A Fast 7Li-based Quantum Simulator
A dissertation presented
byJesse Amato-Grill
toThe Department of Physics
in partial fulfillment of the requirements
for the degree ofDoctor of Philosophy
in the subject ofPhysics

Harvard University
Cambridge, Massachusetts
May 2019© 2019 Jesse Amato-Grill
All rights reserved.

Dissertation Advisor: Wolfgang Ketterle

Jesse Amato-Grill

A Fast 7 Li-based Quantum Simulator
The work presented in this thesis is the design and characterization of a new apparatus forquantum simulation using bosonic Lithium in optical lattices, along with proposals for simulationsof the spin-1 Heisenberg model and the results of initial exploratory experiments. The topic itself
falls under the general category of adiabatic analog quantum computing with neutral atoms: aphysical instantiation of a simple quantum mechanical Hamiltonian is built out of 7 Li atoms nearzero temperature and results are measured directly from the model system. While the techniques
employed in this work are inspired by progress and discoveries in atomic physics, its applications
are to the study of condensed matter systems, and similar approaches may one day be used toaddress a wider range of computational problems.
The strategy of adiabatic analog quantum computing comprises evolving the system through achain of different ground states until the desired state is reached, and the structure of this thesis
reflects that progression. Chapter 1 situates this work in the context of the larger effort to build
computational devices, and introduces the approach followed in this work. Chapter 2 documents
the design and construction of the apparatus for cooling 7 Li to quantum degeneracy, with emphasison the apparatus’ unique features, including fast evaporative cooling and tunable magnetic bias
fields with low curvature. Chapter 3 discusses the role and implementation of the optical lattice, as
well as continuing efforts to characterize and mitigate decoherence mechanisms. Finally, Chapter
4 reports the results of a new lattice-based spectroscopic method for determining the strengthof atomic contact interactions and further suggests how these results can be used to perform asimulation of the spin-1 Heisenberg model.

iii

Contents
Titlei

Copyright

ii

Abstractiii

Table of Contentsiv

Acknowledgments

viii

1 Introduction1

1.1

Analog Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

A Quantum Analog Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.1

The Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.2

Analogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Adiabatic Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3

1.4

1.3.1

Reaching a Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.2

Transforming the Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Why Lithium? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Initialization

13

2.1

Why Cooling 7 Li is Difficult . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2

Overview of the Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1

Cooling and Imaging Laser System . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2

Chamber Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22iv

2.2.3

Coil Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.4

Computer Control and Data Collection . . . . . . . . . . . . . . . . . . . . . 36

2.3

Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4

Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.5

2.4.1

Magnetic Trap Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4.2

Optical Dipole Trap Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . 53

Summary of Cooling Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 A Platform for Simulations
3.1

3.2

3.3

3.4

57

An Atom in an Optical Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1.1

Optical Lattice Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1.2

Wavefunctions in a Lattice: The Momentum Basis . . . . . . . . . . . . . . . 59

3.1.3

Wavefunctions in a Lattice: The Position Basis . . . . . . . . . . . . . . . . . 63

Many Atoms in an Optical Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.1

The Bose-Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2.2

Ground State of the Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . 68

Lifetimes and Coherences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3.1

Inter-band Heating Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3.2

Intra-band Heating Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 82

Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4 Towards Simulations of Spin-1 Heisenberg Chains
4.1

4.2

4.3

91

Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.1

Effective Hamiltonians and the Two-component Bose-Hubbard Model . . . . 92

4.1.2

Ingredients for an atomic spin-1 model . . . . . . . . . . . . . . . . . . . . . . 100

Interactions in 7 Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.1

Interactions on a Lattice Site . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2.2

Interaction Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Considerations for Spin-1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1

Coherent Control and State Preparation . . . . . . . . . . . . . . . . . . . . . 111

4.3.2

Magnetic Field Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112v

4.3.3

Decoherence Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3.4

First Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5 Conclusion and Outlook

119

Bibliography

122

vi

Acknowledgments
None of the work included in this thesis (nor any that has been left out of it) would have been possible without the encouragement, support, and collaboration of a number of remarkable individuals,
whom I am lucky enough to call friends, mentors and family. Two physicists in particular deserve
special mention: Ivana Dimitrova and Niklas Jepsen, my labmates and physics siblings. Throughcountless days and nights of close collaboration, they could always be counted on to join in brainstorming sessions, debugging debacles, and marathons of data-taking. They have always been thefirst people to celebrate my best ideas, and to graciously explicate my worst, while the combined
impetus of Ivana’s get-it-done attitude and Niki’s perfectionism helped us scale to ever-greater
heights of experimental complexity and scientific insight. Many thanks are owed to Wolfgang Ketterle, my Ph.D. research advisor, for granting me the opportunity to participate in the bleedingedge of quantum mechanics research as a graduate student, and for creating and stewarding theintellectually fertile and emotionally supportive environment in which I came of age as a physicist.
He has ever been my teacher, my mentor, and my sponsor, and his unique insight into, and enthusiasm for physics and problem-solving will always continue to inspire my research and renew mycuriosity.
I have been lucky to have many other close collaborators, who have spent a great deal of time
discussing physics with me and a great deal of effort to help design and construct the experimental
platform described in this work. I would especially like to thank David Weld, who gave me myfirst on-the-job crash course in experimental atomic physics; Michael Messer, whose helped theexperimental build during its first year and whose energetic presence can still be felt in the lab;
Yichao Yu, who even as an undergraduate provided invaluable help getting the experiment up andrunning, and with whom I got to share the excitement of our first 7 Li BEC; and Will Lunden, whosevii

talents and effort came to our aid just at a moment when we were really struggling to master thetechnical challenges of working with 7 Li in an optical lattice. Many thanks also to Mark Belanger,master machinist of the MIT Edgerton Shop, who nurtured my love of making things, and without
whose mentorship the experimental hardware would not be nearly as sophisticated. The HarvardMIT Center for Ultracold Atoms has been my home for many years, and my work there would nothave been nearly as fruitful nor quite as fun without the many colleagues and friends from I learned
and borrowed, and with whom I commiserated celebrated. Lawrence, Timur, Colin, Sebastian, theThomases, the Borises, Tarik, Waseem, Claire, Julian, Matt, Vinay, Peter, Alex, Sepehr, and Woo
Chang, thanks for all the illuminating discussions and all the laughs!
I would also like to thank David Grier, my undergraduate research advisor, for starting me on
this path and for providing sage and salient advice along the way, even up to the present. And I
am deeply grateful for the support and encouragement of my friends and family, including Camilleand Joey, who first taught me to be curious; Shonte, who shares my opinion on nearly everything,
and reassures me that I’m not completely crazy; Anik, who has patiently put up with all of mycomplaints about the vicissitudes of lab work and kept my spirits up; Tarun, who has shared andpromoted my desire to be a well-rounded scientist; and Ben and Kelly, who took special care that
I didn’t work too hard. To all of you, a heart-felt “thank you.”viiiChapter 1

