F-state.
85
Chapter 5. Cold Chemistry with Single Particles
cases, we have not detected the presents of Rb+
2 in any of our measurements.
Position along trap axis [Pixel]
In the case of very high neutral atom densities, three-body processes have to be considered,
8
a
6
4
2
0
-2
-4
Axial trap frequency [kHz]
-6
51.6
b
51.5
51.4
51.3
51.2
51.1
0
4
8
Time [h]
12
16
20
Figure 5.2: Search for Rb+
2 molecules. Over the course of 20 hours we successfully prepare
487 binary ion crystals of 176 Yb+ . The ions in their 2 S1/2 electronic ground state interact
with a magnetically trapped cloud of 87 Rb atoms in the |2, 2ia state. a The pixel positions
of the ions along the trap axis after interaction. Two bright ions (light blue), single ion loss
(dark blue) and the presence of a bright and a dark ion (orange) are clearly distinguished
by their position. b Axial trap frequency measurements for the data shown in a. Single
ions and binary crystals of 176 Yb+ have the same centre-of-mass frequency. The dark ions
are identified as 176 Yb+ in the 2 F7/2 state and two possible cases of 176 YbH+ . The orange
line indicates the expected trap frequency for particles with mass m = 174 amu. We
do not observe the formation of Rb+
2 molecular ions. Both ion positions and axial trap
frequencies drift over time, since loading induced offset fields are not compensated during
this measurement.
that give rise to further inelastic collision processes. Recent experiments in Ulm [138] have
+
demonstrated that Rb+
2 molecules and Rb ions can also be created from ultracold clouds
of atoms alone, if neutral Rb2 , created from three-body recombinations in the ultracold
gas is ionized in a multi-photon process by dipole trapping laser light at 1064 nm. For our
dipole trap parameters we do observe the occasional loading of an additional Rb+ ion for
our largest and most dense Rb clouds.
Further studies by the group [77] have also shown that ions can act as a three body reaction
centre that functions as a catalyst for the formation of Rb2 . The release of the binding
energy has been observed to impart kinetic energies on the order of several hundred meV
86
5.1. Chemistry in the Yb+ + Rb Ground State
Table 5.1: Ionization energies Ei of selected neutral atoms and dissociation energies De of
relevant molecules. The dissociation energies are given for the deepest molecular potential
of the electronic ground state dimers. The full information about the Born-Oppenheimer
potentials of the dimers can be found in the listed references.
Ionization Energy Ei /cm−1
Rb
Yb
Ba
Li
33691
50443
42035
43487
[140]
[140]
[140]
[140]
Dissociation Energy De /(cm−1 )
Rb2
Rb+
2
YbRb+
4269 [141]
5816 [142]
3496 [76]
Figure 2.1
on the ion5 , which in our trap could lead to the loss of the Yb+ ion from the trap.
These three-body effects are avoided in our experiments by working with thermal clouds
with low neutral atom densities. We demonstrate the binary nature of the observed inelastic
collisions by measuring their linear dependence on the neutral atom density in Figure 5.6
and in [35].
One possible reason why we do not detect the predicted RbYb+ ions in our trap is the
dissociation of the molecule by cooling light at 369 nm (⇔ 3.36 eV). The electronic ground
state molecules with binding energies of up to ≈ 2000 cm−1 (⇔ 0.25 eV) can be excited into
the region of densely populated excited state of RbYb+ shown in Figure 5.3. By driving
transitions to anti-binding states [139] or to the repulsive wall of binding states the presence
of 369 nm light could potentially lead to a rapid dissociation of the RbYb+ molecular ion
and loss of the charged particle. The significant fraction of ion loss events observed in our
experiment would in this case correspond to dissociated molecules.
In order to test this hypothesis the presence of molecular ions without 369 nm light could
be shown indirectly in the following way. After immersion of a binary ion crystal in the
neutral gas, the interaction outcomes are subjected to a strong electronic excitation at the
axial COM frequency corresponding to a binary ion cyrstal with an RbYb impurity ion. If
indeed a molecular ion is in the trap both ions should be resonantly heated out of the trap.
After the excitation the 369 nm cooling light is turned on and the ion fluorescence images
are taken. In comparison to a reference sequence without the electronic excitation, this
measurement should lead to a sharp increase in double ion losses from the trap, thereby
confirming the presence of RbYb+ ions in the trap6 .
5
6
The study was conducted with Rb+ ions.
The idea about the possible dissociation of RbYb+ molecules by UV light was developed after the completion of our experiments on cold chemistry. It has therefore not been investigated in this thesis and will
have to be addressed in future experiments.
87
Chapter 5. Cold Chemistry with Single Particles
Hyperfine Spin Dependence
We can compare these results now to inelastic electronic ground state collisions in the
absolute lowest hyperfine state |F = 1, mF = 1ia of the neutral atom. For this purpose
the atoms are prepared in the optical trap by an adiabatic passage microwave transition
(Section 4.1.3). The rest of the preparation sequence is identical to one for |F = 2, mF = 2ia
atoms described above. The hyperfine energy difference between the |F = 1, mF = 1ia and
the |F = 2, mF = 2ia states of Rb is 30 µeV, which is larger than the collision energy
but negligible on the scale of the molecular potentials or the trap depth (250 meV). We
find a significantly enhanced probability for inelastic collisions with the ion in the electronic
|1,1ia
ground state ǫS
|2,2ia
= (35±11)×ǫS
, which highlights the important role of the hyperfine
interaction. We also observe a significantly increase in the fraction of reactive events that
result in a trapped Rb+ ion compared to collisions with |F = 2, mF = 2ia atoms. The
large differences in the inelastic collision behaviours for the two hyperfine states are not
understood [143] and will require further theoretical efforts. In the next Chapter we will
shed further light on the role of hyperfine and Zeeman structure in cold atom-ion collisions.
5.2
Inelastic Collisions in Electronically Excited States
After the analysis of electronic ground state reactions we turn our attention to the electronically excited states of the Yb+ + Rb system. The study of electronic states is interesting
both for fundamental and practical reasons. The control over the internal states can be
used to steer chemical reaction and provides an additional experimental degree of freedom
to further the understanding of chemical reactions at low temperatures. The stability of
excited state is also of importance for buffer-gas cooled ion clocks [144] and for the prospect
of quantum information protocols in hybrid traps.
We start by determining the inelastic collision loss rate coefficients for the long-lived 2 D3/2
(radiative lifetime 52 ms) and 2 F7/2 (radiative lifetime 10 years) states of the ion. By optical pumping, both states are prepared as “dark” states, in order to study pure two-body
collisions without the presence of light. This approach differs fundamentally from other
experiments in atom-ion[41] excited state collisions, which always have been in the presence
of near-resonant laser light, where effects of photo association can complicate the picture.
For the measurement the ion and the neutral cloud are first overlapped in the 2 S1/2 ground
state. The ion is then prepared in the 2 D3/2 state in less than 10 µs by optical pumping
from the 2 S1/2 state using the 369 nm cooling laser. To end the interaction interval the
369 nm is switched off and the 935 nm repump laser light is turned on to repump the
ion back into the ground state within microseconds. The fast pumping times, which are
88
3
}
5.2. Inelastic Collisions in Electronically Excited States
[3/2]1/2
E (eV)
2
935 nm
P1/2
2
2
D5/2
D3/2
369 nm
3
2
F7/2
2
2
411 nm
P3/2
2
P1/2
1
2
Yb
+
S1/2
2
S1/2
0
Rb
Yb++Rb
-2
Yb+Rb+
Figure 5.3: Level scheme of the ion-neutral system. a) Left: Yb+ level scheme with
the transitions used for optically pumping the ion (to scale). Middle: Level scheme of
a Rubidium atom. Right: Asymptotic level schemes of the two channels Yb+ +Rb and
Yb+Rb+ . The collision is initiated in the Yb+ +Rb manifold and the Yb+Rb+ manifold
can be populated by charge-exchange processes.
89
Chapter 5. Cold Chemistry with Single Particles
significantly faster than the inverse of the Langevin collision rate of a few 10−3 s are needed
|2,2ia
because of the high inelastic collision loss efficiency of ǫD
collisions of
174 Yb+
= 1.0 ± 0.2 that we measure for
with Rb atoms in the |F = 2, mF = 2ia hyperfine ground state. The
constant ǫ = 1 corresponds to the largest allowed inelastic collision rate in the semiclassical
model, and even in near-resonant charge-exchange between equal elements of atoms and
ions [40] it was expected and approximately found to be ǫ = 1/2. For the inelastic rate
measurement of the extremely long-lived 2 F7/2 state (radiative lifetime 10 years) we first
populate the state outside the neutral cloud by optical pumping with the 411 nm laser
on the 2 S1/2
2D
5/2
line [127] with spontaneous decay on the 2 D5/2
2F
7/2
transition and
then ramp the ion into the centre of the cloud. The end of the interaction is defined by the
release of the atomic cloud from its trap. The inelastic collision loss efficiency of the F-state
|2,2ia
is ǫF
= 0.018 ± 0.004. We also perform the mass spectrometry for the two excited states
and find ion loss as well as significant Rb+ production in both cases (see Table 5.2).
It is important to emphasize that in the case of electronically excited states both reactive
collisions and quenching collisions can contribute to ion loss (see Figure 2.4). The large
number of densely packed molecular levels make a multitude of inelastic pathways possible.
However, we do not observe the formation of molecular ion in the case of excited state
collisions either.
|2,2ia
In addition to ǫD
|1,1ia
we have also determined ǫD
for collision of with atoms in the
|F = 1, mF = 1ia state, which is identical to the former within experimental errors. Once
calibrated in this way, D-state inelastic loss rate measurements can be used to directly
measure the local density of neutral atoms at the position of the ion without resorting
to time-of-flight imaging of the atomic cloud. We will make use of this technique for the
experiments in Chapter 6.
5.2.1
Collisional Quenching
With regards to collisional quenching between different electronic levels of the ion, we are
principally able to detect two different quenching scenarios: 2 F7/2 → 2 S1/2 and 2 D3/2 →
2F
7/2 .
We directly observe quenching of 2 F7/2 → 2 S1/2 as shown in Figure 5.4, where we
exemplarily show two different experimental runs, with and without quenching. In this
measurement we prepare two ions in a small Coulomb crystal at time t = 0 in the 2 S1/2
ground state. The optical pumping into the 2 F7/2 state using light at 411 nm is switched
on at t = 500 ms for 500 ms and typically within 100 ms the ion is pumped into the desired
state. As a result, the fluorescence counts on the 369-nm transition drop to zero and the
369-nm cooling laser light is then turned off. After the interaction with the neutral atoms
(effective duration typically 16 ms) and the subsequent removal of the neutral atom cloud,
the laser is switched on again and the fluorescence on the 2 S1/2 − 2 P1/2 transition (together
90
5.2. Inelastic Collisions in Electronically Excited States
Table 5.2: Measured proportionality constant ǫ and branching ratios. Loss events of the
charged particle are always chemical reactions for the S−states, whereas for D− and F −
states quenching and chemical reactions can contribute.
87 Rb 2 S
1/2
174 Yb+
ǫ
Charged particle lost
Rb+ identified
Dark Yb+ identified
Hot ion (unidentified)
Number of events
|F = 2, mF = 2ia
2S
1/2
−5±0.3
10
65%
35%
283
2D
2F
7/2
3/2
1.0
±0.2
0.018
±0.004
754
1%
225
87%
12%
< 1%
84%
15%
|F = 1, mF = 1ia
2P
1/2
0.1
±0.2
2S
1/2
(35 ±
|2,2i
11)ǫS a
50%
48%
< 1%
2%
236
2D
3/2
1.0
±0.2
with repumping at 935 nm) is probed from t = 5000 ms onwards. If the ion is still in the
2F
7/2
state it will not scatter photons, however, if it has been quenched to 2 S1/2 , we observe
fluorescence. At t = 6000 ms, the ions are optically pumped from the F -state back into
the S-state to ensure that no ions have been lost. The fluorescence count rate during the
preparation part of the sequence is lower owing to the presence of an offset field from the
magnetic atom trap, which is turned off when the atoms are released. The dips in the ion
fluoresecence after the quenching event in Figure 2.4 indicate a large temperature of the
ion due to the release of kinetic energy in the quench. The only way the ion can still remain
in trap after the quench from the F- into the S-state is if the Rb atom gets excited and
takes up most of the internal energy of the ion. Given a trap depth of 250 meV the only
possible Rb states are 4d 2 D5/2 and 6s 2 S1/2 just below the F-state level (see Figure 5.3).
The quenching rate from 2 D3/2 →
2F
7/2
was observed not to be detectable above our
background rate. The background rate for transfers into the 2 F7/2 results from collisions
with background gas during the fluorescence detection cycle.
