Atom-by-atom control and readout for studying
spin-motional dynamics and entanglementin neutral atom arraysbyBrian James Lester
B.S., California Institute of Technology, 2010

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2016This thesis entitled:
Atom-by-atom control and readout for studying spin-motional dynamics and entanglementin neutral atom arrays
written by Brian James Lester
has been approved for the Department of Physics

Prof. Cindy A. Regal

Prof. Ana Maria Rey

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Lester, Brian James (Ph.D., Physics)
Atom-by-atom control and readout for studying spin-motional dynamics and entanglementin neutral atom arrays
Thesis directed by Prof. Cindy A. Regal

In this thesis, I describe early experiments with a new platform that harnesses control overthe full quantum state of individual 87 Rb atoms to study out-of-equilibrium states of a few atoms
placed in tailored optical potentials. We employ an enhanced loading technique that fills each well in
90% of loading attempts, image the configuration of the atoms, and then perform Raman sideband
cooling that results in a 90% three-dimensional ground state fraction. Then, after initializing thespin of each atom, we can reconfigure the traps to initialize dynamics in a final optical potential ofinterest. For example, we can form a double well potential and observe the quantum interferenceof two atoms tunneling between the wells. Additionally, we have demonstrated the ability tocoherently transfer atoms between wells and, by preparing two atoms in opposite spin states, haveobserved spin-exchange oscillations that periodically entangle the two atoms. I will also discussplans and ongoing work to combine these capabilities with new techniques to gain more informationfrom systems containing more atoms. In such systems, we wish to study how the spin-motional
coupling of independently prepared atoms, in the presence of interactions, will lead to complex
dynamics, such as in the Kondo lattice model.

Dedication

This work is dedicated to my parents.

v

Acknowledgements

Most of all, I want to say a big thank you to Cindy for all of the support and guidance that
she has provided over the past six years. I have really enjoyed my time working with and learningfrom her, and I am very grateful for the freedom that she allowed during the design, testing, andconstruction of the apparatus.
Of course, the work presented here would not have been possible without the help of everyone
I have had the pleasure of working in the lab with! First and foremost, the knowledge and work ethicthat Adam brought to the lab was incredible. His ability to bring into focus the most important
aspects of the system and get things working at their best was invaluable, but most of all it was apleasure to work with him for my first five years (and, of course to go grab a Duchess and somewings after a good day in lab). I am also happy to know that the experiment is being transferred
to capable hands and I have really enjoyed working with (and, at times, relearning parts of theexperiment with) Yiheng, Mark, and Randall. I am grateful for all of the discussions and questions
you have brought, and look forward to continuing them! And, athough it was shorter, I also enjoyed
the time I got to spend working Collin, Junling, Niclas, Kathrin, Ludovic, and Mika over the years.
I also owe a huge thanks to Ana Maria Rey and all the members of her group for their
help to fully model and understand both current and future experiments discussed in this thesis.
In addition to Ana Maria, I want to specifically thank Michael Wall and Leonid Isaev for thenumerous discussions we have had over the past couple of years.
I also want to thank everyone who I shared coffee breaks, beer tastings, and physics discussions
with throughout the years! In particular, I want to mention Nir Kampel, who was always up forvi
discussing potential problems with electronics or general physics, or just going to the Goat to chatand enjoy the company. And, aside from beverages, thanks also to the rest of the “other’ side” ofthe lab - Bob, Max, Thom, Tom, Ran, and Ben for fostering an great atmosphere in which I wasable to live and learn during my time.
And to everyone else I got to share my time with in Boulder, helping me to get my mind outof lab when it was necessary. I don’t think it would be possible to name everyone here, but I will
always appreciate the brewery touring, food exploring, and beer brewing adventures - and that is
not to mention all of the ski trips - throughout the years with Andrew, Alejandra, Craig, Monica,
Daniel, Steven, and Rabin. Thank you for everything!

Contents

Chapter
1 Introduction
1.1

1Quantum gas assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Background on control of quantum gases. . . . . . . . . . . . . . . . . . . .

1

1.1.2

Using single-atom control to build complex many-body states . . . . . . . . .

2

1.2

Work performed as part of this thesis. . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82 Experimental Apparatus for Control of Single Atoms
2.1

2.2

10

Trapping and Control of Cold Atomic Gases . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1

Lasers for cooling and trapping 87 Rb . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2

Loading directly from a MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.3

Optical pumping and spin-polarization of 87 Rb . . . . . . . . . . . . . . . . . 14

2.1.4

Single-atom spin addressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Optical System for Single-Atom Trapping . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1

High-numerical aperture lenses . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2

Sources of light for optical trapping . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.3

Optical Rail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Quantum Gas Assembly:
A Platform to Study Tailored Many-Body Systems

25

viii
3.1

3.2

3.3

3.4

Enhanced Loading of Single Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1

Background on light-assisted collisions . . . . . . . . . . . . . . . . . . . . . . 27

3.1.2

Experimental implementation of enhanced loading . . . . . . . . . . . . . . . 29

3.1.3

Collisional beam parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.4

Limitations to the enhanced loading . . . . . . . . . . . . . . . . . . . . . . . 35

Manipulation of Optical Tweezers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1

Overall intensity stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2

Acousto-optic control of optical traps . . . . . . . . . . . . . . . . . . . . . . 42

3.2.3

Generation of radio-frequency drive

3.2.4

Alternative options for arbitrary trapping potentials . . . . . . . . . . . . . . 46

. . . . . . . . . . . . . . . . . . . . . . . 46

Removing entropy from the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.1

Building uniform atom arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.2

Distilling motional state preparation . . . . . . . . . . . . . . . . . . . . . . . 50

Improving readout of the final state of the system . . . . . . . . . . . . . . . . . . . . 50
3.4.1

Spin-resolved imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.2

Raman-cooling imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Preparing and Interfering Indistinguishable Atoms
4.1

4.2

Raman sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1.1

Non-degenerate pulsed Raman sideband cooling . . . . . . . . . . . . . . . . . 61

4.1.2

Efficient cooling in optical tweezers . . . . . . . . . . . . . . . . . . . . . . . . 64

4.1.3

Raman sideband spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Quantum interference of indistinguishable particles . . . . . . . . . . . . . . . . . . . 67
4.2.1

Measuring indistinguishability via tunneling in a double well potential . . . . 67

4.2.2

Using indistinguishability to improve state preparation . . . . . . . . . . . . . 70

5 Entangling Independently-Prepared Neutral Atoms
5.1

60

72

Dynamical entanglement via spin-exchange . . . . . . . . . . . . . . . . . . . . . . . 72

ix
5.2

Spin-entangled states in spin-dependent traps . . . . . . . . . . . . . . . . . . . . . . 78
5.2.1

Adiabatic preparation of entangled states . . . . . . . . . . . . . . . . . . . . 80

6 Assembling the Kondo Lattice Model with Optical Tweezers
6.1

The Kondo Lattice Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1.1

6.2

Limiting cases of the Kondo lattice model . . . . . . . . . . . . . . . . . . . . 87

The Kondo-Hubbard Model for Atomic Systems
6.2.1

6.3

84

. . . . . . . . . . . . . . . . . . . . 88

Studying the Kondo-Hubbard model with bosons . . . . . . . . . . . . . . . . 89

Implementation with optical tweezers . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3.1

Calculating the expected spin dynamics . . . . . . . . . . . . . . . . . . . . . 92

6.3.2

Weakly interacting regime (Veg  Je ) . . . . . . . . . . . . . . . . . . . . . . 94

6.3.3

Interaction dominating regime (Veg  Je ) . . . . . . . . . . . . . . . . . . . . 98

6.3.4

Summary of ideal theoretical investigations . . . . . . . . . . . . . . . . . . . 102

Bibliography

107

Appendix
A Alignment and Characterization of Optical System

116

A.1 External lens testing and alignment procedure . . . . . . . . . . . . . . . . . . . . . . 118
A.1.1 Measurement of the point spread function (PSF) via pinhole imaging . . . . . 118
A.1.2 Wavefront measurement testing . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.2 Characterization of the trapping potential . . . . . . . . . . . . . . . . . . . . . . . . 120
B Derivation of eigenstates in minimal Kondo model

123

Figures

Figure1.1

Operational procedure for quantum gas assembly . . . . . . . . . . . . . . . . . . . .

3

2.1

Level diagram for 87 Rb and laser addressing schematic . . . . . . . . . . . . . . . . . 12

2.2

Measurement of the polarization purity of the optical pumping light . . . . . . . . . 15

2.3

Microwave manipulations of the spin state of a single atom . . . . . . . . . . . . . . 16

2.4

Microwave spectrum taken with light shift beam used to address a single well . . . . 17

2.5

Schematic of the optical setup for controlling single atoms . . . . . . . . . . . . . . . 18

2.6

Diagram of optical rail for beam preparation . . . . . . . . . . . . . . . . . . . . . . 21

2.7

Schematic of Optical Relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1

Operational procedure for quantum gas assembly . . . . . . . . . . . . . . . . . . . . 26

3.2

Schematic for light-assisted collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3

Enhanced loading summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4

Loading efficiency versus length of loading sequence

3.5

Loading efficiency versus relevant enhanced loading parameters . . . . . . . . . . . . 32

3.6

Depump beam power “resonance”

3.7

Effects of trap proximity on loading efficiency . . . . . . . . . . . . . . . . . . . . . . 39

3.8

Calibration of the optical tweezer position as a function of drive frequency . . . . . . 44

3.9

Distillation of imperfect motional state preparation . . . . . . . . . . . . . . . . . . . 51

. . . . . . . . . . . . . . . . . . 31

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.10 Histogram of photon counts during both PGC imaging and Raman cooling imaging . 55

xi
3.11 Imaging during Raman cooling, varying drive strength . . . . . . . . . . . . . . . . . 58
4.1

Schematic of experimental setup used to perform Raman cooling . . . . . . . . . . . 62

4.2

Diagram of the Raman cooling procedure . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3

Raman sideband spectra using gaussian pulsed spectroscopy . . . . . . . . . . . . . . 67

4.4

Schematic of tunneling procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5

Observing two-particle interference while tuning distinguishability . . . . . . . . . . 69

4.6

Improving state preparation by optimizing indistinguishability

5.1

Generation of spin-entanglement via local spin-exchange . . . . . . . . . . . . . . . . 73

5.2

Experimental procedure for the observation of spin exchange and initial data . . . . 75

5.3

Verification of spin entanglement after separation of atoms . . . . . . . . . . . . . . . 77

5.4

Spin exchange in traps that have less spin-dependence . . . . . . . . . . . . . . . . . 80

5.5

Effect of spin-dependent trapping potentials on spin-exchange contrast . . . . . . . . 81

5.6

Eigenstate energy levels in the presence of differential spin-dependent shifts . . . . . 82

6.1

Schematic of the Kondo-Hubbard lattice model . . . . . . . . . . . . . . . . . . . . . 86

6.2

Schematic of the Kondo-Hubbard lattice model . . . . . . . . . . . . . . . . . . . . . 90

6.3

Implementation of the microscopic Kondo lattice model with optical tweezers . . . . 93

6.4

Single spin basis dynamics in the weakly interacting regime . . . . . . . . . . . . . . 95

6.5

Joint spin-state basis dynamics in the weakly interacting regime

6.6

Alternative measurements of the spin dynamics in weakly interacting regime

6.7

Motion of atom in excited band with strong interactions . . . . . . . . . . . . . . . . 100

6.8

Spin state dynamics with strong interactions . . . . . . . . . . . . . . . . . . . . . . 101

6.9

Dynamics of the Kondo lattice model for realistic trap parameters . . . . . . . . . . 104. . . . . . . . . . . . 71

. . . . . . . . . . . 97
. . . . 99

A.1 Schematic of testing setup to measure the point-spread-function . . . . . . . . . . . . 118
A.2 Schematic of the setup to directly measure wavefront aberrations . . . . . . . . . . . 121
A.3 Optimizing trap frequencies via alignment of objective lens . . . . . . . . . . . . . . 122

xii
A.4 Measurement of trap depth via light shift of cycling transition . . . . . . . . . . . . . 122

Chapter 1
Introduction

1.1

Quantum gas assembly

1.1.1

Background on control of quantum gases

Modern atomic physics experiments have alternately been gaining impressive experimental
control in order to understand properties of complex systems, then using this understanding tocreate additional experimental tools for a new generation of experiments. The study of quantum
gases builds on earlier work developing the techniques to cool and trap clouds of atoms at temperatures well below 1 mK, eventually leading to the observation of Bose-Einstein condensates andthe production of a degenerate Fermi gas [1, 2, 3, 4]. Shortly thereafter, these clouds of atoms
started to be used to study the complex interactions of ground-state atoms and molecules and,eventually, to the development of techniques to control the strength of interactions, such as thes-wave scattering length via Fano-Feschbach resonances and dipole-dipole interactions via Rydberg
excitation [5, 6, 7, 8, 9, 10, 11].
The ability to control interactions and study the effects on bulk properties of clouds of many
atoms has been extremely successful and now it has become routine to use this control in thepresence of an external potential to study different Hamiltonians with the density and interaction
parameters as tunable parameters [12, 13, 14, 15, 16, 17]. Even more recently, there has been apush toward observing these systems with resolution of individual atoms (or sites of a lattice),
leading to the development of quantum gas microscopy, where ultracold atoms in an optical lattice2
can be observed over a large range of interaction parameters, allowing the direct measurement ofcorrelations in the lattice [18, 19, 20, 21, 22, 23].
In parallel, there has been a significant amount of progress in the trapping, manipulation, andinteraction of well-separated atoms, with numerous demonstrations of single and two-qubit control
both with trapped atomic ions and neutral atoms via Rydberg blockade [24, 25, 26, 27]. And more
recently, these experiments have been building up to larger systems of strongly interacting atoms,
while maintaining the single-atom control developed for use in quantum information processing [28,29, 30, 31]. In particular, these systems have the benefit of being able to initialize each atom
independently and, because of the strong interactions, to keep the atoms well-separated throughout
the experiment and read out each atom at the end of the experimental sequence. These systems also
have the flexibility to generate nearly arbitrary trapping geometries that can be tailored for a variety
of experiments and enable the transport of atoms during the course of a single experiment [32, 33,
34, 35].

1.1.2

Using single-atom control to build complex many-body states

The work described in this thesis is a part of the development of a platform that combines
the advantages of experiments that maintain individual-atom control in an array of atoms, butwhich also can be used to build a degenerate quantum gas from the ground-up in a custom opticalpotential. In particular, we wish to initially set the full quantum state of each atom (including
both the spin and motional degrees of freedom) and allow the atoms to delocalize and interact ina projected optical potential. This builds on the capabilities afforded in quantum gas microscopeexperiments, but turns the process around, such that we could, in principle, build a small degenerate
Bose (or Fermi) gas from individually prepared atoms. Schematic representation of this style ofexperiment, highlighting many of the capabilities it would afford and have been developed in mythesis work, is shown in Figure 1.1.
To start with, atoms are initially loaded into an array of traps using an enhanced loading
technique that results in a 90% probability of loading an atom in each trap [36, 37, 38]. After

3
Load atoms from MOT:

Enhanced loading, 90% filling:

Rearrange atoms:

Image array,determine configuration:

Isolate filled portion of array:

Laser cool atoms to indistinguishability,
prepare individual spins:

-

(

)

-

)
Detuning ((kHz)Filter imperfect preparation,
place into final potential:

-

(

)

Figure 1.1: A vision of the initialization of experiments via quantum gas assembly, depicting
several of the important capabilities that the optical tweezer platform and the work in this thesis
will afford.

4
loading, the array is imaged to measure the occupation of each of the trap, gaining knowledge of
how many atoms are available in that run of the experiment, as well as their location in the arrayof traps. This knowledge can either be used during processing of the data set, when it is useful toperforming the same experimental procedure in multiple initial atom configurations, or in real-time
to perform different experiments based on the initial state [39, 31]. For the information to be used inreal-time, it is likely that rearranging the atoms, either to fill holes in an array, or even loading into
a completely separate, customized potential, will allow for faster data taking in configurations thatare of most relevance [40, 41, 42]. After constructing a desired pattern of atoms for initializing aparticular experiment, the entire array is spin-polarized and Raman sideband cooling is performed,
bringing each of the atoms to the three-dimensional ground state of the trapping potential [43, 44].
With all of the atoms in the motional ground state, the spin-state of each atom can then be
initialized by taking advantage of the ability to address individual sites with a beam to shift the
spin flip resonance [45, 39, 46].
After the entire state-preparation protocol has been performed, the atoms are ready to beplaced into the final potential in which the dynamics of interest will be observed. However, thisoverlooks the possibility to manipulate the system during this procedure in such a way that imperfections in the state preparation can be detected. For example, the process shown in Figure 1.1
will “distill” the motional state preparation by performing an adiabatic passage (discussed morein Section 5.1) in such a way that atoms in an excited motional state will transfer to a second
well. This extra well is then separated from the remainder of the system and not included in the
final potential; then, by not detecting an atom in the secondary well, it can be inferred that theatom was in motional ground state. The traps containing these fully initialized atoms are thendynamically repositioned to place the atoms into a final potential where the many-body quantum
state can evolve. After a period of dynamics, the traps are again rearranged to separate the atoms
for readout. This procedure does not need to be identical to the procedure that assembled thestate, but the goal is to separate the atoms and to measure both the spin and population dynamics
over time.

5
In our experiment, we have already demonstrated the ability to control all of the internaland external degrees of freedom of a single atom and are actively studying the dynamics of systemsof few interacting particles from a pure initial state (in the single-particle basis). In particular, wehave shown that we can create indistinguishable atoms via laser cooling, and observe two-particle
quantum interference in the dynamics of two atoms tunneling in a double well potential [43, 39].
We also have demonstrated the ability to perform enhanced loading, significantly increasing theprobability of uniformly filling arrays of traps. Additionally, the experiments we have performed
with two atoms already demonstrate our ability to dynamically reconfigure atoms and transfer
between trapping potentials while maintaining the purity of the motional state [38]. We have also
used our ability to selectively flip individual spins and coherently transfer atoms between the wells
to observe dynamical spin-exchange and verified that the entanglement generated between the two
spins is maintained upon separation of the atoms [47]. The ability to image a partially filled arrayand use optical tweezers to generate a smaller filled array has very recently been demonstrated in
one dimension in Ref. [41] and for two dimensions in Ref. [42], and is a technique that we will be
implementing as we scale to larger arrays of atoms.
We are currently studying what happens when we combine these capabilities to observe spinmotional coupling in systems of a few interacting atoms. For example, with just three atoms in twobands of a double well potential, we will be able to construct minimum instances of the bosonic
Kondo-Hubbard model and observe differences in dynamics as we tune the relative strength of theon-site interactions to the tunneling-rate between the wells. Specifically, in the strongly-interacting
regime, we expect to be able to see that, for certain initial states, the tunneling rate will be reduced
by a factor of two; this is a direct analog to the observation of mass-enhancement in heavy-fermion
materials. On the other hand, for weak interactions, we expect to be able to measure the effect of anonlocal spin-spin coupling between the separated ground-state atoms mediated by local exchangeinteractions.

6

1.2

Work performed as part of this thesis
During my time in Cindy’s lab, I have seen this experiment grow from a single table witha recently baked vacuum cell and the laser systems for producing a magneto-optical trap (MOT),
to a fully functional optical tweezer system. For my first five years here, I had the pleasure ofworking closely with Adam Kaufman to help get this system off of the ground and through thatexperience learned a lot of atomic physics. When I arrived, he was getting the MOT set up andvery shortly thereafter began characterizing the details of the setup and implementing the basic
systems required for manipulating the internal states of clouds of atoms.
In parallel, my focus was on designing, testing, and setting up significant portions of the highnumerical aperture (NA) optical system that both generates our optical tweezer potentials andcollects fluorescence scattered from the atom to form an image on our camera. The most important requirement for this system is that the optical traps needed to provide sufficient confinement
to enable the ground-state cooling protocol developed in our lab, which means maintaining neardiffraction limited performance, even with the significant challenges presented by mounting the lensoutside of the science chamber. We started out this process with a first-generation lens and imaging
system that achieved a NA of 0.25, that allowed us to gain experience trapping single atoms andmanipulating their state in tight optical traps, not to mention the debugging and characterization of
the sources of aberrations in the experimental setup, which provided useful information before moving up to a higher NA system. Once this had been set up and was trapping atoms, I got my hands
on our (first) custom made objective lens and was able to start the process of both characterizing
and optimizing its performance in an external testing setup. Eventually, it was this lens that wasable to provide us with the confinement sufficient for three-dimensional Raman sideband cooling,followed by coherent tunneling between two optical tweezer potentials and quantum interference
between two atoms. We have more recently used the system to observe dynamical spin-exchange
between the atoms and verify that entanglement is maintained after separation of the atoms.
After helping with the data taking and characterization of the trapping potentials for the7
two-particle quantum interference experiment, I was able to take the lead on characterizing ourimaging system more thoroughly and demonstrating the ability to image single atoms via lightscattered during continuous Raman sideband cooling. While this technique has proven to be lessuseful for our system (because sub-doppler cooling is so efficient for 87 Rb), it is a technique that isbeing used in several quantum gas microscopes for lighter fermionic atoms, and provided us with a
lot of information about the performance of our imaging system [48, 49, 50, 51].
I also worked with a visiting student Niclas Luick to implement an enhanced loading techniquethat had not previously been used with traps that are as tightly confining as in our system, as wellas to demonstrate its utility in loading small arrays of traps. Importantly, we were able to optimize
the technique such that we loaded a single atom in ∼90% of attempts, while the length of eachloading cycle was not significantly increased compared to the standard techniques that achieve
closer to 60% loading [38]. This technique is going to play an integral role in improving the datacollection efficiency of future experiments where the system size is larger than a few atoms, andis likely to be implemented in similar systems that load an array of traps directly from a thermal
cloud of atoms.
Aside from the published work listed in Section 1.3, I have worked to address some of thelimitations present in our initial experiments. Specifically, improving the characterization and control of the light used for our Raman cooling procedure, switching to using temporally incoherent
light for generating optical tweezers, increasing the dynamic range of our trap intensity control toaccess trap depths that vary over 4 orders of magnitude, and implementing the use of completely
arbitrary waveform generator to allow for arbitrary generation of square arrays of optical tweezers(within the limitations of the optical system). We have additionally tested numerous methods forcorrecting aberrations of our objective lens in-situ, which should lead to better tunneling performance, but none have been to satisfaction. Therefore, we are planning to switch to a new objective
lens whose aberrations have been measured and corrected externally.
In the meantime, I have also developed an understanding of the physics and experimental
protocols for several upcoming experiments that take advantage of the control that our experiment8
has either already demonstrated or is expected to have in the near future. For example, studies ofthe microscopic implementation of the Kondo experiment, which will be discussed in much moredetail in Chapter 6, is an example of an experiment that will take advantage of the improved controlof our optical tweezer potentials, as well as the ability to initialize both the spin and motional statesof atoms and then coherently transfer the atoms between. I will also discuss our plans to use control
over the state-dependence of a double well potential to adiabatically prepare pure singlet or triplet
spin states, which could then be separated to use as a source of maximally entangled atoms thatis less sensitive to experimental fluctuations than the dynamical entanglement demonstrated inRef. [47].
Most of the other unpublished ideas are related to gaining more information from our systemthan in our current experiments. Specifically, we have recently implemented the ability to performspin-selective imaging of our atoms; this can then be used for post-selecting our data to account for
single-atom loss, whereas our current imaging protocol registers loss as a spin-up atom. Additionally, we can use extra tweezers to read out more information about the final state of the system,
similar to the method depicted in Figure 1.1, but which can isolate imperfections in other stages ofpreparation or in the readout of the spin state at the end of an experiment. The ability to use thesetechniques to gain more information about the state of a system, both before and after a period of
evolution is one of the most important features of this platform for quantum gas assembly.

1.3

List of publications
For reference, I am including a list of the published work that I have been involved in during

the coarse of my thesis work.
(1) A. M. Kaufman, B. J. Lester, and C. A. Regal, ”Cooling a Single Atom in an OpticalTweezer to Its Quantum Ground State”, Phys. Rev. X 4, 041014 (2012).
(2) A. M. Kaufman, B. J. Lester, C. M. Reynolds, M. L. Wall, M. Foss-Feig,
K. R. A. Hazzard, A. M. Rey, and C. A. Regal, ”Two-particle quantum interference in9tunnel-coupled optical tweezers”, Science 345, 306 (2014).
(3) B. J. Lester, A. M. Kaufman, and C. A. Regal, ”Raman cooling imaging: Detecting
single atoms near their ground-state of motion”, Phys. Rev. A 90, 011804(R) (2014).
(4) B. J. Lester, N. Luick, A. M. Kaufman, C. M. Reynolds, and C. A. Regal, ”Rapid
Production of Uniformly Filled Arrays of Neutral Atoms”, Phys. Rev. Lett. 115, 073003(2015).
(5) A. M. Kaufman, B. J. Lester, M. Foss-Feig, M. L. Wall, A. M. Rey, and C. A. Regal,”Entangling two transportable neutral atoms via local spin exchange”, Nature 527, 208(2015).

Chapter 2
Experimental Apparatus for Control of Single Atoms

In this chapter, I will give an overview of our implementation of some of the atomic physics
techniques that are made use of in this thesis to initially trap, cool, spin-polarize, and detect atoms
in the experiments performed. For all of the experiments in this thesis, we use 87 Rb, which has
been the workhorse of the cold atomic gas community and along with 133 Cs has been the subject ofmost neutral atom Raman cooling experiments in free space [52, 53, 54, 55, 56]. By choosing thisspecies, we were able to focus our attention on the challenges related to cooling and manipulating
ground-state atoms in optical tweezers.
After covering this background information, I will go into more detail on the operational
principles of optical tweezers, including the optical setup used to generate our traps, and how they
are used specifically for single atom trapping experiments. Additionally, I will detail the methods
we have used to carefully align the optics for generating these potentials, as well as methods forcharacterizing the trapping potentials that are generated using the atoms. Further informationon the details of the experimental setup, such as the vacuum system and how particular laser
frequencies are generated and referenced to each other, can be found in Adam Kaufman’s thesis [57].

2.1

Trapping and Control of Cold Atomic Gases
The experimental setup is built around a single vacuum chamber that is made from 9 fusedsilica windows attached to a machined quartz frame (assembled by Precision Glassblowing). Thereis a pair of larger diameter (2.3”) windows (that provide a large optical access to the cell), seven110.9” diameter windows along the octagonal edge of the cell (which act as ports for additional beams
to address the atoms), and the final port being occupied by a glass-to-metal transition connecting
to both a rubidium source and a 20 liter-per-second ion pump. The experiment is operated byloading atoms from the background vapor into optical traps and then manipulating them all in thesame chamber.
Surrounding this glass chamber are four pairs of coils to control the magnetic field environment for atoms located in the center of the vacuum chamber. There are three pairs of coils in
the Helmholtz configuration, which provide control over the scalar magnetic field, in addition to asingle pair in the anti-Helmholtz configuration, which allows us to apply a tunable magnetic field
gradient through the trapping volume. Importantly, we can tune the scalar magnetic field from 0
to approximately 10 Gauss directed along an arbitrary axis. We can also turn up the gradient coils
to achieve field gradients suitable for loading a magneto-optical trap of 87 Rb.
Aside from the vacuum chamber and magnetic field control, the experimental setup consistsof an arrangement of near-infrared (near-IR) laser beams that are used to trap, cool, image, andspin-polarize the atoms. The majority of these lasers are frequency stabilized via an offset-lock froma reference laser (which is locked directly to a rubidium vapor cell), which operates by measuring
the beatnote between the laser being locked and the reference, and stabilizes that beatnote to aparticular frequency. Details about the lockpoints for the majority of these lasers are given in Adam
Kaufman’s thesis, Ref. [57]. Additionally, there is a sawed-off waveguide that delivers microwave
radiation to the atoms, allowing us to perform precise rotations of the spin state.

2.1.1

Lasers for cooling and trapping 87 Rb

To provide a reservoir of cold atoms, we use standard techniques for generating a magnetooptical trap (MOT) of 87 Rb atoms. Cooling and confinement of atoms in a MOT is supplied by the
combination of a damping force (provided by doppler cooling) and a restoring force (provided by
the magnetic field gradient). A diagram showing the alignment of the MOT beam, as referenced
to the vacuum chamber and other beam paths is shown in Figure 2.5.