Introduction

1

1.1

Analog Computing

The work reported in this thesis concerns the design, construction and evaluation of an adiabatic
analog quantum computer. Before discussing parameter requirements, tolerances, sample problems
and results, it is worth taking a short detour in order to explain precisely what is meant by thosewords. To begin, there exist two sorts of computers: analog and digital. The former is far older,with archaeological examples dating back thousands of years, while the second has become soprevalent that it now dominates the very meaning of computer. As a result, the word analog has
itself picked up a new meaning in this context. The Oxford English Dictionary, a venerable sourceof wisdom on such topics, suggests the following definitions [1]:adjective
Relating to or using signals or information represented by a continuously variable physical quantity such as spatial position, voltage, etc.
noun
A person or thing seen as comparable to another.
and also, somewhat amusingly,
adjective
Not involving or relating to the use of computer technology, as a contrast to a digitalcounterpart.
Despite its colloquial usage, the third definition is not an entirely fair characterization. As thefirst two definitions might suggest, the term analog, as it applies to computing, really has two
distinct meanings: analog in the sense of continuous and proportional, as opposed to discrete,
and analog in the sense of analogous, as opposed to algorithmic. The former is a categorizationof the underlying hardware components, while the latter is a computing paradigm. In a stored
program computer, the internal structure of the device does not change when the computer istasked with solving one problem or another; instead, the device iterates over a sequence of inputs
stored in memory, which constitute an algorithm. The operation of an analog computer is markedly
different, for the internal structure of the device reflects the structure of the problem, and must bereconfigured to model each class of problem to be solved. Given that most computers with digitalcomponents are stored-program computers and most computers built of analog parts are modelbased, it is not surprising that the distinction between problem-solving paradigm and technological2

implementation has been lost in the vernacular. It is nonetheless possible to have computers builtof digital components, which solve problems in an analog fashion (for example, on an FPGA),
and it may be possible to build stored-program computers out of analog components (such as agate-based quantum computer) [2].1
The design of an analog computer is instead inspired by the simple but powerful idea thatsystems may be described by the same mathematics without actually sharing any of their physicalcomponents. For example, the New Zealand economist Bill Phillips noticed a correspondence between the set of coupled differential equations in a certain model of national economies and thosedescribing fluid flow through a system of connected pressurized reservoirs and orifices, and the
MONIAC hydrodynamic computer he designed in 1949 saw much use over the following decade,demonstrating and predicting aspects of the economy of the United Kingdom [3]. Throughout thefirst half of the twentieth century, there were increasingly sophisticated attempts to modularlize
the components of analog computers, especially in order to build general-purpose differential equation solvers. There are mechanical, electrical, and even hybrid devices which integrate, scale anddifferentiate their inputs, and these can be connected in series and parallel in order to performa particular computation. At MIT, the university which hosted the work reported in this thesis, Vannevar Bush and colleagues constructed in 1923 an electromechanical “differential analyzer”
that could solve systems of coupled differential equations, which proved so useful that it was soon
replicated at research institutions around the world [2].
The first example of such a device, however, is an astrolabe recovered from the remains of a
shipwreck off the Greek island of Antikythera. Its thirty-seven brass gears are supposed to havemodeled the universe as the Greeks saw it, with stars traveling in fixed orbits around the earth,and the known planets traveling in epicycles within their own orbits. Gear ratios have nothing to
do with the functioning of the heavens, even in the ancient Greeks’ perspective, and yet the device
reproduced the motions of the planets while relying on completely different physical principles.
The Antikythera device’s likely date of origin, c. 100 BCE, is made all the more amazing by thefact that no computer of similar complexity is thought to have been made at any time during the1

It can be argued, for example, that finite state machines and neural networks employing Heaviside activation
functions are digital-analog computers: the nodes exist in one of two possible states, but rather than behaving asa stored program computer, they implement some functional “in hardware.” Either can, of course, be simulated on
stored program computers, with significant overhead.

3

(a)

(b)

Figure 1.1: (a) A detail of the Antikythera device and (b) Bush’s differential analyzer at MIT.
(a) is a figure in the public domain, reproduced from [4]. (b) is reproduced courtesy of the MIT
Museum archives.
following millennium [2, 5, 6]. Happening upon the device in an archaeological museum in Athensin 1980, none other than Richard Feynman was so struck with its sophistication that he wrote tohis family that he had seen “among all those art objects [...] one thing so entirely different andstrange that it is nearly impossible.” He concluded: “I wonder if it is some kind of fake” [7].

1.2

A Quantum Analog Computer

One year later, it was again Richard Feynman who proposed, in a landmark paper, that a computer
which solved quantum mechanical problems ought to itself operate quantum mechanically at a
hardware level– that is, that the quantum rules of one system can be used to build an analog
computer which models the behavior of some other quantum system [8]. The kind of computer (or“simulator,” in the paper’s language) to which Feynman referred is one that is analog in at least thesense of proportionate. A collection of n two-level systems (or qubits) can exist in any superpositionof 2n possible states. The existence of such superposition states is the advantage which quantum
computers hold over their classical counterparts, and is the very reason that quantum computers arenecessary to solve some problems in quantum mechanics. Yet the superposition state is encodedin the phase relationships between the qubits, and phase is an analog quantity. Whether thissituation is fundamentally the case or not is well outside the scope of this thesis, but is nonetheless4

Figure 1.2: Detail of the central vacuum chamber of a modern analog computer (in fact, the one
discussed in this thesis), photographed with magnetic coils installed but with all of the optics
removed.

an important question: the growth of digital classical computing power has been a direct result of
the fact that digital logic is implemented by using the saturated (i.e. digital) regime of transistors,
which relaxes manufacturing tolerances enough to allow for the creation of billions of microscopic,
logical (that is, error-free) bits on a single device. At least for now, the development of quantum
computers lacks this distinct advantage. While in principle it is possible to design a stored-program
computer out of analog qubits (and such an approach is being actively pursued), the manufacturing
tolerances remain tight enough that it is not yet possible to create meaningfully large numbers oflogical qubits [9]. It is also possible, however, to have the qubits interact directly, via physicalprocesses that are analogs of the processes in the system to be modeled. That is the approach
followed in this work, and it is therefore in the spirit of the Antikythera device and of Feynman’s
proposal that the apparatus described in the next chapter was conceived.