5.2.2
Inelastic Collision during Fluorescence Detection
Having established the inelastic collision parameters of the metastable D- and F -states
without resonant laser light, we now turn our attention to inelastic collisions in the presence
of laser light on both the 2 S1/2 − 2 P1/2 (369 nm) and the 2 D3/2 − 3 [3/2]1/2 (935 nm)
transitions. The purpose of the light is to experimentally tune the rates and to observe
the occurrence of inelastic collisions in real time. The radiative lifetime of the 2 P1/2 state
(8 ns) is too short compared to the collision rate to provide pure P-state measurements.
Here and in the following, we assume that the F -state is unoccupied since the 411-nm light
91
Fluorescence counts S1/2 → P1/2 (kHz)
Chapter 5. Cold Chemistry with Single Particles
Optical pumping
2
S1/2 → 2F7/2
Optical pumping
2
F7/2 → 2S1/2
Time (ms)
Figure 5.4: Collisional quenching from 2 F7/2 to 2 S1/2 . A two-ion Coulomb crystal is prepared in the 2 F7/2 dark state by optical pumping, then interacts with the neutral atoms,
and is subsequently probed by laser fluorescence. An early appearance of laser fluorescence
indicates collisional quenching (red curve) as compared to no quenching (blue curve). The
large fluorescence dips indicate a high temperature of the ion crystal.
is turned off and collisional quenching into the F -state has been measured to be negligible.
Therefore, the overall inelastic collision rate is determined by a mixture of S-, P -, D-, and
D[3/2]-states
γℓ = 2π
p
C4 /µ na pS ǫS + pP ǫP + pD ǫD + pD[3/2] ǫD[3/2] .
(5.2)
Here, px is the occupation probability of state x, which we determine experimentally for
different settings of laser intensities and detunings. The relative state population measurement is performed with the method described in Section 4.2.3. In Figure 5.5a we show the
state populations of the S, P , D and 3 D[3/2]1/2 states as we vary the frequency of the
laser at 935 nm on the 2 D3/2 − 3 D[3/2]1/2 transition. Figure 5.5b shows the associated
change in the reaction rates in the presence of the neutral atoms. We demonstrate tun-
ing of the reaction rate by one order of magnitude and we fit our data with the model of
Equation 5.2. The only fit parameter is the inelastic collision efficiency ǫP = 0.1 ± 0.2 of
the P-state, which is small and consistent with zero. Figure 5.6 shows the linear scaling of
the inelastic collision rate with neutral atom density over almost three order of magnitude
in density, confirming the picture of binary collisions.
92
Inelastic collision rate (s-1) State population px
5.2. Inelastic Collisions in Electronically Excited States
a
D3/2
S1/2
D[3/2]1/2
P1/2
b
Figure 5.5: Inelastic collision control by laser light. a State population measurement of the
ion in the absence of neutral atoms as a function of the detuning of the repump laser at
935 nm. b Inelastic collision rate for the same detuning data in presence of neutral atoms
at na = 1 × 1018 m−3 . The solid line shows the theoretical result of Eq. 5.2.
5.2.3
Error Estimation for the Survival Method
Since every single repetition of the experiment takes ∼ 80 s, the optimization of the inelastic
loss rate estimation is of considerable importance for our measurements. The single ion
inelastic rate estimation method used so far calculates the unknown rate γℓ from the survival
statistics of many binary outcomes after a fixed interaction time tf . A sequence of n
independent repetitions of the experiments gives rise to a distribution of k survivals and
n − k losses that is described by binomial statistics with survival probability Ps = e−γℓ tf .
For large numbers of trials and probabilities Ps not too close to 0 and 1, the binomial
proportion and its 100 (1 − α)% confidence interval are well described by their normal
approximation values
k
Pˆs = ,
n
Pˆs ± z1−α/2
s
Pˆs (1 − Pˆs )
n
(5.3)
where z1−α/2 is the 1 − α/2 percentile of a standard normal distribution. For simplicity of
notation we will from now on only consider the 16 and 84 percentiles, that correspond to
a 1σ confidence interval. The estimated inelastic loss rate and its confidence interval then
93
Inelastic collision rate (s-1)
Chapter 5. Cold Chemistry with Single Particles
1017
1018
-3
Neutral atom density (m )
Figure 5.6: Density dependence of the inelastic collision rate with the repump laser on
resonance. The exponent of the power-law fit (solid line) is 0.98 ± 0.02.
follow from linear error propagation
1
γ̂ℓ = − ln Pˆs
τ
1
∂γℓ ˆ
∆Ps = −
∆γ̂ℓ =
∆Pˆs
∂Ps
τ Pˆs
s
(1 − Pˆs )
1
1 ∆Pˆs
∆γ̂ℓ
=
=
γ̂ℓ
Pˆs · n
ln Pˆs Pˆs
ln Pˆs
(5.4)
In this approximation the relative error on the estimated rate exhibits a minimum at
P̂s,opt ≈ 0.20 independent of n. In order to obtain the smallest error for the least number of
runs, the interaction time tf should be set such that the survival probability is Ps ≈ Ps,opt .
5.2.4
Error Estimation for the Fluorescence Method
In the case of fluorescence scattering during inelastic collisions, we use a more efficient way
to determine the unknown inelastic rate at the single particle level. The method makes use
of the detected photon counts of the ion’s 369 nm fluorescence to extract the time when
the ion undergoes an inelastic reaction.
If the exact times tion,0 , ..., tion,j ..., tion,n of the inelastic events in a series of n experimental
runs can be determined without error, then the maximum likelihood estimate (MLE) of the
94
5.2. Inelastic Collisions in Electronically Excited States
2.00
Bootstrap
Number of
Summed Samples
50
25
5
2
1
100
80
60
40
Theory
Median
Mean
16 %
84 %
1.75
Normalized Rate
Summed Counts (50μs bins)
120
1.50
MLE Exp. Fit 16%
MLE Exp. Fit 84%
Ps=Ps,Opt Binomial 16%
Ps=Ps,Opt Binomial 84%
Ps=0.5 Binomial 16%
Ps=0.5 Binomial 84%
1.25
1.00
20
0.75
0
0
2
4
6
8
0
10
20
Time [ms]
40
60
80
100
120
Number of Summed Samples (n)
Figure 5.7: Inelastic rate estimation from the decay of ion fluorscence. a We fit the 369 nm
fluorescence curves that result from the summation over n identical runs of the experiment
with the exponential decay in Equation 5.10 to retrieve the inelastic loss rate. b Theoretical and experimental parameter estimation confidence intervals for the fluorescence
decay method and binomial survival method as a function of the number of experimental
repetitions n.
inelastic loss rate γℓ in the ion survival probability function Ps (t) = e−γℓ t is given by [145]
γˆℓ =
n
n
P
(5.5)
tion,j
j=1
The 100 (1 − α)% confidence interval of the rate estimate [145]
γˆℓ
2n
χ21−α/2,2n
< γℓ < γˆℓ
2n
,
χ2α/2,2n
(5.6)
is limited by the sample size n. Here, χ2a,v is the 100 (1 − a) percentile of the chi squared
distribution χ2 (v) with v degrees of freedom. By approximating the chi square distribution
by a normal distribution and considering the 16 and 84 percentiles (1σ error interval), we
obtain
γˆℓ
2n
2n
√ < γℓ < γˆℓ
√
2n + 2 n
2n − 2 n
(5.7)
for the uncertainty of the inelastic loss rate after n experimental runs.
In practice, however, the accuracy by which the exact ion loss times tion,j can be determined
is limited by the Poissonian nature of the fluorescence photons. At a detected photons count
rate R from the ion the uncertainty of a single loss time is on the order of 1/R with back
ground counts further complicating the estimation error. Instead of explicitly extracting
95
Chapter 5. Cold Chemistry with Single Particles
the times tion,j , we construct a fluorescence decay curve by binning the photon arrival times
and summing the photon histograms of many consecutive runs of the experiment.
For a single run the photon detection probability is uniform
f (t) =

R,
0 ≤ t ≤ tion
0,
(5.8)
tion < t
until the ion is lost. Over a large number of experimental runs, the summed photon
histogram will approach the normalized photon detection probability (see Figure 5.7a)
f (t) =
1 −γℓ t
e
.
γℓ
(5.9)
We therefore fit the total binned light curve with the function
f (t) = A · e−γℓ t + C,
(5.10)
which contains the desired inelastic loss rate γℓ and considers the amplitude A(R, n) and
the back ground count level C(n) as additional parameters (see Figure 5.7).
To take into account the Poissonian rather than Gaussian statistic of the binned photon
counts we fit the histogram data (tph,i , f (tph,i )) minimizing the function [146]
χ2mle = 2
nBins
X
j=1
f (tph,i ) − tph,i − 2
nBins
X
tph,i ln(f (tph,i )/tph,i )
(5.11)
j=1,tph,i 6=0
instead of the more familiar least square residuals.
In order to assess the efficiency of the method, we investigate how the parameter estimation
error of the fluorescence decay method scales with the number of experimental runs n for
typical experimental conditions. We compare the experimental result to the theoretical
values for the ideal scenario of expontial maximum likelihood estimation in Equation 5.7.
For this purpose we determine the parameter estimation errors for the inelastic loss rate by
bootstrapping [147] a sample of 131 experimental runs. The uncertainties in the inelastic
loss rate as a function of the number of experimental runs n are shown in Figure 5.7a for
the ideal exponential maximum likelihood estimation (Equation 5.7), the experimental fluorescence decay fit method, the optimal binomial survival method, and a binomial survival
method with survival probability Ps = 0.5.
√
For a large number of experimental runs n ≫ 1, it follows from Equations 5.7 and 5.11
that the confidence intervals for the inelastic loss rate determined by the binomial methods
are (Ps = Ps,opt ) 1.24 and (Ps = 0.5) 1.44 times larger than for the fluorescence decay
96
5.2. Inelastic Collisions in Electronically Excited States
method. In other words, the binomial method with Ps = 0.5 requires three times more
experimental runs than the fluorescence decay method to obtain the same precision.
For inelastic collision rates γℓ on the order of the photon count rate R, the fluorescence
method will clear not be efficient any more, since barely any photons are detected before
the ion is lost.
The method can, in principle, be extended to inelastic collisions without fluorescence. Instead of probing the survival of the ion after a single long interaction interval, the ion can
be immersed for shorter intervals, several times (nc ) during the 8 s interaction window of
a single experimental cycle. If the probing times and repetitions are chosen such that the
individual interaction intervals have a large survival probability Ps,c , but the total survival
probability Ps = (Ps,c )nc is low, then the times of the inelastic loss can be reconstructed
and used for the computation of the inelastic loss rate.
a
b
c
d
e
Figure 5.8: Monitoring of inelastic atom-ion collisions. a-c Recorded fluorescence for selected events. After a quick initial loss, the fluorescence reoccurs after random times when
the kinetic energy released in the inelastic collision has been removed by cooling. Then,
a second collision occurs and the ion disappears again. The curves are vertically offset
for clarity. d Sum of 343 repetitions of the experiment, which is fitted with an exponential decay for short times. e Zoom-in on the initial decay of plot d. The solid line is an
exponential fit to the data.
Reoccurence of fluorescence
Finally, the presence of fluorescence during collisions can also be used to observe the kinematics of collision products. The Subplots a-c of Figure 5.8 show individual experimental
97
Chapter 5. Cold Chemistry with Single Particles
runs, in which the initial sharp loss of detected 369 nm fluorescence results from an inelastic collision. In 4% of all experimental runs7 we observe that the fluorescence of the ion
reoccurs after a certain time, before it undergoes a second inelastic collision and is dark
again. The relatively slow increase of the reoccurring fluorescence indicates that the ion
has significant amounts of kinetic energy and is on a large trajectory in the trap (much
larger than the size of the atomic cloud). From the number of photons scattered during
the recooling we determine a lower bound for the released energy of ≈ 8 meV.
Since we keep the Yb+ ion in the trap, these processes cannot be charge-exchange reactions.
One possibility are quenching processes with a kinetic energy release for the ion of less than
the trap depth ≈ 250 meV. Since the ion is colliding in highly excited electronic states, such
as P1/2 and D3/2 with internal energies of ≈ 3 eV, this suggests a mostly radiative decay
into the ground state S1/2 .
A further possibility could be the dissociation of a potential molecular ion by the 369 nm
cooling light suggested in Section 5.1.1. We have also investigated if the reoccurrence events
are linked to the dissociation of a potential molecular ion in a secondary collision with neutral atoms [42]. To this end, we have performed this measurement also with extremely short
interaction times between neutral atoms and the ion (on the order of a few collision times)
and performed mass spectrometry on the dark ion after the collision confirming the absence
of a molecular ion. In light of a possible dissociation of molecules by laser cooling light
in our current mass measurement these results should be reconfirmed using the electronic
excitation mass spectrometry suggested in Section 5.1.1.
7
We consider this number a lower estimate since very short periods of reoccurence might go undetected.
98
Chapter 6
Spin Dynamics of a Single Ion inside
a Polarized Bath of Neutral Atoms
In addition to the changes in the internal electronic quantum states considered in the
previous Chapter, the interaction between ions and atoms also affects the spin degree of
freedom (see Section 2.3.2). The spin state of the ion impurity during immersion in the
ultracold spin-polarized atomic cloud has been in fact one of our longest standing questions
since the inception of our hybrid system.