12
Rb Level Diagram and Laser Addressing Scheme (not to scale):87

52P3/2MOT

267 MHz

Probe

F=3

F=2
D2 Line (780 nm)

157 MHz
F=1815 MHz
F=1

Raman (2-4)

795

D1 Line (795 nm)

F=2
52P1/2

RP

OP
Raman 1

RepumpF=0

Depump

72 MHzF=252S1/2

6.83 GHz
F=1

Figure 2.1: Schematic showing all of the relevant electronic levels in 87 Rb and their splittings.
Additionally, there are lines indicating the levels being addressed (or detuned from) for each beam
used to gain control of the spin and motional degrees of freedom in our experiment: The two red
beams indicate the cycling (or MOT, solid) and Repump (dashed) beams used during loading andimaging, the orange and blue lines represent the Depump and 795 beams, respectively, used forenhanced loading (see Section 3.1), the two purple lines are the Raman beams (the dashed beamis RB1 and the solid line represents RB2, 3 and 4), the two green lines are the optical pumping (orOP, solid) and sigma repump (or RP, dashed) beams, and the yellow beam is the probe beam. The
dashed purple, green and yellow beams are all on the same beam path, as shown in Figure 2.5.

13
To generate a three dimensional (3D) trapping potential, we use three beam paths that havea projection on all three dimensions, each path reflected back on itself in the σ + -σ − polarization
configuration. Each beam path is the trap light, which is red-detuned from the cycling transition(|F, mF i = |2, 2i → |F 0 , m0F i = |3, 3i), which is the source of the doppler cooling and generates anoptical molasses in the absence of a magnetic field gradient. The magnetic field gradient ensuresthat scattering rate is minimized at the center of the trap and as the atom moves toward any
given beam, that beam is shifted closer into resonance, thus providing a restoring force to the
middle of the trap. In addition to the trap light, we need use a repumper beam on two of thebeam paths (the paths in the x̂ − ŷ plane), which is used to keep atoms in the manifold of thecycling transition (the F = 2 ground-state manifold). This beam is tuned to be resonant withthe |F, mF i = |1, 1i → |F 0 , m0F i = |2, 2i transition, ensuring any atom off-resonantly scatteringout of the F = 2 manifold (which occurs roughly once every 105 scattering events for our typicaldetunings) is rapidly pumped back back out of the F=1 manifold.

2.1.2

Loading directly from a MOT

To load single atoms, we simply overlap the optical tweezer potential with the trapping regionof our MOT, which provides a reservoir of atoms around the optical tweezer. Then, we perform subdoppler cooling on the cloud of atoms by further detuning the MOT beams, turning off the magnetic
field gradient, and zeroing the total magnetic field. Sub-doppler cooling techniques, such as thepolarization-gradient cooling (PGC) that we use, can cool atoms to very near the single-photon
recoil limit, which will be very near the ground-state of our deep optical tweezer potentials [58]. In
our case, the cooling brings the atoms to roughly 20 µK, such that any atoms loaded into the trap
ends up near the ground-state of the potential.
Additionally, the same light used for cooling will induce light-assisted collisions between pairsof atoms [59], which can result in the loss of both atoms when the kinetic energy gained during
the collision is larger than the depth of the trap. Due to the small trapping volume of the opticaltweezer and the relatively low-density of our MOT, we are in the collisional blockade regime, where14
only a single 87 Rb atom can remain in the trap and pairs of atoms are lost [60]. This effect is thebasis of parity imaging in many neutral atom experiments [61, 18, 19].

2.1.3

Optical pumping and spin-polarization of 87 Rb

An additional beam path is aligned along the typical quantization axis (−x̂), as shown in
Figure 2.5, for which particular care is taken to achieve a very high σ + polarization purity. This
beam path contains other frequencies of light, but in order to spin-polarize atoms that are trappedin the optical tweezers, we use two frequencies in particular: The “optical pumping” (OP) beamis tuned to the F = 2 → F 0 = 2 transition, while the “sigma repump” (RP) beam is tuned to the
F = 1 → F 0 = 2 transition. With this pair of beams on, the atom will continue scattering photons
until reaching the |F, mF i = |2, 2i electronic state, which is dark to both of these beams.
This optical pumping procedure is limited by the polarization purity of the beam path, which
is very important for the Raman cooling procedure discussed in Section 4.1.2. The fact that the
optical pumping procedure is limited by the polarization purity also provides us with a tool forboth characterizing and optimizing the polarization purity of this beam. Specifically, we measure
the scattering rate of the OP beam with a tilted quantization axis (and thus completely mixed
polarization), as shown in Figure 2.2(a), to characterize the scattering rate by measuring how longit takes for the atoms to leave the F = 2 manifold. Separately, we can measure how quickly
atoms that start in the “dark state” (|2, 2i) are scattered out of the F = 2 manifold with thequantization in its optimal alignment, which happens because of the residual polarization impurityof the beam (and is thus a measurement of the “darkness” of the dark state). The ratio of these
two rates is an estimate of the polarization purity of this beam path [62, 57]. This measurement isshown in Figure 2.2(b), which suggests that the ratio of σ − and π polarization to σ + is better than
1:1250. Note that by fixing the pulse length, this same measurement can be used to optimize theparameters relevant for improving the polarization purity, such as the angle of the quantization axis,
by minimizing the survival probability (which would correspond to a lower scattering rate) [57].

15

=

b)

=

a)

(µ )

(

)

Figure 2.2: These two plots are taken using the same experimental procedure, but in (a) thequantization axis is intentionally misaligned, effectively mixing the polarization of the light at theatoms, while the in (b) the quantization axis is optimally aligned to the atoms. In both plots, theatom is initially prepared in the |F, mF i = |2, 2i state before shutting off all other beams, adjusting
the quantization axis, and then applying only the OP light for a variable pulse length. Afterward,
the probability for the atom to have been depumped to the F = 1 manifold is measured and theexponential decay time fitted. The ratio of these decay times (20 µs to 25 ms) is an indication of
the polarization purity of the beam with the quantization axis well-aligned.

2.1.4

Single-atom spin addressing

After spin-polarizing the atoms in optical tweezers, we need to be able rotate them to arbitrary states to initialize different experiments. Fundamentally, we need the ability to rotate
individual spins in an array of traps, so we take a two-part approach to this: Performing arbitrary
global microwave rotations using a microwave drive (coming through the sawed-off waveguide)and localized vector light shifts, which can move the microwave resonance for individual traps by
hundreds of kHz [45].
The signal for the global microwave drive is provided by a Rhode and Schwarz microwave
generator (SMF100A), which passes through a voltage-controlled variable attenuator, and finallyis amplified to around 10 W of microwave power before being coupled into a sawed-off guide thatlaunches the microwave signal toward the atoms in free-space. With this setup, we can drive
Rabi oscillations between the |2, 2i and |1, 1i hyperfine states at a rate of 2π × 40 kHz, as shownin Figure 2.3. The variable attenuator also allows us to shape microwave pulses, in particular toachieve Gaussian-profile pulses, which results in a narrower spectral feature at the expense of longer

=

=

16

(

)

(

)

Figure 2.3: This figure demonstrates our ability to drive coherent rotations between the |2, 2i
and |1, 1i spin states of 87 Rb by applying a microwave drive at 6.840996 GHz (shifted from thebare transition because of the quantization axis). By varying the length of a square pulse of thisresonant microwave drive, we observe coherent oscillations between the two spin states of the atom
at a rate of approximately 2π × 40 kHz.

π pulses. The Gaussian pulse shaping is key to the single-atom addressing because relatively small
shifts of the microwave resonance (100 kHz or less) can result in a very small probability to flip
the spin of the atom whose resonance is shifted. This is in contrast to the spectral features of asquare-pulse, which extend much farther from the resonance and leading to a higher probability offlipping the spin of the atom whose resonance is shifted.
The localized vector light shifts are produced by adding a λ = 785.1 nm beam (which wecall the “light shift’, or LS, beam), which is deflected backwards on the imaging path toward theobjective lens, as shown in Figure 2.5. This wavelength is chosen so that with pure σ − polarization,the |2, 2i state remains unshifted while the |1, 1i level is shifted.1 This is useful because the energyof the initial spin state is not perturbed, which minimizes the effects of the LS beam (which
could provide an additional trapping or anti-trapping potential, if that energy was perturbed). An
example spectrum with the light shift beam shifting the resonance of the atom in one of the wells
is shown in Figure 2.4, demonstrating that a single spin can be flipped with high efficiency.
In order to align the LS beam to the atom, we can directly measure, and maximize, the vector
1

Note that this is a change from earlier experiments that used a much closer detuned beam (approximately 50
GHz) and much lower powers to shift the resonance. Importantly, by going to a larger detuning, we can perform
longer microwave rotation pulses (including gaussian shaped pulses) to improve the fidelity of our site-selective spinrotations.

17

-

-

-

(

)

Figure 2.4: This is a microwave spectrum taken the light shift beam focused onto one of the wells(green points). The beam induces a large vector light shift of the atom in the right well (red arrow)that allows spin of the unshifted atom (orange) to be flipped with high efficiency and with nosignificant effect to the spin of the atom shifted out of resonance. The peak of the orange spectrum
is less than unity due to a combination of factors including shot-to-shot magnetic field fluctuations
and single-atom loss to background, but spin flip probabilities of about 95% are achievable [57].
Additionally, the broadening of the shifted atom resonance is likely due to a small intensity noiseon this laser and is not a fundamental limitation.

light shift induced between the |2, 2i and |1, 1i spin states. To measure this, we perform a microwave
spin-echo pulse sequence ( π2 − π − π2 ) and turn the LS beam on during the hold between the firsttwo pulses, but not between the second and third pulses. Thus, this is a differential measurementthat cancels out any systematic effects due to stray magnetic fields or vector shift induced by thetrapping light itself. By varying the alignment of the LS beam during this measurement (using a
piezo mirror to adjust the angle), we can maximize the light shift and thus maximize the intensity
at the atom we align the beam to. Once aligned, we can tune the power of the LS beam to reliably
shift the microwave resonance for the selected atom far enough that the spin is not rotated by theGaussian pulse, but keeping the power as small as possible to minimize the scattering rate fromthat beam (and thus minimizing heating or spin-depolarization).

2.2

Optical System for Single-Atom Trapping
Optical tweezers are a type of optical dipole trap that provide very tight (typically sub-wavelength) confinement of atoms. Of particular interest are traps containing only a single atom,

18

{

RB1,
OP + RP,
Probe

λ/4

f=10 cm

f=10 cmλ/4x

λ/4 λ/2

DepumpRB2795yf=10 cmB

σ+
GT

Side view

RB4

λ/4

λ/4
λ/4
MOT3

PBS

MOT2
MOT1

RB3

Objective
z

λ/4
λ/2
Dichroicx

Window
CCD

y
780 nmIF

852 nmFrom optical rail

Top view

λ/4 λ/2
PZT

LS

Figure 2.5: Schematic of the optical setup, showing the relative alignment of all the beams forcontrolling the full quantum state of atoms trapped in optical tweezers formed by the custom
objective lens outside of the vacuum chamber. For imaging, the same objective lens is used tocollect fluorescence from the atoms, which is then deflected by a dichroic mirror and down animaging path with an f = 1 m lens onto our Andor 897 EMCCD camera. In order to manipulate
the spin in a site-selective way, we send another beam back toward the objective lens from theimaging path (via reflection off of a window blank). The remaining beam paths are used for state
manipulation, addressing the transitions depicted in the level schematic shown in Figure 2.1.

19
in principle giving us the control to organize single atoms in arbitrary configurations [32, 63, 35]
and even combine separated atoms into the same trapping potential [64, 65, 47]. The simplicityof the optical trap also offers many advantages in that many electronic states can be trapped
simultaneously (in contrast to magnetic traps) and the same trap system can generally used formany different atomic species (possibly with slight modifications to the trapping wavelength).
The optical dipole trap uses light that is far detuned from resonance with any groundstate transitions and relies on the ability to shift the electronic energy levels of the atom without
scattering a significant number of photons (which would heat the atoms and destroy any coherence
between electronic states). Specifically, we take advantage of the AC Stark shift, which lowers theground-state energy of the atom when the light is below resonance, or red-detuned, because thelight induces a dipole moment that is in phase with the electric field of the light. The confinementof the potential is then formed by the intensity gradient (in all three dimensions) at the focus of thetrapping beam, where the intensity of the light is at a maximum and thus the ground-state energy
is minimized. Alternatively, light that is far above the same transitions, or blue-detuned, can beused when an intensity minimum is created at the focus (such as with higher order Laguerre-Gauss
beams) because, in this case, the light increases the ground-state energy of the atom.
In our system, we use 852 nm light, which is far detuned from both the wavelengths of theD1 and D2 transitions in 87 Rb (which are around 795 and 780 nm, respectively). We also collectfluorescence scattered during PGC (at 780 nm), which sets many of the requirements for our opticalsystem. Ideally, the optical system will be diffraction limited for both of these wavelengths and be
capable of generating a 5×5 array of diffraction-limited traps, each spaced by 1.5 µm.

2.2.1

High-numerical aperture lenses

In the generation of our optical tweezers, the most important element is the high numericalaperture (NA) objective lens that brings the beam of light to a sub-micron focal point, which,when used as an optical dipole trap, provides the tight confinement we want. In order to simplify
the system, we use an infinity-corrected objective lens, which will take as input a collimated beam

20
(and, for imaging, will collimate light scattered from a point source in the focal plane of the lens).
Because we want to place the lens outside of the vacuum chamber, there are some constraints inthe physical design of the lens, which are set by the dimensions of the glass vacuum chamber andthe diameter of the coil mounts. Specifically, the working distance of the objective lens needs to belarger than 20 mm to have some space between the front of the lens and the vacuum window andto preserve the optical access, the half-angle of the cone that the lens can occupy cannot exceed
50◦ from the center of the vacuum chamber.
We are currently using a custom objective that was designed and built by ASE Optics thatprovides an effective NA of approximately 0.6. The objective is designed to be achromatic for both
780 and 852 nm and to operate at a NA of 0.64, with a long working distance of ∼ 21 mm andcorrecting for the aberrations introduced by focusing through a 6.35 mm thick fused silica window.
The housing of the objective is made out of Ultem because it can be precisely machined and isnonmagnetic, which is desirable because of the proximity of the field coils. The objective was notdiffraction limited, but we found that we could get acceptable performance only at a NA of 0.6.
Some more details on the characterization of this lens will be given in Section A.1 and there isadditional discussion of this particular lens in Ref. [57].

2.2.2

Sources of light for optical trapping

We have used a number of different lasers to produce light for trapping single atoms, but recently we have switched between two high power laser sources: A highly coherent Titanium:Sapphirelaser (an M2 SolsTiS system) operating at 852 nm and a temporally incoherent laser system consisting of a 1 W tapered amplifier chip (from Eagleyard Photonics) seeded in a double-passed
configuration with light from a super-luminescent light emitting diode, or SLED (supplied by Superlum). Both of these systems can reach similar intensity levels at the atoms, which are sufficientfor the experiments being performed, but have different advantages. While the SolsTiS system isquite a standard unit, the temporally incoherent source is rather new (we learned about its use in
Markus Greiner’s lab and decided to try it out based on the success in their system) and so I am

f2

λ/2

iris

f2

21

To obj./cell

AOV

4.5 cm 4.5 cm AOH

λ/2
EFL=15.1 mm

f=300 mmFigure 2.6: Schematic of the optical rail that takes input light from a fiber and then generates ahigh-quality beam that is used to generate sub micron optical traps. Note that due to the opticalrelays, the two AODs are in conjugate planes of the optical system, which is then passed to theback focal plane of the 20x telescope, and finally projected onto the back of the objective lens.

adding some of the details of our construction here [66, 67, 68].
Our switch from the SolsTiS system to the temporally incoherent light source was motivatedby our concern that reflections off of any of the beam preparation optics, the cell, or even within theobjective lens itself, could lead to residual amplitude modulation at the focus of our lens, leading
to small (and slow) time-variation of the relative depth between multiple traps. We have someexternal data suggesting that there was residual drifitng that has improved by switching to theincoherent light source, but it is not conclusive and we have yet to verify the improvement withthe atoms directly. I am sure more details on this testing will be discussed in an upcoming thesis
by our new graduate student, Mark Brown.
The main concern about using a broadband source is that there could be “blurring” of thetrap due to the combination of chromatic aberrations, polarization optics working over a narrow
wavelength range, and, potentially most importantly, the acousto-optic deflectors (AODs) having
a slightly different deflection angle for each wavelength component.

22
2.2.3

Optical RailIn order to achieve the expected performance of the optical tweezers, it is important to startwith a high-quality and unaberrated beam. The challenge is that we need a beam with a 1/e2 radiuson the order of 15 mm to fill the input aperture of the long working-distance, high NA objectives
that use. Additionally, we want the beam to pass through two acousto-optic deflectors (AODs),
which will enable some amount of beam steering (more details on the capabilities afforded by these
AODs is given in Section 3.2), which means that the optical system will need several componentsthat are well referenced to each other.
To achieve these goals, we have designed a custom built “optical rail”, which minimizes thenumber of adjustable degrees of freedom such that it becomes a simpler task to align all of the
optical elements using references that are referenced to fixed post locations. Additionally, the rail
is designed to be compact (fitting in the area available near the vacuum chamber) with the output
beam height fixed to be centered on the optical axis of the objective lens and the vacuum cell.
Aside from the mechanical stability and benefits of fixing the location of as many of the optics as
possible, the optics on the rail are designed to provide the best performance for the objective lens.
Fundamentally, the optics of the rail consist of a small set of features that give us the flexibility
and performance required, which are all included in the diagram shown in Figure 2.6. At the input
is a standard fiber launcher setup for a flat-faceted (FC/PC) output fiber connector, which has
been anti-reflection (AR) coated to prevent interferences, and where we have taken special care to
make sure that the output fiber is centered on the collimation lens. Immediately following this is apolarizing beamsplitter cube (PBS), as well as a λ/2 waveplate (after the first AOD) that is used toproperly align the polarization into each of the AODs (which achieve a higher deflection efficiency
when the polarization is transverse to the accoustic wave propagation). After the first AOD is an
f = 4.5 cm 4-f “optical relay” that brings the deflected beam from the center of the first AOD tothe center of the second AOD. An optical relay is simply a 1:1 telescope that maintains the angle
and position (up to a sign) of the beam between the input and output planes of the telescope,

23

f

iris

f

2f
Figure 2.7: Diagram to highlight the utility of the optical relay (also known as a 4f -relay or a 1:1
telescope) to keep the beam deflections on-axis through the optical rail (and eventually into theobjective lens). Specifically, by positioning two lenses of equal focal length, f , by twice that focal
length, any two planes located a total of d = 4f apart (with both lenses in between the two planes)
will be conjugate planes. Because the magnification factor is unity, this means that in both planes
both the location of the beam and the angle of the beam (with respect to the optical axis) will be
the same, up to a sign flip (both the angle and the position will be opposite).

as is demonstrated in Figure 2.6.2

Then, there is a longer f = 15 cm optical relay that brings

the deflections from the second AOD to the back focal plane of the final M ' 20 telescope thatmagnifies the beam and brings it to the back of the objective lens.
However, before this final telescope is a set of polarization optics that allow us to pick off
a fixed fraction of the total beam power. Specifically, we use the sequence of a PBS (to initiallyclean up the polarization after the other optics) followed by a λ/2 waveplate and a second PBS,
which together pick off a fixed fraction of the light transmitted by the first PBS. The first PBS
may not be necessary, but add stability to long-term drift of the polarization after passing throughthe two AODs (which are both birefringent and thermally sensitive), two optical relays, and two
λ/2 waveplates. The signal picked off by the second PBS is then used to monitor the power going
into the objective lens, which is directly proportional to the trap depth of a single optical tweezer.
Further, by feeding back on this measured signal, we can stabilize the trap depth (and use thecomputer control to ramp this depth to different values during a single experiment).
2

These optical relays are particularly important for our beam preparation because of the large magnification
telescope, which would take a slightly off-axis beam and cause it to completely miss the objective lens.

24
In the context of stability, building a completely fixed rail would be ideal, but we do needto make sure there are enough degrees of freedom to achieve the desired performance. Becauseof the sensitivity of our lenses, this performance requires the precision alignment far beyond what
we are likely to achieve with a completely fixed system. For clarity, we have the ability to tune
the alignment of the beam axis and to align with references to the physical axis (built-in pinholes)of the optical rail, as well as to tune the spacing between lenses in the optical relays (and thetelescope) to maintain a well-collimated beam, which provides the required flexibility to optimize
and align the output beam from the rail to the objective lens. A brief outline of how we performed
this alignment is included in Section A, where we discuss the full alignment procedure.

Chapter 3
Quantum Gas Assembly:
A Platform to Study Tailored Many-Body Systems

As introduced in Section 1.1, the goal of quantum gas assembly is to build up interesting
quantum systems with single neutral atoms as the building blocks. There is a tremendous amountof control afforded by the use of optical tweezers, which not only allows us to build these systems
from the ground-up, but also gives us the opportunity to access more information about the system
we are preparing along the way, as depicted in Figure 3.1. It is the goal of this chapter to discuss
some of the methods we have developed to take advantage of this control, as well as to highlightthe utility of the information that we have access to, which has been inaccessible, or otherwise notfully utilized, in many earlier quantum gas experiments.
In our experiment, we have already developed the ability to load atoms in tight opticaltweezers, rearrange the potentials during the coarse of a single experiment, laser cool these atoms to
the three-dimensional ground state, individually manipulate the spin-state of separated atoms, andthe ability to analyze data based on the measured initial and final configurations. The experiments
to date have all been with small arrays of traps and this chapter attempts to address some of
the challenges that need to be overcome to enable these experiments to expand to more than a
few atoms. Specifically, we need to be able to reliably trap a set of atoms that are individually
addressable, can be arbitrarily transported and transferred between potentials, and can each beinitialized into a pure quantum state of both the spin and motional degrees of freedom.

26
Load atoms from MOT:

Enhanced loading, 90% filling:

Rearrange atoms:

Image array,determine configuration:

Isolate filled portion of array:

Laser cool atoms to indistinguishability,
prepare individual spins:

-

(

)

-

)
Detuning ((kHz)Filter imperfect preparation,
place into final potential:

-

(

)

Figure 3.1: A copy of the schematic vision showing the initialization of experiments via quantum
gas assembly. Highlighted are a few of the important capabilities that the optical tweezer platform
affords, which will be discussed in more detail throughout this chapter.

27

3.1

Enhanced Loading of Single Atoms
The loading procedure discussed in Section 2.1.2, where atoms are efficiently laser cooledin the vicinity of optical traps, is a reliable method to stochastically load atoms into tightlyconfining optical traps. However, the loading efficiency with these small-volume traps is typicallylimited to 50% [60], with the highest efficiencies achieved in optical tweezer experiments reachingefficiencies of as high as 64% [69, 70]. The understanding of the light-assisted collision process
between tightly confined atoms has allowed us to understand these limitations and dramatically
improve the efficiency of loading techniques [36, 37, 38].
Note that all previous work studying the use of light assisted collisions to improve the loading efficiency were performed using both larger volume optical traps, where multiple atoms could
initially be loaded before initiating light-assisted collisions, and using 85 Rb, which has both a different level structure and ground-state scattering properties [71, 36, 37]. Additionally, these studies
found that the collisional process to prepare a single atom in the trap required several hundred
milliseconds, which would have been prohibitive to increasing the repetition rate of these optical
dipole trap experiments. More recently, loading efficiencies around 80% have been demonstrated
with 85 Rb in a micron-scale optical tweezer and requiring only 30-60 ms collisional pulses [72].

3.1.1

Background on light-assisted collisions

The process of light-assisted collisions is successfully described by onset of resonant opticaltransition between different molecular (interaction) potentials at specific interatomic distances RC ,as shown in Figure 3.2. The different potentials correspond to the different electronic configurationsof the colliding atoms valence electrons. When the electrons of both atoms are in the s-orbital, thenthe atoms are on an attractive electrostatic C6 /R6 potential, but when one of the atoms is excited
to a p-orbital, the dipolar interaction (which can be positive or negative) varies as C3 /R3 [59]. The
difference in these interactions means that for any reasonable detuning from the bare (single-atom)
resonance frequency, there will be a particular distance, defined as the Condon radius, where the

U

δ

Rc

Rc

|S1/2+P1/2

28

P3/2 F΄=3
F΄=2|S1/2+S1/2
Rd

P1/2 F΄=2|S1/2+P1/2

S1/2|S1/2+S1/2

795
2"2΄

b)

Cyc.
Rep.

a)

δ

F=2
F=1

Rd

r

w0

Figure 3.2: a) Interaction potential for two 87 Rb atoms colliding in the presence of near-resonant
light. For reasonable detunings, there will be an internuclear separation, RC , where the light isresonant with the excitation between the interaction potentials, resulting in a light-assisted collision.
The character of that collision changes depending on the sign of the detuning: For positive (blue)
detuning, the excited potential is repulsive, allowing us to control the maximum kinetic energy
gained by the pair of atoms undergoing to collision. For negative (red) detuning, the excitedpotential is attractive. b) Schematic of the relevant energy levels in 87 Rb, indicating the detuning(from the light-shifted energy levels) of the beams used to perform enhanced loading (not to scale,for full level diagram see Figure 2.1).

transition becomes resonant with the driving frequency. When atoms are in close proximity the
detuning from the bare transition can be quite large (such that the scattering rate from a single
atom is low), even while resonantly driving this collisional process.
Of particular relevance for loading via efficient cooling is that the interaction potential towhich atoms are excited by the red-detuned light that is used for cooling has a strong curvature,
meaning that the atoms will be strongly attracted to each other. The atoms will be accelerated
toward each other on the long-range attractive potential, but the “quasi-molecule” can emit a
photon (now of a different frequency) bringing both atoms back to the s-orbital and giving the pairof atoms a large amount of kinetic energy. This is the dominant process that will result in the lossof both atoms from the trap during cooling in the collisional blockade regime. This pair-loss effectis what leads to the typically 50% loading efficiency (which can be pushed to 60%, in some cases)

29
and to parity imaging [70, 38].
Fortunately, there also exists a repulsive interaction potential that is resonant when thelight is blue-detuned (higher in energy) from the bare atomic transition. Importantly, atoms on
the repulsive interaction potential cannot gain more kinetic energy during the collision than the
detuning of the beam (converted into energy units) because the potential curves asymptoticallyapproach the bare levels. This additional control, combined with our knowledge (and control over)the depth of the optical trap is used to preferentially expel a single atom at a time (rather thanpairs of atoms). Specifically, by setting the detuning to be equal to the depth of the trap, themaximum kinetic energy gained by the pair of atoms is only enough for a single atom to be lost!
This enhancement is demonstrated in Figure 3.3(a), where the collisional process allows us to
improve the probability of fully filling of an array with four sites by a factor of four compared toour best PGC loading procedure. While the process is still stochastic, this is a drastic improvement
for beginning to scale to larger arrays of atoms.

3.1.2

Experimental implementation of enhanced loading

However, there remain some complications to make the repulsive light-assisted collisions
useful for enhanced loading in the experiment: First, we need to be sure that the rate at whichphotons are scattered from a single atom in the trap is small enough that a single atom in thetrap is not lost. Additionally, we must be sure that the scattering/collision rate induced by theblue-detuned collisional photons is much larger than that of any other light that is red-detuned of
the transition. In the end, these effectively require us to use very deep trapping potentials so thata collisional beam with detuning equal to the trap depth results in a single-atom scattering ratethat is quite small, even with the power necessary to rapidly drive these collisions. In our traps,
using such a deep trap has the side-effect of shifting the D2 transitions used for the MOT well outof resonance once the atoms are loaded into the trap; this helps in balancing the scattering rates
between the residual MOT light and the blue-detuned collisional light.1
1

I do not think that this is actually necessary for the enhanced loading to work, but it does help to order the
scattering rates as desired. In principle, one should be able to use a trap that uses a magic wavelength for a particular

30

a)

Photons on Pixel

b)

0

1 2
3
4
Atoms Loaded

Figure 3.3: This figure shows the results of the enhanced loading. a) For a single site, we canlook at the number of photons collected and plot a histogram of the number of photons collectedduring an image after each loading attempt; we see two peaks in the histogram corresponding tozero and one atoms loaded into the trap, with a much larger (approximately 90%) probability for
an atom to be loaded. b) For a small array of optical tweezers, we find that we can achieve similarloading efficiency at each site. On the right, we compare the probability to load n atoms into the
4 site array when using enhanced loading (blue) and PGC loading (red). Specifically, we alreadysee over a factor of 4 enhancement in the probability to load 4 atoms compared to using standard
PGC loading.

As mentioned, the most important concern for implementing this enhanced loading scheme
is to consider the rates of the various processes that are happening. Because we have such smallvolume optical potentials (and are thus in the collisional blockade regime), we cannot rely on loading
many atoms and then sequentially applying a collisional pulse to reduce the remaining number to
1. Thus, we need near-resonant (in free-space) beams on to generate a reservoir of atoms that are
being cooled into the trap.2

In our case, these are simply the MOT beams, which address theD2 F = 2 → 30 cycling transition and the D2 F = 1 → 20 repump transition. These beams are
sufficient for loading a single atom in 50-60% of trials, but we will add two additional beams that aretransition and design a scheme around that trap which can have all the rates work out. However we do not currently
have a trap in which to try this.
2
In principle, this could be a blue-detuned set of beams, which could help alleviate some of the requirements forminimizing two-atom loss events (and possibly improving the final loading efficiency further), but this would require
adding another set of cooling beams on the MOT beam paths (or adjusting the locking setup to allow us to tune
farther blue-detuned than we currently can go), which prohibited exploring this possibility.