1.2.1

The Heisenberg Model

While the apparatus is intended to be a multi-purpose platform for quantum simulation with bosons
in optical lattices, it will prove useful to keep in mind a particular quantum problem to be solved,
in order to provide context for the design and to begin to evaluate its capabilities. One such class5

of quantum problems involve the spin-spin interaction

H=∑

Jijσ Siσ Sjσ ,

(1.1)

i,j
σ=x,y,zwhere Si is a three-dimensional spin vector with components S α located on a lattice site i, and Jα isa potential energy which in general depends on the distance between sites i and j. This deceptively
simple Heisenberg Hamiltonian features an incredible range of ground states and excitation spectra.
Depending on the sign of J (which reflects the quantum statistics of the particles and whether theinteractions between them are attractive or repulsive), the coupling may be ferromagnetic or antiferromagnetic, leading to magnetically ordered ground states with gapless excitation spectra of spinwaves. In a square lattice, the anti-ferromagnetic ground state is predicted and observed to haveextremely shallow overlap with the classical anti-ferromagnet, due to quantum fluctuations[10, 11].
The magnetic order may be frustrated, leading to glassy phases, depending on the coordination
number of the lattice, and also on whether the interactions are long range or restricted to nearest
neighboring sites. Anisotropy in the exchange coupling leads to competition and ultimately phasetransitions between ordering on different axes. Not least surprising is the prediction that for integerspin particles in one dimension, the Heisenberg model features a gapped ground state with stringorder in the bulk and degenerate edge states, which is not what one would expect from mean fieldcalculations– i.e., a magnetically ordered state, as is found in spin-1/2. This prediction, first madeby Duncan Haldane in 1983 [12], suggests that the ground state of the spin-1 Heisenberg model
is one of the simplest examples of a symmetry protected topological state, featuring order that isrobust against perturbations that preserve the symmetry of the Hamiltonian [13].
With the addition of simple extra terms, the Heisenberg model can further describe a vast arrayof real-world behaviors. For example, with the addition of tunneling between sites, it captures theessential behaviors of electrons in strongly interacting materials, and can serve as design guides formaterials with novel properties, such as high temperature superconductors[10, 11, 14–16]. It has
also been shown that problems in the difficulty class NP map to a subset of the Heisenberg modelwith exchange coupling only between the z-component of the spins and with additional tunablebias fields on each site, i.e. the quantum Ising model [17, 18]. While some of these variations are
solvable in certain parameter regimes on classical digital hardware, analog quantum hardware– such6

as the ultra-cold atoms platform presented here– offers a unique opportunity to directly simulate
the most intractable cases.

1.2.2

Analogs

The work described in this thesis rests on many analogs– physical systems that are made of different
parts, or whose parts are arranged differently, but whose behavior is comparable in some way. The
components out of which this computer can be built–in effect, the analogs– are the interactions of
the atoms with electromagnetic fields (both DC and AC, resonant and far-detuned) and interactions
between the atoms themselves. This section is intended to provide a flavor of these analogs andsuggest how they may be used to simulate the Heisenberg model and other models like it; each will
be described in greater detail later on.

Phase space density
It is often said that quantum mechanics is the physics of the very small. This is only part of alarger truth: it is the physics of the very small or the very cold. The essence of this statement is
captured by the thermal de Broglie wavelength

λT =

h
h
=√
,
p
2πmkB T

(1.2)

which shows how the size of the wavefunction of a particle will increase with decreasing temperature(or kinetic energy).2 It is the overlap of the wavefunctions which determines the behavior ofquantum particles, characterized by the phase space density

ρ = nλ3T .

(1.3)

where n is the number density of particles. Thus there is already reason to suspect that the
behavior of electrons in a solid packed at 1021 cm−3 near the Fermi temperature TF ≈ 10, 000 K is
as strongly influenced by quantum effects as is the behavior of dilute clouds of atoms, with densitiesof 1013 cm−3 and cooled to within 100 nK of absolute zero.
2

Assuming there are no internal degrees of freedom.

7

Conservative external potentials
The potential energy landscape experienced by the atoms is crafted using a combination of DC
magnetic fields and off-resonant light, both of which exert conservative forces. While these provide
the trapping geometries needed to initially confine and cool the atoms, they are also responsible for
implementing the external potential of a model Hamiltonian. For example, a far-detuned standing
wave of light with wavevector k ≡ 2π/λẑ creates an AC Stark shift potential proportional to sin2 kz,
which mimics the Coulomb potential of the lattice of nuclei in a solid material, at least in the tightbinding approximation. As a result, the dynamics of an atom in such a standing wave are describedby the same band structure model as a tightly bound electron moving in a real crystal (see section
3.1.1).

Pseudospin
The spin-1/2 variant of the Heisenberg Hamiltonian introduced in the previous section describes
the behavior of quantum mechanical particles which can exist in one of two possible states, or asuperposition between the two. At least in the absence of interactions between the states (i.e. ina single-particle picture), these states might as well be any quantum mechanical two-level systemwith a driving term. The description of the system would be the same if it were two hyperfine states
connected by an RF transition, or two electronic states connected by an optical transition. The
descriptive equivalence of isolated two-level systems even extends to external states the particles,
such as localization in two potential wells connected by tunneling (e.g. in [19]). The same argument
applies in general to n-level systems.

Interactions
Unlike electrons in a solid or magnetic dipoles, neutral atoms do not have any long-range interactions. At most, the electronic polarizability that arises from admixing the spherically symmetricelectronic ground state with higher angular momentum orbitals gives rise to an induced dipoledipole, or van der Waals interaction. The strength of this interaction scales as 1/r6 , and caneffectively be treated as a contact interaction in dilute atomic systems.3 Despite neutral atoms’
3
The effective range of this interaction can, however, be dramatically increased by promoting the outermost
electron to a highly excited, or Rydberg state [20]. This approach, distinct from the work discussed in this thesis,8Figure 1.3: Adiabatic state preparationlack of long-range interactions, it is possible to implement exchange interactions of the type featured in the Heisenberg Hamiltonian by arranging the system so that the pseudospins can physicallyexchange places. This super-exchange process [23] is discussed in greater detail in section 3.2.2.

1.3

Adiabatic Quantum Computing

A computing device intended to find the ground state of some quantum mechanical Hamiltonian
requires both a way to implement the model, and a way to reach the ground state. The strategy
of adiabatic quantum computing is to perform all the cooling necessary to reach degeneracy insome system where cooling is easy, and then evolve the system into a system of interest [24, 25].
For example, a system amenable to cooling may feature large bandgaps, increasing the criticaltemperature, or may have a small number of degrees of freedom that easily exchange entropy.
The target system may inhabit a much larger Hilbert space, the low energy portion of which
sis interesting but difficult to reach because of small energy gaps or entropy distributed between
different, uncoupled sectors. Under certain conditions, it is possible to create the ground state of
the target system by creating the ground state of the original system, rather than by attempting
to cool directly into the ground state of the target system.
comes with its own set of trade-offs and is an area of active research and development [21, 22].

9

Consider the Hamiltonian
H = (1 − δ) · H1 + δH2(1.4)

whose ground states are protected by a bandgap h̄ωbg (δ). As long H1 does not commute with H2 ,it is possible to vary δ in such a way that a system which starts in the quantum mechanical groundstate of H1 when δ = 0 remains in the ground state for all δ (figure 1.3). The adiabatic theorem
supplies the speed limit, which is that ωbg should vary slowly enough to prevent Landau-Zenertransitions out of the ground state:∂ωbg (t)≪ (ωbg (t))2 .
∂t

(1.5)

This condition is fulfilled by a exponential ramp of ωbg with a time constant that depends onthe difference between the initial and final bandgaps. For systems with small bandgaps, this cantake a very long time; nonetheless, it provides a something of a side door into what may prove anotherwise inaccessible ground state.