The spin dynamics and decoherence of the ion are indeed of wider interest as the hybrid
system realizes the coveted concept of an isolated, controllable single qubit that couples to
an adjustable environment. The evolution of single spin-1/2 particle interacting with its
surroundings is a problem that has been intensely studied in theory [60, 61]. It represents
a model system for the comprehension of the quantum to classical transition [62] and is
important for the suppression of qubit decoherence in practical applications of quantum
mechanics, like atomic clocks and quantum information processors.
In this Chapter we investigate the properties of the all-atomic spin-bath system that is
formed by the immersion of the single Yb+ ions into the ultracold 87 Rb cloud in the hybrid
trap1 . In the first part we introduce the qubit preparation, manipulation and detection
methods, we have developed for controlling two different ion spin qubits. The description
of the ideally isolated and tunable two-level Zeeman qubit in the electronic ground state
of
174 Yb+
is followed by the discussion of the
171 Yb+
clock-transition hyperfine qubit. We
use these two complementary single spin qubits to study spin dynamics and decoherence in
the tunable environment of spin-polarized ultracold atoms and find spin coherence times
T1 and T2 on the time scale of the Langevin collision rate. In addition to the spin-exchange
interaction, we identified an unexpectedly strong spin-nonconserving coupling mechanism
as the main source of spin decoherence. We further detect a strong ion-heating mechanism
for atomic clouds prepared in the upper hyperfine manifold that we attribute to the release
of energy due to
1
87 Rb
hyperfine spin relaxation.
This Chapter is mainly based on the publication [47].
99
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
a
b
174
Yb+
171
Yb+
F=1
12.6
GHz
F=0
mF -1 0
mJ -1/2 1/2
c
87
1
Rb
F=2
6.8 GHz
F=1
mF -2
-1 0
1
2
Figure 6.1: a Illustration of the trapped single ion spin coupled to a spin-polarized neutral
atom cloud. b Level structure of the Yb+ electronic ground state in a weak magnetic field.
To implement the spin-1/2 system, we use either the Zeeman qubit |mJ = ±1/2i in the
isotope 174 Yb+ or the magnetic field insensitive hyperfine qubit |F = 0, mF = 0i and |1, 0i
in the isotope 171 Yb+ . c The cloud of neutral 87 Rb atoms is prepared in one of the four
atomic spin states |F = 2, mF = 2ia , |2, −2ia , |1, 1ia or |1, −1ia . A quantitative description
of all three spin systems is given in Appendix A.1.
6.1
Methods
The long trapping times, excellent coherence properties, and precise control over the internal and external quantum states make single trapped ions an excellent physical system for
the realization of qubits [22]. The internal level structure of Yb+ ions offers the possibility
to encode spin qubits in two levels of the Zeeman and hyperfine structure of the electronic
ground state2 for the isotopes 174 Yb+ and 171 Yb+ , respectively (see Figure 6.1). The experimental challenge part is to realize state preparation, state manipulation and state detection
for the two different spin qubits in a way that is compatible with the requirements of the
hybrid system. There are several factors that need to be considered for the implementation
of single ion qubit operations that are unique to the hybrid environment. They include the
relatively high temperatures of the ions (see Section 6.2.3) and the low repetition rate of
pulse sequences due to the long timescales of atom-ion interaction. During overlap with
the atoms, the high inelastic losses efficiencies of electronically excited states (Chapter 5)
furthermore prevents the use of protocols that require the occupation of metastable states.
In some of our experiments also the strength and direction of the magnetic offset field is
2
We do not consider optical qubits for the investigation of spin dynamics because of the high reactivity
of electronically excited states in the hybrid system. Moreover, due to its relatively short (D-state) or
extremely long (F-state) metastable state lifetimes, Yb+ ions is currently a rather inconvenient candidate
for the implementation of an optical qubit.
100
6.1. Methods
imposed by the magnetic trap of atoms.
6.1.1
Zeeman Qubit in
174
Y b+
The first qubit we investigate is the Zeeman qubit, formed by the |mJ = 1/2i ≡ | ↑i
and the |mJ = −1/2i ≡ | ↓i states of the 2 S1/2 electronic ground state of
absence of hyperfine structure in the nuclear spin-0 isotope
174 Yb+
174 Yb+ .
The
offers the advantage of
studying an ideally isolated two-level spin system. By changing the external magnetic field
the energy separation between the two qubit states can furthermore be tuned according
to ∆E = 2.8 MHz/G. The qubit’s susceptibility to the first order Zeeman effect, however,
also limits its coherence time due to dephasing by fluctuations in the magnetic field. For
our typical experimental magnetic field stabilities this dephasing time is only on the order
of hundreds of microseconds (see Section 6.1.2) and too short compared to the time scale
of atom-ion collisions. We therefore conduct our investigations of coherent spin evolutions
with the first order magnetic field insensitive hyperfine qubit in 171 Yb+ (see Section 6.1.2).
Preparation of Zeeman Qubit
In order to initialize the S1/2 ground state zeeman qubit we employ polarization selective
optical pumping on the S1/2 -P1/2 cycling transition. Near resonant light of left (right)
circular polarization, aligned along the direction of the magnetic field, drives the σ + (σ − )
transition. In this way, the ion is pumped within a few scattered photons into the upper
or lower Zeeman state, which is then a dark state for ideally polarized pump photons (see
Figure 6.2). In practice the fidelity of the initialization is limited by imperfections in the
174
Yb +
P 1/2
g J = 2/3
gJ = 2
S1/2
mJ
-1/2
1/2
Figure 6.2: State initialization of the Zeeman qubit by optical pumping. In the example
shown the lower state is prepared by σ − polarized light.
polarization preparation of the 369 nm Pol. Pump. beam. In order to characterize the
101
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
efficiency of the preparation in our setup we determine the fluorescence rate as a function
of the motorized quarter wave plate angle in the polarization pumping beam. We fit the
observed fluorescence rate in Figure 6.3a with the steady state solutions of the 4-level (S1/2
mJ = ±1/2, P1/2 mJ = ±1/2) rate equation model that considers the offset magnetic field
strength of 13.8 G, the frequency detuning of ≈ −Γ/2 = 2π · 10 MHz and the imperfect
retardance (0.2374) of the quarter waveplate at 369.5 nm. The free parameters are the
misalignment angle of the beam propagation with respect to the magnetic field, and a
small amount of beam displacement as the wave plate is rotated. The model allows us
to calculate the preparation fidelity (see Figure 6.3b) and lets us test its robustness for
different ion temperatures, magnetic fields, and laser detunings. The significantly higher
state preparation efficiency of 99.8% for the lower (mJ = −1/2) as compared to 97% for the
higher (mJ = +1/2) Zeeman state is caused by the additional frequency pumping effect of
the Doppler detuned polarization pump light. Whereas this is not an issue for our current
experiments, significant improvements to the state preparation fidelities can be made by
optimizing the frequency detuning, increasing the magnetic field during preparation and
using a better matched quarter-wave plate in the future.
Counts in 2[s]
15000
12000
9000
6000
3000
a
State Occupation
Probability
0
1.0
0.97
0.998
0.97
0.5
b
0.0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
Quarter Wave Plate Angle (rad/ )
Figure 6.3: Fidelity of the state initialization by optical pumping. a The photon scattering
rate exhibits minima, when the alignment of the quarter wave plate results in σ + ( 14 π and
5
− 3
4 π) or σ ( 4 π) polarized light. The horizontal line (100 ± 5) shows the background counts
without ion fluorescence. b The state populations of the qubit states after preparation,
determined from the fit of the rate equation model to a.
102
6.1. Methods
Detection of Zeeman Qubit
Internal state detection in ions is based on state selective fluorescence detection [148, 149].
For this purpose the ion is addressed with a laser light on a cycling transition such that it
will scatter many photons if it is in one internal state (| ↑i) and practically no photons if it
is in the other internal state (| ↓i). A small fraction of the scattered photons will reach a
photo-detector during the detection interval and will be recorded. Based on the distinctly
different photon detection behaviours for the two quantum states, the original state of an
ion can be inferred from a specific set of photon detection events by statistical arguments.
For qubits encoded in the Zeeman split electronic ground state 2 S1/2 , the energy difference
between the two levels is only on the order of the linewidth of the dipole-allowed cycling
transition and therefore both states will resonantly scatter photons when illuminated by
the fluorescence laser. In order to distinguish the two states the electron shelving technique [150] is employed3 , whereby one of the two states gets selectively transferred into a
metastable state not occupied during the ion’s fluorescence cycle before the fluorescence
detection is applied. In the case of the 2 S1/2 Zeeman qubit, the “dark” qubit state is
mapped into the 2 D5/2 manifold by a laser that coherently excites the narrow electronic
quadrupole transition [149, 152].
In order to reconstruct the original state of the ion for a given number of recorded photon
counts n during the detection interval tdet we need to known the probability distributions
p̃↑ (n) and p̃↓ (n) for detecting n photons in our detection setup if the ion starts in the qubit
state | ↑i and | ↓i, respectively. Formally, we can decompose the probability distributions
p̃↑ (n) = ηDark,↑ p̃Dark (n) + (1 − ηDark,↑ ) p̃Bright (n)
p̃↓ (n) = ηDark,↓ p̃Dark (n) + (1 − ηDark,↓ ) p̃Bright (n)
(6.1)
into the probabilities ηDark,↑ , ηDark,↓ that the ion gets mapped into the metastable (dark)
state from a particular initial state and the probability distributions p̃Dark (n) and p̃Bright (n)
in the (dark) mapped and (bright) original state. In order to infer the state of an ion as
| ↓i, if the number of detected photons n is smaller than a discriminator value nd or as | ↑i,
if n ≥ nd , with high fidelity, the probability distributions in Equation 6.1 are required to
have minimal overlap. This requires both a good distinguishability between the probability
distributions p̃Dark (n) and a high state selectivity of the shelving sequence p̃Bright (n) and a
ηDark,↑ ≈ 0 and ηDark,↑ ≈ 1.
In ion trap experiments dedicated to quantum optic experiments the shelving errors have
been shown to robustly reach the 10−3 or 10−4 level [152, 149] level. As will be explained
in the following, the experimental conditions in the hybrid system do not permit electron
3
A notable exception is the method based on electromagnetically induced transparency in [151].
103
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
shelving at this level. Whereas high fidelity state detection is essential for fault tolerant
quantum computation and quantum limited metrology, we are interested in investigating intrinsically probabilistic processes in the hybrid system. This purpose requires the
detection-error corrected spin state occupation probabilities, but does not demand perfect
single shot readout.
Experimental photon number distributions, determined from many identical repetitions
with identical spin preparation, will be described by
p̃Det (n) = p↑ · p̃↑ (n) + p↓ · p̃↓ (n)
= (p↑ · ηDark,↑ + p↓ · ηDark,↓ ) p̃Dark (n)
+ (p↑ · (1 − ηDark,↑ ) + p↓ · (1 − ηDark,↓ )) p̃Bright (n),
(6.2)
where p↑ and p↓ are the occupation probabilities of the ion in the states | ↑i and | ↓i before
the state measurement. Figure 6.4 shows the example of a histogram of the number of
0.07
0.06
Probability
0.05
0.04
0.03
0.02
0.01
0.00
0
20
40
60
80
100
120
Photons counted in detection interval
Figure 6.4: Normalized histogram of the detected photon numbers during 174 Yb+ fluorescence detection. The number of photons is counted during a 2 ms fluorescence interval after
electron shelving. The dashed line is a fit assuming the sum of two Poissonian distributions.
The fit shown as a solid line, considers Gaussian intensity noise of 8% standard deviation
in the detection laser beam4 . pDark = 508/1968.
photons that are detected during a 2 ms fluorescence interval following a mapping into
the 2 D5/2 state. The histogram is well-described by the sum of two clearly separable
Poissonian distributions for p̃Dark (n) and p̃Bright (n) with average photon numbers nDark =
2.5 and nBright = 88.2 that represent photon scattering in the dark and bright state (in
Equation 6.2). The detection events within p̃Dark (n) do not originate from the ion but are
caused by stray light photons from the detection laser and detector dark counts (≈ 20 s−1 ).
4
This measurement predates the implementation of an efficient PID for the UV laser light.
104
6.1. Methods
By introducing the discriminator level nd , we define bright and dark events and their
<n
∞
P
Pd
p̃Det (n) .
probabilities are pBright =
p̃Det (n) and pDark =
n=nd
n=0
From Equation 6.2 it follows that the probability of dark detection events is determined by
pDark = p↓ · ηDark,↓ + p↑ · ηDark,↑ ,
where the error made by the approximations
<n
Pd
n=0
negligible for our discriminator nd = 20.
p̃Dark (n) ≡ 1 and
(6.3)
<n
Pd
n=0
p̃Bright (n) ≡ 0 is
If we rewrite Equation 6.3 as
p↓ = 1 − p↑ =
pDark − ηDark,↑
ηDark,↓ − ηDark,↑
(6.4)
it becomes evident that the spin occupation probability can be reconstructed for any measurement from the experimentally determined fraction of dark fluorescence detection outcomes pDark and the independently calibrated shelving efficiencies ηDark,↑ , ηDark,↓ .