31
tuned to address atoms that are already loaded into the optical tweezers: The main beam is used
to induce the collisions as described above and is blue-detuned from the D1 F = 1 → 20 transition.
The final beam is nearly resonant (in the trap) with the D2 F = 2 → 20 depumping transition. Fora discussion of the reasoning behind the choices of these transitions, refer to Section 3.1.4.1.
For simplicity, we choose to add these collisional beams during the MOT loading stage, whichmeans we don’t need to add a multi-stage loading procedure that is long and complicated. This
procedure results in one atom remaining in the trap more often than our standard MOT loading
procedure because the primary loss mechanism is single-atom loss induced by the blue-detuned
light-assisted collisions. As seen in Figure 3.4(b), the loading efficiency saturates to around 90% in
about 170 ms, this time being primarily limited by the loading time of our MOT (which could be
improved, but without larger scale changes to the setup would require sacrificing the trap lifetime)rather than any effects related to the blue-detuned collisions.

a)

b)

(

)

(

)

Figure 3.4: The loading process is split into two sections: The MOT fill time, which provides thereservoir of atoms to load into the optical tweezers, and the collisional pulse, which ensures that
there is never more than a single atom at the end of the loading sequence. a) Here, we measure
the loading probability while varying the MOT fill time for a fixed collisional pulse length of 35
ms. The probability for each well in a 2×2 array (blue, red, orange, and purple) is plotted alongwith the data taken when loading a single well via PGC (green). b) Here, we look at the effect of
the collisional pulse length on the final loading probability. The blue points were taken using the
optimal collisional beam power, while the purple points were taken with 30% of that beam power.
The black dashed lines indicate a 90% loading efficiency.

32

a)

b)

(

c)

-

)

d)

(

)

(

)

(

)

Figure 3.5: a) Loading efficiency as a function of the “795 beam” detuning with the trap depth
highlighted as a green dashed line. The optimum detuning is slightly less than the trap depth, andlarger detunings results reduced efficiency due to pair loss. b) Loading efficiency as a function of the795 beam power. c) Loading probability as a function of the depump beam detuning. d) Loading
probability as a function of the depump beam power, indicating the relatively high power required
for enhanced loading. Note the drop in efficiency at low, but nonzero, powers (see Figure 3.6).
In all of these figures, the colors (blue, red, orange, and purple) represent the measured loading
efficiency for each of the four sites of the 2×2 array shown in Figure 3.3(b).

33
3.1.3

Collisional beam parameters

For our particular implementation of the enhanced loading procedure, we use our standard
MOT beam configuration, as outlined in Section 2.1.1 and add the two additional beams discussed
here. The primary collisional beam, called the “795 beam”, is detuned from the D1 transition(which is around 795 nm) and is aligned with the atoms along the same path as the top Raman
beam (discussed in Section 4.1.1). The polarization of the 795 beam is linear and aligned along the
propagation axis of the optical tweezer (orthogonal to the polarization of the top Raman beam).
The second additional beam, called the “depumping beam”, is used to quickly depump atoms thathave been loaded into the optical tweezer to the F = 1 manifold so that it becomes dark to the
MOT light and can be addressed by the 795 beam. This beams is effectively unpolarized dueto the beam being incident to the optical tweezer axis at an approximately 50◦ angle, and being
retroreflected in the lin⊥lin configuration.
The 795 beam is the primary driver of these light assisted collisions, so it is not surprisingthat both the power and detuning are important parameters for achieving the best possible loading
efficiencies. The effect of these parameters is highlighted in Figure 3.5(a,b), which show a clear
optimum in both parameters. It is important to note that these parameters are coupled: The
heating of single atoms due to off-resonant scattering of the light limits both the loading at higherpower and smaller-detuning. However, the global optimum will occur when the detuning is very
nearly equal to the trap depth, and then the power is tuned such that the loss due to heating of
single atoms is negligible (at least in comparison to the cooling rate from the far-off resonant MOT
beams). Fundamentally, this is because we are maximizing the rate at which the blue-detuned lightassisted collisions will occur, which allows this rate to compete favorably to the pair-loss induced by
the MOT beams that are required to initially cool atoms into the optical tweezers. This behavior isconsistent with earlier work in larger volume dipole traps where the collision rate was smaller (andwhich were not in the collisional blockade regime, so more than one atom could be initially loaded
into their traps), and thus much longer collisional pulses at lower powers were required to minimize

34
single-atom loss [37], but because we are in the collisional blockade regime, both of the light-assisted
collision rates are much faster and collisional pulses only need to be tens of milliseconds before weare guaranteed to be left with either one or zero atoms, which is demonstrated in Figure 3.4(b).
While the role of the depumping beam is conceptually simple, it plays an important part inpromoting the processes we care about during our loading procedure. Importantly, we find that
there is a rapid drop in loading performance when moving outside of the range of values that work
well for the detuning and power of this beam, as demonstrated in Figure 3.5(c,d). For the detuning,problems only occur when detuning red of the light-shifted transition (meaning, red-detuned of thetransition for atoms loaded into the trap) up to at least +50 MHz detuning. However, for the power,
we find that there is a broad “sweet-spot” where optimal loading occurs. Surprisingly, if the power
is too low, we see that the atoms are always kicked out of the trap, in addition to problems when
the power is too high (likely due to the scattering rate becoming high enough to heat single-atoms
out of the trap).
This low power problem is highlighted in Figure 3.6(a), which plots the loading probability as
we vary the depump beam power for a fixed detuning. Notice that at very low powers (or when the
beam is off), we get approximately 50% loading efficiency, which is equivalent performance to nothaving a blue-detuned collisional beam on because, without depumping, the atoms are unlikely toundergo blue detuned collisions. However, as the power is increased, the loading efficiency rapidly
drops to very nearly to 0, before increasing again and saturating at much higher powers. The
resulting conclusion is that we need very large depumping beam powers to achieve optimalloading efficiencies, however the exact reason is not fully understood. It is likely that it is related tothe competition of rates between various scattering processes, but what is particularly interestingis that this is a single-particle effect!
That this is a single particle effect is highlighted in the data shown in Figure 3.6(b), where we
see two curves that both exhibit the same dip in the survival: The black points are data taken where
two atoms are placed in the same well before applying the collisional light, while the blue points
demonstrate the same effect when the same experiment starts with only a single atom. Because

35

a)

Enhanced
795 Blocked

(µ )

b)

-

One Atom
Two Atoms(µ )Figure 3.6: a) Measurement of the loading efficiency as a function of depump beam power withfiner resolution than in Figure 3.5(d), comparing the behavior seen when the enhanced loading
procedure is performed as usual (red) and when the 795 beam is blocked (purple). b) Measurementof the probability for one atom to remain in a single trap, given that it was initially prepared withone (blue) or two (black) atoms before a collisional pulse (consisting of only the 795 and depumping
beams) was applied to the trap.

this is a single particle effect, it is worth mentioning that this could be due to the exact orientationof the beams we are using in our setup; we do know that effect requires both the 795 and depumping
beams to be on in order for atoms to be kicked out.

3.1.4

Limitations to the enhanced loading

In the best case, we are currently limited to an approximately 90% loading efficiency using
blue-detuned light assisted collisions, but its worth asking two questions: What are the limitations

36
to improving this optimum rate? And are there any limits when scaling up to larger arrays oftraps?

3.1.4.1

Limitations to the maximum loading efficiency

As has been the theme throughout this section, there will be a fundamental limitation to
the loading efficiency given by the competition of the two-atom loss rate to the single-atom loss
rate when multiple atoms remain in the trap. The fundamental limitation here comes from thelevel structure of the atom being loaded into the traps and the relevant interaction potentials forundergoing light assisted collisions [37]. In 87 Rb we benefit from well-isolated hyperfine structure
and the ability to, ideally, isolate a single interaction potential, but this is never perfect for any
real case. Without changing atomic species, this means one has to carefully consider the choiceof beam(s) used for driving these light assisted collisions to minimize the total probability of offresonant red-detuned, far blue-detuned, or spin-changing light-assisted collisions from occurring,which could each result in both atoms being lost from the trap. Because these can never becompletely turned off, the important consideration then becomes maximizing the rate of the desired
single-atom loss channel in comparison to these two-atom loss channels.
Specifically, our scheme in 87 Rb is chosen to utilize the D1 F = 1 → 2 transition because
it has no nearby higher-energy transitions that could lead to driving collisions on an attractive
interaction potential. Additionally, the excited hyperfine structure is well-resolved, which meansthat we are an extra 816 MHz off-resonant from any blue-detuned collisions that would lead tolarge gain in energy (which means that the atoms need to be much closer together to drive thistransition, so the probability is much smaller).3 An additional benefit of using this transition isthat there trap induced light shifts of both the ground and exited state energy levels are mainly
scalar shifts, which means that the detuning is the same for all Zeeman sublevels and, because we3

To be explicit, even for quite large blue-detunings there will be an internuclear separation where the transition
will be resonant. For such large detunings, this distance is very small and thus the probability of driving this transitionin comparison to the desired transition is likely small. However, when this additional detuning becomes on the orderof the detuning from the desired transition (i.e., the trap depth), which could certainly be the case with different
transitions, other atomic species, or by going to much larger trap depths , then this concern becomes much more
relevant.

37
are not spin polarized during loading, this is particularly important for efficient loading [73, 74].
Another major limitation to controlling single versus two particle loss comes from the uncertainty related to the initial conditions of colliding atoms. In particular, if the temperature of
the atoms is not well known, then the spread in initial energies will lead to a higher probabilityof both atoms being lost, even when colliding in the desired channel. If the energy is too high tobegin with, then it is possible for both atoms to be kicked out of the trap, even when the energy
added is only enough for one atom to be removed. This also cannot be mitigated by using a smaller
detuning: If you err on the side of not enough energy, then there will be a larger probability ofneither atom getting kicked out, which means you start with a much higher initial energy during
the next collision.
An additional limitation to the maximum loading efficiency will be set by the competitionof the single-atom loss rate (when only a single atom remains in the trap) to the length of thecollisional pulse required to ensure that there are never multiple atoms remaining in the trap. This
source of single-atom loss will fundamentally be limited by the background (vacuum) collision rate,
but for now is negligible and is mainly limited by the heating rate associated with scattering thecollisional beam photons from the atom in the absence of cooling beams. Standard methods can beused to mitigate this (larger detunings or lower powers) by lowering the scattering rate, but this has
to be balanced with the primary limitations dictated by the balance of the light-assisted collision
rates. In the end, the combination of both of these concerns points to going to deeper traps suchthat larger detunings (as long as they are not so large that the rate of collisions leading to two atom
loss becomes significant) can be used to optimize the single-atom loss rate, and then higher powers
can also be used (thus improving the ratio of the light-assisted collision rates) without increasing
the scattering rate off of single atoms.
It is certainly worthwhile to pursue an even better overall loading efficiency, which in ourcase would mean going to larger trap depths (and thus larger blue-detunings), but this pushed us
into two other limitations that complicate efforts to significantly improve the loading efficiency.
The first are a set of technical limitations because going to much higher trap depths requires a38
proportional amount more optical power per trap, as well as a proportionally larger tuning range
for any lasers that need to be resonant with atoms loaded into the trap. Additionally, when goingto larger trap depths, the initial temperature of any atoms initially loaded into the trap will likely
increase because of the larger light shifts at the center of the trap, which, as discussed, also limits
the maximum loading efficiency. In the end, we used several different trap depths ranging from 2
to 4 mK and found that above 3 mK the efficiency saturated around 90%.

3.1.4.2

Limitations when scaling to larger arrays

While our movable tweezers mitigate some of the challenges associated with loading overlapping optical tweezers, it is important to consider proximity effects that become relevant when
scaling to larger arrays of traps with a finite accessible field of view (as discussed in Section 3.2.2.1).
In particular, even once we implement the ability to rearrange atoms after loading, the ability toload atoms with high efficiency will be critical for scaling up to larger arrays. However, this loading procedure relies on our ability to selectively induce light-assisted collisions between atoms inthe same trap without affecting atoms in nearby traps. In the end, we do find that the loading
efficiency becomes compromised as the traps are brought into close proximity, as demonstrated in
Figure 3.7 [38].
The experimental loading probability drops off quickly as the spacing of two wells is reduced
below ∼ 1.7 µm [Fig. 3.7(a)], at which point the barrier between the two wells is reduced. By
comparing the outcome to a separate experiment where we load two atoms into one side of the
double well potential before applying a collisional pulse, we observe that this leads to an increasein the probability of hopping of at least one of the atoms to the second well. Importantly, we
note that these collisions resulting in the atom hopping to the other well reduces the probabilityof completely removing a single atom during each collision event. Using the numbers measured in
the experiment, we can perform a simple Monte-Carlo simulation to estimate the loading efficiency
we would expect if we simply increase the hopping fraction. We find that the reduction in final
loading efficiency is reasonable and compare the results of this Monte-Carlo to two sets of loading

39

a)

a

b)

(

)

(

)

Figure 3.7: Effect of tweezer spacing on loading efficiency. a) Maximum loading probability
achieved (per tweezer) after optimizing the loading parameters at each well spacing (dark- blue
dashed line) and the loading probability when using the isolated-tweezer beam parameters at eachspacing (light-blue solid line) versus the spacing between the center of two tweezers a. The loading
probability calculated from a Monte Carlo simulation of the loading process (gray band) using themeasured probabilities of the four possible outcomes [three of which are shown in part (b)] during
each collision event and neglecting any single-particle effects. (b) Controlled experiment studying
the effect of collisional light on two trapped atoms in proximity to a second trap. Measured probabilities of a single atom remaining in the right well (dash-dotted green line), one atom remaining
in each well (dashed orange line), a single atom remaining in the left well (dotted red line), andthe total probability for an atom to remain in the right well (solid purple line) as a function of thespacing between the optical tweezers. Note that this image is reproduced from Ref. [38].

data measured (with different collisional and cooling beam parameters) in Figure 3.7(a).

40

3.2

Manipulation of Optical Tweezers
As discussed in the Section 3.1, we use relatively deep traps (typically around 3.5 mK depth)that are far apart (meaning they are not overlapping) in order to promote processes that enhance
the probability of loosing a single atom at a time for all of the traps in an array. After thisloading process is complete, we need to dynamically modify the trapping potentials so that theatoms locations can be imaged, the spin and motional degrees of freedom prepared, and then theatoms transferred into a configuration relevant for the experiment of interest. Within our current
experiments, this can be achieved with the ability to change three quantities: 1) The spacing
between neighboring rows or columns in a rectangular array, 2) The relative intensity between
neighboring rows or columns (the “bias”), 3) The total optical power used to generate all of thepotentials. The remainder of this section discusses how we gain control over the remaining two
quantities within our experimental setup, as well as how it can be improved or extended for future
experiments.

3.2.1

Overall intensity stabilization

To control and stabilize the depth of the optical tweezer array, we control the total opticalpower entering the rail for beam preparation. Specifically, this sets the overall depth scale for thetraps, but its important to note that it does not control the actual depth of any given optical tweezer(which is modified by the total number of traps and their relative intensities, as discussed further in
Section 3.2.3). In the case of our system, there are two concerns for the intensity stabilization: First,
we need to be sure that the shallow science potentials are consistent shot-to-shot (and thus there
is no long-term drift in the absolute power), such that the tunneling and interaction parameters
remain constant. Second, we need to be sure that there is no resonant intensity noise on the trap
light that will lead to heating of the atoms (and thus lead to distinguishability between our atoms,
as discussed in Chapter 4).
The optical power entering the rail is controlled by modifying the RF power sent to an41
accousto-optic modulator (AOM) that deflects light into the optical fiber at the beginning of the
optical setup described in Section 2.2.3. This intensity is stabilized with standard feedback electronics via the power measured at the pickoff just before the beam expanding telescope in the
optical rail. There are two main considerations for designing this feedback system: We need tomaintain a stable intensity at DC, up to about 10 kHz in bandwidth (for ramping the intensity
during experiments on millisecond timescales) without adding noise at frequencies resonant withparametric excitation processes (in the range of 10-300 kHz). Additionally, we need to be able tostabilize the intensity over a very large dynamic range of approximately four orders of magnitude(e.g., ranging from just a few microwatts to 40 milliwatts of power).
Standard techniques can easily achieve the desired bandwidth requirements, so I will not
go into details about the feedback controller (typically a loop filter containing some amount ofproportional and integral gain). As a part of my thesis work, it was important to go beyond thelimitations of trap depth (both to go to deep traps for enhanced loading and shallower traps formore reliable tunneling) that were limited by the dynamic range of our previous detection system.
Namely, we were previously completely saturating a linear photodetector for cooling trap depths,
and could not go to shallower traps because we were running into the noise floor of the photodetector
at tunneling depths.
However, to achieve a large dynamic range is a bit of a challenge and is typically limited by
how the response of the system changes at various setpoints. In particular, we have begun using
a logarithmic transimpedance amplifier, which has a very large dynamic range (in principle, theAD8304 can operate over 8 orders of magnitude in photocurrent). Additionally, to preserve ourability to use a standard feedback controller, we use a “linear-in-dB” variable gain amplifier thatlinearizes the response of the system [75]. Unfortunately, the response of this system changes quite
dramatically as a function of the DC offset, preventing us from simply setting the loop filter as if
this was a truly linear system. Currently, we have found a way to tune the system to give acceptableperformance at most signal levels, but are working on ways to improve this using adaptive feedback
systems, which will be discussed in more detail in future work.

42
Another method that we have explored (and may come back to in the future) is to use morethan a single linear photodetector, with separate photodetectors optimized for different, overlapping, dynamic ranges. This has the benefit of a lower noise floor than a single logarithmic amplifier,and returns to a “simple” feedback system for each segment of the dynamic range. However, thecomplication is that we need a smarter controller than can either take more than a single photodiode input, which should be possible by switching to the use of a digital feedback controller.
Alternatively, one could use an external control system to switch between separate control loops,
each containing a single photodetector and feedback system.

3.2.2

Acousto-optic control of optical traps

The actual optical setup that is used to generate the optical tweezer potentials was describedin detail in Section 2.2.3, so we will not go through the entire optical system here. However, it
is important to note that the acousto-optic deflectors (AODs) located in the optical rail are in
the Fourier plane of the trapping potentials (and are in conjugate planes with each other and theentrance aperture of the high numerical aperture objective lens).4

Importantly, this means thatchanging the beam deflection angle with either of these AODs directly translates to motion of thetweezer in the focal plane of the lens. Further, because the AODs are oriented orthogonal to eachother, we have the ability to steer the optical tweezer potential in both X and Y at the focus bychanging the deflection angles in each of the AODs. However, simply changing the angle of a single
beam is not enough of a reason to use AODs; in principle, we could do the same thing with piezo
mirrors at the same locations with minimal change to the optical rail.
The use of AODs is necessary because of the ability to change the deflection angles very
rapidly, as well as to deflect multiple beams simultaneously (and to change the number of deflections
in real time), all without physically changing the optical setup to scale to larger arrays. Some
4
The main difference between an acousto-optic deflector (AOD) and an acousto-optic modulator (AOM) is in
its design: An AOD is designed to optimize for deflection efficiency and to have a large bandwidth over which thedeflection remains efficient, which allows for efficient beam steering. An AOM is optimized for speed, such thata beam can quickly be turned on and off, or have its power modulated, but the frequency is expected to remain
relatively constant in order to maintain the optimum deflection efficiency.

43
alternative options we have considerred are discussed in more detail in Section 3.2.4. To demonstrate
the utility of this system, we will start by reviewing the operational principles and importantcharacteristics of these AODs.5
Acousto-optic devices operate by driving a crystal, whose index of refraction is particularly
sensitive to stress, with a piezoelectric transducer and thus coupling an acoustic wave into thematerial. This acoustic wave leads to a spatially varying index of refraction, which is to say thatit imprints a periodic index grating into the crystal. When the input angle of an optical beam ismatched to the (acoustic) wavevector of this grating, then efficient diffraction of the input beam
to a different mode (with angle related to the frequency of the acoustic wave and the material
properties of the crystal itself) can be achieved. Because this deflection angle is directly dependent
on the frequency used to drive the AOD, then by changing this driving frequency we can tune thedeflection angle, and thus the location of the tweezer at the focus of out objective lens. Additionally,
this is simply a Fourier optics problem, so by overlapping multiple frequencies into the AOD, wecreate a more complex diffraction grating with multiple spatial frequency components, and thus
we end up with multiple deflections out of the AOD. Using the imaging magnification calibrated
in Section A.1.1, and by measuring the spacing of the images of two atoms as a function of thefrequency difference between the two traps, we can calibrate the change in position as a functionof the frequency into the AOD, an example of which is shown in Figure 3.8.

3.2.2.1

Technical design considerations

The ultimate limit to our ability to position optical traps at the focus of the objective is set
by the bandwidth of the AOD, the specific material properties of the crystal, and the details of
the optical system used to move the beam from the AODs to the focal plane (specifically, the total
5
The AODs we use are Gooch and Housego model #: 46080-3-.85-LTD. They have 3 mm aperture longitudinalmode TeO2 crystals (4.2 mm/µs acoustic velocity), which allows for the use of a collimated beam in generating
deflections (which is important to put the AOD in the Fourier plane of the focus of the lens). Importantly, they canachieve about a 90% deflection efficiency into the first mode at the peak response of 80 MHz (the specification is for> 70%, but measured 88 and 92% for the two in the rail when set up in an external test). They are designed to havea 30 MHz RF bandwidth, which achieves a ±3 mRad deflection angle about the 16 mRad deflection angle of the 1storder at 80 MHz (and all at 852 nm).

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Figure 3.8: Plot of the measured spacing between two atoms held in two traps as a function of thefrequency difference between the traps. This in-situ measurement of the spacing calibration gives(0.204 ± 0.003) µm/MHz and is consistent with both external measurements and the expectation
based on the properties of the AO crystal and the magnification of the rail telescope.

magnification of the telescopes and the effective focal length of the objective lens). Our opticalsystem is designed so that we can, in principle, scale up to a 5 × 5 array of tweezers, each spaced
by 1.5 µm, which is sufficiently spaced for resolved imaging (although may not be sufficient forenhanced loading with the current aberrations). This is a somewhat conservative choice made toensure that we are not limited by the frequency difference between adjacent deflections in partiallyoverlapping traps.6
This is an extremely important consideration: The frequency difference corresponding to thedesired minimum trap spacings should be much larger than the expected trap frequencies that might
be used at these spacings. In our system, we chose to use the AODs discussed in the previous section(which result in a relatively small available field of view) because we are able to get to 0.5 µm spacing(on the order of the minimum spot radius we might be able to achieve) at >2 MHz in frequency
difference, which is more than an order of magnitude larger than the trap frequencies we would
encounter at these spacings.7
6

The reason this is an important consideration is that these beams

Recall that the frequency of the beam deflected by the AOD will be shifted by the acoustic frequency, so two
adjacent traps will have optical frequencies that differ by the difference these frequencies.
7
If the frequency of modulation of the trap (due to interference of beams at slightly different frequencies) is large
compared to the trapping frequency of the potentials, then there is unlikely to be significant heating induced by thismodulation of the trap.

45
will be copropagating and spatially coherent, so the frequency difference between these beams willlead to a beating of the intensity of the trapping potential wherever the potentials overlap! In
particular, we need to bring our traps close enough together to couple atoms between individual
wells, which requires the traps to overlap without significantly affecting the atomic temperature.
This requirement is in contrast to other experiments using optical tweezers, such as experimentswith Rydberg atoms, where the atoms remain separated and the overlap between the traps can be
kept to a minimum.
If desired, the maximum array size could be improved in a number of ways, but all of themwill require some modification of the optical system. For a relatively small change, adjusting thebeam size within the AOD would give a slightly smaller trap size, which could allow us to use
more spots within the same field of view (at the expense of more power being clipped into theobjective lens). Alternatively, a larger initial beam size would allow us to use a slightly smaller
magnification in the final telescope, resulting in a larger deflection range (at the expense of a larger
trap spacing for a given frequency difference), which would enable larger arrays of traps. However,for even larger arrays we would probably need to use a different AOD whose crystal properties
or bandwidth enabled an even larger increase the range of deflection angles available (and thus alarger number of resolvable spots at the focus of the objective).
As you can see, it is important to balance these design considerations when building thesystem initially, but when properly designed, we transfer the complication of controlling the opticalpotentials into control of the drive signal for these AODs, which is simply an electronics problem(and excellent control can be achieved, as discussed in Section 3.2.3). Specifically, by changing thefrequencies in each of the AODs, we can move rows and columns of an array of tweezers. Further,by changing the relative driving power in each of these frequencies, we adjust the deflected intensityof the traps in each of these rows or columns (thus, tuning the bias between rows or columns). This
does not give completely arbitratry control over individual traps in larger arrays, but it will be
sufficient in our typical use cases (and, for one-dimensional arrays it does give completely arbitrary
control).

46
3.2.3

Generation of radio-frequency drive

We have historically used a few different methods to generate the appropriate frequencies
to drive the AODs, starting with voltage controlled oscillators and then function generators withmore built-in functionality. These were convenient and relatively simple, but each channel could
only provide a single trap and, unless we sacrificed the stability of the signals, we were unable totrap positions in real-time.8

Further, in order to get more than a single trap, we needed to use

multiple output channels and combine them onto the same line using external RF components (e.g.,
a power combiner), which means there were more components before the amplifier, which could
lead to interferences, and the scheme is not arbitrarily scalable!
With the goal of improving both scalability and stability of the RF intensities, the sourceof the RF signals is now a single (dual-channel, one for each AOD) arbitrary waveform generator(AWG). This AWG has a large onboard memory (2 GB), which can store a set of completely
arbitrary waveforms (at 400 MS/second) along with a script that determines the order of the
waveforms.9

Critically, the script will repeat a base waveform until the generator receives a

trigger to initiate some action, then hold at another waveform before being triggered to go toanother position.

3.2.4

Alternative options for arbitrary trapping potentials

Many methods for generating projected optical potentials have been explored, including theuse of time-averaged potentials [33], as well as spatial light modulators (SLMs) [76, 35] and digital micromirror devices (DMDs) [77], but none of these can alter the potential with sufficient
bandwidth to fully replace the AODs in our system. Therefore, the AODs described above are an
important and necessary component of the real-time control required for loading individual atoms,
8

In order to change the trap position in real-time, we needed to use a DAC voltage to control the frequency of a
VCO (or, equivalently, as an input to certain function generators), which resulted in a less stable and noisier output
RF frequency.
9
While we designed the system around an AWG that is designed (and specified) to work at 400 MS/s, thegenerator we are using (a National Instruments PXIe-5451) cannot actually meet these full specifications while alsousing triggers in scripting mode, so we have been operating at 350 MS/s as we work to find an option that will fulfill
all of our specifications.

47
performing ground-state cooling, and observing dynamics of atoms tunneling between optical potentials. However, there are limitations (arbitrary two-dimensional control, aberration correction)that have caused us to consider alternatives for generating the final “science potential” used toobserve dynamics, where we are most affected by these limitations, an dI will outline some of thebasic advantages and disadvantages of these systems here.
There are two main capabilities that we do not currently have in our system that lead us tosearch for alternatives to AODs. First is the ability to project nearly arbitrary two-dimensional
potentials onto the atoms, as have been demonstrated in other groups [76, 35, 77], which is not possible with static potentials constructed with AODs arranged in a square array (and time-averaged
potentials are hard when the trap frequencies are as large as we have in our tweezers). Second, these
holographic beam shaping methods immediately provide the ability to correct for nearly arbitrary
wavefront aberrations in the optical system, which will distort the focus, weaken the tightness ofthe traps, and lead to asymmetric overlap of the optical potentials.
A liquid-crystal spatial light modulator (LC SLM) is a device that has many pixels (for thecase of the one we were looking at, 512×512), each of which applies a voltage across that segmentof the LC display. There are multiple ways to operate these devices, but in our case we want tooperate in phase-only modulation mode, which means aligning the polarization of the light in the
direction that is modulated by the birefringent liquid crystal material.10

This is important very

important feature of their operation, because a small misalignment of the polarization to this axis
directly leads to a small polarization rotation of the light, which couples to amplitude modulation
after going through polarizing optics farther down the beam path. Importantly, one technical detailof this style of SLM is that it has to refresh the voltage on each pixel at a rate of about 1 kHz, which
we found coupled to a small, but measurable, amplitude modulation at 1 kHz. Unfortunately, after
carefully characterizing a test SLM with the best achieved in-situ alignment, we found that the10

An LC SLM operated by applying a different voltage at each pixel located in a grid across a thin sheet of liquid
crystal. The different voltage leads to different alignment of the LC molecules, which changes the effective index ofrefraction of that portion of the SLM. The modulators that we were looking into worked in reflection, where the lightpassed through this sheet to liquid crystal twice. The spatial variation of the index of refraction is what is then used
to either holographically generate patterns or gratings, or to shift the wavefront to correct for wavefront aberrations.