1.3.1

Reaching a Ground State

In order to make use of this scheme, it must be possible to produce the ground state of some quantum
system– that is, to make it Bose- or Fermi-degenerate, depending on the quantum statistics of its
components. In 1995, it was shown by Eric Cornell and Carl Wieman [26] and by Wolfgang
Ketterle [27] that it is possible to create the quantum ground state of a dilute gas of composite
bosons (atoms with integer total spin), via a combination of laser cooling and forced evaporative
cooling in a conservative potential. The critical temperature for this phase transition can be derived
in the context of the Grand Canonical Ensemble [28] by considering the difference between the total
number of atoms N and the number of condensed atoms N0 :N − N0 =

∫ ∞
dE
0

g(E)
,
exp [β(E − µ) − 1]

(1.6)

where g(E) = 0.5 E 2 /(h̄ω̄)3 is the density of states in a harmonic trap with geometric mean trap
frequency ω̄ ≡ (ωx ωy ωz )1/3 , and the denominator is the Bose-Einstein distribution function forenergy E, chemical potential µ and inverse temperature β = 1/(kB T ). When µ = 0, none of the N10atoms are in the condensed state, leading to the relation
(
N=

kB T
h̄ω̄

)3
Li3 [exp(βµ)](1.7)

and a critical temperature ofkB Tc ≈ 0.94h̄ω̄N 1/3 ,(1.8)

which works out to be about 500 nK for 106 7 Li atoms in a 100 Hz spherical trap. The condensed
fraction is N0 /N = 1 − (T /Tc )3 , so that a 99% pure condensate requires T /Tc ≈ 1/5. Chapter 2
will describe a new apparatus and strategy for producing a Bose-Einstein condensate (BEC) of 7 Li
atoms, as a starting point for adiabatic state preparation of eigenstates of spin models.

1.3.2

Transforming the Ground State

The strategy of adiabatic quantum computing suggests that these massively occupied quantum
degenerate systems can be used as a computational resource. By ramping on a three-dimensional
optical lattice, for example, one transforms the external confinement of the trap into a grid of
approximately harmonic traps connected by tunneling, which introduces a band structure, andsimultaneously increases the on-site interactions relative to other energy scales in the system bysqueezing the atomic wavefunctions. In precisely this way, it is possible to transform a superfluid
BEC– the ground state of many weakly-interacting atoms in a large harmonic trap– into a Mott
Insulator, which is rather the ground state of a strongly interacting system in a periodic potential[29, 30]. While the final ground state looks nothing like the original ground state, it is nonethelesspossible to connect them adiabatically by turning up the strength of the lattice at an appropriate
rate. This very step will be the first in a chain of several such adiabatic transitions proposed in thiswork, which connect a single-component BEC to the ground state of the spin-1 Heisenberg spin
chain.

1.4

Why Lithium?

Despite the fact that 7 Li was one of the first atoms to be cooled to degeneracy (after Rb andNa) [31], it has not, to this author’s knowledge, been previously used in a lattice-based quantum
11

simulator. The difficulties both in cooling lithium to degeneracy, discussed further in chapter 2,
and in keeping it cool in a lattice, discussed in chapter 3, may both be contributing factors. The
fundamental motivation to use 7 Li in an adiabatic quantum simulator, however, is the enabling
features it brings to simulations of strongly interacting systems. Firstly, the Feshbach resonances
within and between hyperfine states of 7 Li [32–36], discussed in detail in chapter 4, mean that theinter- and intra-state interactions vary with respect to one another as a function of the external
magnetic field. For example, for some states a and b and characteristic interaction energy U , theratio Uaa (B)/Ubb (B) may vary between ±∞ as a function of the magnetic field B. This tunability
will prove to be instrumental in transforming the system being modeled between different groundstates.
Secondly, the Hamiltonian description of atoms in optical lattices is made dimensionless when
all distances are in units of the lattice spacing λ/2 and all energies are in units of the atomic recoil
energy ER ≡ h̄2 k 2 /(2m) (see section 3.2.1 for a full derivation), which means that all temperatureand timescales vary as the inverse of the atomic mass. The particle exchange coupling introducedin the previous section therefore generates dynamics which are faster in Lithium than in nearly
any other atom; for example, particle exchange is enhanced by about a factor of ten compared towhat can be achieved with 87 Rb, and the ground state in a lattice is protected by a bandgap tentimes larger. In principle, choosing a light atom is extremely beneficial in overcoming the time
constraints imposed by technical sources of heating that might cause a system to leave its groundstate during an adiabatic sweep.

12Chapter 2

Initialization
Making a 7Li BEC

13

F Shift
0 +11.180
1 +8.347
2 +2.457

2P3/2

3 - 6.929
2P1/2

2 +34.517
1 - 57.529

D2D1

2 +301.314

2S1/2

1 - 502.190
Figure 2.1: Hyperfine structure of 7 Li . Shift frequencies are given in megahertz. The center ofgravity of the D1 and D2 transitions are 446,800,132.006 MHz and 446,810,184.005 MHz respectively(D-line data taken from [37]). The natural linewidth Γ = 2π × 5.87 MHz [38].

2.1

Why Cooling 7 Li is Difficult

Despite the advantages 7 Li offers for quantum simulation, several of its unique features constrain
both the design of the apparatus and the path to degeneracy. 7 Li was the third atomic species
in which Bose-Einstein condensation was demonstrated, and then only after overcoming significant (and unforeseen) difficulties [31]. The particular challenges posed by cooling these atoms aredescribed in this section.

Nearly-degenerate 2 P3/2 hyperfine levels
Unlike most other alkali elements, the hyperfine levels of the 2 P3/2 excited state of 7 Li are separated
by less than a natural linewidth (see figure 2.1). This coincidence of atomic structure has twoconsequences. Firstly, the transition which would usually provide the cooling and trapping in amagneto-optic trap (MOT) is not closed, as a transition to the F′ = 2 manifold is almost as likely
as the transition to F′ = 3 and the former can decay to the F = 1 manifold in the ground state. As
a result, nearly as much power is necessary on the transition out of F=1 in the ground state as outof F=2 (hereafter, these transitions will be referred to as F1 and F2), rendering nearly pointlessthe usual labels of “repumping” and “cooling” light, although for convenience the convention is14

maintained here. This condition requires extra laser amplifiers and forces the repumpers to be
configured as cooling beams. The second consequence of the excited state hyperfine structure isthat it precludes the use of standard sub-Doppler cooling techniques using the D2 transition, which
rely on coherences within hyperfine levels and thus on the distinguishability of the excited state
manifold. Fortunately, the two hyperfine levels of the 2 P1/2 state are separated by nearly 16Γ,
which leaves the D1 line as an option for sub-Doppler cooling.