Electron Shelving by Optical Pumping
The state-selective electron shelving is performed in our experiment by driving the narrow
2S
1/2
- 2 D5/2 electric quadrupole transition [127] at 411 nm. The Zeeman level structures
of the S1/2 and D5/2 states are depicted in Figure 6.5. The strengths of the ten possible
transitions between these states are described by the Rabi frequencies [106]
ΩE2
0
eE
′ = hS, m|(ǫ · r)(k · r)|D, m i
2~


2
X
(2) eE
1/2 2 5/2 (q)
 c ǫ i n j ,

= hS1/2 |C |D5/2 i
ij
2~
m′
q=−2 −m q
(6.5)
(6.6)
where hS1/2 ||C (2) ||D5/2 i denotes the reduced matrix element, the 2x3 matrices are the
Wigner 3-j symbols and we have for now neglected the motional state of the ion in the trap5 .
(q)
The terms cij ǫi nj contain the dependence on the relative orientation of the polarization
vector ǫ of the light, the wave vector of the light n and the magnetic field vector defining
(q)
the quantization axis. The 2nd order tensors cij are explicitly stated in [106]. In our
experiments the geometry is chosen such that all three vectors are mutually orthogonal,
resulting in the suppression of transitions other than ∆m = ±2.
In contrast to previous experiments [152, 149], however, we do not use these transitions
for coherent population transfer but instead perform a scheme that is based on incoherent
5
This corresponds to a vanishing Lamb-Dicke parameter ηLD , see Equation 6.7.
105
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
174
Yb +
D5/2
g J = 6/5
S1/2
mJ
gJ = 2
-5/2
-3/2
-1/2
1/2
3/2
5/2
729 476 869 [MHz]
-1/2 to 3/2
1/2 to -3/2
-1/2 to - 5/2
1/2 to 5/2
Figure 6.5: Zeeman level structure of 2 S1/2 −2 D5/2 quadrupole transition for 174 Yb+ .
The linear polarization and the wave vector of the 411 nm light exciting the transition are
aligned mutually orthogonal with the direction of the magnetic field, so that only transitions
with ∆m ± 2 are excited. The relative transition strength and the Zeeman splitting of the
individual components of the line are shown below.
optical pumping.
I will first briefly explain, why the standard approach to electron shelving on the electric
quadrupole transition does not work well in our hybrid setup. A minor complication in this
context is the fact that the natural lifetime of the 2 D5/2 in Yb+ ions is only τD5/2 = 7.2 ms.
The branching ratio from the 2 D5/2 state is 0.17 for decay back into the 2 S1/2 ground state
and 0.83 for spontaneous decay to the long lived 2 F7/2 . For typical experimental conditions
the decay into the ground state during the fluorescence detection interval will therefore
lead to a detection error at the few percent level.
The main factor preventing coherent population transfer are the high temperatures of the
ions during the qubit measurement. As we will see in Section 6.2.3, neutral atom flip
heating can lead to ion equilibrium temperatures that are on the order of 200 mK.
The light-ion interaction on the narrow quadrupole transition takes place in the strong
binding regime (Γ ≪ ωtrap ), where the spontaneous decay rate is much smaller than the
trap frequency and the motional side-band spectrum can be resolved. The coupling of the
light to the ion, this time considering both its internal (Equation 6.6) and motional states,
leads to the Rabi frequencies [100]
†
iηLD (a+a )
Ωn+∆n,n = ΩE2
|ni
0 hn + ∆n|e
106
(6.7)
6.1. Methods
for transitions between the initial motional state n and the
q final state motional state n+∆n.
The Lamb-Dicke parameter ηLD = k · z0 =
2π
λ
· z0 =
2π
λ
~
2mωr
takes the value 0.21 for the
transition wavelength λ = 411 nm, the radial trap frequency ωr ≈ 2π · 150 kHz and the
mass of the Yb+ ion m. For ion temperatures T ≈ 200 mK ⇔ n =≈ 27700 the light-ion
2 ·(n+1) ≈
interaction is far outside the Lamb-Dicke regime hΨmotion |k 2 z 2 |Ψmotion i1/2 = ηLD
2500 ≫ 1. The great variations in the Rabi frequencies Ωn,n (Equation 6.7)6 of the many
thermally occupied initial states n, even for the carrier transition (∆n = 0) averages out
any coherent behaviour on timescales much shorter than the π-Rabi flip time.
In order to exceed the transfer efficiency limit of 0.5 for saturated incoherent excitation,
174
Yb +
D5/2
F7/2
S1/2
mJ
-7/2
-5/2
-3/2
-1/2
1/2
3/2
5/2
7/2
Figure 6.6: Electron shelving in the F-state by optical pumping. We frequency selectively
drive the 2 S1/2 (m = −1/2) - 2 D5/2 (m = −5/2) transition to optically pump the population
of the lower Zeeman level into the long-lived F-state.
we employ the optical pumping scheme depicted in Figure 6.5. For this purpose we apply
a magnetic offset field of 13.8 G and frequency selectively drive the 2 S1/2 (mJ = −1/2) 2D
5/2 (mJ
= −5/2) transition with a intermediate linewidth laser (see Figure 6.6). Spon-
taneous decay from the D5/2 state back into the S1/2 can only return the ion into the
mJ = −1/2 state, thereby avoiding the leaking of population into the other qubit state.
The timescale for optical pumping into the long lived 2 F5/2 is set by the radiative lifetime τD5/2 = 7.2 ms state and the branching ratio BF D5/2 = 0.83. The fidelity of the
shelving is generally limited by our finite optical pumping time and for high ion temperatures also off-resonant pumping on the Doppler broadened transitions S1/2 (mJ = 1/2) D5/2 (mJ = −3/2) (17 MHz detuned) and the S1/2 (mJ = 1/2) - D5/2 (mJ = −1/2)
(∆m 6= ±2 polarization suppressed and 10 MHz detuned) starts to contribute as an error
source.
Since the shelving efficiencies are temperature dependent, we need to determine the detection errors experimentally for different spin-bath configurations. Another important aspect
6
The explicit relations for evaluating the Rabi frequencies outside the Lamb-Dicke regime are given in [153].
107
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
of
174 Yb+
qubit detection is that it needs to be performed in the absence of neutral atoms.
The high inelastic loss efficiencies of electronically excited shelving states will otherwise
lead to loss of the ion during the qubit detection.
3
3
D[3/2]1/2
2.2095(11) GHz
F=0
F=1
b
D[3/2]1/2
2.2095(11) GHz
S
=1
1/2
F=1
a
F=0
F=1
2
2
P1/2
2.105 GHz
2
P1/2
2.105 GHz
F=1
F=0
2
S1
/2
F=0
=1
=0
2
D3/2
0.86(2) GHz
F=2
F=1
'
F=2
F=1
2
D3/2
0.86(2) GHz
&
!
%
$
#
"
!
2
S1/2
1
2
S1/2
F=1
F=1
12.643 GHz
12.643 GHz
F=0
F=0
Figure 6.7: Initialization and Fluorescence detection of the hyperfine qubit in 171 Yb+ . a
Initialization of the qubit by frequency selective optical pumping into the dark |F = 0, mF =
0i hyperfine ground state. b Qubit state detection. The large hyperfine splittings enables
the efficient distinction of the states in the upper hyperfine manifold (F=1, “bright”), from
the single lower hyperfine state (F=0, “dark”) by direct fluorescence detection on the UV
transition without the need of electron shelving. Adapted from [68].
6.1.2
Hyperfine Qubit in
171
Yb+
The second ion qubit we employ for our studies of spin dynamics and decoherence is the
hyperfine qubit in electronic ground state of
171 Yb+
ions. The nuclear spin-1/2 ion has
the simple hyperfine structure shown in Figure 6.7. In recent years
171 Yb+
ions have
found widespread use in ion trap quantum metrology and quantum information experiments [154, 155]. Several 1st order magnetically insensitive (mF = 0 → m′F = 0) tran-
sitions in
171 Yb+
are currently investigated for use in future ion-based optical frequency
standards [156, 157, 158]. In our experiment we encode the qubit in the hyperfine states
|F = 0, mF = 0i − |F = 1, mF = 0i of the 2 S1/2 electronic ground state that are connected
by the microwave “clock” [159] transition at 12.6 GHz (see Figure 6.1 and Appendix A.1).
Due to its low magnetic field sensitivity, the spin dephasing times of this qubit are significantly longer than the typical collisional timescales in our experiment.
108
6.1. Methods
Initialization
The initialization of the ion in the |0, 0i state is performed by the frequency selective
optical pumping scheme shown in Figure 6.1. With the microwave repump field turned off,
a short (250 µs) 369 nm Pol. Pump. laser pulse resonantly excites the ion on the S1/2 |F =
1i → P1/2 |F = 1i transition and pumps it into the dark |0, 0i state [68]. Microwave
radiation resonant with the 12.6 GHz hyperfine transitions is used for the coherent qubit
manipulation on the clock transition and for Rabi π-flips on the |0, 0i → |1, −1i and
|0, 0i → |1, 1i transitions. Before we describe the coherent manipulation of the qubit in
more detail, the qubit detection scheme needs to be introduced.
Detection
In contrast to the
174 Yb+
Zeeman qubit discussed in Section 6.1.1, the large hyperfine
splittings and transition selection rules (F = 0, mF = 0 6→ F = 0, mF = 0) in 171 Yb+ enable
state detection without the need for electron shelving into metastable electronic states [68].
For direct fluorescence detection the ion is probed on the 2 S1/2 |F = 1i →2 P1/2 |F = 0i UV
transition, which together with the 2 D3/2 |F = 1i →2 [3/2]1/2 |F = 0i 935 nm transition
forms a closed fluorecence cycle for ions initially in one of the three states |1, −1i,|1, 0i,
|1, 1i. The probability for detecting n photons when starting in one of the F = 1 states
is [148]:
p̃|F =1,mi (n) =
Here, P (a, x) ≡
α2 /η
e−(1+α2 /ηDet )λ0 λn0
+
P (n + 1, (1 + α2 /ηDet )λ0 )
n!
(1 + α2 /ηDet )n+1
1
(a−1)!
Rx
(6.8)
e−y y a−1 is the incomplete Gamma function and λ0 denotes the
0
average number of detected photons that would be recorded during the detection interval,
if the ion did not leak into the dark state. α2 is the probability per scattered photon to
leak from the |1, −1i, |1, 0i and |1, 1i into the |0, 0i state. Finally, ηDet is the detection
efficiency of the photon counting system. The derivation [148] of Equation 6.8 does not
consider the effect of background counts due to stray light and detector dark counts, which
play a negligible role for the bright state.
In contrast, when the ion starts in the lower hyperfine state |0, 0i, the probability for
109
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
recording n photons7
p̃|0,0i (n) = e
λ0
−α1 η
Det
α1
α1
e−λD λD
α1
ηDet
+
)λ0 )−P (n+1, (1−
)λD )
P (n+1, (1−
1
n!
(1 − ηαDet
ηDet
ηDet
)n+1
(6.9)
is dominated by background counts λD with an additional contribution that results from
the leakage probability α1 from the |0, 0i into the |1, −1i, |1, 0i, |1, 1i states during the
detection window.
Figure 6.8a and b show the experimentally determined photon number distributions for
ions prepared in the |0, 0i and the |1, 0i, respectively. Under the assumption that the
state preparation errors should be negligible compared to other detection errors due to the
very high efficiency of the optical pumping and the microwave Rabi π-pulse, we can fit
both histograms directly by Equation 6.9 and 6.8, respectively. In direct analogy to the
1.00
0.10
0.08
Probability
Probability
0.99
0.98
0.02
0.01
0.06
0.04
0.02
0.00
0.00
0
5
10
15
0
20
5
10
15
20
25
30
Photons counted in detection interval
Photons counted in detection interval
Figure 6.8: Normalized histograms of the detected photon numbers during 171 Yb+ fluorescence detection. a Distribution of recorded photon counts for ions prepared in the |0, 0i
state and b for ions initialized the |1, 0i state during a fluorescence detection interval of
1 ms. The red curves represent fits of the data with the functions in Equation 6.9 and 6.8 for
a and b, respectively. The fits use ηDet = 0.002 and yield α1 = 2.3· 10−6 and α2 = 3.8· 10−5
(see text for details).
discussion of fluorescence detection in
174 Yb+ ,
and such that the detection error Err =
<n
Pd
n=0
<n
Pd
0
we can introduce a discriminator value nd
<n
∞
P
Pd
p̃|F =1i (n)+
p̃|F =1i (n)+(1−
p̃|0,0i (n) =
n=nd
n=0
p̃|0,0i (n)) = ηDark,|F =1i +(1−ηDark,|0,0i ) is minimal. The optimal discriminator is nd = 2
and for the measurement in Figure 6.8 the detection errors are ηDark,|F =1i = 0.038(6) and
7
Here we have extended the treatment in [148] to include the contribution from background counts.