48
residual amplitude modulation was not small enough to perform tunneling experiments; the refresh
rate becomes near resonant with the trap frequency for very shallow traps and an increased atom
loss rate was observed. This has led us to consider other more static options that will allow us toobserve dynamics of ground-state atoms.
A digital micromirror device (DMD) is a device with many individual mirrors that can beturned on or off and which can be used to define a diffraction pattern or image, depending on
the optical setup (most DMD chips come from projectors and so have standard screen resolutions,e.g. 1024×768) [77]. This overcomes the limitation of heating due to the refresh rate because
the mirrors remain fixed for any given pattern. However, the efficiency of the deflection into thedesired mode is very small (on the order of 5%), which means that we will need a laser that isa factor of 20 larger for any given array size. Additionally, while the DMD is relatively fast (incomparison to SLMs), the update rate is still below (or on the order of) the trap frequency of ourtraps, so we would only be able to use this as a static diffraction grating (that could be changed
experiment-to-experiment).

3.3

Removing entropy from the system
After the loading procedure is complete, we immediately take a fluorescence image by scat-tering the light used for PGC cooling, which allows us to determine how many atoms have been
loaded into the array [78, 79, 80, 18]. This image provides information about the number of atoms
that have been loaded into an array and their locations, which can be used to post-select experiments based on the atomic configuration or even to rearrange the atoms in real-time. As discussed
in Section 1.1, this is a key technique for quantum gas assembly in that we can gain information
about the system that does not lead to atom loss and then can use this information during final
state preparation to construct a state of interest using the information gained [41, 42].
Specifically, this process of gaining information about the configuration of atoms before performing state preparation and then an experiment with knowledge of the initial configuration, allows
us to remove entropy from the system that would otherwise be inaccessible. For instance, experi-

49
ments that start with a Bose condensed gas and generate an array by going to the Mott insulating
regime of the Bose-Hubbard model cannot measure whether there are imperfections in the array
before running an experiment, which is a significant source of entropy in many of these systems. In
our experiment, this means that we can measure the configuration of the atoms, then perform laser
cooling to the ground state of the optical potentials, produce an array of (independently prepared)
ground-state atoms that can then be studied in much the same way as Mott insulators have beenstudied in quantum gas microscopes in recent years [18, 19, 81, 22].

3.3.1

Building uniform atom arrays

In past experiments, the information we have gained has strictly been used to separate datataken in a single run into separate datasets; data starting with a single atom in the left well of adouble-wee potential versus data starting with one atom in each well, for instance [43, 47]. However,this becomes increasingly inefficient as the number of wells used in a given experiment is increased
and a significant subset of the possible configurations does not provide useful data. Using enhanced
loading, as discussed in Section 3.1, improves the data collection rate, but given the capabilities ofour setup that have been discussed in Section 3.2, we can do better!
Specifically, we can use the information about the number of atoms available to performdifferent experiments depending on what we have. For instance, if we are loading 3 wells for aparticular experiment (e.g., one described in Chapter 6), we can perform certain calibration experiments when only one or two of the atoms were loaded, and then only perform the “real” experimentwhen all three atoms are present. We can even go a step further by using the configuration information provided by the same initial image to rearrange the atoms into a specific desired configuration
for different experiments, as has recently been demonstrated in one dimension in Ref. [41] and two
dimensions in Ref. [42]. Importantly, this style of rearranging will allow us to start each experiment with precisely the desired configuration of atoms, as long as the required number of atoms
was initially loaded.

50
3.3.2

Distilling motional state preparation

Aside from the entropy associated with a lack of knowledge of the presence of atoms when asystem is assembled, there is also entropy associated with the imperfect ground state preparation.
When planning techniques for future studies with larger arrays of atoms and, in particular, methods
to gain as much information as possible about the state of our system, it became clear that we
could use our ability to transfer atoms in different motional states between traps to selectively
isolate atoms that are not in the ground state of the optical trap, thereby performing a “controlled
distillation” of the quantum state.
While improving the ground-state preparation will always be the goal, some amount of imperfection is inevitable and my vision for future experiments is to use our ability to dynamicallyrearrange the optical tweezers to read out imperfect preparation. Specifically, by introducing an
empty trap near a trap containing an atom that has been cooled, we can pull off atoms that arelocated in excited states by dynamically shifting the bias between the two wells (a process that will
be described in more detail in Chapter 5). Importantly, this extra well can then be pulled off tothe side and read out at the end of the experiment; an atom in one of these wells indicates thatthe cooling was imperfect and the result of that run of the experiment should be ignored.11
In this process, depicted in Figure 3.9, we are purifying the motional state of the single atomby gaining as much information as we can. Thus, with this new technique, we will be able toimprove our ability to generate indistinguishable atoms after postselection, which is important not
only for the experiments described in this thesis, but also for all future experiments where quantum
statistics play a significant role in the dynamics of the system.

3.4

Improving readout of the final state of the system
So far, the theme of this chapter has been to initially prepare the system as well as we can,then gain as much information as possible about the system we do prepare before performing the11
An alternative is to simply drop these extra wells after pulling off atoms in excited states. In this case, we aretransferring imperfection in the cooling to effective loss of that atom, so for experiments where we want to post-selecton atom survival, we loose no significant information in this process.

51

Figure 3.9: This is a simple schematic showing how we would be able to further purify the motional
state of a single atom in a tweezer before using that atom in an experiment. By using a second
tweezer to pull excited motional levels out of the original trap, we can distill the state and postselectthe data from experiments where the atom was in the ground state.

experiment. Now, we turn our attention to trying to extract the most information about the stateat the end of an experiment. Specifically, at the end of all experiments, the atoms are separated
into resolved traps and some form of spin readout process is performed. Typically, this consists ofpushing out one of the spin states and performing standard fluorescence imaging on the remaining
population, but two alternative imaging methods that have been implemented will be discussed
here.

3.4.1

Spin-resolved imaging

For future experiments, it will be much more important for us to be able to resolve theindividual spin states at the end of an experiment compared to keeping the atoms, but only being
sensitive to the population. In particular, we would like to expand the imaging sequence to takea picture of the location of all “spin-up” atoms, immediately followed by the location of all “spindown” atoms because this will give us more information about the system compared to kicking out

52
all atoms that are in one spin state and imaging the location of the remaining atoms. The ability
to know whether an atom was actually in the spin-up state or lost due to imperfect preparation(or a collision with a thermal atom) makes a big difference when analyzing data. Specifically,
this spin-sensitive imaging provides additional information during each run of the experiment topostselect the data on runs of the experiment where known errors did not occur (such as loss orimperfect spin preparation), giving us a better signal-to-noise in the final dataset.
Aside from the benefits related to postselection, one version of this imaging method is, inprinciple, lossless, meaning that we can read out the spin of the atom without loosing the atomfrom the trap [82, 83]. In this case, we also would benefit by being able to reuse atoms in several
experiments before reloading, saving a significant amount of time because our loading procedurecurrently dominates our cycle time. There are two keys to properly implementing this method:High polarization purity to preserve the cycling transition (scattering photons only off of one of thespin states). And a large detection efficiency, since we can only scatter a few hundred photons offof the atom before it will be heated out of the trap. In our case, we require the high polarization
purity during the optical pumping cycle for Raman cooling, so we can easily achieve this by adding
another beam along the same optical path. Additionally, the large collection efficiency is “for free”
with the high numerical aperture lens we are using to collect the fluorescence. The limitation isthen only the signal-to-noise, which is limited by the readout noise of the EMCCD camera we areusing to detect the collected photons.

3.4.2

Raman-cooling imaging

The work discussed in this section was detailed in Ref. [48] and much of the information
below has been reproduced or adapted from this work.
Imaging 87 Rb via the fluorescence from standard PGC cooling light, as discussed in Section 2.1.2, is extremely efficient. The signal to noise is very high, even for relatively short images,
and the temperature of the atom scattering these photons typically remains below 50 µK. However,there may be situations where it would be beneficial to keep the atom very near the ground state

53of the traps (or if using a different atomic species where standard fluorescence imaging is not as
efficient) and for these cases one can consider collecting and imaging the scattered photons during aperiod of continuous Raman sideband cooling. This work in this section was published in Ref. [48]and several groups have more recently used this in quantum gas microscope experiments [84, 49, 50].
In such situations, efficient continuous Raman sideband cooling can be used to maintain asignificant ground-state fraction of trapped atomsand during the cooling process there are photons
scattered that can be collected to form an image. However, the operational principle of sideband
cooling relies on the existence of a dark state, composed of a spin degree of freedom and the motional
ground state, that is decoupled from the cooling light. Thus, to achieve a sufficient signal to noise
during a period of imaging requires some method by which the dark state is tunably compromised
to allow continuous photon scattering. There are numerous methods which could be chosen, but
for this work we applied a parametric modulation to the trapping potential to couple the atom outof the dark state during the imaging process. By varying the parameters of this drive, we increased
the detection fidelity and speed of the imaging procedure when a smaller ground-state fraction
could be tolerated.
The full optical tweezer apparatus is shown in Figure 2.5, with the most important elements
highlighted in Figure 4.1(a) for implementing imaging via Raman cooling [Fig. 4.2] [43, 85, 52,
53, 86]. The efficiency of the cooling cycle is determined by our ability to optically pump back to
the initial spin state while preserving the reduced motional state, which is related to the Lambp
Dicke parameter η OP = x0 k, where x0 = ~/2mω is the oscillator length for a particle of mass

m with trap frequency ω, and k is the wavenumber of the optical pumping (OP) light. For our

optical tweezer traps in this work, U/kB = 1.1 mK and the radial (axial) trap frequency is ωr =2π × 140 kHz (ωa = 2π × 30 kHz), which gives ηrOP = 0.16 (ηaOP = 0.35) [43]. This work uses
continuous Raman cooling, where the OP beams remain on during the entire procedure [39].
Each experimental cycle begins with a single atom that has been stochastically loaded byoverlapping a magneto-optical trap (MOT) with an optical tweezer trap [60, 70]. After 10 ms ofpolarization-gradient cooling (PGC), the atom is cooled to near the bottom of the trap, with a final

54
temperature of ∼ 15 µK [58, 87, 43]. The trap occupancy is then measured with a standard PGC
fluorescence image [78, 43], which works very well for 87 Rb. We use this as a pre-selection image toindicate the presence of an atom for the studies we present. After the pre-selection image, two more
images are taken in each experimental cycle. The second image is the exposure during continuous
Raman cooling. The final image is another PGC fluorescence image (the post-selection image) thatis used in combination with the pre-selection image to distinguish experiments where the atom waslost. For all of our data analysis, the recorded signal is the number of photons detected in a single
pixel on which the atomic signal is centered. This represents ∼ 1/3 of the total detected signal.
While the best SNR is achieved when all of the signal is binned onto a single pixel, we operate thecamera with a smaller pixel size to fully separate the signal from atoms in different traps. However,for absolute comparison of the measured scattering rates (discussed below), we integrate the signal
on the surrounding 3×3 pixels to ensure that the majority of detected photons are counted.
Figure 3.10(a) shows an example of the signal from 25 ms of PGC fluorescence imaging, which
provides a useful benchmark for imaging single neutral atoms [78, 79]. For this comparison, we use asingle pair of PGC beams in the σ + -σ − configuration [shown in Fig. 4.1(a)] with 0.61 mW of power ina beam with a gaussian waist of 1.66 mm 12 . This configuration minimizes interferometric intensity
gradients near the atom, which reduces fluctuations in the atomic scattering rate [58]. We red
detune the PGC light 38 MHz from the trap-shifted cycling transition, which corresponds to 13 MHzfrom the free-space transition [73, 88, 89, 90]. Taking into account the 6% collection efficiency of our
optical system, the measured scattering rate during PGC imaging is (2.6 ± 0.4) × 105 photons s−1 .
This is reasonable in comparison to the estimate of 4.5×105 photons s−1 [71], which assumes perfectalignment of the peak intensity of the PGC beams to the atom, balanced circularly-polarized beams,
and the ideal steady-state spin distribution for an atom at rest in one-dimensional (1D) PGC
light [58].
During Raman cooling imaging, 3D cooling must be performed even if our controlled paramet12

Note that the pre- and post-selection images used a six-beam PGC configuration, which we find achieves acomparable SNR.

55(a) 350

Single Atom in Trap

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Figure 3.10: Photon detection histograms. The red (blue) bars on the right (left) indicate images taken with (without) an atom in the trap, as determined by the pre-selection image. The
dashed (solid) black gaussian curve approximates the expected distribution for the atomic signal(background). (a) Fluorescence signal from 25 ms of PGC imaging. (b) Signal collected during
1 second of continuous Raman cooling. The presence of an atom is triggered using the pre- andpost-selection PGC images. (c) same as in (b), but now applying a sinusoidal parametric drivewith ∆U/U ∼ 0.055, resonant with the radial trap frequency, to the optical tweezer potential.

ric drive affects only one motional axis. This requirement arises from the optical pumping process,which can heat any motional axis via spontaneous emission. We use three beams to simultaneously
perform continuous Raman cooling along both the radial (RB1+RB2) and axial (RB1+RB4) dimensions of the trap. For the radial cooling, we take advantage of the near degeneracy of the tworadial trap frequencies of our trap to cool with only one beam pair and still achieve the necessary

56(albeit slower) cooling along both dimensions.
Figure 3.10(b) shows a histogram of the number of photons detected during 1 second ofcontinuous Raman sideband cooling. The red (blue) data represent experiments where an atom
was present (absent) in both the pre- and post-selection images, from which we infer that there
was an atom present (absent) during the entire period of continuous Raman cooling. Selecting
experiments where the atom is not lost allows us to directly characterize the imaging procedure,
independent of imperfections (such as the vacuum lifetime) that are specific to our setup. We notethat the signal peak is smaller than the background peak because the trap loading rate was smaller
than experiments in Fig. 3.10(a) (roughly 25% instead of 60%) due to a lower MOT density duringloading. This does not affect the results other than to reduce the available statistics for the signal
data points compared to background.
In principle, we should expect no signal when no parametric drive is applied to the trap
because the atom should remain in the dark state and thus scatter no photons. The main contributions to the signal in Fig. 3.10(b) are from off-resonant carrier [∆n=0 in Fig. 4.1(b)] Raman
transitions, which initiate an optical pumping cycle, and, to a lesser extent, residual coupling of thedark state to the optical pumping beams. Off-resonant carrier transitions are relevant in these experiments because the ratio of the carrier Rabi rate (the transition linewidth) to the trap frequency(the detuning from the transition) is non-negligible, leading to a finite population transfer. The
measured scattering rate for the data in Fig. 3.10(b) is (1.9 ± 0.3) × 103 photons sec−1 . We estimate
the maximum scattering rate from off-resonant carrier transitions to be 2 × 103 photons s−1 bysumming the scattering rates from two 1D calculations of the axial and radial rates.
The fidelity of the imaging is determined by a combination of the signal amplitude and theability to separate the signal and background distributions. The expected standard deviation in thenumber of detected photons is a combination of shot noise (Poisson statistics) and the readout noise√of our detector (σRO = 6.6 photons). The expected distributions for the atomic signal (background)pixel

are approximated by the dashed (solid) gaussian curves on the histograms in Fig. 3.10. Based on
the overlap integral of two normalized gaussian distributions with means and standard deviations

57
from the data in Fig. 3.10(b), we estimate a minimum detection error rate of 1.4×10−2 for 1 secondof integration time [82] 13 . Note that this is a measure of the achievable SNR, but the effects ofatom loss are removed based on the pre- and post-selection images.
To improve the signal detected during continuous Raman cooling, we apply a parametricdrive by modulating the total optical power generating the optical tweezer trap at twice the radial
trap frequency (2ωr = 2π × 280 kHz). This drives the atom out of the motional ground state,allowing the cooling cycle to restart, and thus more OP photons are scattered during the exposure.
Figure 3.10(c) clearly shows an increase in the atomic signal during Raman cooling imaging with aparametric drive applied to the trapping potential. The increased signal corresponds to a reductionof the minimum detection error rate to < 10−4 for the same 1 second integration time. For
comparison, we estimate that 20 ms of our PGC imaging, which has a much higher scattering rate,
can achieve the same minimum detection error rate.
The Raman cooling imaging procedure can be simply understood as the competition between a variable parametric excitation rate and the (constant) continuous Raman cooling. The
competition of these rates rapidly yields an equilibrium ground-state fraction, and hence non-dark
fraction, which largely determines the measured scattering rate. This simple description is effective
because the parametric drive does not directly interfere with the cooling and the Raman coolingcycle occurs on sub-millisecond timescales compared to the 1 second image.
The average number of photons detected during Raman cooling imaging is observed to increase as a function of the applied parametric drive amplitude [red circles in Fig. 3.11(a)]. Hence,increasing parametric drive amplitude reduces the equilibrium ground-state fraction, allowing us
to control the atomic fluorescence rate. The atom signal can be compared to the average numberof background photons detected (blue squares) measured at each drive amplitude. The amplitudeof the parametric drive is defined as the ratio of the peak-to-peak variation of the trap depth ∆U
to the trap depth in the absence of a drive U .
13

The minimum detection error rate δ is half the overlap integral of the two normalized gaussian distributions,which requires an optimal choice of threshold to achieve. The fidelity of the measurement is then defined to be
F = 1 − δ (i.e., δ = B = D in Ref. [82]).

Photons Detected
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Figure 3.11: (a) Varying the amplitude of the parametric drive applied to the optical tweezer
potential during 1 second of Raman cooling imaging. The green diamonds are the atom survival
probability, as determined using the pre- and post-selection images, and the orange dashed line is
the measured vacuum lifetime limit. The red circles (blue squares) are the signal collected with
(without) an atom in the trap during the entire imaging period. The blue line indicates the averagebackground value of 16.5 detected photons. The inset displays the equilibrium radial ground-state
fraction (PGS radial) measured via Raman sideband spectroscopy immediately following a periodof Raman cooling imaging [43]. (b) Signal as a function of the parametric drive length, for fixed
total imaging length (1 s) and drive amplitude (∆U/U ∼ 0.04). The red line is a linear fit to thedata. All error bars indicate the standard error of the measurement.

Up to intermediate drive amplitudes, we characterize the equilibrium radial ground-state
fraction during imaging by performing radial sideband thermometry immediately after a period of
Raman cooling imaging. We verify that the equilibrium ground-state fraction varies with the applied

59
parametric drive amplitude [inset of Fig. 3.11(a)]. For the lowest drive amplitudes (∆U/U . 0.02),
thermometry is limited by the resolution of our sideband spectroscopy [43], suggesting a radial
ground-state fraction PGS > 80%. We further find that a significant ground-state fraction PGS ≥55% is maintained for drive amplitudes up to ∆U/U ∼ 0.06 (near the maximum drive withoutloss). The scattering rate observed at this drive amplitude is (4.9 ± 0.7) × 103 photons s−1 . We
estimate an expected scattering rate of ≤ 7×103 photons s−1 by weighting the scattering rates fromeach of the motional excited states undergoing Raman cooling, assuming a thermal distribution
corresponding to the measured motional occupation at ∆U/U ∼ 0.06 [43].
Further, the survival probability of the atom is measured as a function of the parametric drive
amplitude [green diamonds in Fig. 3.11(a)] by recording the probability for an atom to appear in
the post-selection image, given that it was present in the pre-selection image. The orange dashedline represents the expected survival probability due to the vacuum lifetime (τ = 7.1 s) over thecourse of the 1.023 seconds between the pre- and post-selection images. The atom loss is initiallyconsistent with background collisions, but at larger drive amplitudes the parametric excitation rate
exceeds the Raman-cooling rate and the atom is quickly heated out of the trap.
In Fig. 3.11(b), we perform 1 second of continuous Raman cooling and apply a parametricdrive for a fraction of the image. The constant drive amplitude of ∆U/U ∼ 0.04 realizes anequilibrium radial ground-state fraction of ≥ 70% [inset of Fig. 3.11(a)]. The measured data
(red circles) are well fit by a linear increase (red line), which is further evidence that the radial
ground-state fraction rapidly reaches equilibrium. For the same drive amplitude, we estimate thata minimum detection error of < 10−2 can be reached after about 0.5 seconds of imaging. This iscomparable to the site-resolved imaging used in recent optical lattice experiments [18, 19].

Chapter 4
Preparing and Interfering Indistinguishable Atoms

In Chapter 3, we reviewed the methods by which we can generate an array of atoms for use
in an experiment, as well as ideas for how to read out these atoms efficiently, gaining as muchinformation as possible about the initial state prepared and the final state after an experimentfor use in the data analysis. However, we left out a crucial step for performing real experiments:
Bringing the atoms to the ground motional state of the traps before initiating dynamics. In this
Chapter, I will review the basics of Raman sideband cooling and how we have implemented it tocool single atoms trapped in optical tweezers [43]. The Raman cooling is an important technicalaspect of the experiment, for which I provide details here that are not necessarily in any of ourother published work, though reviewed some in Adam Kaufman’s thesis [57]. This work led to theobservation of two-particle quantum interference of independently prepared atoms, verifying theability to make atoms indistinguishable via laser cooling alone. This work is described in Ref. [39, 57]
and, hence, I do not cover the details here. However, I do briefly discuss the application of ourability to measure the indistinguishability of two atoms to improving our initial state preparation.

4.1

Raman sideband cooling
Reliably bringing atoms to the motional ground state of a trapping potential is a hard prob-

lem. In many ultracold atomic gas experiments, a large cloud of atoms is trapped, cooled to thelimit of traditional laser cooling techniques (in the few µK regime), and then initiating some formof evaporative cooling reach sufficiently small temperatures for Bose condentation (or the genera-

61tion of a degenerate Fermi gas) with the remaining atoms. While this is effective and has led tosignificant progress in the field of ultracold gases, it is inefficient both in terms of the number ofground-state atoms versus number loaded and in terms of the length of the cooling process, and is
not amenable to quantum gas assembly as envisioned in Chapter 1 [1, 2, 3, 4, 12, 91, 92, 18, 19].
For cooling single atoms to the ground state of tightly confining traps, Raman sideband cooling has been extremely successful, and quite generally applicable, being used for trapped ions [93],atoms in optical lattices [52, 53, 54, 56, 86, 94], optical cavities [95, 96, 97], and optical dipole
traps [43, 44]. The operational principle is to drive transitions that manipulate both the spin andthe motional degrees of freedom, decreasing the motional quantum number in each step, and eventually reaching a state that is dark to the cooling light, which occurs in the motional ground-stateof the trapping potential. Below, I will briefly review the form of pulsed Raman sideband coolingthat we use in our system, but there are many other forms of Raman sideband cooling, such as continuous sideband cooling and projection sideband cooling, in addition to a large amount of detailed
derivations that I am not going to repeat in this thesis because they are covered very thoroughly
elsewhere, in particular in Refs. [98, 99, 57].

4.1.1

Non-degenerate pulsed Raman sideband coolingFirst, note that much of the content of the next two sections is detailed in Refs. [57] and [43],but certain details have changed since the original publication and are discussed here. Here, we
focus on the current implementation of a nondegenerate, pulsed Raman cooling scheme. Briefly,
this is a two-step process that alternates between driving a coherent, energy-reducing spin-flip
and then driving spontaneous transitions to bring the atom back to its original spin state, while
maintaining the reduced motional level. By repeating such a procedure many times, the population
accumulates in the ground state of the trap with the original spin state.
The key component of this procedure is the ability to drive a coherent, energy-reducing spinflip; this is achieved using a two-photon Raman process, where two far-off-resonant beams havea frequency difference that (nearly) matches the microwave transition frequency for the spin-flip

62

(b)

(a)
Axial view:

RB4
B

RB1
OP

OP

RB1

RB2
B

x

z
y

PGC

y

RB3

Δn =
+1
Δn =
0
Δn =
-1

Top view:

n

n+1
n
n-1

Figure 4.1: a) The content of this diagram is contained in Figure 2.5, but here we highlightthe beams used during Raman cooling (RB1-RB4, and OP), as well as the beam used for bothsub-Doppler cooling and imaging (PGC). The PGC beam is used to initially cool the atoms suchthat the temperature of the atoms is amenable to Raman cooling. The polarizations of each ofthese beams is also indicated in the figure with a dashed line (or circle). b) Diagram depicting two
ground-state spin levels the beams used to perform a Raman transition. The black vertical arrow
represents one of the linearly polarized beams (RB2-RB4), and is far-detuned from the excitedstate energy level. The three arrows on the right indicate three possible transitions selected by theσ + Raman beam (RB1), all of which flip the spin of the atom, but address transitions between
different motional states (red for ∆n = −1, black for ∆n = 0, and blue for ∆n = +1).

that we want to drive. These two beams then cause the atom to undergo a virtual process, wherea photon is absorbed from one beam and emitted into the other, changing the spin-state of theatom in the process. As depicted in Figure 4.1(b), by adjusting the frequency difference between
the beams, we can simultaneously change the motional state of the atom in the trap, meaning thatfor particular frequency differences, we drive coherent, energy-reducing, spin-flips that are the firststep in the Raman cooling procedure. Then, the spontaneous process that brings the atom backto its original spin state is simply the optical pumping that was described in Section 2.1.3.
Figure 4.1(a) highlights the relevant beams for performing the Raman transitions in our setup.
In particular, we highlight that the optical pumping beams (OP) are aligned along the quantization
axis and are copropagating with the Raman beam, RB1, which is used when addressing all threeaxes (this ensures that all three beams have the same high polarization-purity). The remaining
three Raman beams (RB2, RB3, and RB4) are all aligned orthogonal to RB1 and the three pairs are

63
a)

...

b)
OP

Top Radial

OP

Axial Raman

OP

Bottom Raman

OP

Figure 4.2: This is a schematic representation of the pulsed Raman cooling procedure. a) Alonga single axis, this shows the pulse sequence and how the population of atoms is brought to a lower
temperature. by repeating a single Raman cooling cycle repeatedly (each two-step sequence ishighlighted by the dashed box around it). b) Pulse sequence demonstrating how we sequentiallycool all three axes (each completing a full Raman cycle before the others have a chance to accumulate
population in the three dimensional ground state of the trap.

used to address three orthogonal axes of the trap. Because these beams are pointing along different
axes, there is a net momentum kick imparted to the atom during a two-photon Raman transition,
which then allows us to couple different motional states of the atom. Assuming a nonzero total
momentum kick, then we can selectively address transitions that change the motional state by aknown number of quanta (typically either ∆n = ±1 for the blue and red sidebands, or 0 for the
carrier transitions) while flipping the spin of the atom, as depicted in Figure 4.1(b).
A schematic representation of the full cooling procedure is shown in Figure 4.2(a). Afterloading of the optical tweezer with a single 87 Rb atom, sub-doppler cooling is performed to lower
the initial temperature of the atom such that the atom starts near the bottom of the trap; this isimportant for ensuring that the cooling procedure is effective, as discussed in Section 4.1.2. Then,before the Raman cooling procedure is initiated, optical pumping is performed to set the spin stateof the atom to the |F, mF i = |2, 2i = | ↑i Zeeman sublevel. Then, a pair of Raman beams (RB1plus RB2, RB3, or RB4) is used to drive a coherent rotation of the spin to the |1, 1i = | ↓i Zeeman
sublevel, while changing the motional level by ∆n = −1 (to be explicit, this is a Raman transition
between the states | ↑, ni → | ↓, n − 1i), as shown in Figure 4.2. The spin state is then optically
pumped back to the original spin state | ↑i, while preserving the reduction in the motional state,

64
and the final two steps can be repeated many times to ensure that the atoms end up in the groundstate at the end of the process. Note that this procedure is shown for a single motional axis, which isdetermined by the choice of the pair of Raman beams and the relative frequency difference, but weactually switch between the different axes for each cycle to bring the atom to the three-dimensional
ground state.
Near the end of a successful Raman cooling procedure, the majority of atoms are in the “darkstate”, meaning that the atom is not affected by the any of the light applied. Specifically, when
applying light for a Raman transition that is designed to reduce the motional state by one quantaduring the spin flip, then no resonant transition will occur when you are already in the groundstate of the trap. Additionally, because of the design of the optical pumping process, the | ↑ spinstate is already dark to the optical pumping light. Therefore, the state | ↑, 0i remains the darkstate for the entire Raman cooling procedure.