Small and attractive triplet scattering length
The strength of the scattering process between two low-temperature atoms is characterized by ascattering length a [39, 40]. This length scale arises from considering the asymptotic form of the
wavefunction in the relative coordinate r and reduced mass µ:ψ(r) ∝ eikz + f (θ)

eikz
,
r

(2.1)

where the first term describes an incoming plane wave in the ẑ direction, and the second term
describes a spherical wave scattered off the spherical atomic potential V (r). The symmetry of
the outgoing wave can be used to write an expansion in terms of spherical waves with angularmomentum l( 2iδl)∞∑e−1f (θ) =
(2l + 1)Pl (cos θ) ,
2ik

(2.2)

l=0

where Pl (cos θ) are the Legendre polynomials and δl is the spatial phase shift added by eachpartial wave. At temperatures below which the kinetic energy of the atom Ekin = h̄2 k 2 /(2µ)
does not overcome the centrifugal barrier Vef f = V (r) + h̄2 l(l + 1)/(2mr2 ), only the l = 0 partial
wave contributes to the scattered wavefunction. This limit is safely reached for all the collisions
considered in this work. It can then be written thatf (θ) ≈

e2iδ0 − 1sin(kr + δ0 )= eiδ0
,
2ik
kr

(2.3)

tan δ0 (k)
.
k→0
k

(2.4)

while scattering length is defined by
a = − lim

15

In 7 Li , this scattering length a is approximately -27a0 , where a0 is the Bohr radius [41]. This
relatively small magnitude (three times lower than 87 Rb) limits the scattering cross-section σ =8πa2 , which in turn limits the two-body collision rate nσv. Thermalization in a trap depends onelastic collisions, so evaproative cooling proceeds more slowly in the case of small scattering length.
One can attempt to mitigate this issue by compressing the cloud to high densities, e.g. by using amagnetic quadrupole trap.
The sign of the interactions further requires that Bose-Einstein condensation take place in thevicinity of a Feshbach resonance, so that the interactions can be made repulsive and the condensate
prevented from collapsing. As the Feshbach resonances of 7 Li are found at magnetic fields above
700 G, one must work at high magnetic field, requiring large magnetic coils placed near the atoms
and ruling out the use of a magnetic quadrupole trap for condensation.

Temperature-dependent zero-crossing of the scattering cross-section
The difficulties caused by the small magnitude of the background scattering length are compoundedby the fact that atoms with negative background scattering lengths exhibit a zero-crossing of thetwo-body cross-section as a function of temperature. It is straightforward to see where this effectcomes from in the context of scattering off a rectangular well potential of depth V0 and width R(the same argument applies to the pseudopotential that more closely describes the interactions ofneutral atoms) [39]. The wavefunction at r > R is proportional to the scattered wavefunction (2.3),
while inside the potential
ψin ∝ sin k ′ r(2.5)

for k ′ determined by E − V0 = h̄2 k ′2 /(2m).1 The amplitude must further go to zero at r=0, while′ψtot and ψtot
must be continuous at R, which determines the behavior of the wavefunction for

all r. Either by increasing k or making V0 deeper, it is possible to reach a situation in whichδ0 = π, resulting in complete destructive interference of the incident and scattered wavefunctions
for all r > R and consequentially zero scattering cross-section (figure 2.2). For 7 Li , the k atwhich this Ramsauer-Townsend effect takes place corresponds to a temperature of approximately
10 millikelvin, which is just an order of magnitude higher than the lowest temperature achievable
1

For V0 > R, ψin ∝ sinh k′ r and k′ is rather determined by V0 − E = h̄2 k′2 /(2m)

16

E

0

r

R

V0

Figure 2.2: Cartoon of scattering off a rectangular well. The scattered wavefunction (solid blueline) is phase-shifted by δ0 with respect to the unscattered wavefunction (dashed red line). The
inset shows the point at which the wavefunction crosses the potential.
in the MOT,2 necessitating some form of sub-Doppler cooling.

High three-body loss coefficient
The three-body coefficient of 7 Li is 1 × 10−27 cm6 /s [42], compared to 4.3 × 10−29 cm6 /s for 87 Rb[43], or 1.1 × 10−30 cm6 /s for 23 Na [44]. At densities of 1013 cm−3 , typical for magnetic trap
evaporation in other atoms, the characteristic loss time is approximately one second– a time much
shorter than the length of evaporative cooling in most cold atoms experiments. As the inelastic
collision rate scales with n2 but the elastic collision rate only with n, one might be tempted toreduce the density of the cloud in order to avoid losses, but sacrifices overall speed of evaporation.

Path to degeneracy
After laser cooling, the cloud should be compressed in a tight magnetic trap in order to increase
the two-body collision rate, yet doing so without sub-Doppler cooling would be futile because thetemperature of the compressed cloud would approach the zero-crossing of the scattering crosssection. A simple polarization-gradient cooing step will not work due to 7 Li ’s level structure, so
it is replaced by scheme on the D1 line involving a motional dark state, which is known in the2

Although the Doppler limit h̄Γ/(2kB ) ≈ 140 µK in 7 Li , the high optical density of the laser-cooled cloud requires
large detunings and engenders rescattering, both of which contribute to raising the temperature of the MOT to about
1.2 mK.

17

literature as a Gray Molasses. These two steps require significant laser power on both the D2 andthe D1 lines. Once in the magnetic trap, the density is constrained on one side by three-body loss,
and on the other by background gas collisions. The framing of this problem is not unique to 7 Li ;rather, it is the stringency of the conditions that causes unusual difficulty, as the range of acceptabledensities and initial temperatures is narrow. Once the phase space density is increased to neardegeneracy, the atomic cloud must be transferred to an optical dipole trap so that a large magnetic
bias field may be introduced and the scattering length tuned to a positive value by proximity to a
Feshbach Resonance.

2.2

Overview of the Apparatus

Before detailing how the unique cooling challenges presented by 7 Li are solved in this experiment,a brief description of the apparatus is in order, with particular emphasis on design decisions thatenable unique features of the experiment. A description of the IR laser system and its associatedcontrol hardware, which is used for both the optical dipole trap and lattice potentials, will be saved
for Chapter 3.

2.2.1

Cooling and Imaging Laser System

Lithium laser systems are often power-starved; not only does the atom require high optical power
for cooling schemes, but 671 nm lasers are both comparatively weak and expensive. These considerations strongly influence the design of the laser system. Laser light is provided by three external
cavity diode lasers (ECDLs) and five tapered amplifiers (TAs). The ECDLs are Toptica models,
while the TAs are a combination of Toptica models and home-built designs.

Atomic line locks
A Saturated Absorption Spectroscopy (SAS) locking scheme is employed to control the frequenciesof the ECDLs which provide D1 and D2 seed light (the two locks are operated in parallel). Approximately counter-propagating pump and probe beams are passed through a nitrogen-bufferedcell containing a gas of Lithium atoms heated to 400◦ C. The pump beam saturates the transitionof atoms with a certain longitudinal velocity, resulting in peak transmission of the probe beam
18