110
6.1. Methods
(1 − ηDark,|0,0i ) = 0.018(4). In most measurements the detection errors are even somewhat
larger as we do not individually optimize the orientation of the detection light polarization
vector [131] for the different atom trap configurations. The magnetic bias fields for our
various measurements with the hyperfine qubit are 5.64(1) G in the optical trap, 6.91(1) G in
the magnetic trap of the 87 Rb |2, 2ia and 7.56(1) G in the magnetic trap of the 87 Rb |1, −1ia
atoms and they are approximately aligned with the ion trap axis. As in the case of 174 Yb+ ,
the imperfections in the detection scheme limit the direct correspondence of the detection
of a bright or dark state, as determined from the statistics of photon events, and the
occupation of a specific quantum spin quantum state. However, the statistically correct
value of the spin probability
p|0,0i = 1 − p|F =1i =
pDark − ηDark,|F =1i
ηDark,|0,0i − ηDark,|F =1i
(6.10)
can be calculated from the measured probability pDark and the detection errors.
The direct fluorescence detection can only distinguish the |0, 0i state from the three hy-
perfine states in the F = 1 manifold. During collisions with neutral atoms, also the |1, −1i
and |1, 1i state can get populated. In order to detect a specific hyperfine state |1, mi we
perform a π Rabi flip on the |0, 0i - |1, mi microwave transition just before the fluorescence
detection.
For quantum hybrid experiments, the
171 Yb+
ion has the great advantage that at no point
during the state initialization, evolution or detection, significant amounts of time are spent
in the electronically excited states. This opens the possibility to study ion spin dynamics
in the atomic bath without the need to repeatedly merge and separate both species for
successive interaction and detection cycles. We have not taken any quantitative data on
the inelastic loss efficiencies of the various electronic states of the
values similar as for
detections as Ps =
174 Yb+ ,
Nc
Ps,c
≈ e
171 Yb+ ,
but assuming
we estimated the ion survival probability after Nc successive
−
n·BP D γLangevin
(TP ǫP +TD ǫD )·Nc
ηDet
≈ 0.99Nc , where we have used
γLangevin ≈ 200s−1 , an average number of detected photons n ≈ 15 and the results of
Chapter 5. This allows us to perform on the order of 100 qubit detections before the ion is
lost.
Coherent Manipulation
The coherent manipulation of the qubit is performed with 12.6 GHz microwave radiation
resonant with the hyperfine |0, 0i → |1, −1i clock transition. Due to the extremely small
Lamb-Dicke factor of the microwave transition, the homogeneous magnetic fields at the
position of the ion and the low magnetic field sensitivity of the clock transition, we do not
observe any ion temperature dependent effects for the coherent qubit manipulation.
111
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
1.0
0.8
pBright
0.6
0.4
0.2
0.0
0
20
40
60
80
100
Rabi pulse time ( s)
Figure 6.9: Resonant microwave Rabi oscillations on the qubit clock transitions |F =
0, mF = 0i - |1, 0i in the optical for the maximum microwave power available in our system.
Rabi frequency is Ω0 = 2π· 17.9 kHz, corresponding to a Rabi π flip time of 28.0 µs. Each
point is the result of more that 2000 measurements.
The coherent evolution of the two-level system during interaction
with
the microwave
√
2 +∆2 ·t
Ω
Ω20
0
radiation gives rise to Rabi oscillations p|0,0i (t) = Ω2 +∆2 sin2
in the qubit state
2
0
population, where ∆ is the detuning of the microwave frequency from the atomic transition
frequency, Ω0 is the resonant Rabi frequency and the system is initially prepared in the state
√
|0, 0i at t = 0. Figure 6.9 shows resonant (∆ = 0) Rabi oscillations (Ω0 ∼ P ) between
the two qubit states for the maximally available microwave power P in our experiment.
In order to probe the coherence time of our qubit in the absence of atomic collisions, we
perform Ramsey spectroscopy. The spin population after a Ramsey sequence is described
by [160]
!2
q
"
∆T
2
2
2
sin
p|1,0i (∆, T ) = p 2
Ω0 + ∆ · τ /2 cos
cos
2
Ω0 + ∆2
!#
p
Ω20 + ∆2 T
∆T
∆
sin
−p 2
sin
,
2
2
Ω0 + ∆2
Ω0
!
p
Ω20 + ∆2 T
2
(6.11)
where τ is the duration of the first and second microwave pulse, T the time in between
the pulses and the ion is initialized in the state |0, 0i before the first microwave pulse. For
short Rabi π/2-pulses (τ · Ω0 = π/2) and long waiting times T ≫ τ in between, the central
section of the Ramsey fringe pattern in Equation 6.11 is well-approximated by
1
p|1,0i (∆, T ) = (1 + V · cos(∆ · T )),
2
112
(6.12)
6.1. Methods
where we have introduced the visibility contrast V in order to account for the effects of
p(|0,0>)
decoherence. In order to determine the coherence properties of the ion under the exact
0.9
0.7
0.5
0.3
0.1
0.9
0.7
0.5
0.3
0.1
0.9
0.7
0.5
0.3
0.1
a
b
c
0.9
0.7
0.5
0.3
0.1
-10
-5
0
5
10
Frequency Detuning [Hz]
Figure 6.10: Ramsey fringes for an ion in the magnetic field of the optical trap without
neutral atoms. The decay of fringe visibility with increasing Ramsey time T a 83 ms,
b 133 ms, c 258 ms. In a single experimental cycle one point can be measured Nc ≈
5s/T times and every point shown in a-c is the average over several experimental cycles.
In d we have plotted the Ramsey fringe for 258 ms only including data from a single
experimental cycle for each point. The comparison with c shows that the long term drift of
the magnetic field between one experimental cycle and the next is the largest contribution
to the dephasing.
conditions relevant for the later interaction measurement with the atoms, we perform the
complete experimental cycle but without atoms8 . To this end, we prepare the ion in
the |0, 0i hyperfine ground state and apply a π/2-pulse on the 12.6 GHz clock transition,
√
creating a superposition state (|0, 0i + i|1, 0i)/ 2. After a varying waiting time we apply
a second π/2-pulse, followed by a readout of the |F = 1i spin state. Figure 6.10 shows
the Ramsey fringes of the ion measured for the optical trap configuration. We determine
the transverse coherence time from the decay of the visibility with waiting time T and
find T2 ≈ 130 ms. To identify the dominant source of qubit decoherence in our system
we have additionally performed spin echo measurements and obtained coherence times of
TRephase ≈ 700 ms. We can therefore exclude the phase stability of our microwave oscillator
and 369 nm leakage light as important decoherence contributions. The limiting factor for
the coherence time T2 in our experiment is dephasing due to the poor stability of the
8
The magneto-optical trap light is blocked for this purpose.
113
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
magnetic field. We can estimate the magnetic field stability by considering
1 dE
1 (gJ − gI )2 µ2B B
1
∼ ∆ν ≡
(B) · ∆B =
∆B
T2
~ dB
~
E HF S
∆B
~E HF S
⇒
≈ 10−4 ,
∼
2
2
2
B
T2 (gJ − gI ) µB B
where we have used Equation A.10 to evaluate the linear Zeeman shift
(6.13)
(6.14)
dE
dB (B)
at the mag-
netic bias field value of 5.64(1) G in the optical trap. The magnetic field stability is
consistent with the specifications of our high current power supplies. For the magnetic
trap configurations the coherence times of the qubit are about a factor of 3 shorter due
to the higher magnetic bias field values and the fact that several power supplies contribute to generate the magnetic field. The dephasing effect is particularly strong in the
1st order magnetically sensitive transitions |F = 0, mF = 0i − |F = 1, mF = −1i and
|F = 0, mF = 0i − |F = 1, mF = 1i as shown in Figure 6.11, where it is limiting the
fidelity of the Rabi π-flips that we use for the detection of the individual states in the
F = 1 manifold.
1.0
0.8
pBright
0.6
0.4
0.2
0.0
0
20
40
60
80
100
120
0
20
40
60
80
100
Rabi pulse time [
120
0
20
40
60
80
100
s]
Figure 6.11: Resonant microwave rabi flips on the transitions between the hyperfine ground
state |F = 0, mF = 0i and the |1, −1i a, |1, 0i b and |1, −1i c states. The measurements
were performed in the magnetic trap configuration for the |F = 1, mF ia atoms, hence the
different Rabi frequency for the clock transition compared to Figure 6.9. Decoherence is
starting to limit the fidelity of our Rabi π-flips on the 1st order magnetically sensitive
transitions.
114
120
6.2. Experiments
6.2
6.2.1
Experiments
Spin-Exchange and Spin-Relaxation in
174
Yb+
We start our investigation of ion impurity spin dynamics in the neutral cloud with measurements on the
174 Yb+
Zeeman qubit, which represents an ideal two-level system. As
a first step, we determine the steady state distribution of the ion spin for all four atomic
bath configurations |2, −2ia , |1, −1ia , |1, 1ia and |2, 2ia . For this purpose the neutral
87 Rb
atoms are loaded into the optical trap and prepared in one of the four polarization states
using the techniques described in Section 4.1.3. During this time a homogeneous magnetic
bias field pointing along the trap axis is ramped up to 13.8 G. The single
174 Yb+
is then
moved into the centre of the neutral cloud and one of two different sequence shown in
Figure 6.12a is applied.
The first sequence (case b) serves as a direct measurement of the detection errors. After an
ion-atom interaction time of 40 ms, that is chosen such that the ion acquires both its spin
and motional energy steady state, the neutral atom cloud is displaced from the interaction
position by 45 µm in the vertical direction. Once the atoms have been removed in this way
and the ion-atom interaction has ceased, the ion is prepared in the upper or lower spin state
by a 500 µs long polarization pumping pulse. Since only a few UV photons are scattered
during the optical pumping the temperature, that the ion has acquired during interactions
with the atoms, is not changed during state preparation. This is followed by the quantum
state detection sequence described in Section 6.1.1 that includes electron shelving with the
quadrupole laser at 411 nm and subsequent fluorescence scattering. After the detection
repump light at 638 nm is used to return the population in the shelved F-state back into
the ground state and only then is the neutral cloud ramped on top of the ion again. The
entire pulse sequence takes 430 ms and is repeated 8 times during a single experimental interaction cycle in order to increase the data acquisition rate. Significantly more repetitions
could be achieved in the future by ramping the ion from the interaction region9 instead of
the current method that moves the atoms in the dipole trap using the piezo mirrors. As
shown in Figure 6.12b, the measured spin distributions pBright differ between the neutral
atom bath states |1, −1ia , |1, 1ia and |2, −2ia , |2, −2ia . Given the known preparation efficiencies from Section 6.1.1 we can calculate the detection errors from Equation 6.4. For
measurements with
87 Rb
|F = 2ia , the detection errors are (1 − ηDark,↓ ) = 0.19 ± 0.01
and ηDark,↑ = 0.03 ± 0.01, and for measurements involving the |F = 1ia manifold we find
8
We repeat 500 µs long 411 nm pulses every 5 ms for a total optical pumping time of 60 ms. Numerical
simulations have since shown that a continuous excitation would lead to higher transfer fidelities.
9
Since this requires in practice photon collection and laser addressing of the ion at two different positions
along the ion trap we have chosen to move the atoms
115
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
(1 − ηDark,↓ ) = 0.10 ± 0.01 and ηDark,↑ = 0.00 + 0.01. The larger detection errors of the
Zeeman qubit, when the atoms are in the |F = 2ia state result from the increased ion
kinetic energy due to release of atomic hyperfine energy (see Section 6.2.3).
c
369nm Pol. Pum p.
b
c
b
c
369nm Det ect ion
411nm
638nm
0[ µm]
At om
Posit ion -45[ µm]
a
0.0
0.2
0.4
0.6
1.0
0.8
Tim e[ s]
1.0
0.6
0.6
p ↑,
8
pBright
0.8
0.4
0.2
0.0
b
c
|2, -2>a |1, -1>a |1, 1>a |2, 2>a
Rb state
|2, -2>a |1, -1>a |1, 1>a |2, 2>a
Rb state
0.5
0.4
d
|2, -2>a |1, -1>a |1, 1>a |2, 2>a
Rb state
Figure 6.12: Measurement of the equilibrium spin states of the 174 Yb+ Zeeman qubit for
all four atomic bath configurations |2, −2ia , |1, −1ia , |1, 1ia and |2, 2ia . a Timing diagram
of the two different sequences that are used to determine the detection errors b and the
equilibrium spin states c for each bath state. The sequence starts with the ion immersed
in the centre of the neutral cloud. After 40 ms of interaction, the atoms in the dipole trap
are ramped away from the position of the ion in a 80 ms ramp and the lower Zeeman state
is shelved during a 60 ms long optical pumping sequence10 . After 2 ms of fluorescence
detection, the ion is repumped from the F-state for 160 ms and the atoms are ramped
back on top of the ion for the next interaction cycle. For the measurement of the detection
efficiency b and the steady state spin state c the preparation of the qubits in the | ↑i (full
symbols) or | ↓i (open symbols) with the 250 µs UV pulse is performed after and before the
interaction with neutral atoms, respectively. b The detection errors of the ion spin state
are different for the |2, 2ia , |2, −2ia states compared to the |1, 1ia , |1, −1ia bath states due
to the unequal ion temperature in the two cases (Section 6.2.3). c and d show the ion spin
steady-states for all four atomic bath configurations |2, −2ia , |1, −1ia , |1, 1ia and |2, 2ia
before and after the correction for the detection efficiencies according to Equation 6.4.