4.1.2

Efficient cooling in optical tweezers

One of the most important considerations for the successful implementation of ground-state
Raman sideband cooling is to have sufficiently strong harmonic confinement of the atom (or atleast approximately harmonic for the relevant energy scales). Specifically, this confinement isneeded in order to resolve transitions between individual motional states, as well as to ensure thescattering of a small number of photons (during optical pumping) will not result in the atom’s
motional state being increased more often than it is reduced. The parameter describing used todescribe the confinement of a trap is called the Lamb-Dicke parameter [100]. The Lamb-Dicke
parameter is defined as η = kr x0 , with kr being the recoil momentum imparted to the atom by
the light addressing a given transition and x0 the harmonic oscillator length of the ground state
wavefunction. Equivalently, this can be written as η 2 = ωωr0 , with ωr the recoil frequency (where theenergy gained by the atom emitting a photon is given by Er = ~ωr ) and ω0 is the trap frequencyof the harmonic potential.
Note that the probability to preserve the reduced motional state is dependent on the average

65
number of photons scattered during the optical pumping stage, the Lamb-Dicke parameter η OP
for scattered optical pumping photons, and the motional state that the atom is in when scattering
photons. There is a fundamental minimum to the number of photons scattered during opticalpumping that is set by the decay pathways from the excited state. Conservatively, we assume 3
scattering events per optical pumping cycle, which would say that if we increase the energy level ofthe atoms less than once every three scattering events, then we will have a net cooling effect during
each cycle. Importantly, if there are no mechanisms for heating or scattering out of the dark state,then, if a net cooling is achieved, arbitrarily high ground-state fractions can be generated with along enough period of cooling.
However, there is always some limit to the fidelity of the dark state, either from a mechanismthat can directly heat the atom out of the ground state, or due to the polarization of the opticalpumping light not being perfectly σ + and causing an undesired spin flip. Heating of the atom cancome from the elastic scattering of photons, possibly from the trap or Raman light, or from intensity
noise modulating the trap at a frequency resonant with motional state changes. In either case, thesewill never be completely gone, but can be minimized with relatively standard techniques. On theother hand, preventing unwanted spin flips from the optical pumping light is more challenging
because this light is designed to bring the atom back to the | ↑i spin state quickly, and therefore
has a relatively high scattering rate of ΓOP ∼ 100 kHz. The “darkness” of the dark state thendepends on our ability to prevent unwanted scattering events, which relies on high polarization
purity. To be specific, from the | ↑i = |2, 2i spin state, the OP light (from Figure 2.1) would beresonant with transitions to both the |F 0 , m0F i = |2, 2i and |2, 1i states, if the polarization were
not purely σ + . As discussed in Section 2.1.3, we are able to achieve a polarization purity to better
than a part in 1000, which means the scattering rate out of the dark state is less than 100 Hz.
As mentioned above, the probability of the atom heating during a photon scattering event isalso dependent on the motional state of the atom, which we can quantify by defining an effectiveOP = (2n̄ + 1) η OP 2 . This modification simply means that to achieve
Lamb-Dicke parameter as ηeff

efficient Raman sideband cooling, we need to start with a low thermal occupation before initiating

66
the Raman cooling process which the in-trap PGC cooling provides.
The coupling between the motional states is also parameterized by a Lamb-Dicke parameter,
but in this case the momentum is replaced by ∆k, the amount of momentum imparted to the atom
during a Raman transition (the difference in the momenta of a photon absorbed from one beam andemitted into the other). Therefore, the Raman Lamb-Dicke parameter is defined as η R = ∆kR x0 .
This parameter determines the ability to drive sideband transitions with a given set of beams and,along with the motional occupation, defines the relative rate at which sideband transitions will be
driven in comparison to the carrier (∆n = 0) transition.1 The strength of this coupling will affectthe spectra resolution of the sidebands from the carrier transitions because it changes the ratioof the Rabi rates, and thus the widths of the spectral features. This also affects the final coolingefficiency because a reduction in the spectral resolution leads to a higher probability of driving
an off-resonance carrier transition (instead of a sideband). This directly leads to the scattering of
optical pumping photons without the benefit of reducing the motional state first, directly leading
to heating of the atom.
This highlights another important concern for maximizing the efficiency of the cooling procedure, in that improving the spectral resolution between the sidebands and the carrier is beneficial.
Specifically, there are two things we have changed that improved this in our setup: Changing thealignment of the axial Raman beam (thus increasing the Lamb-Dicke parameter η R ) and using
Gaussian shaped pulses to narrow the spectral extent of each pulse.

4.1.3

Raman sideband spectroscopy

After performing a sequence or Raman sideband cooling, we can estimate the motional state
occupancy (or, more specifically, the ground-state fraction) by performing Raman sideband spectroscopy. In this procedure, we perform a π-pulse for the blue sideband (assuming the sideband
Rabi rate associated with atoms from the ground-state) while varying the relative frequency be1
This also has some significant consequences, where atoms with too high a motional occupation will undergo a
2π rotation rather, getting some fraction of the population stuck in higher harmonic oscillator levels. In the end, thisenforces a similar requirement that we need to start with a low enough initial temperature.

67

b)

=

=

a)

-

-

(

-

)

(

)

Figure 4.3: These plots show (a) the radial and (b) the axial Raman sideband spectrum taken after
a three-dimensional cooling process that has been optimized by improving the indistinguishabilityof the atoms, as discussed in Section 4.2.2. The dashed black lines indicate the expected locationof the red (energy reducing) sideband based on the measured trap frequencies. The lack of a red
sideband (on the right) indicates a large ground state fraction. In (b), the central peak is theoverdriven carrier transition, which adds noise that partially overlaps with the location of both the
sidebands.

tween the two Raman beams. The resulting spectra are shown in Figure 4.3for the radial and axial
dimensions after the cooling procedure. In both of the spectra, the energy-reducing red sidebands
are not noticeable, suggesting that the atoms are in the ground-state of the trapping potential.
The large asymmetry between the sidebands (and especially the absence of the red sideband) issuggestive of the large ground state fraction after this cooling process.

4.2

Quantum interference of indistinguishable particles
The details of our experiment that led to the observation of two-particle quantum interference

are detailed extensively in Refs. [39] and [57], hence I will only briefly introduce the experimental
sequence and relevant observable before discussing how we have more recently used the results ofthis experiment as a tool for optimizing our initial preparation procedure.

4.2.1

Measuring indistinguishability via tunneling in a double well potential

A schematic of the tunneling procedure is shown in Figure 4.4, along with a representative
set of recent single-particle tunneling data that has been post-selected on the single atom not being

68

b)

c)
0.9 µK

0.9 µmd)
Probability in Bottom Well

PLeft:

1.0

0.14µK

1 mK

a)

Postselected Tunneling Data

0.8
0.6

FromBottom
Left Well
Well
From
From
Well
FromTop
RightWell

0.4
0.2
0.0
0

2

4

6

EvolveTime
Time
(ms)
Tunnel(ms)

8

10

Figure 4.4: Schematic of the tunneling procedure used to observe interference between indistinguishable atoms. a) Atoms are loaded into ∼1 mK deep traps that are in separated by 1.57µm to
perform Raman cooling to the motional ground-state. b) The traps are dynamically reconfigured
to both partially overlap and be shallow enough that there is a tunnel coupling between the sidesof the double well potential, in the example shown this means a 0.9 µm separation and 0.9 µK
single-well depth. c) The atom is allowed to evolve in this final trap configuration for a periodof time, during which the atom can coherently tunnel between the two wells. d) The traps are
separated and imaged to measure the population in each well. An example of data taken with thetrap parameters listed above is shown, where the probability for the atom to end in the left well isshown as a function of time for a single atom initially in the left (red) and right (blue) wells.

69

P11
P11

0.5
0.4
0.3
0.2

0

πΠ

22π
Π

3Π
3π

44π
Π

Single spin rotation pulse area

55πΠFigure 4.5: In this figure, we observe two-particle quantum interference as a function of the
distinguishability of the two atoms. The black points on this plot show the measured probability
for two atoms to end up in separate wells after tunneling for a period that splits each atom into
both wells. The two-particle quantum interference is the reduction of this probability below the
purple points. The purple points show the expected probability for distinguishable atoms to end
up in separate wells, calculated from the single particle data (as shown in Figure4.4), which would
ideally go to 0.5. In this figure, we are specifically varying the spin state of one of the atoms
before this tunneling procedure occurs. When the atoms are in opposite spin states, they are thendistinguishable and there is no interference “dip” below the distinguishable limit, but when they
are the same spin state, we do see this interference.

lost before the end of the experiment (as discussed in Section 3.4.1). The coherent tunneling of
single atoms between the wells forms the basis of our observation of quantum interference between
two atoms. In this sequence, two optical tweezers are stochastically loaded when separated andthe trapped atoms are prepared in the motional ground state, as described in Section 4.1.1, before
repositioning the wells to initiate a period of tunnel-coupling. After a variable period of evolution,
the wells are separated and the population measured.
When two atoms are initially loaded, one in each of the two tweezers, then they will simultaneously experience a tunnel-coupling between the two wells, and periodically both atoms will be
split evenly between the two wells. However, because they started in opposite wells, the phases
between the two atomic wavepackets will be different, which leads to quantum interference that
results in the atoms always being observed in the same well when the atoms are indistinguishable
in all other degrees of freedom [101, 39, 57, 102]. Figure 4.5 demonstrates the change in the natureof the interference between two atoms split evenly between the two wells, as the distinguishabil-

70
ity between the two atoms (in the form of the initial spin state) is tuned via site-resolved spin
addressing.

4.2.2

Using indistinguishability to improve state preparation

As demonstrated in Figure 4.5, the depth of the two-particle interference between particles
evenly split between the two wells of a double-well potential is a good measure of the distinguishability of the two atoms. Therefore, because the goal of our initial state preparation is to put all
atoms in the motional ground state of their respective traps, we can use this measure of the distinguishability to optimize our state preparation. This is equivalent to the experiment demonstrated
in Figure 4.5, except that instead of purposefully rotating one of the spins (and not the other), we
will change the global parameters used during state preparation, such as the cooling frequencies
and Raman beam powers, and choose to use the values which give a minimum in the interference.
An example of the signal from this optimization procedure, as applied to the parameters usedduring the Raman cooling procedure, is shown in Figure 4.6. Specifically, for this run, the axial
Raman beam frequency used during cooling was varied to find where the axial cooling rate was
maximal. After tuning this and several other cooling parameters, we achieved the sideband spectra
shown in Figure 4.3.

71

1.0

P11

0.8
0.6
0.4
0.2
0.0

-20

-10

0

10

20

30

Detuning (kHz)
Figure 4.6: In this figure, we are observing the quantum interference between two atoms as wevary the axial Raman beam frequency used during the initial state preparation. We can clearly seethat the interference is maximized for a particle frequency for Axial sideband cooling. This allows
us to set the exact frequency used to the best of my abilities, and is often the best signal aroundfor optimizing any given Raman cooling parameter.

Chapter 5
Entangling Independently-Prepared Neutral Atoms

We have already seen that the quantum statistics for identical particles can have drastic effects
on the dynamics of indistinguishable atoms in a double well potential. However, in this chapter wewill study effects in a system where we purposefully prepare distinguishable atoms and allow themto interact in a single tweezer. In particular, we observe how the quantum statistics of the atoms
imposes additional restrictions on the total quantum state, which leads to the correlated degrees of
freedom will become coupled in such a way that they can no longer be described independently [103].
Specifically, by preparing two atoms in different motional states of a single trap and with opposite
spin states, the symmetrization of the total wavefunction will lead to spin-exchange dynamics,
which periodically generated spin-entanglement between the two atoms [39].
We note that this type of coupling forms the basis of many exciting areas of research, including quantum information protocols with quantum dots (where dynamics similar to those observed
here are used to entangle and manipulate qubits), to effects observed in unconventional superconductors, which will be discussed further in Chapter 6. The information presented here thus lays thefoundation for the discussions in Chapyer 6, but certain details will be left out and can be foundin Ref. [47] and [57].

5.1

Dynamical entanglement via spin-exchange
The work discussed in this section was published in Ref. [47] and for for any additional

information see discussions there, as well as in Ref. [57].

73

(b)

|ψT (x1 , x2 )|22x2(r0)

|T 

(c)

Time

| ↑g L | ↓g R
z

0

-2
-2

Jex
|e

0 2
x1(r0)|ψS (x1 , x2 )|2x2(r0)

2

|g
x

|S

y
x

φ

0

| ↑g L | ↓g R + eiφ | ↓g L | ↑g R
√
2

-2
-2

0 2
x1(r0)| ↓g L | ↑g R

Figure 5.1: a) Starting on the left, two atoms are loaded into separate optical tweezers and preparedin their motional ground state and in opposite spin states. Then, the tweezers are reconfigured to b)
The on-site contact interactions shift the energy of triplet spin-state (which will be in a symmetricspatial wavefunction), but due to the symmetrization of the total wavefunction, there is no contact
interaction in the singlet state (which must be in an antisymmetric spatial configuration). The
differential phase accumulation between these spin states thus leads to the spin exchange dynamics
that we wish to observe. (Note that this figure is adapted from Ref. [47].)

74
In this section, I will discuss our demonstration of the use of local interactions to entangle
the spins of two atoms that are initially trapped in separate wells. When the two atoms areplaced in the same well, the symmetry of the two-particle spatial wavefunction determines theinteraction energy of the particles. However, assuming the atoms are identical in all other degrees
of freedom, the spatial symmetry of the two-particle wavefunction directly determines the spinsymmetry, which means that the singlet and triplet spin wavefunctions will experience difference
interaction strengths, as shown in Figure 5.1. It is this difference in interaction strength that leads
to a differential phase accumulation between the singlet and triplet spin states, resulting in theexchange of spin between the two atoms.
In the experiment, we can observe this effect by preparing an initial state with two atoms
in two optical tweezers, bringing both to the ground state, and then flipping the spin of one of
the atoms. The remainder of the procedure is depicted in Figure 5.2, where we sweep the wells
together in an asymmetric configuration such that we are not resonant with tunneling, and slowly
reconfigure this tilt to couple the ground motional state of the left well to the first excited motional
state of the right well [Fig. 5.2], adiabatically transferring the atom into the right well in a process
I will refer to as an “adiabatic passage”. After a period of evolution, this process is reversed toseparate the two atoms, and the the spin state of the atoms in the two separate wells is measured.
The observed dynamics are shown in Figure 5.2, exhibiting anti-correlated oscillations between the |↑ie |↓ig (green) and |↓ie |↑ig (purple) spin configurations. We also verify that, as wevary the depth of the well during the exchange period (and only during this hold to ensure thatthe remaining process is identical), the oscillation rate changes in a way that is consistent with the
expected Hubbard parameters [Fig. 5.2(c)], as calculated using a full three-dimensional model ofour system, as discussed in Refs. [39] and [104].
These measurements show correlations in single-particle spin states. However, to use theentanglement generated in this process for future quantum information protocols, we must verify
that the phase of the entangled state is preserved after separation of the particles. The entanglement verification protocol for separated atoms is summarized in Figure 5.3(a). For explanatory

75

a)

ExchangeRb)

Probability

0.8
0.6
0.4
0.2
0.0

10

20
30
Exchange time (ms)40

Spin exchange frequency (Hz)

L

c)
250
200
150
100

0.0
100
150
200
250
Single tweezer depth (kHz)

Figure 5.2: a) Schematic representation of the experimental procedure for observing spin exchange
oscillations. Starting with atoms in separate wells and in opposite spin states, we then proceed byrearranging the traps to couple the atom from the ground-state of the left well to the first excitedstate of the right well. Then, we let the system evolve for some period of time before separating
the wells and observing the final spin configuration of the system. b) Time-evolution of the spin
configurations over time. We see the anti-correlated oscillations of the spin states that is indicativeof spin-exchange dynamics. c) Varying the depth of the trap during the period of evolution, we
see that the oscillation rate varies in agreement with the theory calculated for the trap parameters
being used. (Note that this figure is adapted from Ref. [47].)

76
purposes, we first focus on the case when the particles are separated after an exchange time oftent = nπ/2Jex where n is an odd integer. The entangled state after the second adiabatic passage is|ψ± i = √12 (| ↓iL |↑iR ± i| ↑iL | ↓iR ), omitting from now on the ground-state (g) motional subscripts
to simplify notation. The |ψ± i states correspond to the gray and orange Bloch vectors, respectively, in Figure 5.3(b). We then apply a magnetic-field gradient that imposes a difference, δ~, inthe | ↑i ↔ | ↓i single-atom-transition energy between the left and right optical tweezer. By applying
the gradient for a time tg , a transformation |ψ± i → √12 (| ↓iL | ↑iR ± ieiδtg | ↑iL | ↓iR ) is achieved. As
a function of tg , the state rotates between the singlet [pink in Figure 5.3(b)] and triplet (blue) withfrequency δ. We then apply a global π/2 pulse in the {| ↑i, | ↓i} sub-space. This pulse maps the
singlet back to itself, while it maps the triplet to a Bell state √i2 (| ↑iL | ↑iR + | ↓iL | ↓iR ). Therefore,by measuring the probability that the spins are aligned or anti-aligned as a function of tg , we canobserve singlet-triplet oscillations whose amplitude characterizes the two-particle coherence. We
Pquantify this probability with the parity Π(tg ) = j Pj (−1)j , where Pj is the likelihood to mea-

sure j atoms in the spin-down state [105, 106, 27, 26]. The parity is equivalently the projectionof the Bloch vector in Figure 5.3(b) onto the x-axis of the effective two-particle Bloch sphere (inthe Sz = 0 subspace) prior to the π/2-pulse, and hence the gradient is essential because, thoughentangled, the states |ψ± i (gray, orange) exhibit zero parity after application of a π/2-pulse.
We demonstrate the outcome of the verification protocol on the state |ψ+ i in Figure 5.3(c).
We plot Π(tg ) after the microwave π/2 pulse, and observe oscillations in the parity signal as thegradient time tg is scanned. The contrast of these oscillations is consistent with what is expected
given the exchange oscillation contrast in Figure 5.2, and non-vanishing parity oscillation would
certify entanglement in the ideal case of perfect spin preparation. However, we have imperfectspin preparation, and the erroneous spin populations outside the {| ↓iL | ↑iR , | ↑iL | ↓iR } manifold
could lead to parity oscillations even in the absence of entanglement. We have derived a conditionon the parity oscillation contrast that is necessary and sufficient to certify entanglement and isexperimentally the most accessible way to see there is entanglement in our system (for a fullderivation of this condition, see Ref. [47]). We relate the measured parity contrast, C, to the

77

a)a)

Spin-exchange operations

|ψ+ 
y

+1
-1
-1
+1

0.8
x

x

0.4

R

d)

y

c)

L

Time

| ↑L | ↓Rzb)

Parity detection|ψ− | ↓L | ↑R

0.6
0.4
0.2

Parity
Π

0.2
0.0

-0.2
-0.4

0

2

4
6
tg (ms)

8

10 0

5

10
15 20 25
Exchange time (ms)Figure 5.3: a) Procedure for performing spin exchange, followed by the entanglement verification
procedure, which consists of apply a magnetic field gradient to the separated atoms and thenperforming a spin parity readout sequence. b) Two-particle Bloch sphere representing the Sz = 0
subspace of the spins. The eigenstates of the spin-exchange interaction are along the x-axis (singlet
and triplet), while the eigenstates in separate wells and in the presence of a gradient are along thez-axis. Then, spin exchange becomes a σx rotation, while the gradient evolution is a σz rotation.
Then, upon application of a microwave π/2 pulse, we map the projection of the Bloch vector onto
the x-axis to parity. c) Evolution of the parity after performing spin-exchange for a time thatshould result in an entangled state and then varying the length of the gradient evolution. The
gray bar represents the calculated oscillation bound of oscillation amplitude that would verify thepresence of spin entanglement. d) Varying the length of spin-exchange evolution before performing
the readout sequence for a fixed gradient evolution time. (Note that this figure is adapted from
Ref. [47].)

78
measured probabilities (P ↑↑ ,P ↓↓ ) that the spins are erroneously prepared in the same spin-state: If
C > Cbnd = 4(P ↑↑ P ↓↓ )1/2 , then the state is entangled. By directly measuring the spin populations
and their associated uncertainty, we ascertain Cbnd = 0.133(25) as indicated by the dashed lines
in Figure 5.3(c). The observed parity oscillation contrast C = 0.49(4) exceeds Cbnd by morethan 7σ, certifying the presence of entanglement in the final state of the separated spins. We verify
entanglement without correcting the measured parity for experimental imperfections, such as single
atom loss due to background collisions.
While in Figure 5.3(c) we varied the parity detection parameters via tg , in Figure 5.3(d) wemeasure the dependence of the parity on the exchange time at fixed tg , thereby observing oscillations
as the exchange interactions periodically entangle and unentangle the two atoms. We fix tg in theparity detection such that the entangled state |ψ+ i [gray lines in Figure 5.3(b,d)] is rotated to apeak in Π, corresponding to the creation of the triplet [blue lines in Figure 5.3(b,c)]. Because thistg amounts to a π/2 rotation about the z-axis of the Bloch sphere, it will also rotate |ψ− i to thesinglet, which corresponds to maximal negative parity. In the lower panel of Figure 5.3(d), we show
how the parity measured under these conditions oscillates at the exchange frequency Jex /(2π~) .
For comparison, in the upper panel, we show the measured exchange oscillations (purple, green)
without the parity detection. At the linear points of the exchange oscillations, one expects maximal
entanglement corresponding to states |ψ+ i (gray) and |ψ− i (orange) and thus the extremal pointsof the parity. At the minima and maxima of the exchange oscillations, the atoms are unentangled
and the parity vanishes.

5.2

Spin-entangled states in spin-dependent traps
In the verification protocol discussed in the previous section, we purposefully break thedegeneracy of the |↑L , ↓R i and |↓L , ↑R i states by using a magnetic field gradient between the two
wells, which initiates a σz rotation on the effective Bloch sphere representing the Sz = 0 subspace ofthe two atoms (as shown in Fig. 5.3). The breaking of this degeneracy is useful for such operations,but when not controlled or zeroed at other times, the fidelity of the initial state preparation can be79degraded due to the splitting of the eigenstates, which should be degenerate during the adiabatic
passage procedure.
This becomes especially important when considering that our trapping potential during thisstate preparation is intentionally unequal between the two wells in order to couple atoms between
the ground band in the left well to the first excited band in the right well (as shown in Fig. 5.2). In
this situation, any imperfection in the state independence of a single trap (meaning, the differential
energy level shifts between the |↑i and |↓i single particle states on a single site), will directly turninto a position-dependent spin-selective shift between the two wells, which is equivalent to adding
a magnetic field gradient. In particular, an abundance of σ + (or σ − ) polarization in the trap lightwill act as an effective magnetic field for atoms trapped in that potential, but because we are
purposefully making the intensity of light unequal between the traps, this turns into an effective
magnetic field gradient.
Thus, having an extremely pure linear trap polarization (we prefer it to be π polarized
for other reasons, but for this purpose other linear polarizations can work) is very important tomaintain the degeneracy of the |↑L , ↓R i and |↓L , ↑R i states when initiating spin-exchange dynamics.
To achieve the largest spin-exchange contrast, we add a quarter-wave plate to the trapping lightand use it to minimize the spin dependence of the trapping light. Specifically, we measure themicrowave resonance for the |2, 2i → |1, 1i transition as a function of trap depth to get a measureof the spin-dependence of the traps and then adjust the quarter-wave plate to minimize the shift
in the resonance as a function of trap depth. For reference, the data shown in Fig. 5.2 was taken
when the shift in the microwave from a zero-depth trap to that in a kB × 1 mK depth trap of
approximately h × 120 kHz. After adjusting the quarter-wave plate, we were able to null the state
dependence of the trap to better than a h × 1 kHz shift between the free-space transition and akB × 1 mK depth, as demonstrated by the microwave spectra in Fig. 5.4(b). The resulting spin
exchange dynamics are shown in Fig. 5.4(a) and have a contrast more than 50% larger than thatseen without polarization correction.
To confirm that these changes are correlated, we used the same quarter-wave plate to make

80

b)

|
|

=

a)

(

-

)

(

)

Figure 5.4: a) Observation of spin-exchange dynamics in a double well potential after compensating
for imperfect trap polarization and, therefore, improving the state dependence between the two wells
during the spin exchange procedure. Notice that after correcting for the spin dependence of thetrapping potential, we have improved the spin exchange contrast by over 50%. b) Microwave spectra
taken for both deep (red) and shallow (blue) traps, showing that there is no longer a significant
vector light shift induced by the trapping light. For comparison, this same measurement for theoriginal data shown in Figure 5.2, these peaks were separated by over h × 120 kHz.

the polarization worse. Specifically, we tune the shift from the zero-depth transition to be h × 140
kHz, which translates into a roughly h × 70 Hz difference in the differential shift of the spin states
between the two traps during the adiabatic passage process (at tunneling depths). The resulting
spin-exchange dynamics are shown in Fig. 5.5, demonstrating a reduction of contrast by 50% fromthe optimized dynamics. The reduction observed is consistent with a picture where breaking thedegeneracy of these states results in an improper transfer of state population from the state |↑L , ↓R i
into an equal superposition of the localized singlet and triplet spin-states. Thus, using this state
dependence, we preferentially end up in one of the singlet or triplet spin states.

5.2.1

Adiabatic preparation of entangled states

After observing how large this effect can be, we consider a method that could take advantage
of such a state dependence to reliably generate pure entangled states, without relying on the
dynamics of spin-exchange. In Figure 5.6, we plot the eigenenergies of the system of two atoms in atilted double-well potential, both with an external magnetic field gradient (which introduces a spindependent shift between the two wells) and without the magnetic field (such that all spin-states are

81

QWP angle +1.5º
Probability in State

1.0
0.8
0.6

|↑,↓〉
|↓,↑〉

0.4
0.2
0.0
0

5

10

15

20

25

Exchange Time (ms)Figure 5.5: A plot of the spin exchange dynamics in the presence of a small circular componentof the trap polarization, which is achieved by turning a λ/4 waveplate by a small angle from its
optimum position, such that the tilt between the two wells leads to breaking of the degeneracy of|↑ie |↓ig and |↓ie |↑ig by ∼ h × 70 Hz, which is on the order of the interaction strength in the samewell. We can see that this has the effect of reducing the contrast of the spin exchange oscillations
by roughly a factor of two compared to the optimum case, where this shift is 0.

degenerate when the atoms are in separate wells). We notice that with the magnetic field gradienton, the |↑iL |↓iR (|↓iL |↑iR ) eigenstate is adiabatically connected to the singlet (triplet) spin state
localized in the right well. Thus, we can transfer from an easy-to-prepare state (|↑iL |↓iR ) into amaximally entangled state by performing an adiabatic passage.
The goal is then to separate the two atoms without projecting back into the original state.
Thus, if we can shut off the state-dependent shift between the two wells, a singlet spin stateshould not preferentially map into one of the spin eigenstates (due to their being degenerate). This
procedure is depicted in Fig. 5.6, where in part (a) we use an external magnetic field gradient topreferentially transfer from |↑iL |↓iR into a singlet state and then, in part (b), we turn off thegradient and reverse the adiabatic passage, resulting in a singlet spin state, in separate wells.
The requirements for this procedure are simply that we need to be able to break the degeneracy of the |↑iL |↓iR and |↓iL |↑iR spin configurations on-demand, as well as the ability to performthe transfer of atoms at a rate that is much slower than the spin-exchange coupling. While weEigenenergies (kHz)

a)

82

0.5

0.0

| ↓, ↑; Lg , Rg
| ↑, ↓; Lg , Rg

- 0.5

Adiabatic
Following

|S

- 1.0

;R

e,

R

g

;R

e,

R

g

- 1.5

- 1.0

- 0.5

0.0

0.5

1.0

1.5

Bias (kHz)b)
Eigenenergies (kHz)

|T

0.0

|S ; Lg , Rg
Adiabatic
Following

- 0.5

|S

- 1.0
- 1.5

;R

e,

R

g

- 1.0

- 0.5

0.0

0.5

1.0

1.5

Bias (kHz)

Figure 5.6: a) Diagram showing the eigenstate energy levels as a function of double-well bias,
highlighting the adiabatic passage procedure (as depicted in Figure 5.2). In particular, this showsthat by sufficiently breaking the degeneracy of the |↑ie |↓ig and |↓ie |↑ig states in separated wells, forinstance by applying a large magnetic field gradient, we can adiabatically follow a single eigenstate
to a pure singlet or triplet state with both atoms in the same well. b) Importantly, if we can thenshut off the state dependence and separate the traps, then we would be able to separate the puresinglet or triplet states after the initial preparation.

demonstrated this effect by changing the polarization of the trap light, it is unlikely that doing thisin real-time is going to be stable or reproducible, but we can easily use the magnetic field gradientthat is used in the verification protocol of Section 5.1 and achieve sufficient splitting; this is themagnitude of the spin-dependence that was assumed in the calculation used to make Figure 5.6.
Alternatively, we may also prefer to use the vector light shift beam localized to one of the wells (atmuch lower intensity than is used for microwave spin flips) because it will provide us with much

83
larger shifts and faster actuation than the magnetic field gradient, which may become necessary toincrease the speed while simultaneously limiting the coupling to other states during the transfer.
In the end, this procedure will likely be slower than using dynamical spin exchange to entangle
the atoms, but may be much more reliable, as it is not sensitive to many day-to-day fluctuationsof the trapping potential. Additionally, if we truly want to make a pure singlet or triplet for use inother simulations, as will be discussed in Section 6.3.3, then this may be the most reliable methodof state preparation.