when the laser frequency is swept over the atomic transition. The frequency of the probe lightis modulated at 30 MHz using an Electro-optic Modulator (EOM), and demodulated by mixing
with a reference signal, as in lock-in amplification. The resulting dispersive feature centered at thepeak transmission frequency is an error signal which can be used to lock the imaging laser with acomputer-controlled frequency offset with respect to the D2 laser (see figure 2.3).
Offset lock
The imaging laser (Toptica DL100) is locked to the D2 transition using a delay-line offset locktechnique described in [45]. The beatnote between the D2 laser and the imaging laser is observed
on a fast photodiode and mixed with a local oscillator whose frequency is tunable between 300
MHz and 1.3 GHz. The resulting signal is low-passed and mixed with itself after a 6 ns delay. This
produces fringes spaced by about 100 MHz as the frequency of the imaging laser is scanned overthe frequency of the D2 laser. The linear region of one such fringe constitutes an error signal which
can be used to lock the laser.
Optical table layout
The MOT requires similar intensities of both cooling and repumping light (D2-F2 and D2-F1),
while the use of Gray Molasses requires high intensity D1-F2 light. At the same time, while theD1 and D2 transitions in 7 Li are relatively closely spaced compared to other alkalis, their 10 GHz
detuning is still beyond the range of Acousto-optic Modulators (AOMs). These constraints canbe elegantly satisfied by a modular system in which the same TAs can be seeded by either of two
different ECDLs, locked separately to the D1 and D2 transitions. One TA provides F2 light, whose
seed frequency is adjustable by means of a tandem AOM pair, while the other TA has its seedfrequency shifted by a double-passed AOM so that it provides tunable F1 light. These seed AOMs
are responsible for all the frequency tuning of the laser cooling system; the rest of the AOMs may
operate at fixed frequency. Imaging light comes from a separate DL whose frequency is controlled
by the previously described offset locking scheme. This design has the advantage that all frequency
tuning happens before the amplification stage, so that minimum power is sacrificed in the least
efficient AOMs in the system. Additionally, there is no dependence of fiber-coupling efficiency onlaser frequency.
19

ISO

TA

TA
ISO

NPBS

To Zeeman Slower

-8
0

0
-8

To Vapor Cell

From D1 Laser

+80

-80

AOM

AOM

BoosTA

BoosTA

30

30

EOM
EOM

Vapor Cell

+803.5AOM
From D2 Laser

ISO

+

80

80
+

To MOT and
Gray Molasses

ISO

AOM
AOM

-600

0

0
-3

ISO

-6
0

AOM

AOM
AOM

TAPro
D1 Laser

ISO

NPBS

ISO

AOM
To Imaging

80

+

80
+

ISO

80

+

2x6 Fiber
Splitter

80
+

NPBS

LegendFaraday Isolator
Lens

Beamsplitter CubeTo Optical Pumping

To Imaging

Waveplate

Photodiode

Dielectric Mirror

APC PM Single Mode Fiber

20

DLPro
D2 Laser

From D2 Laser

DL100
+300

PD

AOM
AOM

AOM
AOM

PD PD

803.5

AOM

Figure 2.3: Schematic of the optical table layout, including cooling and imaging light generation, atomic line lock, and offset lock.
Beamsplitter cubes are polarizing, except where marked as non-polarizing beam splitters (NPBS). The direction of propagation of thelight is indicated with arrows. The Topotica TAPro laser contains both a DLPro ECDL and a tapered amplifier in a single unit. Not
pictured are beam-shaping telescopes, coupling lenses, folding mirrors, and polarization cleanup optics.

Imaging Laser

To Offset LockPD

EOM
To Vapor Cell

Beam delivery

Because the requisite MOT repumping light intensity is almost equivalent to the cooling lightintensity, the repumping light must also be configured as a MOT, with six beams and appropriate
polarizations. The Gray Molasses light should have the same geometry and polarization as the
MOT light, but whereas the MOT should use the largest possible beams in order to increase
trapping volume, the GM should have smaller beams in order to maximize intensity and therefore
cooling rate. To solve these dual problems, we use a polarization-maintaining 2x6 evanescent wave
coupled fiber splitter (Evanescent Optics). Each input fiber receives light from either the F1 or F2
TA, and the six output fibers are arranged in counter-propagating pairs along the spatial axes ofthe vacuum chamber. At the outputs, polarization-dependent optics send the beams through two
different telescopes depending on which axis of the fibers is used. These axes are in turn selectedby turning on one of two different AOMs, which couple light through different ports of a polarizing

P
R
P
R

M

+

G

T

P

O

R

M

+

+

M

G

To Camera

beam splitter (PBS), as in figure 2.3.

P

M

R

O

T

+
Zeeman Slower
+ RP

P
R

M

Imaging

M

P

O

R

T

+

+

T

P

R

O

G

+

+

R

M

P

G
M

Figure 2.4: Horizontal cooling and imaging beams, and their repumpers (RPs), viewed from above
against a horizontal cutaway of the chamber. Not pictured are the vertical MOT and Gray Molasses(GM) beams, or the vertical and auxiliary imaging beams.

21

2.2.2Chamber Design

The effusive oven source is separated from the main chamber by two stages of differential pumping,with aspect ratios of 10 and 25 (the second stage includes the Zeeman slower tube itself). Vacuum
pressures of about 5 × 102 times higher than in the main chamber are permitted in the ovenchamber, which is important for operating the experiment with fast loading times and low atomloss. The atomic beam is collimated by an orifice before it enters the first differential pumping
tube. The Zeeman slower is a spin-flip slower design, which has the simultaneous advantagesof detuning the slowing light far from zero-field atomic resonance and minimizing the stray fieldcreated by the slower coils at the position of the MOT. The length of the slower is constrainedby the maximum deceleration experienced by a 7 Li atom due to saturation-limited scattering,3√amax = hΓ/λ/(2m), and the RMS velocity of the atoms leaving the oven v̄ = 2 kB Toven /m, sothat Lmin = v̄ 2 /(2amax ) ≈ 1 m for the parameters of 7 Li and assuming an oven temperature of 700
K [46].
Cycle time is minimized by employing a single-chamber design. Laser cooling light is periscoped
into the chamber from below. After the atoms are laser-cooled and transferred to a magnetic trap,
the top mirrors of the periscopes retract on stages with compressed air actuators to make room
for lattice beams propagating along the same axes. This design replaces the standard techniqueof moving the atomic cloud from a chamber with laser cooling beams to a chamber with lattice
beams, saving time and reducing complexity. Beams for optical dipole traps have independent
access through windows offset at ±22.5◦ on either side from the atomic beam axis. An imaging
axis along the perpendicular direction supports 1” lenses with an f -number of 1 using a re-entrant
viewport on one side.
In addition, the top and bottom chamber windows are recessed in so-called “buckets” so thatmagnetic coils may be placed close to the atoms, ensuring that sufficiently high gradients andfields can be generated without in-vacuum coils. A “chimney” for titanium sublimation pumping
is appended to the main chamber; this chimney has the dual requirement that it should havean interior surface area similar to that of the main chamber, and that the titanium sublimation
filament should not have line-of-sight to any windows. There are also assorted access ports for
3
Assuming an optimized magnetic field profile that matches the Doppler shift of the slowed velocity group everywhere in the slower.

22

23
Ion Pumps

Atomic Beam
Shutter
Zeeman Slower Tube

Flexible
Bellows

Re-entrant
Imaging
Viewport

Angle
Valve

Residual Gas
Analyzer

TiSub
Chimney

Main Chamber

Ion Gauge

Ion Pump

Figure 2.5: A diagram of the vacuum apparatus, viewed from the top. Not pictured are magnetic coils, support structures, or beam
optics.