The second sequence (case c) is used for the actual measurement for the ion spin steady
state. It only differs from the first sequence by the fact that the ion is prepared in the upper
or lower state before the atom-ion interaction. The fact that the measured spin distributions are identical for the two different initial state preparations demonstrates that indeed
the collisional steady state distribution is measured. The corrected steady state spin dis-
116
6.2. Experiments
tributions p↑,∞ , shown in Figure 6.12d are calculated from the measured spin distributions
pBright (Figure 6.12c) and the previously determined detection errors, with Equation 6.4.
In order to determine the timescale of the spin dynamics of the ion due to interactions with
the neutral atoms, we measure the coherence time T1 for two different bath configurations
|F = 1, mF = 1ia and |2, 2ia . To this aim the immersed ion is initialized in the upper or
lower qubit state by a short 50 µs polarization pump pulse. After a varying time t the ionatom interaction is abruptly ended by pushing the neutral atom cloud away with resonant
imaging light in less than 50 µs. During the interaction period, the ion undergoes binary
collisions with the atoms, and we normalize interaction times by the Langevin time constant
tL = 1/γL . The Langevin collision rates of the ion in the centre of the atomic clouds are ex|F,mF ia
perimentally determined (γLangevin = γℓ /ǫD
) from additional experimental runs that
measure the previously calibrated inelastic loss rate in the electronically excited D3/2 state
(see Section 5.2). The results for the two different bath configurations |F = 1, mF = 1ia
1.0
a
b
|1,1>a
|2,2>a
0.8
pBright
0.6
0.4
0.2
0.0
0
2
4
6
8 10
0
2
4
6
8
10
t/tL
t/tL
Figure 6.13: Spin-relaxation in the 174 Yb+ Zeeman qubit. The probability pBright of the
ion to occupy the bright | ↑i state after preparation in | ↑i (full symbols) or | ↓i (open
symbols) in a bath of a |1, 1ia or b |2, 2ia atoms. Error bars denote one standard deviation
uncertainty intervals resulting from approximately 3000 measurements per spin state.
and |2, 2ia are shown in Figure 6.13a and b, respectively. The spin relaxation behaviour
towards the steady state is well described by solutions of the general two-level rate equation
model
dp↑ = −p↑ γ↑ + p↓ γ↓ = −p↑ γ↑ + (1 − p↑ )γ↓
⇒ p↑ (t) = p↑,0 e−t/T1 + p↑,∞ (1 − e−t/T1 )
(6.15)
with the longitudinal coherence time T1 = 1/(γ↑ + γ↓ ) and the equilibrium spin state
p↑,∞ = γ↑ /(γ↑ + γ↓ ). The decay rates γ↑ = γ↑,SE + γ↑,SR and γ↓ = γ↓,SE + γ↓,SR can
117
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
be decomposed into a spin-exchange part depending on the atomic bath spin and a spinrelaxation part affected by spin-orbit coupling. For collision with the maximally polarized
bath state |2, 2ia at energies much larger than the Zeeman splittings the relation γ↓,SR =
γ↑,SR should hold true, and since the atoms are in a spin stretched state, the spin-exchange
rate form the upper state is γ↑,SE = 0. From our fitted steady-state population data
p↑,∞ = 0.609±0.015 and T1 = (2.50±0.39)tL for atoms in the |2, 2ia state, we can determine
the spin-exchange and spin-relaxation rates to be γ↓,SE = (2p↑,∞ −1)/T1 = (0.22±0.03)/T1
and γ↑,SR = γ↓,SR = (1 − p↑,∞ )/T1 = (0.39 ± 0.02)/T1 . An identical result within errors
follows from the measurement on the |2, −2ia state. The steady-state spin values for the
|1, −1ia and |1, 1ia bath states in Figure 6.12 can be qualitatively understood by looking
at the allowed spin-exchange processes in the two different spin environments. For this
purpose, we consider the collision of an ion in both the | ↑i and the | ↓i with atoms in
the |1, 1ia bath state and expand the atom’s hyperfine state in the electronic spin basis
(Equation 2.23):
√
3
3 3
1
| ↑i ⊗ | ↓ia | , ia + | ↑i ⊗ | ↑ia |3/2, 1/2ia
2
2 2
2
√
3
3 1
3 3
1
| ↓i ⊗ | ↑ia | , ia + | ↑i ⊗ | ↑ia | , ia
⇒−
2 |{z} |
2 2 } 2
2 2
{z
| ↑i ⊗ |1, 1ia = −
SE
√
✟
|2,2i
✟
3
3 3
1
| ↓i ⊗ | ↓ia | , ia + | ↓i ⊗ | ↑ia |3/2, 1/2ia
2
2 2
2
√
3
3 3
1
3 1
| ↓i ⊗ | ↓ia | , ia + | ↑i ⊗ | ↓ia | , ia
⇒−
2
2 2
2 |{z} |
2 2 }
{z
| ↓i ⊗ |1, 1ia = −
SE
(6.16)
√
3
✟2
✟
✟
|2,1i + 12 |1,1i
Spin-exchange collisions leading to a transition of the atomic hyperfine state from the
|F = 1ia to the |F = 2ia manifold are energetically strongly suppressed. We therefore
expect γ↑,SE < γ↓,SE for |1, 1ia and analogous γ↑,SE > γ↓,SE for |1, −1ia , which is in
accordance with the observed ordering of the steady state spin values in Figure 6.4.
All measurements so far have been performed at an energy splitting of the ion Zeeman
qubit of 37.5 MHz (at a magnetic field of 13.8 G). In order to investigate if the spin behaviour
varies with different Zeeman splittings, we measured the steady-state spin distribution of
the ion in the |1, 1ia environment for Zeeman splittings in the range from 0.39 MHz to
139 MHz, as shown in Figure 6.14. Following the interaction with the atoms the Zeeman
qubit splitting is ramped from the interaction value to a state detection value just above
or below the ion trap drive frequency for the values 139 MHz and 0.39 MHz, respectively.
This is necessary, because the qubit transition is driven by the ion trap radio frequency
118
6.2. Experiments
|1, 1>a
0.60
p↑,
8
0.55
0.50
0
50
100
150
Zeeman Splitting [MHz]
Figure 6.14: Steady-state spin population for different qubit Zeeman splittings. The measurement is taken in the neutral |1, 1ia environment. We do not observe any dependence
on Zeeman energy splittings within the statistical uncertainties (1σ) of our measurements.
field if it becomes resonant with the Zeeman splitting. We do not observe any dependence
of the steady-state spin population on the qubit energy splitting within the statistical
uncertainties (1σ) of our measurements.
The observed spin dynamics of the Yb+ ion in a spin polarized cloud of
87 Rb
atoms is
inconsistent with the picture of dominant spin-exchange and negligible spin-relaxation,
that has previously been reported in He+ + Cs collisions [161], and with alkali-metal atomatom collisions, where for various regimes the hierarchy γc ≈ γSE ≫ γSR is fulfilled (see
Section 2.3.2). In both cases the dominant spin-exchange interaction leads to a spin steady
state of near perfect polarization, if the environment is polarized.
The measured spin-exchange rate γSE = (0.09 ± 0.02) γLangevin of the ion is factor
of 2.5 smaller than the theoretically estimate γSE =
1
4
γLangevin in the degenerate state
approximation of Equation 2.25. However, it is the unexpectedly large spin-relaxation rate
that dominates the spin dynamics in the hybrid system and that is responsible for almost
complete mixture of ion spin states. The main reason for this strong spin-relaxation could
be a strong spin-orbit interaction, that is caused by the level crossings of both the incoming
singlet 1 Σ+ and triplet 3 Σ+ channels with the strongly spin-orbit coupled 3 Π channel at
short internuclear separation (see Figure 2.1) [75, 76]. Level crossings of the excited chargeexchange 3 Σ+ state with the incoming collision channel are not unique to the combination
Yb+ +Rb, but also occur, for example, in Ba+ +Rb [50] and Sr+ +Rb [49]. This could
indicate similarly strong spin-relaxation behaviour in several other hybrid-systems, that
are currently under investigation.
6.2.2
Hyperfine Spin-Relaxation in
171
Yb+
Next, we study the spin dynamics in the hyperfine spin qubit of 171 Yb+ . The measurements
are performed with |1, −1ia or |2, 2ia atoms confined in the magnetic trap, which offers the
119
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
possibility of a large atomic reservoir with a low central density. After the immersion of
the ion into the centre of the cloud, the measurements sequence is started by a 250 µs long
369 nm Freq. Pump. pulse (and a microwave π-pulse) to prepare the ion in the |0, 0i (|1, 0i)
qubit state. Following 25 ms of ion-atom interaction time the state of the qubit is detected
during a 1 ms fluorescence interval after which the preparation for the next interaction
interval can start again. In this way we achieve up to 100 measurements during a single
experimental cycle of the experiment which greatly improves the measurement statistics
compared to the
174 Yb+
Zeeman qubit. We vary the effective interaction time t/tL by
increasing and decreasing the number of atoms in the atomic cloud rather than changing
the actual time of the interaction interval. The atomic density at the location of the ion
in the centre of the neutral bath is determined from the atom number and temperature
measured in absorption imaging and the harmonic trap frequencies. Figure 6.15a shows
1.0
a
pBright
0.8
|1,-1>a
0.6
0.4
0.2
0.0
1.0
b
pBright
0.8
|2,2>a
0.6
0.4
0.2
0.0
0
2
4
t/tL
6
8
1.0
pDark
0.35
c
|1,-1>
0.30
0.35
d
0.8
|1,0>
0.30
0.25
0.6
0.25
0.20
0.4
0.20
0.2
0.15
0.15
0.10
0
2
4
6
8
10 12
0.0
10
0
2
4
6
t/tL
t/tL
8
10 12
0.10
e
0
|1,1>
2
4
6
8
10 12
t/tL
Figure 6.15: Hyperfine spin relaxation in 171 Yb+ . a The probability pBright after preparation in the |1, 0i (full symbols) or the |0, 0i (open symbols) state vs. the interaction
time for atoms in the |1, −1ia state. Error bars denote one standard deviation uncertainty
intervals resulting from a total of 5000 measurements. The fit values at t = 0 are limited
by detection errors. b Similar data for collisions with 87 Rb atoms in the |2, 2ia state and
a total of 19000 measurements. c-e Zeeman-resolved detection within the F = 1 manifold after preparation in |1, 0i (atoms in |2, 2ia ) with 1200 measurements per state. The
measurements are performed by applying resonant π pulses, exchanging the population of
the dark state |0, 0i with |1, −1i, |1, 0i or |1, 1i immediately before the detection of the
probability of the ion being in the dark state pDark .
120
6.2. Experiments
the results for ions in a neutral |1, −1ia bath. We observe an exponential decay of the
population in the upper hyperfine manifold towards the |0, 0i ground state with a time
constant T1 of a few Langevin collision times. This is in strong contrast to the case of the
neutral |2, 2ia bath (Figure 6.15b), where interactions result in a steady-state occupation
P
in the excited |F = 1i manifold of p1 =
m p|1,mi = 0.16(1). Here, we have already
corrected the probability pBright for the detection errors of (1 − ηDark,|0,0i ) = 0.02 ± 0.01 for
the hyperfine ground state |F = 0i and ηDark,|F =1i = 0.07 ± 0.03 for the |F = 1i hyperfine
states (see Equation 6.10).
6.2.3
Spin Relaxation Heating
We can assign to this non-zero steady-state occupation
of
the energetically excited states a
spin-temperature Ts by considering
p1
p0
HF S
= 3 exp − EkB Ts , where E HF S = h × 12.6 GHz is
the hyperfine splitting of the qubit and the factor 3 considers the multiplicity of the F = 1
manifold. The steady-state of the ion for the interaction with atoms in the |2, 2ia state
corresponds to a spin temperature of Ts ≈ 200 mK. Since spin selection rules do not play an
important role in the spin dynamics of the ion due to the dominance of spin relaxation, this
spin temperature will correspond to a similar kinetic energy of the ion. The micromotion
heating effect discussed in Section 3.2 can be neglected at the high temperatures under
consideration. Instead the kinetic energy Ekin of the ion results from the balance of elastic
and inelastic collision processes between the ion and the atoms. The average kinetic energy
intake of the ion per Langevin collision due to a hyperfine flip in the neutral atom is
δEheat = ǫEaHF S
ma
ma + mi
(6.17)
Here, we have considered the probability 0 ≤ ǫ ≤ 1, that an atomic hyperfine flip from
the |2, 2ia to the F=1 manifold occurs during a Langevin collision, which releases internal
hyperfine energy of the atom EaHF S = h × 6.8 GHz in the centre-of-mass frame of the two
particles. The ion receives the fraction
ma
ma +mi
as kinetic energy, where ma is the mass of
the atom and mi the mass of the ion. The average ion energy loss by elastic ion-atom
momentum transfer on the other hand is
δEcool = −
2ma mi
Ekin
(ma + mi )2
(6.18)
per Langevin collision, which follows directly from Equation 3.31 by neglecting the micromotion v = 0 and integrating over the isotropic scattering angle. As a result, the
121
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
steady-state average kinetic energy of the ion is given
hEkin i = ǫEaHF S
ma + mi
= ǫkB × 240 mK.