Chapter 6
Assembling the Kondo Lattice Model with Optical Tweezers

In this chapter, I will discuss ongoing work that will combine many of the capabilities discussed in this thesis toward the goal of implementing a microscopic realization of the bosonic
Kondo-Hubbard model. A future demonstration of this control would be a significant realization of
the utility of this platform for studying the interplay of the spin and motional degrees of freedom
in systems of a few atoms with local interactions, when the individual degrees of freedom are nolonger separable and thus the full quantum state must be considered. It will be particularly interesting to observe how the symmetrization of the many-body wavefunction in the presence of local
interactions manifests in the out-of-equilibrium dynamics of states assembled from a initially puresingle-atom states.
First, we will introduce the Kondo lattice model along with some background and a discussionof the utility of this model for explaining interesting phases of matter in heavy-fermion systems [107].
Then, we will take a closer look at the bosonic analog of the model and, in particular, its minimalversion that we propose to implement in optical tweezers. For this system, we will take a lookat some preliminary simulations we have been performed to understand the observables related tointeractions of interest and discuss some of the challenges related to implementation, as well as
highlighting regimes in which we will, realistically, be able to work.

85

6.1

The Kondo Lattice Model
The development of the Kondo lattice model has a long history in condensed matter physics,

starting with the understanding of the “Kondo effect”, which describes the observations that metals
doped with small fractions of magnetic impurity atoms had a minimum in their resistance asa function of temperature.1

In this system, the exchange interaction between localized f -shellelectrons (bound to the impurity atoms) and the conduction band electrons result in the creationof localized quasiparticles when the thermal energy is not enough to overcome the binding of theenergetically favorable singlet states to the localized electrons.2

However, the presence of these

localized states reduce the mobility of the conduction electrons and therefore increase the resistanceat low temperature [108].
The Kondo lattice model is an extension of this model of the Kondo effect, with many ofthese impurities arranged in an ordered fashion (typically a square lattice) to study the interplayof the competing local interactions with the quantum statistics and motion of conduction band
electrons [109, 107]. This model was has the minimum ingredients to explain the complex behaviors
present in heavy-fermion materials, where the spin and motional degrees of freedom of electrons
are coupled via the Coulomb interaction [110, 111]. In particular, the propagation on a lattice ofthe type quasiparticles that are described in the Kondo effect can explain the very large effective
masses that these heavy-fermion materials exhibit. The specific form of this model imagines anarray of spins that are fixed, which can interact with mobile electrons that can move between (andinteract with) spins localized in different locations, as shown in the schematic representation given
in Figure 6.1(a). The interaction Hamiltonian is given by
HKLM = −t


X  †
X f
ciσ cjσ + c†jσ ciσ + J
Si · Sichi,ji,σ(6.1)

iWhere here we use the notation defined in Ref. [109], where ci,σ (c†i,σ is the annihilation (creation)
1
These observations were at low-temperature, where the resistance of a normal (not superconducting) metal
would approach a constant minimum value with power-law dependence.
2
This description assumes that the exchange interaction is negative, which will shift the singlet state to lower
energies for fermions. However, for impurities where the interaction coefficient is positive, the resistance decreases
faster at low temperature, as mentioned in Ref. [108].

86

a)
t

Valence spins (c):Lattice sites
J

Localized spins (f):b)

t >> J: Fermi sea in excited bandkF-......
i-1

j-1

j

)

j+1

bil
mo

mass enhancedeJ >> t: Flattened, hybridized bands, quasi-particles
Energy

c)

i i+1

(

localizedkFigure 6.1: a) Diagram to depict the Kondo lattice model written out in Equation 6.1. Noticethat the model contains only two terms: The on-site sin-interaction term J (indicated in red), andthe tunneling term t (indicated in green). Note that this is a general model that does not specify
the underlying realization, but is meant to describe, for example, a solid material where each
localized atom contains an inner-shell electron that interacts with the mobile valence electrons.
b) Schematic showing how, in the weakly-interacting regime, the perturbative local interactions
can lead to a second-order coupling between distant fixed spins via virtual excitation of electrons
above the fermi sea. This virtual excitation leads to a distance scale that is set by the Fermi
momentum, kF (indicated by the green dashed line). c) Depiction of the band structure in thestrongly interacting regime, where the flat band representing the fixed spins, and the cosinusoidal
band represents the mobile electrons. The effective mass of the quasiparticles formed in this system(which is proportional to the slope of the band) will become very large when the lattice filling factor
is near 1/2, such that there can be a single atom per site in the excited band (and thus all groundspins are paired into singlets).

87
operator for the conduction band spins on site i, Sic is the total spin operator for the conduction
band spins on site i, and Sif is the total spin operator for the localized spins (defined by the creation
operator fi† ) on site i. In this model, J represents the interaction energy between the localized andconduction band spins, while t represents the energy scale for tunneling of the conduction band
spins between sites. It is important to note that this notation will differ from notation used in
Section 6.2 and beyond (see text after Equation 6.2 for details), but I want to use the original
notation in this introduction.

6.1.1

Limiting cases of the Kondo lattice model

In this model, it is interesting to consider the two limiting regimes of strong and weakinteractions compared to the tunneling term (J  t and J  t, respectively), where interesting
phases of the system develop. In particular, I have already mentioned the strongly-interacting
regime, where the formation of quasiparticles on the lattice leads to measurements of the effective
mass of the conduction band electrons becoming extremely large (up to a factor of 1000 larger
than the bare mass of an electron) [112, 113]. In this regime, the strong interaction between
the localized spins on the lattice and the conduction band electrons results in a hybridization of
the band structure that results in a flattening of regions of the dispersion relation, as depicted
in Figure 6.1(b). As described in band theory, the flattening of this band is a signature of theincreasing effective mass of the electron quasiparticles because the effective mass is defined by the
local curvature of the dispersion relation (specifically, the second derivative of the energy withrespect to the momentum) for a given electron momentum.
In the weakly interacting regime, the interactions are a perturbative effect, so it is appropriate
to expand the Hamiltonian in orders of the on-site interaction energy. At second-order, the groundband atoms interact with each other in the same manner as that described by the RudermanKittel-Kasuya-Yosida (RKKY) interaction that was originally derived when studying the magnetic
ordering of nuclear spins in a metal [114, 115, 116]. Specifically, this effective interaction between
the localized spins will lead to the long-range ordering of these spins in a way that depends heavily

88
on the electrons in the conduction band because the process that drives these interactions requires
an effective flipping of a conduction band electron. However, to flip a conduction band electron, it
has to virtually be excited out of the fermi-sea, propagate to the other local spin, and then flip back;this process, as depicted in Figure 6.1(c), sets the energy (and length) scale of this interaction, whichis the Fermi energy EF (or the inverse of the Fermi momentum kF ). Thus, at low filling fractions,this direct coupling of the localized spins leads to a ferromagnetic interaction, while at larger filling
fractions there is an oscillatory coupling between local spins that depends on the separation ofthese spins. This second-order interaction between the localized spin states is thought to stabilize
long-range ordering of the localized spins in materials that have weaker interactions between thelocal and conduction band electrons [111].

6.2

The Kondo-Hubbard Model for Atomic Systems
Despite decades of research invested into studying the Kondo lattice model, its general phase

diagram remains unknown even for a simple case of a 2D square lattice [111], which points to thecomplexity of the spin-motional coupling, especially as the array size, and number of particles, isincreased. Hence, any insight into its physical properties obtained within a controlled setting isvaluable. Such an environment is offered by cold atomic gases that have proven to be excellent
candidates for studying condensed matter models [18, 117, 19, 81]. In our work, we will leverage
the additional control afforded with the optical tweezer platform, this seems to be a particularlyinteresting model to study with our system.
The Kondo-Hubbard model is a representation of the Kondo lattice model that can be
replicated in a relatively simple system, but describes the complex behaviors present in certain
strongly correlated materials where the spin and motional degrees of freedom of electrons are coupled [110, 111]. The interaction Hamiltonian for the Kondo-Hubbard model, as implemented in a
two-band lattice, isHKHM = −Je

X †X
Ue X  b  2
bi bj +
ni + 2Veg
Sia · Sib
2
hi,ji

i

i

(6.2)

89
Here a†i (ai ) are the creation (annihilation) operators for atoms in the ground-band at site i, b†i(bi ) are the same for atoms in the excited-band, nbi is the operator indicating the number of atoms
in the excited band at site i, hi, ji indicates summation over all pairs of neighboring wells i andj, Siσ is the total spin operator for atoms in the σ band at site i, and the coefficients Jg , Ug , Je ,Ue , and Veg set the relative tunneling rates and on-site interaction energies for the system.3 Notealso that to be general, we have added a term for on-site interactions within the excited band,
because multiple bosonic atoms in the same state can spatially overlap (unlike the case of fermionsin Equation 6.1, where only one particle per spin state can exist in a single band at each site).
The Hamiltonian in Equation 6.2 also assumes that each site in the ground-band can have
only a single atom per site, that we can neglect the tunneling of atoms within the ground-bandof the lattice, and that we can neglect the presence of any band-changing collisions (which would
require taking into account higher bands of the lattice). Note that the Hamiltonian is writtenwith simplified notation that may suggest a one-dimensional chain of atoms, but many systems
of interest will actually be two- or three-dimensional, in which case the indices i and j simply
represent coordinates in a higher-dimensional array.4

6.2.1

Studying the Kondo-Hubbard model with bosons

Of course, in our system, we are using spin-1/2 bosons, rather than the fermions considered
for condensed matter systems. However, many of the same effects can be seen in the bosonic
Kondo-Hubbard model, and the understanding of these interactions gained with bosons will only
give more information to aid in understanding the fermionic case (which has been studied extensively in condensed matter systems, albeit without the level of control afforded to cold atom
3

The notation I am using here will be familiar to those used to working with cold, bosonic quantum gases.
However, the notations differ quite dramatically in the condensed matter literature, where the exchange energy (here
Veg ) is written as J and the tunneling rate (here Je ) is given by t. Because I am writing this with the simplified model
we work with in our system (and in comparison to past work with effective Hubbard parameters), I have chosen to
stick with this notation, but want to include a warning because this can lead to significant confusion.
4
While the model works with no other modifications, care must be taken in real implementations of the model
to take into account degeneracies between different axes (the first excited band of the lattice have different motional
states along each dimension) or different couplings in each dimension (if you break the degeneracy, then the tunnel
coupling will likely be altered as well).

a) Microscopic model:

90

Je

Ue
b

Veg

a

b) Je>>Veg: Delocalized excited band, long-range interactionsc) Veg>>Je: Localized spin-singlet (triplet) quasi-particles
|S

|T↓ 

|T0 

|T↑ 

Figure 6.2: a) Schematic showing the components of the Kondo lattice model, as implemented
in a two-band optical lattice. The diagrams here as shown in one dimension, but can be extended
to two or three dimensions, where the full phase diagrams remain unsolved. In this model, theexcited band (red) contains atoms that can tunnel around the lattice potential with tunnelingcoefficient Je . The ground band of the lattice (green) contain atoms that are effectively fixed (thetunneling rate Jg , is much smaller than all other timescales in the experiment), but which caninteract with the excited band particles through the on-site exchange interaction, characterized by
the interaction coefficient Veg . b) When the tunneling timescale, Je dominates, then the excited
band particles effectively become delocalized, which will lead to mediated exchange interactions
between the localized ground band atoms. For fermions, the different quasimomenta of the atoms
in the excited band can lead to periodic spin-coupling terms, analogous to the RKKY interaction
term. c) When the interaction strength, Veg , dominates over the tunneling, then the energies related
to different on-site spin configurations (singlet vs. triplet) will be split. In this situation, the excited
band tunneling rate will be modified based on these splitting. This modification to the tunnelingrate is an analog to the heavy fermion physics observed certain condensed matter systems.

experiments) [118, 119, 120, 121]. The fundamental difference in these two models comes from
how the quantum statistics affect the configurations of atoms (and the distribution of spin states)in the excited band: Due to Pauli blocking, there can be no more than two fermionic atoms oneach lattice site (and they would have opposite spin), while bosons will happily bunch many atoms
onto the same site. Thus, the number of particles per site will depend heavily on the ratio of the91interaction energy Ue to the tunneling rate Je (in addition to the density of carriers) for the bosoniccase. For example, when Je  Veg , this results in the magnetic ordering becoming ferromagnetic
no matter the filling fraction of the lattice, as opposed to the spin ordering in the fermionic casethat depends on the filling fraction (via the Fermi momentum kF ), where at low filling fractions
the ordering is ferromagnetic, but at higher filling fractions becomes oscillitory (as described by
the RKKY interaction, introduced above) [120, 121]. Importantly, this means that we can apply
studies of low filling-fractions with bosons to the general problem, but certain behaviors that are
observed in condensed matter systems at higher filling fractions will not be apparent with bosons.

6.3

Implementation with optical tweezers
Typical theoretical studies of this model assume are carried out for infinite systems using.

However, in our case, we have the ability to directly study small arrays of atoms placed into alattice that is projected into the focal plane of our objective lens. In such systems, the Hilbert
space is already very large, such that detailed theoretical study becomes very challenging, if not
impossible. For initial experiments with just a few atoms, this will be accomplished by reorganizing
and overlapping the optical tweezers themselves, but could also involve transferring the atoms toan arbitrary potential generated via a DMD, as discussed in Section 3.2.4. Using the techniques
for quantum gas assembly, we have the ability to create and study out-of-equilibrium states of thesystem (by preparing a pure state in well-separated potentials and quenching into the final potential
to initiate dynamics) and then by pulling the atoms back to the separated potentials, we can follow
the dynamics of the system from initial state to final state.
For our first experiments, we will concentrate on the minimum requirements to implement
the microscopic Kondo lattice Hamiltonian, which is to use three atoms in a double well potential.
Two of these atoms will occupy the ground-state of the two wells and the third atom will be mobile
and in an excited band of the potential; for the purposes of this discussion, we will use the firstradial excited band along the axis connecting the two wells. As in previous work, we will encode
the spins in the ground state hyperfine levels of 87 Rb, and for the study of this system will prepare

92
the total spin of the system to have a projection of Sz = ±1/2 (meaning the spin of one atom
will be opposite the other two atoms). The specifics of how we prepare these atoms, as well as theshape of the final potential we use, will have drastic effects on the dynamics of the system.
The process for initializing these states is depicted in Figure 6.3. Part (a) shows the standard
steps to trap a set of atoms and initialize the spin and motional states before assembling the stateof interest from single atoms, but the crucial differences are highlighted in Figure 6.3(b-d), where
both the shape of the final trapping potential and the method for initiating dynamics are altered(and in the case of (d), a different spin configuration is used, as discussed in Section 6.3.3). Whilethese minimum realizations of the Kondo lattice model can be solved analytically, the observationof these dynamics in our system will serve as a verification of the techniques used to study spinmotional coupling with optical tweezers, and provides a stepping stone to studying larger systems
where an analytic solution may not be available.

6.3.1

Calculating the expected spin dynamics

Before discussing the different parameters regimes that we want to study, as well as how we
can implement them with our system, I want to recognize the productive collaboration we have hadwith Ana Maria Rey’s group for studying this system. In particular, Leonid Isaev derived the analytic result for a simplified three-spin model, which has been extremely useful for quickly exploring
different parameter regimes, and Michael Wall has performed full three-dimensional calculations of
the potential to extract effective Hubbard parameters that we can use in these calculations [104].
The plots in the remainder of this chapter that show the expected spin dynamics are generatedusing the analytic solutions to the restricted three-spin model that we are implementing by taking
advantage of the simplifying assumption that the ground-state atoms are fixed in their respective
wells, as well as knowledge of the symmetries associated with the trapping potential and the spin
interactions.5
Many of these systems were also numerically solved by propagating the initial states in the full
5This derivation is contained in a set of notes written by Leonid Isaev and is briefly summarized in Appendix B.

93

a)

b)

c)

Ramp bias todelocalize atom

Je*

Sweep wells together
to initiate dynamics

Veff

Figure 6.3: Diagram showing how to implement the microscopic Kondo lattice model with opticaltweezers. a) Starting with three isolated atoms, we bring all atoms to the ground state of their
optical tweezer and prepare the spins of each atom according to the requirements for any given
experiment (one specific preparation is shown). The traps are then repositioned to form two double
well potentials and, as was done for a single pair in Section 5.1, we perform an adiabatic passage
that transfers atoms in the outer wells into the excited band of the inner wells. Note that this sameprocess results in motional distillation on the two ground-band atoms (as described in Section 3.3.2),
which will improve the final state preparation. b,c) After the initial state preparation from (a),we can rearrange the two central wells (of the four total wells) and initiate dynamics a central
double well potential. For different trap parameters, we will initiate the dynamics in this potential
differently: b) In the weakly-interacting regime, we will want to delocalize the excited atom, whichis achieved by performing half of an adiabatic passage after the wells are close. c) In the stronglyinteracting regime, we start with a localized excited particle, and thus will sweep the traps together
at zero bias to initiate dynamics.

94
Hilbert space via the Schrödinger equation (with the full Hamiltonian given in Equation 6.2). The
numerical propagation produced nearly identical results, aside from the presence of the superexchange interaction between the ground band atoms, which is neglected in the analytic solution.
However, the timescale for superexchange is much longer than all other dynamical timescales ofinterest, which justifies neglecting that term initially. While the analytic solution is much fasterfor this particular system, the parallel development of these two models is important because theanalytic solution is specific to this system and will not hold for larger systems of atoms; that is not
to say an analytic solution cannot be found, but it becomes less likely (and certainly much more
complex to connect the eigenstates of the system back to the experimentally prepared states). On
the other hand, the numerical propagation can be very simply extended to larger arrays with more
atoms, where the complexity of the system is dramatically increased.

6.3.2

Weakly interacting regime (Veg  Je )To understand the parameter space of this problem, we will first look at extreme cases, where
one energy scale dominates, and observe the resulting dynamics. In these cases, with the appropriate
choice of the initial state we can effectively restrict ourselves to a smaller subspace of the system,
where the observed dynamics have clear interpretations. In the case where the interaction energy
Veg is small compared to the tunneling rate Je , the separated subspaces are the delocalized statesof the excited atom, one that has positive (and the other negative) inversion symmetry; these are
the eigenstates of a single atom under a tunnel coupling, as discussed in Ref. [39]. Each of thesesubspaces is spanned by three spin states of the two ground-state atoms (assuming no external
forces that can change Sztot ). For now, we will work in the subspace with with Sztot = − 21 and use
the single-particle measurement basis for the ground-state spins: |↑L , ↓R i, |↓L , ↑R i, and |↓L , ↓R i.
Figure 6.4(a,b) shows the population of the |↑L , ↓R i and |↓L , ↑R i spin configurations over time. We
can immediately see that the spin dynamics in this system are relatively complex, so let’s lookcloser at the Hamiltonian to see how to extract the information we want.
In this regime, we are looking for evidence of a second-order, direct-coupling term between

Symmetric State

a)

(

c)

b)

Anti-Symmetric State

)

(

95

)

|+
|+ - |- |Ground spins:

-

(

)

Figure 6.4: a) Plot of the population in the |↑L , ↓R i (red) and |↓L , ↑R i (blue) ground state spin
configurations as a function of time in the weakly interacting regime. The initial state was in
the |↓L , ↑R i ground-state spin configuration (the ground-state spin configuration indicated in theblue box), with the excited band atom in the symmetric parity state. b) Plot of the same spin
population dynamics, but not with the excited band atom in the anti-symmetric parity state. c)
Looking at the difference in the time evolution of the spin populations between the symmetric (|+i)
and antisymmetric (|−i) parity states of the excited band atom. The envelope of these oscillations
is a manifestation of the direct spin-interaction term in Equation 6.3.

the ground-state spins (e.g., SR · SL ), which is due to the third atom being in the excited band.
The effective spin Hamiltonian, up to second-order in the perturbative spin-interaction, is written
in Equation 6.3 where Se represents the spin- 21 operator for the excited atom and the sign ofthe second term is positive (negative) for the symmetric (antisymmetric) delocalized state of theexcited atom. In order to derive this Hamiltonian, we look at how the presence of an atom in the
excited band perturbs the spin eigenstates of the ground-band atoms.6
6

Notice that the second

The symmetric and antisymmetric delocalized states of the excited atom are the parity eigenstates related tothe tunnel-coupling between the two wells. A localized atom in one well of the excited band becomes a superpositionof these two parity states, and since they have different energies, the phase evolution results in tunneling. The
addition of interactions further splits these manifolds, and it is a difference in the sign of this splitting, as indicatedin Equation 6.3, that we want to measure in this experiment.

96
term of the effective Hamiltonian is simply the Heisenberg model on a triangle, which includes the
direct coupling term that we wish to detect. The problem is that the dynamics will be dominatedby the direct coupling between the two ground-state atoms and the excited state atom at a rate
∝ Veg  Je . Thus, to isolate the relevant signal we will use the change of sign and compare thedynamics between identical initial spin states with opposite parity of a delocalized excited atom,
as shown in Figure 6.4(c); this will reveal the shift of the energy levels due to the second-ordercoupling term.

Hef f = 2 Veg Se · (SR + SL ) ±2
Veg[SR · SL + Se · (SR + SL )]
4Je

(6.3)

Specifically, we start by observing the dynamics of the ground state spins for both parity
states, as shown in Figure 6.4(a,b). The simulated evolution is performed using “idealized” parameters of Veg = h × 50 Hz and Je = h × 1000 Hz, to ensure that the perturbative model isjustified. The initial spin state is prepared using the combination of the pure single-atoms spinstates |↓R , ↑L ; ↓e i, and we measure joint probability for the ground state atoms to have swapped tothe state |↑R , ↓L i during the evolution. Figure 6.4(c) shows the difference in the dynamics between
the two initial states and it is the slowly varying envelope signal that signifies the presence of thesecond term in Equation 6.3. The faster oscillation is the result of a direct coupling between multiple spin states, which we will explore in more detail below, and because it is not a pure sinusoidal
term, indicates that there are at least three levels involved in these dynamics.
While this initial state (and measurement basis) will be the easiest to use experimentally, we
will look at the more natural spin basis of the singlet [ √12 (|↑L , ↓R i− |↓L , ↑R i)], triplet [ √12 (|↑L , ↓R i−|↓L , ↑R i)], and down (|↓L , ↓R i) states to understand the spectrum of the energy levels. This information also gives us the information to explore ways to expose a cleaner signal, even if it is harder
to implement in the experiment. Specifically, we take a closer look at the case where we initialize
a triplet state between the two ground-band atoms in Figure 6.5, measuring the probability to bein the down state as a function of time. Importantly, this gives a single frequency of oscillation,

Symmetric State

a)

(

c)

b)

)

Anti-Symmetric State

(

97

)

|+
|+ - |- |Ground spins:

-

(

)

Figure 6.5:
These are an analogous set of plots to those in Figure 6.4, but now both measuring and initializing in the typical basis for a two-spin system. Note that the singlet state[ √12 (|↑L , ↓R i− |↓L , ↑R i)] is an eigenstate and will remain constant on these plots. Hence we havechosen to initialize the two ground band atoms in the triplet state [ √12 (|↑L , ↓R i+ |↓L , ↑R i], indicated in the orange box) and are plotting the triplet and down (|↓L , ↓R i, indicated in the cyan box)
state fractions over time. This plot demonstrates that when preparing the right initial state (andmeasuring in the same basis), we can simplify the dynamics such that we are effectively measuring
the difference in exchange frequency for initial states with the excited atom in the symmetric (a) orantisymmetric (b) parity state. The difference signal (c) shows the same envelope as in Figure 6.4,
but the frequency of this demonstrates that the frequency of this envelope is simply the differencein frequencies of the oscillations in (a) and (b).

which is a measure of the splitting between the triplet and down states, which will be changed as
the sign of the second term in Equation 6.3 is changed. Importantly, this means that rather thansubtracting amplitudes, which are affected by noise and dephasing experimentally, we would be able
to measure a change in the frequency of oscillations to detect the presence of this second-order term
and this would likely be a more robust signal to the noise typically encountered in our experiments.
While simply measuring the difference of frequencies is sufficient to detect the shifts (andthus the presence of the second-order interaction), we can also directly look at the analogous

98
difference in the dynamics, as shown in Figure 6.5(d), which exhibits the same envelope as the
dynamics in Figure 6.4. Further, we observe that when initially preparing the singlet state, we arein an eigenstate of the spin-interaction Hamiltonian, so we will not observe any dynamics. This
explains the more complex dynamics in Figure 6.4 as being due to the interference of all threeof the singlet, triplet, and down states. It also suggests that, while the oscillation amplitudes
will be reduced by a factor of two, we can measure the down state when starting with the initialstate |↓R , ↑L ; ↓e i, as shown in Figure 6.6. This gives us the option of measuring the difference inoscillation frequency even when starting with the easier-to-prepare initial state, which may prove
to be useful experimentally.

6.3.3

Interaction dominating regime (Veg  Je )At the other extreme, we can observe dynamics in our system when the interaction energy
between the atoms dominates the tunneling rate. In this regime, there is a separation of the energy
spectrum for the spin states within a single well, so we choose the natural basis in the Sztot = − 12spanned by the singlet [ √12 (|↑g , ↓e i− |↓g , ↑e i)], triplet [ √12 (|↑g , ↓e i+ |↓g , ↑e i)], and down(|↓g , ↓e i)states. In this section, we will also simulate the dynamics using “idealized” parameters, but now
with the scales reversed: Je = h × 50 Hz and Veg = h × 1000 Hz. In this regime, we are looking
for evidence of “mass enhancement” of the excited band particle, which will manifest itself as theslowing of the tunneling rate of the excited band atom. Thus, we initialize states with the excited
atom localized to one of the wells, as depicted in Figure 6.3(d) and observe the tunneling of thisatom for various spin configurations. In particular, we first measure the baseline tunneling rate byusing a spin-polarized state (Sztot = − 23 ) to get a baseline for the excited band tunneling in theabsence of interactions; these dynamics are shown in blue on in Figure 6.7. Equivalently, we canmeasure the tunneling of the excited atom in the absence of ground-band atoms and would observe
the same tunneling rate.
In comparison, we initiate dynamics after preparing the initial spin states of a localized singlet(green) and triplet (purple) states in the right well of the potential. These each have significantlySymmetric State

a)

(

)

Anti-Symmetric State

(

99

)

Ground spins:|+|+ - |- |

c)

b)

-

(

)

Figure 6.6: This is just a proof-of-principle that by mixing the preparation and measurement
bases, we can actually simplify the state preparation, while also potentially improving our ability
to read out the shift by measuring the change in oscillation frequencies. In this set of plots, we
initialize the |↑L , ↓R i state, as in Figure 6.4, while measuring the down state |↓L , ↓R i (cyan). While
we sacrifice a factor of two in the amplitude of the signal because half out state is stuck in the
singlet, shown in red in (a) and (b) (which has the side effect of preventing us from easily measuring
the triplet state as well), we may win because we are less sensitive to further amplitude reduction
due to the dephasing of the state (as long as the dephasing time is long enough that we can get a
precise enough measurement of the frequency).

different dynamics from the baseline tunneling rate, but importantly the singlet state exhibits pureoscillation at half the frequency of the bare tunneling. This is precisely what we expect in the
Kondo lattice model with a single mobile spin, and is a manifestation of the preference to generate
localized singlet states [109]. The dynamics in the case of the localized triplet state is more complex
due to the coupling between the triplet and down states (as seen in Figure 6.8), but qualitatively we
see that there appears to be dynamics occurring at both rates, but the atom never fully transfers
to the other well.
While not experimentally feasible, in the simulation we can also look at the spin states in

100

(

)

Figure 6.7: Probability for the excited band atom to be in the left well (highlighted in red) as
a function of time, plotted for three different initial spin configurations, and artificially choosing
the parameters to be deep in the interaction dominated regime (Je = h × 50 Hz, Veg = h × 1000
Hz). In all three initial states, the excited atom is initially localized to the right well, but theinitial spin states change: In blue, we are in the fully spin-polarized case, where the excited atomtunnels between the wells at rate 2Je , as if there are no ground band atoms. In green, we show thecase where the two atoms in the right well form a spin singlet state [ √12 (|↑g , ↓e i− |↓g , ↑e i)], which
tunnels at full contrast but at half the rate of a bare atom. This reduction in the tunneling rate
can be translated to an effective mass enhancement factor of 2. In purple, we show the excited
atom dynamics when the atoms in the right well form a triplet state [ √12 (|↑g , ↓e i+ |↓g , ↑e i)], andwe can see that there are multiple frequencies due to the mixing of the triplet manifold states, as
discussed in Figure 6.8.

the transferred well to get a better understanding of how the states evolve over time. Specifically,in Figure 6.8 we plot the probability of the three relevant spin states localized to the right well(depicted at the bottom of the figure) as a function of time for different initial spin configurations.
In all four cases, the excited atom is initially localized to the right well and the total spin is set
to Sztot = − 12 ; we then vary only the combined spin state of the two atoms the right well (and, if
relevant, the spin of the atom in the left well to preserve the total spin). In (a,b), we show the
dynamics for atoms starting in the triplet and down state, respectively, demonstrating that thetwo states are coupled as the atom tunnels between the wells. In particular, notice that when theatoms starts in the triplet state, it tends to transfer first to the down state in the right well, andvice-versa; it also is the interference of these couplings that leads to the imperfect transfer between
the two wells. In part (c), we show the spin dynamics when we start with a singlet spin state, which

a)

101

b)

(

)

c)

(

)

(

)

d)

(

)

PState Left:

Figure 6.8: These plots show the evolution of the two-atom spin states in the left well for thesame parameters as in Figure 6.7. This information gives some insight into the different behaviors
for varied spin configurations. a) However, if we start with a triplet [ √12 (|↑g , ↓e i+ |↓g , ↑e i)], we cansee that the probability to be in the triplet and |↓g , ↓e i state is mixed over time. (Note that thesestates are both in the triplet manifold, but for clarity in notation I only refer to the Sz = 0 stateas the triplet state.) b) A similar evolution occurs when we start in the |↓g , ↓e i state, but withslightly different mixing. c) When starting with a singlet state [ √12 (|↑g , ↓e i− |↓g , ↑e i)], the singlet istransfered between the wells at unit peak probability and at a single frequency. d) Now, if we startwith the state |↑g , ↓e i, the dynamics will be a combination of the singlet and triplet states, meaning
we project into all three spin states. This complex dynamics also includes some fast oscillations
that are due to the large splitting between the singlet and triplet manifolds.

demonstrates that the singlet state transfers completely; this suggests that the localized singlet canbe thought of as a quasiparticle that tunnels between the wells, which has an effective mass twicethat of the bare atom in the excited band. Additionally, we show that starting with a mixed initial
spin state, |↑g , ↓e i, we see a superposition of the same dynamics, but it is important to note that, in102
the measurement basis, we would see very fast spin-exchange dynamics whenever there are nonzero
components of both the singlet and triplet states.