20 cm

1 ft

Nozzle

Oven

Turbo
Pump

Ion Gauges

Ion Gauges

Oven

Turbo
Pump

AtomicBeam
Shutter

Pneumatic
Gate Valves

Ion Pumps

Zeeman Slower Tube

TiSub Chimney

RF FeedthroughBucket Window

Bucket Window

Zeeman Slower
Window

Ion Pump

1 ft

20 cm

Figure 2.6: A diagram of the vacuum apparatus, viewed from the side. Not pictured are magnetic coils, support structures, or beam
optics.

24

various vacuum pumps and gauges. Ion pressure gauges and ion pumps are operated continuously.
Aside from the coils in the recessed windows (“main coils”), which carry currents up to 500
Amps and are water-cooled, there are also low-current Helmholtz coils wound about the chamber
along each axis. These “shim coils” can provide bias fields of up to 20 Gauss when powered by a
few Amps of current, and each has several layers which can be controlled independently. They areused to cancel the Earth’s magnetic field (about half a Gauss) and to provide biases for various
pumping and imaging steps.
There is also a an in-vacuum RF antenna, which is arranged so that it creates AC magnetic
fields which are perpendicular to the strong bias field created by the main coils; this orientation ischosen so that it can be used to drive σ ± transitions between hyperfine states at high field. It is also
used for RF evaporation in the magnetic trap (see section 2.4.1). The antenna is a single loop ofuninsulated, 3 mm diameter oxygen-free copper, held in place by MACOR standoffs and powered
via a feedthrough on the top bucket window. At low field (|B| <20 G), this antenna achieves Rabi
frequencies of up to 2π × 20 kHz on hyperfine transitions when powered by a 10 W RF amplifier.

2.2.3

Coil Design

The experiment requires both quadrupole fields for magnetic trapping and bias fields for manipulation of the atomic interactions via Feshbach resonances. Because the Feshbach resonances of 7 Li
are located at high fields, the coils must be capable of generating up to 1000 G using reasonable
currents (< 500 A). Proximity to the atoms is important both for creating bias fields with the least
curvature (fulfilling the Helmholtz condition) and for generating the largest quadrupole fields, so
the same coils will be used in two different electrical configurations in order to supply both typesof fields: the polarity of some of the top layers can be swapped using four IGBTs configured asian H-bridge (figure 2.7b). Because the Larmor precession is much faster than dB
dt /B for atoms inmotion through a trap or bias field with components i = x, y, z, the axis about which the atoms
precess follows the direction of the magnetic field everywhere. Thus the atoms respond to themagnitude of the magnetic field |B|, which is plotted in figure 2.8 for ideal coils in quadrupole andbias configuration.
Curvature (second derivatives of |B|) influences the potential energy landscape experienced
by the atoms by creating trapping (or anti-trapping) forces with characteristic trap frequency
25

Coil 1T(b)(a)

Coil 1B

Coil 3B

IGBT 1
Coil 2B

IGBT 4

Coil 3T

1B
2B
3B

Sorensen(150 A)

IGBT 6

3T
2T
1T

Lambda(500 A)

IGBT 3

IGBT 2

Coil 2T
IGBT 5Figure 2.7: (a) The top and bottom coils are each made from three segmented double-layers,
pictured here in cross-section from the side. (b) A schematic of the high current circuit for generating magnetic fields. Insulated-Gate Bipolar Transistors (IGBTs) in an H-bridge configuration
can switch the circuit between bias field and quadrupole field configurations. When IGBTs 4 and6 are closed (IGBTs 3 and 5 are open), layer 1 generates a bias field (current follows the bluepath). When IGBTs 3 and 5 are closed (IGBTs 4 and 6 are open), layers 1 and 2 generate aquadrupole field as the polarity of the bottom layers has been inverted (current follows the redpath). Layer 3 is configured permanently in quadrupole configuration, to provide a quadrupole
field for magneto-optic trapping and auxiliary gradients.
ωx =√2µB B ′′ /m. Because of its light mass, 7 Li is therefore particularly sensitive to these trapping

potentials, which can limit the minimum achievable density of the BEC or the minimum flatnessof an applied optical potential. To ensure that the bias fields can be generated with minimum
curvature, the coils should be segmented in such a way that the primary layer fulfills the Helmholtz
condition as nearly as possible, while the other layers can be used for tuning the curvature (figure2.7a). The coils should also be able to maintain this field for the duration of the experiment
without overheating, and should quickly approach an equilibrium temperature if the experiment is
run continuously with some duty cycle.

A warning about symmetry
Special care should be taken that the center of curvature of the bias field overlaps with the zeroof the quadrupole field, so that it is not necessary to transport atoms between the two locations.
To meet this condition, the coils need to be wound with inversion symmetry– put another way,26

for every bit of dI in the top coil, there must be an equivalent dI on the bottom coil to which
one can draw a line which passes through the origin. If both coils are wound in the same fashionand one is simply rotated about the x or y axis and then the z axis in order to form a coil pair,
the resulting setup will not be inversion-symmetric. The distance between the centers of the bias
and trapping fields can be quite significant– many millimeters– so that atoms which are to betransferred between a magnetic trap and an optical dipole trap located at the saddle point of alarge bias field must be transported over a large distance, which adds time to the cooling sequenceand introduces opportunities for heating and atom loss.
(a) 0.1

(b)

|B|

z

.05

0

-.05

-0.1
-0.1

-0.05

0

0.05

0.1 -.1

-.05

x

0

.05

.1

x

Figure 2.8: A cut through the XZ plane of the fields produced an ideal coil pair in anti-Helmholtz(a) and Helmholtz (b) configurations. The contours are equipotential lines of |B| and the arrows
are the unit vectors B/|B| which indicate the field direction at each point. The scale of the axes isthe coil radius.

Helmholtz condition
The magnetic field due to a static current, according to the Biot-Savart law, is
µ0
B=
4π

∫

Idl × r ′|r ′ |3C(2.6)

for a current I following a path C made of segments dl [47]. At a point r the field B(r) isperpendicular to both the direction of the current l, and to the vector connecting the point to the27Bz (au)

Bx (au)
-R

0

R

-R

0

R

Figure 2.9: A pair of coils fulfills the Helmholtz condition when the separation distance equals
the coil radius R (blue line). The axial field Bz acquires positive curvature when the separation
distance is greater than R (yellow line) and negative curvature when the separation distance issmaller than R (red line). The opposite is true for the transverse fields Bx and By .

current element, r ′ = r − l. A circular conductor of radius R is rotationally symmetric and will
therefore create a field that points along the axis of symmetry, everywhere along this axis:

B=µ0 I R 2ẑ
2 R2 + z 2(2.7)

This field exhibits a magnetic field gradient Bz′ = ∂B∂z . In order to generate a quadrupole field, one
may add a second such coil with opposite polarity at some distance a along the z-axis so that alleven derivatives of the field cancel everywhere and B = 0 at the origin (see figure 2.8). If instead
another coil is added with the same polarity, it is rather the even derivatives of the field whichcancel and Bz′ = 0 everywhere in the midplane (by reflection symmetry). One may further vary theratio a/R in order to vary the curvature of the field Bz′′ at the midpoint. A quick calculation showsthat a/R = 1 achieves Bz′′ = 0, which is known as the Helmholtz condition. In general it is possible
to add more coils in order to cancel even higher order derivatives of the field. The residual curvature√leads to a trapping (or anti-trapping) potential with the oscillator frequency ω = 2B ′′ /m.