2mi
(6.19)
For ǫ near unity, which corresponds to a neutral atom spin relaxation event in every
Langevin collision, we hence obtain hEkin i ≈ kB Ts , which signals an equidistribution of
energy between kinetic and spin degrees of freedom. The spin dynamics of the neutral
cloud is not directly observable in the hybrid system, because regardless of the spin change
of the atom during a Langevin collision, the atom is always lost as the amount of elastically
transfered kinetic energy exceeds the trap depth.
If the atomic bath is prepared in its hyperfine ground state |F = 1ia , no energy-releasing
hyperfine changes can occur. The steady-state energies expected from the release of Zeeman energy in the |1, −1ia bath is smaller than the ion energies due to micromotion heat-
ing [161, 37] for our experimental parameters. However, in order to reach the s-wave
scattering regime for ion-atom combinations with a similarly large spin relaxation rate, the
atoms will need to be prepared in the absolute spin ground state.
To independently confirm the high kinetic energies of the ion, we perform a temperature
measurement using the
174 Yb+
ion. We have previously observed (Figure 6.12) that the
detection efficiency of the Zeeman qubit is different for the |1, −1ia and |1, 1ia compared
to the |2, −2ia and |2, 2ia and have attributed this to the temperature sensitivity of the
411 nm electron shelving pulse. In order to quantify the effect we perform spectroscopy on
the shelving transition for three different scenarios. The measurement sequence shown in
Figure 6.16 is identically repeated for ions immersed in a neutral bath of |1, 1ia atoms, a
bath of |2, 2ia atoms and no atoms at all. Whereas the transition line shape is dominated by
the laser linewidth for Doppler-cooled ions and ions immersed in the |1, 1ia bath, we observe
significant Doppler broadening for atoms after interactions with |2, 2ia atoms. Assuming a
thermal energy distribution
of the ion we estimate an ion temperature of T ≈ 120 mK from
q
the fit of σ(T ) =
kB T ν411
m
c
to the line shape. This measurement independently confirms
the high temperature the ion acquires during interactions with a |2, 2ia as a result of atomic
hyperfine energy release during inelastic collisions.
Spin dynamics within the Hyperfine manifold
Returning to the spin dynamics of the
171 Yb+
171 Yb+
ion, we can additionally study spin transfer
within the excited |F = 1i manifold. We prepare the ion in the |1, 0i qubit state and
detecting the individual excited states by exchanging the population of the dark state |0, 0i
with |1, −1i, |1, 0i or |1, −1i with a resonant microwave π-pulse as discussed in Section 6.1.2.
The data is shown in Figure 6.15c-e and compared to the four-level rate equation model
122
6.2. Experiments
369nm Pol. Pum p.
369nm Det ect ion
411nm
638nm
0[ µm]
At om
Posit ion -45[ µm]
a
0.0
0.2
0.4
0.6
1.0
0.8
Tim e[ s]
0.4
b
pDark
0.3
0.2
0.1
0.0
-30 -25 -20 15 -10
Δ
-5
0
5
10
15
20
25
30
ν411[MHz]
Figure 6.16: Ion energy estimate from the spectroscopy of the 411 nm electric quadrupole
transition. a The experimental sequence is similar to the one we use for state detection in
Figure 6.12a, but with a single 400 µs short pulse of 411 nm light instead of the extended
optical pumping phase. After an atom-ion interaction time of 40 ms during which the ion
reaches its steady state energy distribution for the respective environment, the quadrupole
transition is probe by the 411 nm light. b The transfer efficiency as a function of the
detuning of the 411 nm light from the resonance of the S1/2 (m = −1/2) − D5/2 (m =
−5/2) transition. The transition is significantly Doppler-broadened for ions after the |2, 2ia
bath (red markers and line) as compared to the |1, 1ia bath (black) or ions at Dopplertemperature (blue).
described in Appendix A.2. The spin dynamics within the hyperfine system of
171 Yb+
is
consistent with the strong spin non-conserving mechanism observed for the Zeeman qubit
in
174 Yb+ .
Interaction with the neutral atoms rapidly transfers population from the |1, 0i
state into both |1, ±1i states, where the occupation correspondingly rises. Eventually, the
|F = 1i states decay to their steady-state populations with roughly the same occupation
in all three excited states.
6.2.4
Spin Coherence of the
171
Yb+ Qubit
With the investigation of clock qubit we answer the question if the transverse coherence
time is limited by the longitudinal coherence time or if it is additionally affected by forward
scattering events. We have shown earlier that spin-exchange and spin-relaxing inelastic collisions happen at the timescale of the Langevin rate. The full quantum mechanical cross
123
Frequency shift [Hz]
p(|0,0>)
Visibility
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
t/tL
Frequency offset [Hz]
Figure 6.17: Spin coherence of the hyperfine clock transition |0, 0i ↔ |1, 0i in 171 Yb+ . a-d
Ramsey fringes recorded for increasing atomic density. e Ramsey contrast as a function
of t/tL displays an exponential decay with a time constant of T2 = (1.4 ± 0.2)tL . f No
frequency shift of the clock transition is observed within the experimental errors. Error
bars denote one standard deviation uncertainty intervals.
section of Equation 2.17, which also includes forward scattering events, however, is a factor
of 10 larger than the Langevin rate for a collision energy of E = kB · 100 mK.
A good understanding of the influence of atomic collisions on hyperfine clock transitions
is also of practical interest for buffer gas cooled ion microwave clocks [159], where clock
shifts have previously been studied for inert buffer gases at room temperature [162]. We
characterize the effect of atom-ion interaction on the transition by measuring its coherence
time and line shift using the Ramsey technique described. The time between the two π/2-
pulses is set to 27 ms and the interaction time in units of the Langevin timescale is again
varied by changing the density of the magnetically trapped cloud of neutral atoms in the
|2, 2ia state. The subplots a-d in Figure 6.17 show sample traces of Ramsey fringes with
increasing atom-ion interaction strength. Even without neutral atoms, the visibility of the
Ramsey fringes is limited to (55 ± 2)% owing to magnetic field fluctuations of the bias field.
We find that the spin coherence decays within T2 = (1.4 ± 0.2)tL (see Figure 6.17e). This
is on the time scale of the population relaxation of the |1, 0i state, which is driven by the
spin-exchange and spin-orbit interaction. The measurement hence identifies spin-relaxation
as the leading mechanism of spin decoherence in our system with a non-detectable contribution from elastic (forward) scattering.
Finally, the phase of the Ramsey fringe pattern as a function of increasing atomic density is used to detect interaction induced frequency shifts in the clock transition. The
124
6.2. Experiments
frequency shift ∆ν of the clock transition is below our measurement resolution of ∆ν ≤
4 × 10−11 EiHF S /h (see Figure 6.17e). It is worth noting that the feasibility to detect fre-
quency shifts is largely determined by the intrinsic properties of ion-atom collisions itself.
On the one hand, the expected frequency shift ∆ν ∼ na from binary collisions between
the ion and atoms is directly proportional to the atomic density. On the other hand the
frequency resolution δν of the Ramsey measurement is proportional to the inverse of spinrelaxation limited interrogation time, and therefore also proportional to the atomic density.
For spin-1/2 alcali metal buffer gases in hybrid setups, the decoherence effect due to spinrelaxation is strong compared to the collision induced frequency shift, in contrast to the
case of inert spin-0 buffer gases.
Table 6.1: Summary of the decoherence times (T1 and T2 ) and steady-state spin distributions of all spin-bath combinations. In addition to the statistical errors quoted, all times
have systematic uncertainties of 40% due to the determination of the absolute atomic densities at the location of the ion. All values are compensated for detection efficiencies.
Ion isotope
Atom spin state
T1 /tL
171 Yb+
|1, −1ia
|2, 2ia
|2, 2ia
|2, −2ia
|1, 1ia
|1, −1ia
1.73 ± 0.17
3.39 ± 0.16
2.50 ± 0.39
171 Yb+
174 Yb+
174 Yb+
174 Yb+
174 Yb+
p↑,∞
1.60 ± 0.24
125
0.609 ± 0.015
0.423 ± 0.026
0.563 ± 0.017
0.457 ± 0.021
P
p|1,mi,∞
0.000 + 0.005
0.163 ± 0.013
T2 /tL
1.4 ± 0.2
Chapter 6. Spin Dynamics of a Single Ion inside a Polarized Bath of Neutral Atoms
126
Chapter 7
Outlook
In this thesis I have presented our investigations into a recently realized [33] quantum hybrid system of single trapped Yb+ ions inside ultracold clouds of neutral Rb atoms. In order
to explore its numerous innovative features we have developed and implemented various
manipulation methods to exert detailed control on the hybrid system’s internal quantum
states.
By applying these techniques to the study of cold reactive collisions, we have demonstrated
the capabilities of our apparatus for the investigation of chemical processes at the most fundamental level [47]. Our results on Yb+ +Rb highlight the strong internal state dependence
of reactive collisions at low energies. We have used the control over the internal quantum
states to tune chemical reactions at the single particle level. From a practical perspective,
the measurements have helped us to characterize the role of electronic state changing collisions for our particular hybrid system combination of Yb+ + Rb. The understanding of
chemical processes in our system has been essential to suppress these events during our
measurements of the more subtle spin degree of freedom.
Our investigations of how a single ion spin impurity interacts with its environment of neutral atom spins represent a significant step forward for the study of spin-bath physics in
atomic systems and for the potential use of hybrid systems in quantum information processing [31]. We study spin dynamics and decoherence in well-understood setting of an
all-atomic spin-bath system at low temperatures [97]. For this purpose, we implement
techniques for the state preparation, controlled manipulation and spin detection of two
different ion qubit systems, the tunable ground state Zeemann qubit in
clock-transition qubit in
171 Yb+ .
174 Yb+
and the
We measure longitudinal and transverse decoherences on
the Langevin timescale and discover in addition to the expected spin-exchange interaction
an intriguingly strong spin relaxation mechanism behind spin decoherence. Neutral atomic
hyperfine spin relaxation is further identified as a significant source of motional heating for
certain states of our hybrid system.
The understanding and control of the internal degrees of freedom complements our earlier
insights into the role of elastic collisions [35, 37] and provides a comprehensive picture of
a atom-ion hybrid system in the semiclassical regime. An interesting open question that
remains in this context is the formation and stability of molecules in collisions. The mea-
127
Chapter 7. Outlook
surements suggested in Section 5.1.1 could confirm or exclude photo-induced dissociation
as the cause for the absence of molecules in our setup.
Future studies in the hybrid system have a full portfolio of experiments to choose from.
They will include the characterization of the atom-ion interaction potential by photoassociation spectroscopy [135] and the search for Feshbach resonances [83]. We have also
started to perform the first measurements in our setup that look at the interactions of ions
with Rb atoms that are confined in three dimensional optical lattices. All of these studies
will benefit greatly from technical improvements that are currently under way. The implementation of a new ion trap with increased ion trapping frequencies and reduced excess
micromotion heating, will extend the accessible collision energy range to lower limits in the
future.
The Route Towards S-Wave Ion-Atom Scattering
The logical next long-term step for quantum hybrid systems is the quest for the s-wave
regime of atom-ion scattering, which is needed to tap into the full potential of coherent
atom-ion interaction. Of paramount importance for such a future hybrid system will be
the choice of the atom-ion combination. The ever increasing list of laser cooled ion species
together with the considerable number of degenerate quantum gases species results in principle in an impressive plethora of possible combinations to choose from. The selection is
limited by the fact that approaching the the s-wave regime by the sympathetic cooling
of an ion in the ultracold neutral cloud demands a heavy ion and a light atom since the
micromotion limited final ion energies need to be smaller than the s-wave energy threshold
E ∗ ≫ Eem , Ef m . A large s-wave energy threshold (Equation 2.11) requires a small reduced
mass, whereas the steady state energies due to excess micromotion Eem (Equation 3.36)
and the fundamental limit Ef m discussed in [39] become small for large ion to atom mass
ratios. Although a large ion-atom mass ratio also reduces the relative amount of energy
transfered per collision and therefore the cooling efficiency for the ion, this is not expected
to limit the attainable temperature during immersion cooling. Assuming collision rates1
on the order of 1000 s−1 the cooling rate should sufficiently dominate the typical heating
rates of several quanta per second in large macroscopic ion traps [163].