6.3.4

Summary of ideal theoretical investigations

The above simulations have been useful to understand how the different interactions affectthe dynamics observed in the minimum instance of the bosonic Kondo-Hubbard model (two sites
with three atoms). Importantly, we see that it is possible to observe interesting dynamics related toboth extremes of the model, given the ability to arbitrarily set the interaction and tunneling energy
scales. Additionally, we note that the observables in such systems are experimentally accessible.
Unless noted above, the simulations shown were either projecting out the population distribution
in the excited band between the two wells (for the mass-enhanced regime), or the observed groundband spin states in the (observable) basis of |↑i and |↓i on individual sites.
With knowledge that these interesting dynamics can be observed in our system, it is also
worth considering the state initialization to understand a specific experimental procedure thatwould result in this observation. In particular, we must be conscious of other interactions anddynamics that will prevent us from quenching into the pure initial state that we want. In general,
this means we must always have spin eigenstates during the preparation (before initiating the
dynamics); for example, in Figure 6.3(c), we would not be able to initialize to opposite-spin atoms
in the left pair of wells because they would undergo spin-exchange during the sweep that initializes
the tunneling dynamics of the excited band atom. Additionally, when delocalizing an atom in the
excited band, the adiabatic ramp would need to be slow compared to the tunneling rate, but fastcompared to the exchange rate, setting a bound on the separation of energy scales that we canexperiment with and dictating an optimal rate for the best possible state preparation.
With this in mind, I would aim for two specific implementations of the experiment, one exploring each of these extremes. To observe the ground-state spin coupling in the weakly interacting
regime, I think simply measuring the probability for both atoms to be in the same spin state, as
shown in Figure 6.6, will give the cleanest signal. While the signal amplitude is smaller than it

103
could be, there are only two frequencies of oscillation, which will simplify the analysis; the mainquestion is what the absolute timescale will be in comparison to the tunneling dephasing.
The most promising experiment, in my opinion, is to first implement the adiabatic preparationof a singlet state, as discussed in Section 5.2 (which is a spin eigenstate), but leave the atoms inthe same well after shutting off the magnetic field gradient. Then, by sweeping the spacing of the
middle two wells [as shown in Figure 6.3(c)], we can initiate mass-enhanced tunneling dynamics,
which can be compared to the exact same procedure, but with spin-polarized atoms (if all spins are
the same, we would observe the bare tunneling rate), which should be a factor of two slower. In thisexperiment the state preparation is more complicated, but the ability to separate the challenge ofpreparation from the initialization of dynamics, combined with the very clear signal of the tunnelingrate, makes this an attractive first experiment to demonstrate the coupling of the spin and motional
degrees of freedom in our system.

6.3.4.1

Experimental limitations and available parameters

Now that we have an idea of the signals we are looking for, its important to discuss theprospects for actually observing them in our experiment with the accessible trap parameters, rather
than the idealized parameters used in the simulations. In principle, we can tune the shape of thetrapping potential to cover the range of parameters extending all the way from the weakly toVstrongly interacting regimes (achieving ratios of Jege from 0.1 up to 10) simply by tuning the overall
trap depth and spacing between the traps. Generally speaking, there are two limitations that weneed to balance: The fidelity for preparing and initializing dynamics with the desired state, andthe stability of the trapping potential setting the interaction parameters in the Hamiltonian. For
example, we can make Veg  Je ; however, the absolute value of each of these will be small because
we can only reasonably achieve Veg ∼ h × 100 Hz (without going to extremely large trap depths)
and then would need to adjust the trap spacing to make the tunneling rate even smaller. However,with these parameter, the dynamics of interest will occur over hundred’s of milliseconds, during
which time noise in the system (e.g., fluctuations of the magnetic field or the bias between the

104

b)

|+
|+ - |- |

a)

-

(

)

(

)

Figure 6.9: a) Spin dynamics shown for realistic trap parameters that should allow us to observe
the ground-state spin coupling in the weakly interacting regime. Specifically, this simulation was
run for a trap separation of 900 nm and single-well depth of h × 75 kHz, trap parameters that are
expected to result in the interaction parameters Veg = h × 48 Hz and Je = h × 310 Hz. b) Excited
atom dynamics for a different set of parameters that should be sufficient to observe dynamics inthe strongly interacting regime. Specifically, the trap separation is 905 nm and the single-well trap
depth is h × 150 kHz, which should result in the interaction parameters Veg = h × 93 Hz and
Je = h × 38 Hz.

wells) will likely cause dephasing of the system.
Fortunately, both of these regimes have very clear observables, which should allow us to see
the effects described above, even without long coherence times or moving far into the limits ofstrong or weak interactions. Specifically, in the weakly interacting limit we simply need to see a
deviation of the difference between the dynamics with a symmetric and antisymmetric excited atom
state (or, equivalently, a difference in the frequency of oscillation of the probability for the groundatoms to be in the |↓R , ↓L i state, as shown in Figure 6.6). And in the limit of strong interactions,
we simply need to see a single oscillation to see the difference compared to the bare tunneling rate.
Examples of the dynamics we hope to observe (in the absence of dephasing), for parameters thatapproach both of these regimes, are shown in Figure 6.9.
For observing the dynamics in the weakly interacting regime, we plan to set the spacing
between the tweezers that define the final double-well potential to 900 nm and a single-well trap
depth of h×75 kHz (using the same definitions given in Ref. [39]). Based on calculations taking into
account the full three dimensional shape of our tweezer potentials, this configuration is expected105
to achieve the effective Hubbard parameters parameters Veg = h × 48 Hz and Je = h × 310 Hz,
putting us very close to the weakly interacting regime. Figure 6.9(a) shows the signal observed
on shorter timescales when initiating the ground atoms in the |↓R , ↑L i state and measuring theprobability to end in the |↓R , ↓L i over time (as demonstrated in Figure 6.6). In this parameter
regime, there are two experimental concerns that might reduce, or even wash out, the signal we aresearching for: First, we must reliably be able to prepare the delocalized excited atom states. The
plan for achieving this is to simply perform an adiabatic passage halfway and stop when the wells
are in resonance. We have received positive results with some initial testing of this procedure in
the experiment, but need to confirm that it is a coherent process (and not simply thermal) before
we can be confident it will work reliably. Additionally, the timescale for the second-order couplingis very long, which likely means that the dephasing will occur before we see anything close to a full
oscillation. However, we note that any significant deviation between the oscillation frequencies (or,equivalently, the deviation of the difference in spin probabilities from zero) is sufficient to verify thesecond order coupling. So, as shown in Figure 6.9, with dephasing times in the tens of milliseconds(as we can typically achieve for tunneling experiments, and even longer is possible for exchange
experiments) we should be able to observe this deviation.
On the other hand, to observe dynamics in the strongly interacting regime, we plan to set
the spacing between the tweezers to 905 nm with the single-well trap depth of h × 150 kHz, whichshould provide the effective Hubbard parameters parameters Veg = h × 93 Hz and Je = h × 38
Hz. As can be seen in Figure 6.9(b), atoms in this trap clearly exhibit tunneling the characteristic
slowing of the excited particles motion, however with some additional fast oscillations on imposedon the tunneling. In this case, the fast oscillations are not going to be problematic for differentiating
between the two cases. However, there are also two concerns in this regime: First, we need to be able
to generate a pure singlet on a single site before initiating the dynamics. This is another experimentthat we think is possible, as discussed in Section 5.2, but the fidelity of this operation needs to behigh, which has not been verified. And second, the dephasing time need to be longer than a single
oscillation, meaning ∼30 ms or longer. While we have certainly achieved these dephasing times

106in specific trap configurations, it is not yet clear that this particular trap configuration can reachthose dephasing times because the larger depth will increases the absolute amplitude of the noisecompared to the tunneling rate (which may cause rapid dephasing).

Bibliography[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198 (1995).
[2] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose-Einstein
Condensation in an Atomic Gas with Attractive Interactions,” Phys. Rev. Lett. 75, 1687
(1995).
[3] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. Van Druten, D. S. Durfee, D. M. Kurn,and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett.
75, 3969 (1995).
[4] B. DeMarco and D. S. Jin, “Onset of Fermi Degeneracy in a Trapped Atomic Gas,” Science
285, 1703 (1999).
[5] J. L. Roberts, N. R. Claussen, S. L. Cornish, and C. E. Wieman, “Magnetic Field Dependenceof Ultracold Inelastic Collisions near a Feshbach Resonance,” Phys. Rev. Lett. 85, 728 (2000).
[6] T. Johnson, E. Urban, T. Henage, L. Isenhower, D. Yavuz, T. Walker, and M. Saffman, “RabiOscillations between Ground and Rydberg States with Dipole-Dipole Atomic Interactions,”
Phys. Rev. Lett. 100, 113003 (2008).
[7] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, “A High Phase-space-density Gas of Polar
Molecules,” Science 322, 231 (2008).
[8] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and
M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110 (2009).
[9] A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet,
A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms
in the Rydberg blockade regime,” Nat. Phys. 5, 115 (2009).
[10] T. Lompe, T. B. Ottenstein, F. Serwane, A. N. Wenz, G. Zürn, and S. Jochim, “RadioFrequency Association of Efimov Trimers,” Science 330, 940 (2010).
[11] R. Schmidt, H. R. Sadeghpour, and E. Demler, “Mesoscopic Rydberg Impurity in an Atomic
Quantum Gas,” Phys. Rev. Lett. 116, 105302 (2016).

108[12] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition
from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature (London) 415, 39(2002).
[13] R. Jördens, N. Strohmaier, K. Günter, H. Moritz, and T. Esslinger, “A Mott insulator offermionic atoms in an optical lattice,” Nature (London) 455, 204 (2008).
[14] L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger, “Creating, moving and merging
Dirac points with a Fermi gas in a tunable honeycomb lattice,” Nature (London) 483, 302(2012).
[15] S. Krinner, D. Stadler, D. Husmann, J.-P. Brantut, and T. Esslinger, “Observation of quantized conductance in neutral matter,” Nature (London) 517, 64 (2014).
[16] R. A. Hart, P. M. Duarte, T.-L. Yand, X. Liu, T. Paiva, E. Khatami, R. T. Scalettar,
N. Trivedi, D. A. Huse, and R. G. Hulet, “Observation of antiferromagnetic correlations in
the Hubbard model with ultracold atoms,” Nature (London) 519, 1 (2015).
[17] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller, and
F. Ferlaino, “Extended Bose-Hubbard models with ultracold magnetic atoms,” Science 352,
201 (2016).
[18] W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, “A Quantum Gas Microscopefor Detecting Single Atoms in a Hubbard-regime Optical Lattice,” Nature (London) 462, 74(2009).
[19] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, “Single-atomresolved fluorescence imaging of an atomic Mott insulator,” Nature (London) 467, 68 (2010).
[20] M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauß, C. Gross, L. Mazza, M. C.
Bañuls, L. Pollet, I. Bloch, and S. Kuhr, “Observation of Correlated Particle-Hole Pairs andString Order in Low-Dimensional Mott Insulators,” Science 334, 200 (2011).
[21] P. Schauß, M. Cheneau, M. Endres, T. Fukuhara, S. Hild, A. Omran, T. Pohl, C. Gross,
S. Kuhr, and I. Bloch, “Observation of spatially ordered structures in a two-dimensional
Rydberg gas,” Nature (London) 491, 87 (2012).
[22] R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, “Measuring
entanglement entropy in a quantum many-body system,” Nature (London) 528, 77 (2015).
[23] M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif, and M. Greiner, “Site-resolved observations of antiferromagnetic correlations in the Hubbard model,” arXiv:1605.02704 (2016).
[24] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstrationof a Fundamental Quantum Logic Gate,” Phys. Rev. Lett. 75, 4714 (1995).
[25] F. Schmidt-Kaller, H. Häffner, M. Rieve, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher,
C. F. Roos, J. Eschner, and R. Blatt, “Realization of the Cirac-Zoller controlled-NOT gate,”
Nature (London) 422, 408 (2003).
[26] L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker,
and M. Saffman, “Demonstration of a Neutral Atom Controlled-NOT Quantum Gate,” Phys.
Rev. Lett. 104, 010503 (2010).

109[27] T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys,“Entanglement of Two Individual Neutral Atoms Using Rydberg Blockade,” Phys. Rev. Lett.
104, 010502 (2010).
[28] J. W. Britton, B. C. Sawyer, A. C. Keith, C. C. J. Wang, J. K. Freericks, H. Uys, M. J.
Biercuk, and J. J. Bollinger, “Engineered two-dimensional Ising interactions in a trapped-ion
quantum simulator with hundreds of spins,” Nature (London) 484, 489 (2012).
[29] R. Islam, C. Senko, W. C. Campbell, S. Korenblit, J. Smith, A. Lee, E. E. Edwards, C. C. J.
Wang, J. K. Freericks, and C. Monroe, “Emergence and Frustration of Magnetism with
Variable-Range Interactions in a Quantum Simulator,” Science 340, 583 (2013).
[30] K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a twodimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[31] H. Labuhn, D. Barredo, S. Ravets, S. de Leseleuc, T. Macri, T. Lahaye, and A. Browaeys,“Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models,” Nature (London) 534, 667 (2016).
[32] J. Beugnon, C. Tuchendler, H. Marion, A. Gaëtan, Y. Miroshnychenko, Y. R. P. Sortais,
A. M. Lance, M. P. A. Jones, G. Messin, A. Browaeys, and P. Grangier, “Two-dimensional
Transport and Transfer of a Single Atomic Qubit in Optical Tweezers,” Nat. Phys. 3, 696(2007).
[33] B. Zimmermann, T. Müller, J. Meineke, T. Esslinger, and H. Moritz, “High-resolution Imaging of Ultracold Fermions in Microscopically Tailored Optical Potentials,” New J. Phys. 13,043007 (2011).
[34] R. Bowler, J. Gaebler, Y. Lin, T. R. Tan, D. Hanneke, J. D. Jost, J. P. Home, D. Leibfried,and D. J. Wineland, “Coherent Diabatic Ion Transport and Separation in a Multizone Trap
Array,” Phys. Rev. Lett. 109, 080502 (2012).
[35] F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Béguin, A. Vernier, T. Lahaye, and
A. Browaeys, “Single-Atom Trapping in Holographic 2D Arrays of Microtraps with Arbitrary
Geometries,” Phys. Rev. X 4, 021034 (2014).
[36] T. Grünzweig, A. Hilliard, M. McGovern, and M. F. Andersen, “Near-deterministic Preparation of a Single Atom in an Optical Microtrap,” Nat. Phys. 6, 951 (2010).
[37] A. V. Carpentier, Y. H. Fung, P. Sompet, A. J. Hilliard, T. G. Walker, and M. F. Andersen,“Preparation of a single atom in an optical microtrap,” Laser Phys. Lett. 10, 125501 (2013).
[38] B. J. Lester, N. Luick, A. M. Kaufman, C. M. Reynolds, and C. A. Regal, “Rapid Productionof Uniformly Filled Arrays of Neutral Atoms,” Phys. Rev. Lett. 115, 073003 (2015).
[39] A. M. Kaufman, B. J. Lester, C. M. Reynolds, M. L. Wall, M. Foss-Feig, K. R. A. Hazzard,
A. M. Rey, and C. A. Regal, “Two-particle quantum interference in tunnel-coupled opticaltweezers,” Science 345, 306 (2014).

110[40] D. S. Weiss, J. Vala, A. V. Thapliyal, S. Myrgren, U. Vazirani, and K. B. Whaley, “Another
Way to Approach Zero Entropy for a Finite System of Atoms,” Phys. Rev. A 70, 040302(R)(2004).
[41] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A. Krajenbrink, C. Senko,
V. Vuletić, M. Greiner, and M. D. Lukin, “Cold Matter Assembled Atom-by-Atom,”arXiv:1607.03044 (2016).
[42] D. Barredo, S. de Leseleuc, V. Lienhard, T. Lahaye, and A. Browaeys, “An atom-by-atom
assembler of defect-free 2D atomic arrays,” arXiv:1607.03042 (2016).
[43] A. M. Kaufman, B. J. Lester, and C. A. Regal, “Cooling a Single Atom in an Optical Tweezer
to Its Quantum Ground State,” Phys. Rev. X 2, 041014 (2012).
[44] J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Coherence andRaman sideband cooling of a single atom in an optical tweezer,” Phys. Rev. Lett. 110, 133001(2013).
[45] C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch,
and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” Nature (London) 471,
319 (2011).
[46] H. Labuhn, S. Ravets, D. Barredo, L. Béguin, F. Nogrette, T. Lahaye, and A. Browaeys,“Single-atom addressing in microtraps for quantum-state engineering using Rydberg atoms,”
Phys. Rev. A 90, 023415 (2014).
[47] A. M. Kaufman, B. J. Lester, M. Foss-Feig, M. L. Wall, A. M. Rey, and C. A. Regal, “Entangling two transportable neutral atoms via local spin-exchange,” Nature (London) 527, 208(2015).
[48] B. J. Lester, A. M. Kaufman, and C. A. Regal, “Raman cooling imaging: Detecting single
atoms near their ground state of motion,” Phys. Rev. A 90, 011804(R) (2014).
[49] M. F. Parsons, F. Huber, A. Mazurenko, C. S. Chiu, W. Setiawan, K. Wooley-Brown, S. Blatt,and M. Greiner, “Site-Resolved Imaging of Fermionic 6 Li in an Optical Lattice,” Phys. Rev.
Lett. 114, 213002 (2015).
[50] L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr, T. Lompe,and M. W. Zwierlein, “Quantum-Gas Microscope for Fermionic Atoms,” Phys. Rev. Lett.
114, 193001 (2015).
[51] A. Omran, M. Boll, T. A. Hilker, K. Kleinlein, G. Salomon, I. Bloch, and C. Gross, “Microscopic Observation of Pauli Blocking in Degenerate Fermionic Lattice Gases,” Phys. Rev.
Lett. 115, 263001 (2015).
[52] S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen, “ResolvedSideband Raman Cooling to the Ground State of an Optical Lattice,” Phys. Rev. Lett. 80,
4149 (1998).
[53] H. Perrin, A. Kuhn, I. Bouchoule, and C. Salomon, “Sideband Cooling of Neutral atoms in a
Far-detuned Optical lLttice,” Europhys. Lett. 42, 395 (1998).

111[54] V. Vuletic, C. Chin, A. J. Kerman, and S. Chu, “Degenerate Raman Sideband Cooling ofTrapped Cesium Atoms at Very High Atomic Densities,” Phys. Rev. Lett. 81, 5768 (1998).
[55] A. J. Kerman, V. V. Vuletic, C. Chin, and S. Chu, “Beyond Optical Molasses: 3D Raman
Sideband Cooling of Atomic Cesium to High Phase-Space Density,” Phys. Rev. Lett. 84, 439(2000).
[56] D.-J. Han, S. Wolf, S. Oliver, C. McCormick, M. T. DePue, and D. S. Weiss, “3D Raman
Sideband Cooling of Cesium Atoms at High Density,” Phys. Rev. Lett. 85, 724 (2000).
[57] A. M. Kaufman, Laser-cooling atoms to indistinguishability: Atomic Hong-Ou-Mandelinterference and entanglement through spin exchange, Ph.D. thesis, University of Colorado
at Boulder (2015).
[58] J. Dalibard and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization
gradients: simple theoretical models,” J. Opt. Soc. Am. B 6, 2023 (1989).
[59] J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold andultracold collisions,” Rev. Mod. Phys. 71, 1 (1999).
[60] N. Schlosser, G. Reymond, and P. Grangier, “Collisional Blockade in Microscopic Optical
Dipole Traps,” Phys. Rev. Lett. 89, 023005 (2002).
[61] M. T. DePue, C. McCormick, S. L. Winoto, S. Oliver, and D. S. Weiss, “Unity occupation ofsites in a 3D optical lattice,” Phys. Rev. Lett. 82, 2262 (1999).
[62] A. Steffen, W. Alt, M. Genske, D. Meschede, C. Robens, and A. Alberti, “Note: In situ
measurement of vacuum window birefringence by atomic spectroscopy,” Rev. Sci. Instrum.
84, 126103 (2013).
[63] D. Barredo, S. Ravets, H. Labuhn, L. Béguin, A. Vernier, F. Nogrette, T. Lahaye, and
A. Browaeys, “Demonstration of a Strong Rydberg Blockade in Three-Atom Systems with
Anisotropic Interactions,” Phys. Rev. Lett. 112, 183002 (2014).
[64] K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein,“Quantum computing in optical microtraps based on the motional states of neutral atoms,”
Phys. Rev. A 66, 042317 (2002).
[65] D. Hayes, P. S. Julienne, and I. H. Deutsch, “Quantum Logic via the Exchange Blockade in
Ultracold Collisions,” Phys. Rev. Lett. 98, 070501 (2007).
[66] V. Bolpasi and W. von Klitzing, “Double-pass tapered amplifier diode laser with an output
power of 1 W for an injection power of only 200W,” Rev. Sci. Instrum. 81, 113108 (2010).
[67] R. Ma, Engineered potentials and dynamics of ultracold quantum gases under the microscope,
Ph.D. thesis, Harvard University (2014).
[68] P. M. Preiss, Atomic Bose-Hubbard Systems with Single-Particle Control, Ph.D. thesis, Harvard University (2015).
[69] A. Fuhrmanek, R. Bourgain, Y. R. P. Sortais, and A. Browaeys, “Light-assisted collisions
between a few cold atoms in a microscopic dipole trap,” Phys. Rev. A 85, 062708 (2012).

112[70] P. Sompet, A. V. Carpentier, Y. H. Fung, M. McGovern, and M. F. Andersen, “Dynamics
of two atoms undergoing light-assisted collisions in an optical microtrap,” Phys. Rev. A 88,
051401(R) (2013).
[71] D. A. Steck, “Rubidium 87 D Line Data,” available online at http://steck.us/alkalidata(revision 2.1.4, 23 December 2010).
[72] Y. H. Fung and M. F. Andersen, “Efficient collisional blockade loading of a single atom into
a tight microtrap,” New J. Phys. 17, 073011 (2015).
[73] F. Le Kien, P. Schneeweiss, and A. Rauschenbeutel, “Dynamical polarizability of atoms inarbitrary light fields: general theory and application to cesium,” Eur. Phys. J. D 67, 92(2013).
[74] Y. H. Fung, A. V. Carpentier, P. Sompet, and M. Andersen, “Two-Atom Collisions and the
Loading of Atoms in Microtraps,” Entropy 16, 582 (2014).
[75] A. Peng, Quantum Gas Microscope With Optical Lattice, Ph.D. thesis, Harvard University(2010).
[76] S. Bergamini, B. Darquié, M. Jones, L. Jacubowiez, A. Browaeys, and P. Grangier, “Holographic generation of microtrap arrays for single atoms by use of a programmable phasemodulator,” J. Opt. Soc. Am. B 21, 1889 (2004).
[77] P. Zupancic, P. M. Preiss, R. Ma, A. Lukin, M. E. Tai, M. Rispoli, R. Islam, and M. Greiner,“Ultra-precise holographic beam shaping for microscopic quantum control,” Opt. Express 24,
13881 (2016).
[78] N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single
atoms in a microscopic dipole trap,” Nature (London) 411, 1024 (2001).
[79] K. D. Nelson, X. Li, and D. S. Weiss, “Imaging Single Atoms in a Three-dimensional Array,”Nat. Phys. 3, 556 (2007).
[80] M. Karski, L. Förster, J. M. Choi, W. Alt, A. Widera, and D. Meschede, “Nearest-Neighbor
Detection of Atoms in a 1D Optical Lattice by Fluorescence Imaging,” Phys. Rev. Lett. 102,053001 (2009).
[81] M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauß, C. Gross, L. Mazza, M. C.
Bañuls, L. Pollet, I. Bloch, and S. Kuhr, “Single-site- and single-atom-resolved measurementof correlation functions,” Appl. Phys. B 113, 27 (2013).
[82] A. Fuhrmanek, R. Bourgain, Y. R. P. Sortais, and A. Browaeys, “Free-Space Lossless State
Detection of a Single Trapped Atom,” Phys. Rev. Lett. 106, 133003 (2011).
[83] M. J. Gibbons, C. D. Hamley, C.-Y. Shih, and M. S. Chapman, “Nondestructive FluorescentState Detection of Single Neutral Atom Qubits,” Phys. Rev. Lett. 106, 133002 (2011).
[84] Y. S. Patil, S. Chakram, L. M. Aycock, and M. Vengalattore, “Nondestructive imaging of an
ultracold lattice gas,” Phys. Rev. A 90, 033422 (2014).
[85] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser Cooling to the ZeroPoint Energy of Motion,” Phys. Rev. Lett. 62, 403 (1989).