In the current generation of atom-ion hybrid setups with degenerate quantum gases, Rb
has been chosen as the neutral component for experimental convenience. The inexpensive
laser system and the relatively small efforts needed to bring Rb to quantum degeneracy
have been the main arguments in favor. Its comparatively large mass, however, prevents
1
The collision rate scales as E 1/2 in the threshold regime and its magnitude depends on the yet unknown
s-wave scattering length.
128
Rb from being used as an ultracold buffer gas in experiments that seek to obtain the s-wave
limit. The most attractive alternative are Li atoms due to their low mass, the choice of
isotopes to create both quantum degenerate Fermi gases and small2 BECs and their large
ionization energy.
On the ion side, the heavy ions Yb+ and Ba+ are the prime contenders. Immersed into
Li both ions should be cooled below their respective s-wave thresholds (Equation 3.36) of
≈ 6400 nK and ≈ 6500 nK for a well compensated ion trap. The combination of Ba+ +Li
has the rare property that the entrance channel is the absolute electronic ground state and
the charge exchange channel Li+ +Ba is energetically closed. As a consequence, electronic
ground state Ba+ can never be lost from the ion trap due to two-body collisions with the
buffer gas. At high atomic densities, three-body collisions could nevertheless amount to
quite significant Ba+ loss and heating rates [77], respectively3
In the case of Yb+ + Rb the entrance channel is well isolated from both the charge exchange
ground state and its first electronic excitation by asymptotic energy gaps of 7000 cm−1 and
10000 cm−1 and should therefore exhibit both small charge exchange and spin relaxation
rates. The large number of available isotopes for Yb+ allows not only flexibility in the
implementation of the spin qubits, but also provides a range of different scattering length
options. Additional control of the scattering properties should be possible through the
predicted presence of Feshbach resonances at low magnetic fields.
In summary, a hybrid system of Li atoms and Yb+ or Ba+ ions with adequate micromotion
compensation should make the investigation of s-wave collisions and the immersion cooling
of ions into the ground state possible.
2
At small magnetic fields the atom number is limited to ∼ 1400 due to the attractive interaction of 7 Li
atoms.
3
The use of a fermi degenerate quantum gas could potentially suppress these three-body effects.
129
Chapter 7. Outlook
130
Acknowledgements
One of the great privileges of pursuing a PhD in experimental physics are the inspiring
people one happens to work alongside every day.
This could not be more true for my supervisor Michael Köhl. In an ever growing research
group, he manages to give his 24/7 support to every single one of his students and postdocs.
His enthusiasm and knowledge about physics and all things technical have been invaluable
during the many discussions and hours of hands-on help in the lab. With his friendly and
relaxed way (and his considerable table football skills) he has been with us during many
after-work fun activities.
Thank you!
The experiments in this thesis build on the excellent work of former PhD students Stefan
Palzer and Christoph Zipkes, who designed and constructed the apparatus that supports
the basic operation of the hybrid trap. Stefan was responsible for the laser system and the
BEC side of the setup, while Christoph took charge of the ion trap. Together with Michael
and the postdoc Carlo Sias, who joined in 2008, they pursued the project with the great
foresight, care and efficiency that was needed to make the idea of placing a single trapped
ion inside a ultracold atomic bath a success.
Christoph and Carlo continued to stay on the experiment with me until autumn 2012
and I fondly remember the passionate discussions the three of us have had about new
measurements, exciting results and the physics behind it all. To Christoph I am particularly grateful for passing on some of his extensive electronics and programming knowledge.
His skills remain evident in many of the devices that are now essential parts of our experimental setup. To Carlo, my long time lab colleague, I not only owe my vocabulary
of Italian/Sardinian curse words, but also a lot of the fun and many of the insights that
developed during the long lab hours. His undefeatable optimism and his practical approach
helped our experiment through the tough times and made the good times even better.
With Jonathan Silver and Leonardo Carcagni’ the Ion-Bec experiment got a new generation of talented PhD students that will carry the experiments with ions and atoms to
Bonn and beyond.
I am very grateful to the current and former members of the neighboring experiments, Bernd Fröhlich, Michael Feld, Enrico Vogt, Hendrik Meyer, Matthias Steiner, Marco
Koschorreck, Luke Miller, Daniel Pertot, Daniel Sigle, Eugenio Cocchi, Johanna Bohn, Tim
Ballance and Alexandra Behrle for the atmosphere of companionship in our group, where
advice and helpful discussions, the loan of an essential piece of equipment and some much
needed after-work entertainment are never far away.
131
Thanks are due also to the staff of the Cavendish laboratory, in particular to our secretary Pam Hadder for her kind assistance with administrative tasks, to Peter Norman for
his help with the realization of mechanical designs as head of the workshop and to Nigel
Palfrey for his supervisions during the work in the students mechanical workshop.
My deepest gratitude goes to my family and friends for their encouragement, trust, love
and friendship throughout the years.
I acknowledge financial support by the Cambridge European Trust and the Postgraduate
Scholarship of the Austrian Bundesministerium für Wissenschaft und Forschung.
Appendix A
A.1
Spins of
174
Yb+ ,
171
Yb+ ,
87
Rb in a Magnetic Field
The following discussion applies to all atomic systems with a single valence electron.
The total Hamiltonian of an atom in a magnetic field B
H = H0 + µ · B
(A.1)
contains the atomic Hamiltonian H0 and a magnetic term that describes the interaction
of the atoms magnetic moment µ with the magnetic field B. The magnetic moment of
an atom µ has contributions from the orbital angular momentum L and the spin angular
momentum S of the electron, as well as, the spin angular momentum of the nucleus I
µ = µB (gL L + gS S + gI I)/~,
(A.2)
with gi being the Landè factor of the respective angular momentum.
For experimentally relevant magnetic fields the spin orbit coupling dominates the effect
due to external fields. Therefore the total electric angular momentum J = L + S is well
conserved and the Zeeman interaction can be treated as a perturbation. The joint Landè
factor for the electron angular momentum is thus
gJ = gL
J(J + 1) − S(S + 1) + L(L + 1)
J(J + 1) + S(S + 1) − L(L + 1)
+ gS
2J(J + 1)
2J(J + 1)
(A.3)
with gL = 1 and gS ≈ 2.0023192. The Zeeman energy splitting is given by
E = µB gJ mJ B,
mJ = mL + mS .
(A.4)
and applies to the spin manifolds of all electronic states of the nuclear spin-0 isotopes of
Yb+ (see Table A.1)
In atoms with nuclear spin the hyperfine interaction, which results from the coupling of
electron angular momentum J to the total nuclear angular momentum I, leads to a further
energy splitting of states.
The total spin Hamiltonian is given by
H = hA I · J − µB (gJ J + gI I) · B,
133
(A.5)
Appendix A.
where gI = −9.95 · 10−4 and A is the magnetic dipole constant. Here, we have ignored
hyperfine orders beyond the magnetic dipole term.
In first approximation a similar argument as for the fine structure coupling can be made for
the much smaller hyperfine structure. If we consider the Zeeman interaction a perturbation
to the total atomic angular momentum F = J + I eigenstates, then the energy splitting is
E = µB gF mF B
(A.6)
with
gF = gJ
F (F + 1) + J(J + 1) − I(I + 1)
F (F + 1) − J(J + 1) + I(I + 1)
+ gI
.
2F (F + 1)
2F (F + 1)
(A.7)
For the case of J = 1/2 an analytic solution of the Hamiltonian of Equation A.5 exists for
all magnetic field strength, that is known as the Breit-Rabi formula
EF =I±1/2,mF
hE HF S
hE HF S
+ µB gI mF B ±
=−
2(2I + 1)
2
x≡
For mF = −(I + 1/2) the expression
interpreted as (1 − x).
µB B(gJ − gI )
hE HF S
q
1+
2mF x
I+1/2
s
2mF x
+ x2
I + 1/2
1
=A I+
.
2
1+
E HF S
(A.8)
(A.9)
+ x2 is an exact square and needs to be
The Expansion to lowest non-vanishing order of the Breit-Rabi formula around zero magnetic field yields a quadratic magnetic field dependence for the |F = 0, mF = 0i − |F =
1, mF = 0i hyperfine “clock” transition of
~ωclock (B) − ~ωclock (0) = EF =I+1/2,mF =0 − EF =I−1/2,mF =0 − E HF S =
134
(gJ − gI )2 µ2B 2
B
2E HF S
(A.10)
A.2. Rate Equation Model for Spin Dynamics in
Table A.1: Summary of spin properties of relevant electronic states in
171
Yb+
171 Yb+ , 174 Yb+ ,
87 Rb.
Atomic
Species
Nuclear
Spin
EHF S /h
171 Yb+
12.6428121185 GHz [164]
174 Yb+
1/2
0
87 Rb
3/2
6.8346826109 GHz [165]
A.2
Electronic
(& Hyperfine) State
2S
1/2 (F =
2S
1/2
2P
1/2
2D
3/2
2D
5/2
2F
5/2
2S
1/2 (F =
2S
1/2 (F =
Low Field
Landè factor
1)
1
2
2/3
4/5
6/5
8/7
-1/2
1/2
1)
2)
Rate Equation Model for Spin Dynamics in
171
Yb+
In order to model the data in Figure 6.15 we start with the general case of a four level rate
equation model
ṗ00 = −p00 · (γ00,10 + γ00,1−1 + γ00,11 ) + p10 · (γ10,00 ) + p11 · (γ11,00 ) + p1−1 · (γ1−1,00 ),
ṗ1−1 = −p1−1 · (γ1−1,10 + γ1−1,00 + γ1−1,11 ) + p10 · (γ10,1−1 ) + p11 · (γ11,1−1 ) + p00 · (γ00,1−1 )
ṗ10 = −p10 · (γ10,1−1 + γ10,00 + γ10,11 ) + p00 · (γ00,10 ) + p11 · (γ11,10 ) + p1−1 · (γ1−1,10 )
ṗ11 = −p11 (γ11,10 + γ11,00 + γ11,1−1 ) + p10 · (γ10,11 ) + p1−1 · (γ1−1,11 ) + p00 · (γ00,11 ),
(A.11)
where γi,j is the rate for a transition from state i to state j and the initial conditions in
our experiment are p10 (0) = 1 , p00 (0) = p1−1 (0) = p11 (0) = 0. Given data quality and the
size of the parameter space, an unconstraint fit will not provide much insight, we therefore
choose to make some assumptions based on our earlier results on spin-dynamics to minimize
the size of the parameter space.
We take into account that the hyperfine rates are dominated by energy considerations and
that spin selection rules play a minor role
γ1−1,00 = γ1−1,00 = γ11,00 ≡ γH
γ00,1−1 = γ00,10 = γ00,11 ≡ γH e
135
(A.12)
HF S
− Ek T
B
,
(A.13)
Appendix A.
where we use T = 200 mK.
For the transitions with the F = 1 manifold we consider the result γ↑,SE ≈ 0.5 γSR that
we have obtained for
174 Yb+
interacting with |2, 2ia
γ10,1−1 = γ11,10 ≡ γSR
(A.14)
γ1−1,10 = γ10,11 ≡ (1 + 0.5) γSR e
Zeeman
−E k T
B
≈ 1.5 γSR
γ1−1,11 = γ11,1−1 ≈ 0,
(A.15)
(A.16)
where we assume that spin relaxation events that change the ion spin by ∆m = ±2 are
strongly suppressed. The hyperfine relaxation rate for the neutral bath |2, 2i has been
obtained in Section 6.2.2; γH = 1/T1 = (0.29 ± 0.02) γLangevin and we make the assumption
that the spin relaxation rate for 174 Yb+ also holds true for the 171 Yb+ isotope γSR = γSR =
(0.1 ± 0.02) γLangevin . As a last step, we take into account the state detection efficiencies
and π-flip efficiencies on the magnetically sensitive hyperfine transitions ηDark,|0,0i ≈ 0.98,
ηDark,|1i ≈ 0.07 and ηπ,1−1 ≈ ηπ,11 ≈ 0.95.
The final model without free parameters describes the evolution of the spin populations in
Figure 6.15 well.
A.3
Axial Trap Frequencies of Small Ion Crystals
Number
of Ions
2
3
4
Impurity
Position
Identical
Ions
171 Yb+
Edge
Centre
Edge
Centre
1.000
1.000
1.000
1.000
1.000
1.00432
1.00287
1.00288
1.00215
1.00216
87 Rb+
Impurity
(174 Yb87 Rb)+
(174 Yb87 Rb87 Rb)+
1.12603
1.07914
1.08694
1.05821
1.06373
0.885861
0.917615
0.921832
0.935862
0.939734
0.796225
0.843392
0.854744
0.87329
0.884995
Table A.2: Centre-of-mass (COM) mode frequencies of small linear 174 Yb+ ion crystals
with relevant impurity ions. The frequencies are given as multiples of the COM frequency
of the pure ion crystal.
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