113[86] L. Forster, M. Karski, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, A. Widera, E. Montano,J. H. Lee, W. Rakreungdet, and P. S. Jessen, “Microwave Control of Atomic Motion in Optical
Lattices,” Phys. Rev. Lett. 103, 233001 (2009).
[87] C. Tuchendler, A. M. Lance, A. Browaeys, Y. R. P. Sortais, and P. Grangier, “Energy Distribution and Cooling of a Single Atom in an Optical Tweezer,” Phys. Rev. A 78, 033425(2008).
[88] M. S. Safronova and U. I. Safronova, “Critically evaluated theoretical energies, lifetimes,
hyperfine constants, and multipole polarizabilities in 87 Rb,” Phys. Rev. A 83, 052508 (2011).
[89] B. Arora and B. K. Sahoo, “State-insensitive trapping of Rb atoms: Linearly versus circularly
polarized light,” Phys. Rev. A 86, 033416 (2012).
[90] C.-Y. Shih and M. S. Chapman, “Nondestructive light-shift measurements of single atoms in
optical dipole traps,” Phys. Rev. A 87, 063408 (2013).
[91] M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, W. D. Phillips, and J. V. Porto,“Controlled Exchange Interaction Between Pairs of Neutral Atoms in an Optical Lattice,”
Nature (London) 448, 452 (2007).
[92] S. Trotzky, P. Cheinet, S. Fölling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov, E. A.
Demler, M. D. Lukin, and I. Bloch, “Time-resolved Observation and Control of SuperexchangeInteractions with Ultracold Atoms in Optical Lattices,” Science 319, 295 (2008).
[93] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and
P. Gould, “Resolved-Sideband Raman Cooling of a Bound Atom to the 3D Zero-Point Energy,” Phys. Rev. Lett. 75, 4011 (1995).
[94] X. Li, T. A. Corcovilos, Y. Wang, and D. S. Weiss, “3D Projection Sideband Cooling,” Phys.
Rev. Lett. 108, 103001 (2012).
[95] P. Maunz, T. Puppe, I. Schuster, N. Syassen, P. W. H. Pinkse, and G. Rempe, “Cavity
Cooling of a Single Atom,” Nature (London) 428, 50 (2004).
[96] A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Cooling to the Ground
State of Axial Motion for One Atom Strongly Coupled to an Optical Cavity,” Phys. Rev.
Lett. 97, 083602 (2006).
[97] A. Reiserer, C. Nölleke, S. Ritter, and G. Rempe, “Ground-State Cooling of a Single Atomat the Center of an Optical Cavity,” Phys. Rev. Lett. 110, 223003 (2013).
[98] A. Kerman,
Raman Sideband Cooling and Cold Atomic Collisions in Optical Lattices,
Ph.D. thesis, Stanford University (2002).
[99] L. Förster, Microwave control of atomic motion in a spin dependent optical lattice, Ph.D.
thesis, Universität Bonn (2010).
[100] D. J. Wineland, C. Monroe, D. M. Meekhof, B. E. King, D. Leibfried, W. Itano, J. C.
Bergquist, D. Berkeland, J. J. Bollinger, and J. Miller, “Quantum state manipulation oftrapped atomic ions,” Proc. R. Soc. A 454, 411 (1998).

114[101] C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between
two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[102] R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic
Hong-Ou-Mandel experiment,” Nature (London) 520, 66 (2015).
[103] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M.
Marcus, M. P. Hanson, and A. C. Gossard, “Coherent Manipulation of Coupled Electron
Spins in Semiconductor Quantum Dots,” Science 309, 2180 (2005).
[104] M. L. Wall, K. R. A. Hazzard, and A. M. Rey, “Effective many-body parameters for atoms
in nonseparable Gaussian optical potentials,” Phys. Rev. A 92, 013610 (2015).
[105] C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A.
Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement offour particles,” Nature (London) 404, 256 (2000).
[106] D. Liebfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of arobust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412 (2003).
[107] A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge studies in magnetism: 2(Cambridge University Press, 1997).
[108] J. Kondo, “Resistance Minimum in Dilute Magnetic Alloys,” Prog. Theor. Phys. 32, 37(1964).
[109] H. Tsunetsugu, M. Sigrist, and K. Ueda, “The ground-state phase diagram of the onedimensional Kondo lattice model,” Rev. Mod. Phys. 69, 809 (1997).
[110] P. Coleman, “Heavy Fermions: electrons at the edge of magnetism,” Handbook of Magnetism
and Advanced Magnetic Materials (2007).
[111] P. Coleman and A. H. Nevidomskyy, “Frustration and the Kondo Effect in Heavy Fermion
Materials,” Journal of Low Temperature Physics 161, 182 (2010).
[112] K. Andres, J. E. Graebner, and H. R. Ott, “4f-Virtual-Bound-State Formation in Ce Al3at
Low Temperatures,” Phys. Rev. Lett. 35, 1779 (1975).
[113] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schäfer, “Superconductivity in the Presence of Strong Pauli Paramagnetism: Ce Cu2Si2,” Phys. Rev. Lett.
43, 1892 (1979).
[114] M. Ruderman and C. Kittel, “Indirect Exchange Coupling of Nuclear Magnetic Moments by
Conduction Electrons,” Phys. Rev. 96, 99 (1954).
[115] T. Kasuya, “A theory of metallic ferro- and antiferromagnetism on Zener’s model,” Prog.
Theor. Phys. 16, 45 (1956).
[116] K. Yosida, “Magnetic Properties of Cu-Mn Alloys,” Phys. Rev. 106, 893 (1957).

115[117] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and
M. Greiner, “Probing the Superfluid-to-Mott Insulator Transition at the Single-Atom Level,”
Science 329, 547 (2010).
[118] L. M. Duan, “Controlling ultracold atoms in multi-band optical lattices for simulation of
Kondo physics,” Europhys. Lett. 67, 721 (2004).
[119] M. Foss-Feig, M. Hermele, and A. M. Rey, “Probing the Kondo lattice model with alkalineearth-metal atoms,” Phys. Rev. A 81, 051603 (2010).
[120] M. Foss-Feig and A. M. Rey, “Phase diagram of the bosonic Kondo-Hubbard model,” Phys.
Rev. A 84, 053619 (2011).
[121] T. Flottat, F. Hebert, V. G. Rousseau, R. T. Scalettar, and G. G. Bartrouni, “Bosonic
Kondo-Hubbard model,” Phys. Rev. B 92, 035101 (2015).

Appendix A
Alignment and Characterization of Optical System

Alignment of an optical system should be a straightforward task, where the goal is simply
to align the light to the optical axis of each element and set the spacings properly to maintaincollimation, where appropriate, as well as generating images that are in focus. However, thisprocedure can be complicated and, in particular for high numerical aperture (NA) systems, thesensitivity of the system performance to this alignment may be extremely high. In our system, we
have worked to separate these problems into the alignment of our optical rail system, describedabove, and then the alignment of the output beam to both the lens and the cell (which must bealigned with respect to each other).
For the optical relay rail, depicted in Figure 2.6, we have designed a system where the opticalaxis is mechanically defined via a precision machined rail, on which we mount fixed-height optics
holders. This mechanical definition of the optical axis works well for the low-NA optics that areused on the rail (aside from the initial collimation lens, the shortest focal length lens lens is f = 15mm for an input beam of w0 = 0.68 mm, which would be an effective NA<0.05) and simplifies
the setup significantly. In particular, because the optical axis is mechanically defined, we can add
reference pinholes along the two telescope arms of the rail (the first with M = 1, the second with
M = 20), as indicated in Figure 2.6, and use the mirrors to align the collimated beam along thisaxis before adding the lenses to these paths. One the beam is aligned through these references,the lenses can be mounted to generate an output beam that is collimated and that will be directed
toward the cell and objective lens using a single mirror.

117
In fact, the most critical alignment in the system is that of the objective lens to the window
through which the beam is focused (and scattered light imaged). This is a fundamental challenge
imposed when focusing a high NA lens through a thick (6.35 mm) window because the objective isdesigned to compensate the optical path lengths for each part of the beam passing symmetricallythrough this window, but when the cone of rays is tilted through the window, these path lengths
become asymmetric and there will be a mismatch of the phase-fronts at the focus. This mismatchprimarily results in the third-order aberration coma, but also includes higher-order terms and, forreal misaligned systems, will likely introduce other aberrations such as astigmatism. Because ofthis sensitivity to angle, it can be challenging to achieve a sufficiently high-precision alignmentof the optical axis to the beam using transmission of the light through the objective, so we haveinstead turned to using a beam reflected off of the flat front-facet of the lens as one of our alignment
references.
Unfortunately, simply adding this alignment reference to the others would result in overconstraining the system and, in the case of our lenses, these alignment references are often not fully
consistent. In particular, this is likely due to the known presence of aberrations in the objective
lens (discussed more in Section A.1, below), which makes the optimal alignment of the lens notoccur when the optical axis is aligned to the cell. In this case, we typically choose two referencesthat are reproducible and can be referred to an alignment that give optimal performance in an
external testing setup, for example: 1) Setting a specific angle between the front facet of the lensand the cell window through which it is focusing (ideally this would be 0◦, but as discussed below
does not have to be) and 2) That a small beam transmitted through the lens remains on-axis at
a given point behind the lens. Unfortunately, for aberrated systems where the angle of the lens isnonzero, this second metric will not be unique (in that it will likely depend on exactly where thereference point is located behind the lens), but the important thing is that it gets the lens very
close to centered on the beam, since there is a larger tolerance to transverse displacement of theobjective than to tilting of the lens.1
1

In principle, a better alignment reference would be to ensure that a small beam passing through the objective

118

Window and objective lensCCD

ø=350 nmf=1m

Figure A.1: In this testing setup, we are directly imaging the point spread function of the objective
lens (and thus the imaging system used in the experiment) by illuminating a sub-wavelength (350
nm diameter) aperture with 852 nm light. This results in a uniform illumination of the objective
lens aperture as a spherical wave diverges from the aperture. The beam that is collimated and thenimaged onto a CCD camera (using the standard imaging system from the experiment) will exhibit
the best-focus that can be generated by the imaging system for the alignment of the imaging system.
This provides the opportunity to adjust the alignment of the objective lens in-situ to optimize theperformance, and then measure the optimal alignment using external references (which can thenbe used to align the objective to the cell).

A.1

External lens testing and alignment procedure
We have relied on two particular types of external testing to optimize the performance of our

optical system, as well as to estimate the remaining aberrations present in the system. Importantly,
we were able to use these testing setups to determine the proper method for aligning the opticalsystem and to achieve the best optical tweezer performance, which has since been verified using bymeasuring trap properties with the atoms while varying parameters of the alignment.

A.1.1

Measurement of the point spread function (PSF) via pinhole imaging

The first measurements we performed (and the basis for our external characterization of the
ASE lens) was to measure the point spread function (PSF) of the objective lens by imaging a
sub-wavelength pinhole (which is small enough that we can effectively consider it a point-source)
through the objective and onto a CCD camera. A simple schematic of this setup is shown inpropagates parallel to the optical axis in the absence of the objective (meaning it is going through the optical axis of
the system, even if that is tilted and imparts a displacement on the beam). However, experimentally observing thisis harder because it requires using multiple reference points along the path after transmission through the lens, andmaking sure that the offset of the beam from the center of each reference is identical at different points.

119
Figure A.1. The main benefit of this setup is that we can directly observe the optimal performanceachievable with the objective lens, assuming a completely aberration-free source, and can change
the alignment in-situ while directly observing the change in the focal shape.
As in the experiment itself, the alignment of this system is critical. However, if we set up theremainder of the system properly, we can leave the degrees of freedom relevant to the objective lens
free to be manipulated while knowing that we continue to observe the effective spot size. Briefly,
this alignment starts by defining the optical axis from the source to the CCD camera and aligningthat optical axis to be normal to the window that acts as our cell. Next, we will align the imaging
lens both to the optical axis, and ensure that the focal plane is in the plane of the CCD array.
Then, we will add the objective lens and roughly align it to the optical axis (in this case, we use
an iris between the objective and the imaging lens, as well as the location of the beam on the CCD
to ensure that the transmitted beam follows the optical axis). Finally, we add a relatively small
focal length lens before the window, which will focus light onto the back of the pinhole, and use
it to recollimate (as much as possible) the beam after the objective lens. Then, we can simply
add the pinhole at the focus between the two lenses, and position it by aligning it to the opticalaxis (when its image on the CCD is where the initial beam was located) and, at this point, weare measuring the point-spread-function (PSF) with the lens optical axis of the lens aligned to
the window. However, to optimize the performance, we can adjust the combination of the lensalignment and pinhole location (using both to keep the focal spot at the same location on thecamera) to find an alignment that gives a focus with minimum aberrations.
At the end of this procedure, once we have found the optimum performance of the lens, we
can remove the pinhole and focusing lens and measure the angle of the lens with respect to thewindow for use in the alignment procedure in the experiment. Note that there are a couple of ways
to measure this angle, but the optimum will use a long focal length lens, placed roughly one focal
length away from the lens, and then the separation between the two foci will indicate the relative
angle with very high precision. This is also the method that we use in the experimental setup to
make sure the alignment is identical.

120This setup also has the benefit of quantifying the magnification of our imaging system, whichallows us to calibrate the spacing of our traps in situ. Specifically, we have used different test
targets that consist of multiple pinholes arranged in close proximity, with known distances between
these apertures (measured with a scanning electron microscope). In particular, our final calibration
for the ASE objective was been performed with a test target that has two apertures, each 350 nm
in diameter, separated by 6.861 µm, giving a magnification of 48.8(5) for the imaging system (theinfinity-corrected objective and an f = 1 m achromat.

A.1.2

Wavefront measurement testingMore recently, we have put together a few different setups that are able to directly measure
the wavefront aberrations introduced by the objective lens. For fully understanding and attempting
to correct the wavefront aberrations present, this is the best method to isolate effects due to theobjective lens itself. Previously, we tried to perform measurements with a Thorlabs Wavefront
Sensor, but the results were unreliable. However, it is possible to set up a Michelson interferometerwith the objective lens in the arm under test, which produces an interference pattern containing
the wavefront aberrations introduced by the objective lens.

A.2

Characterization of the trapping potential
After setting up the system in-situ, we are able to use the atoms to characterize certain

properties of the trapping potential. This is not a direct measurement of any aberrations, but a
local probe of the harmonic potential that is generated for the atoms. The goal of this alignment
procedure is, again, to achieve the highest harmonic confinement for a given power into the backof the lens, which will likely correspond to a minimum in the overall aberrations.
The key part of this procedure is to be able to measure the trap depth that the atom
experiences, which is a measure of the intensity of light at the focus of the trap, as well as thetrap frequency that the atoms experience. Combined, we can estimate the size of the trap, given a
few assumptions about the intensity profile based that is based on the clipping of the input beam

Reference mirror

121

Window and objective lensSiN Sphere

Observe and record
interferogram on CCD

Figure A.2: In this testing setup, an interference pattern that contains information about the
wavefront aberrations of the objective lens is generated using a Michelson interferometer. The
reference path is defined with a reference mirror whose flatness has been verified on a commercial
Zygo interferometer. Thus, the interference pattern arises solely from aberrations introduced on
the lens arm of the interferometer and appear with twice the magnitude because the beam passes
through the lens twice. By recording the interference pattern and calculating the Fourier transform,
we can extract the wavefront curvature, and thus have full knowledge of the wavefront aberrations
that need to be corrected for optimal performance.

and the expected PSF from our Zemax model. Note that the main role of the Zemax model here
is used to estimate an effective gaussian waist that is representative of the actual size of the trap,
given that for a clipped gaussian beam, the trap frequency is noticeably lower than that of an ideal
gaussian beam of similar extent (meaning, if you fit the actual profile with a gaussian profile, a
trap with that waist would have trap frequencies larger by more than 10%).
In the end, we estimate that we have an effective Gaussian waist of w0 = 0.7(1) µm basedon the measured trap depth, trap frequency, and ratio of the harmonic component of the PSF
calculated by Zemax (which is an approximation of the intensity profile) to that of a gaussian fit to
the same calculated PSF. Notably, this estimate is consistent with the waist used in the full threedimensional theory calculations given in Ref. [39] to understand the interactions and tunnelingrates in a double-well potential.

122

70
60
50
40
30
0

5
10
15
20
25
Collimating lens position (mm)64 (b)
62
60
58
56
54
52
0.565

30

Axial Resonance (kHz)(a)

Axial Resonance (kHz)

Axial Resonance (kHz)

80

0.566
0.567
0.568
y from normal (degrees)

0.569

64 (c)
62
60
58
56
54
52
0.186

0.187
0.188
0.189
x from normal (degrees)

0.190

(b)

25

15

0.6
0.4
0.2
0.0

0

5
10 15 20 25
Cycling detuning (MHz)

1.0
Survival Probality

10
5
0
0

1.0

0.8

Survival Probability

20

1.0
Survival Probality(a)|2,2>-| 3',3> light shift (MHz)Figure A.3:
This figure shows how we can optimize the trapping frequencies that the atom
experience by adjusting the alignment of the lens to the cell, as well as the collimation of the beam
into the back of the objective lens. In particular, we note that the angle optimum angle of thelens with respect to the cell window is consistent with the optimum angle measured in the external
testing setup.

0.8
0.6
0.4
0.2
0.0

1

5

10 15 20 25 30 35
Cycling detuning (MHz)2
3
4
5Power in Telescope (mW)

6

0.8
0.6
0.4

5.96 mWin telescope

0.2
0.0

240

260
280
300
320
Parametric Drive Freq (kHz)

Figure A.4: This figure shows the method by which we characterize the intensity of the light at
the atoms, where we measure the cycling transition resonance at different trap depths, then matchthe slope of this line with the expected light shift as a function of intensity of light at the atom.
The key capability here is to use a pure σ + polarized beam, which simplifies the analysis of thelight shift. In this case, we assume a pure linear polarized trapping potential, which is a reasonableassumption based on the additional vector light shifts being at most 100 kHz at full depth (asshown in Figure 5.4).

Appendix B
Derivation of eigenstates in minimal Kondo modelOn the following pages, I am reproducing notes written by Leonid Isaev (from Ana Maria
Rey’s group), in which the exact eigenstates are derived for the model system composed of adouble-well potential with one fixed spin in the ground-band of each well, as well as a single atomin the excited band of the double-well potential. The eigenstates are derived using arguments ofsymmetry, which lead to conserved quantities in the form of the total spin of the system, as wellas the parity the wavefunction under exchange of the two well indices. Note that the notation
for fermionic systems (e.g., the Kondo lattice model) was used when writing these notes, but forreference the energy scales are given by t = Je and V = 2Veg .
The exact solutions can then be used to perform fast simulations with arbitrary interaction
and tunneling parameters from any initial state (by projecting into the eigenbasis of the system,
propagating in time, and then projecting back into observable basis states); this was the process usedin simulations shown in Chapter 6. Importantly, this allows us to rapidly change the parametersof the simulation and calculate the change in the expected dynamics (as compared to numericallysolving the Schrodinger equation, which is computationally intensive).
After deriving the exact eigenstates, the effective Hamiltonian given in Equation 6.3 is foundusing second-order perturbation theory in the weakly-interacting limit (V  t, in this notation, or
Veg  J in the notation of Section 6.2).

124

Kondo box and geometric frustration with cold atoms
(a)

MODEL

We consider a double-well system shown in Fig. 1(a).
The lowest levels are strongly localized, while the higher
band splits due to particle tunneling between the wells[see Fig. 1(b)].
The double-well is populated with three spin- 12 atoms:
one in the excited band, and two localized in the lowest
levels of individual wells. This system can be modeledby a Hamiltonian:
H=t

X †

(c1σ c2σ + c†2σ c1σ ) + V S1 · sc1 + S2 · sc2 . (1)

(b)

t

2t
|ai

V
1

P̂

2

(c)
|f i

P = +1|f i|f i = |1i+|2i|ai = |1i−|2i

(d)
|ai

P = +1|f i

P = −1|ai

P = −1σ

The first term describes the hopping (hybridization) between wells 1 and 2 with an amplitude t which we assume
to be real and positive. The operator c†iσ creates a particle (the statistics does not matter) in the i-th well withspin projection σ: |1iσ i = c†iσ |vaci (|vaci is an empty
state). The second term is the exchange coupling between local atoms in each well and the mobile atom. We
assume that V > 0 but this is not a crucial assumption
because we will not be interested in the ground state
properties. Si is the local spin- 12 operator in i-th well,P
and sci = 12 αβ σαβ c†iα ciβ is the spin of the mobile particle (σ is the vector of Pauli matrices). Below we willuse implicit summation over repeated spin indices.

EIGENSTATES OF THE MODEL HAMILTONIAN

The Hamiltonian (1) has two important symmetries.
First, there is the spin SU (2) continuous symmetry thatguarantees conservation of the total spin Stot = S1 +S2 +sc1 +sc2 of the system. Stot can be either 12 or 32 . The second one is a discrete “lattice” symmetry that corresponds
to a mirror plane interchanging the wells (i.e. 1 ↔ 2) andleads to the conservation of parity P̂ : [H, P̂ ] = 0. The
parity transformation can be defined as P̂ S1 P̂ = S2 andP̂ c†1σ P̂ = c†2σ . Since P̂ 2 = 1, the parity operator has
two eigenvalues P = ±1 (no hat). The corresponding
three-particle states are schematically shown in Fig. 1(c)and (d). With the aid of these symmetries one can easily
determine all eigenstates of the model (1).
In the following it will be convenient to diagonalize thehopping term in H by introducing symmetric (f ) andantisymmetric (a) combinations of the mobile particle:


fσ
aσ




1
= √ c1σ ± c2σ ,2(2)FIG. 1. Panel (a) The double-well setup with two local atoms(red arrows) and one mobile atom (blue arrow). The parity operator P̂ interchanges the two wells. (b) Hybridization
splitting 2t of the mobile band. |f i and |ai denote symmetric and antisymmetric states (2). (c) Schematic illustrationof different states with P = +1. The gray ellipse indicatesa spin-singlet state across the wells. Parallel local spins correspond to a spin-triplet. (d) Same as in (c), but for totally
antisymmetric states with P = −1.

and rewrite the Hamiltonian (1) as
H =t(fσ† fσ − a†σ aσ )+V+ (S1 + S2 ) · (Tf f + Taa )+2
V+ (S1 − S2 ) · (Tf a + Taf ).
2(3)

Here Tf f = 21 σαβ fα† fβ , Taa = 12 σαβ a†α aβ , Tf a =†1†2 σαβ fα aβ and Taf = Tf a .
We will denote local spin states as |↑i i and |↓i i and use
Clebsch-Gordan coefficients of build wavefunctions witha definite total spin out of the mobile S = 12 and local
two-spin states. When writing the three-particle states
we will follow the naming schemez|Stot , Stot
; P ; ai,

where a is the index of a state with other quantum numbers fixed. Since the number of particles in the lowest
levels is exactly one/level, the Hilbert space of Eq. (1)has dimension 2 × 23 = 16.
zSector Stot = Stot
= 21 , P = +There are two ways to have a state with total spin
S = 12 : (i) local particles form a S = 0 singlet, so thespin comes from the mobile atom, and (ii) local spinsare in the S = 1 triplet and we have to couple it to the1252mobile spin using angular momentum addition theorems.
Hence, there are two basis states:1 1
1, ; +; 1 = √ |↑1 ↓2 − ↓1 ↑2 i ⊗ a†↑ |vaci,
2 2
2

1 
1 1
, ; +; 2 = √ 2|↑1 ↑2 i ⊗ f↓† |vaci−
2 26− |↑1 ↓2 + ↓1 ↑2 i ⊗ f↑† |vaci .

The first ket is built from antisymmetric local-spin singlet and an antisymmetric fermion combination, while
the second one is constructed using all symmetric states.
In this basis the Hamiltonian matrix has the form!
√−t
− 43 V√.
H 12 , 12 ;+ =− 43 V t − 12 V
On the diagonal, t originates from the kinetic energy, and− V2 from the interaction of the mobile atom with a mean
local magnetization in the 2nd state. The eigenvalues andcorresponding eigenfunctions are:E1,2 = −with R =qt − V /4

2

+

√

V
∓R
4

2
3V /4 , and



ϑ 1 1
ϑ 1 1
, ; +; 1 + sin, ; +; 2 ,
2 2 2
2 2 2


ϑ 1 1
ϑ 1 1
|ψ2 i = − sin, ; +; 1 + cos, ; +; 2 ,
2 2 2
2 2 2
√

where cos ϑ = t − V /4 /R and sin ϑ = 3V /4Rt.
|ψ1 i = coszSector Stot = Stot
= 21 , P = −In complete analogy with the previous subsection, we
have the basis:
1 1
1, ; −; 1 = √ |↑1 ↓2 − ↓1 ↑2 i ⊗ f↑† |vaci,
2 2
2

1 
1 1
, ; −; 2 = √ 2|↑1 ↑2 i ⊗ a†↓ |vaci−
2 26− |↑1 ↓2 + ↓1 ↑2 i ⊗ a†↑ |vaci .

q

√
2
2
t + V /4 +3V /4 , and


ϑ 1 1
ϑ 1 1
|ψ1 i = cos, ; −; 1 + sin, ; −; 2 ,
2 2 2
2 2 2


ϑ 1 1
ϑ 1 1
|ψ2 i = − sin, ; −; 1 + cos, ; −; 2 ,
2 2 2
2 2 2
√

where cos ϑ = t + V /4 /R and sin ϑ = − 3V /4Rt.
with R =z
Sector Stot = 12 , Stot
= − 12 , P = ±The basis states can be chosen as1 11, − ; +; 1 = √ |↑1 ↓2 − ↓1 ↑2 i ⊗ a†↓ |vaci,
2 2
2

1 1
1 
, − ; +; 2 = − √ 2|↓1 ↓2 i ⊗ f↑† |vaci−
2 26− |↑1 ↓2 + ↓1 ↑2 i ⊗ f↓† |vaci ,11 1, − ; −; 1 = √ |↑1 ↓2 − ↓1 ↑2 i ⊗ f↓† |vaci,
2 2
2

1 
1 1
, − ; −; 2 = − √ 2|↓1 ↓2 i ⊗ a†↑ |vaci−
2 26− |↑1 ↓2 + ↓1 ↑2 i ⊗ a†↓ |vaci .

The Hamiltonian matrices and their eigenstates are thezsame as in corresponding parity sectors with Stot
= 12 .
z
Sector Stot = 32 , all Stot
, P =+z:
There are 2Stot + 1 = 4 states with different Stot
3 3†, − ; + =|↓1 ↓2 i ⊗ f↓ |vaci,
2 2

3 1
1 
, − ; + = √ |↓1 ↓2 i ⊗ f↑† |vaci+
2 23+ |↑1 ↓2 + ↓1 ↑2 i ⊗ f↓† |vaci ,

3 1
1 
, ; + = √ |↑1 ↑2 i ⊗ f↓† |vaci+
2 23+ |↑1 ↓2 + ↓1 ↑2 i ⊗ f↑† |vaci ,3 3, ; + =|↑1 ↑2 i ⊗ f↑† |vaci.
2 2

The Hamiltonian matrix can be obtained from that in
the previous subsection by replacing t → −t:!
√t− 43 V√H 21 , 12 ;− =.
− 43 V −t − 12 V

Of course, these states are not mixed by the Hamiltonian,hence they are already eigenstates with an energy

The spectrum and eigenstates can be written as before,
but with a small change in labeling,z
Sector Stot = 32 , all Stot
, P =−

E1,2 = −V±R
4

1
E = t + V.
4This sector can be obtained from the previous one byreplacing f → a and t → −t.

126

3
RKKY REGIME V  t

It is instructive to build a low-energy effective Hamiltonian for the case of small exchange coupling (comparedto the bandwidth). In this regime, the model (1) can beseen as a frustrated triangular plaquette with two local
and one mobile spin being at the vertices. The sign of theeffective exchange coupling can be controlled by populating either symmetric (f -type) or antisymmetric (a-type)band.
When V = 0, there are two 8-fold degenerate sectors corresponding to the mobile atom being either in asymmetric or antisymmetric state (the degeneracy comesfrom spins). For small but finite V , this manifold is splitby the exchange interaction. Hence, we start from Eq.
(3) whose first two terms are diagonal and do not mix
the hybridized states in the upper band, and the last line
is an off-diagonal perturbation. We will call the diagonal terms in Eq. (3) as H0 and the last off-diagonal
exchange correction – as H1 . The latter can be removed
in each order on V /t by a Schrieffer-Wolff transformation H → eΛ He−Λ ≈ H0 + 21 [Λ, H1 ] with Λ chosen sothat [H0 , Λ] = H1 . To the lowest order in V we haveΛ=V X S12 · P00 Tf a P0
− h.c. = U − U † ,2 0
E00 − E0
E0 ,E0where S12 = S1 − S2 . In the summation, E0 (and E00 )denotes the full set of quantum numbers that characterizea degenerate state of H0 (here only the kinetic energy).

P0 (and P00 ) are projectors on this subspace.
If the mobile particle is in the lowest-energy antisymmetric state [see Fig. 1(b)], the 2nd order correction[Λ, H1 ] = [U, H1 ] + h.c. is1
hvac|aσ0 [Λ, H1 ]a†σ |vaci =2ji
V 2 X S12
S12
i
hvac|aσ0 Taf
P00 Tfja P0 a†σ |vaci ==−4 0
E00 − E0
E0 ,E0V2 i j i j=−
S S (σ σ )σ0 σ =32t 12 12
2

3V
V2
=−
+
S1 · S2 + τ · (S1 + S2 ) .
64t
16t

Adding the linear contribution [second line in Eq. (3)],
we obtain an effective spin Hamiltonian (up to a constantenergy shift)Hef = V τ · (S1 + S2 ) +V2S1 · S2 + τ · (S1 + S2 ) ,16t

where ταβ = hvac|aα Taa a†β |vaci is the spin- 12 operatorof the antisymmetric mode. The quadratic term in thisexpression is an antiferromagnetic Heisenberg model on
a geometrically frustrated triangle.
The case when the mobile atom is in the symmetricstate can be treated in exactly the same way. The resulting effective Hamiltonian can be obtained from Hef
by replacing Taa with Tf f and t → −t. In this casethe effective Heisenberg model is ferromagnetic and thetriangle is non-frustrated.