Ideal coils
In general, it is not possible to fulfill the Helmholtz condition with realistic coils. As a first
approximation, one may consider infinitely thin, circular coils whose separation deviates from the28Helmholtz condition. In free space, magnetic fields obey the Laplace equation∇2 Bz = 0 .

(2.8)

Due to cylindrical symmetry, Bz is the only non-zero component of the field on-axis. Cylindrical222

symmetry therefore requires that ∂∂xB2z = ∂∂yB2z = − 12 ∂∂zB2z , so that the form of the magnetic field at
the origin is a saddle point. If ∂ 2 Bz /∂z 2 is zeroed, the other two second derivatives of Bz must be
identically zero, again because of cylindrical symmetry. Using two sets of coils whose separations
deviate from the Helmholtz condition, it is possible to solve for a ratio of currents I1 /I2 that gives
zero curvature at the origin. It is desirable to have one coil pair spaced by less than the Helmholtz
condition and the second spaced by more– that way, Bz will have the same sign for both coil pairs
at the zero curvature condition, requiring less overall current.

Practical coils
Practical coils offer several obstacles to achieving a curvature-free magnetic field. A pair of coils
made from wire of finite thickness cannot meet the Helmholtz condition for every current pathcontained in the bulk of the wires. Additionally, it is not possible to wind a perfectly symmetriccurrent loop, for at least one must account for leads to bring the current into and out of the loop.
Further, if the coils are to be wound from a wire conductor, it may be advantageous to wind theloops as a continuous spiral rather than as a series of concentric circles. These considerations breakthe cylindrical symmetry of the problem and so it is no longer true that ∂ 2 Bz /∂x2 and ∂ 2 Bz /∂y 2
are zero when ∂ 2 Bz /∂z 2 is zero; nonetheless, (2.8) still requires they have equal magnitude andopposite sign. It is also no longer true that Bx = By = 0 on-axis, although in the high field limit|B| ≈ Bz . The resulting field will have elliptical equipotential lines, where the ellipticity comesfrom the breaking of circular symmetry and the deviation of the angle of the axis of the ellipse
comes from the breaking of inversion symmetry. These problems can be somewhat mitigated bycreating a counter-wound bilayer coil (where each “coil” has two layers, one with each handedness),which also has the advantage that the distance between the leads is minimized, along with theircontribution to the total field.
29

Canceling curvature
Defining the midpoint between the coils to be the origin, one finds
|B| =

√∑
(

Bi0

)2

(2.9a)i∂|B|
1 ∑ 0 l
=
Bi Bi
∂xl
|B|
i
(
)
)
∂ 2 |B|
1∂|B| ∂|B| ∑ ( l m
0 ml
=
−
·
+
Bi Bi + Bi Bi∂xl ∂xm|B|∂xl ∂xm(2.9b)(2.9c)iwhere i, l and m each range over the three dimensions and superscripts indicate a derivative. The
components Bi , Bim and Biml can be easily calculated from (2.6):
dlj rk ϵijk
|r|3
Bi rl dlj ϵijl
∂dBi
= −3 2 +
∂xl
|r|
|r|3
dlj (rm ϵijl + rl ϵijm )∂ 2 dBiBi
Bi rl rm
= −3 2 δlm − 3+ 155∂xl ∂xm
|r|
|r|
|r|4
dBi =

(2.10a)(2.10b)(2.10c)0I
where the initial constants µ4π= 1 and repeated indices imply summation.

The asymmetry of practical coils means that even if the z-curvature is zeroed using a multi-coilscheme, there will be residual horizontal curvature, which can lead to trapping frequencies as large
as several Hertz for 7 Li atoms. In general, it can be said that every pair of coils spaced by a distanced can be characterized by a certain ratio of ∂xx /∂zz (or equivalently ∂xx /∂yy ). It is then clear thatzeroing arbitrary curvature requires at minimum three layers, and the minimization procedure is
as follows:

1. Measure (or calculate) the ratio ∂xx /∂zz for the primary coil pair.
2. Place two more pairs of coils with varying separation on the same axis. Vary the ratio of thecurrent in these two coils until their combined field has the same ∂xx /∂zz as the primary coil.
3. Vary the ratio of the currents between the primary coil and the compensation “super-coil”
until the z curvature is canceled.
4. Vary the overall prefactor of the currents in order to achieve the desired magnetic field.
30

Any remaining curvature should be due entirely to second derivatives of the transverse fields.
This problem can be mitigated by minimizing the transverse fields themselves, for Bi′′ always
appears with the coefficient Bi in the expressions for the derivatives of |B|. The residual transverse
fields are due to the presence of leads and to the helicty of the coil (approximately in equal parts).
It is possible to understand the contribution from the helicty by recognizing that the current
flowing in a helical coil has a radially symmetric component and a transverse component (thetransverse component is proportional to the pitch of the spiral, which is the wire diameter). Boththe contributions of the helicty and of the leads can be simultaneously addressed by winding bilayer
coils, which wind inwards on one layer and then outwards on the other (figure 2.10). The two layers
mostly cancel the effect of their helicties, while also minimizing the distance between the input andoutput leads. Further, specifically at the origin, it would be possible to servo out the small remaining
curvature with bias fields, for ultimate control.
(a)

(b)

(c)

(d)

(e)

Figure 2.10: Schematic of one of the spiral bilayer coils designed for this experiment, with eight
windings, an inner radius of 37 mm and pitch of 5 mm (matching the diameter of the square wire).
This coil and its inverse form the pair closest to the atoms, designated layer 1T and 1B in figure
2.7. (a) through (d) show the coil from various angles, while (e) shows view (c) in cross-section.

31

The fact that ∂ 2 Bz /∂x2 = −∂ 2 Bz /∂y 2 when the z-curvature is zeroed suggests a clever arrangement of coils which uses current most efficiently: if the compensation coils are all similarly wound,rotating one pair of them by 90 degrees will make the horizontal curvatures approximately cancel
and allow total curvature cancellation with far less current than would otherwise be required. A
further caveat is that the symmetry axes of the field of a coil (the major and minor axes of thegenerally ellipsoidal equipotential lines) are not aligned with the axes of the coil itself, as defined by
the leads. In order to fully exploit the curvature canceling strategy discussed above, the symmetry
axes of the fields must be alinged with one another; in this respect, a full numerical simulation of
the coil configuration, including thick wires and leads, proves extremely useful.
From equation 2.10c, it is readily apparent that magnetic field gradients are turned into curvature of |B| in the presence of a bias field. Indeed, if one is bothering to minimize curvature at all,it is quite likely that a gradient will also be necessary to cancel the effect of gravity. In 7 Li at highfield, this gradient is about 1.23 G/cm, leading to residual transverse curvature of 5 × 10−4 G/cm2
at 800 G, which corresponds almost exactly to a trap frequency of 100 mHz. If the trap is madeisotropic by adjusting the curvature compensation, the curvature can be lowered by a factor of√
2/3 and the trap frequency by 2/3. The theoretical limit of curvature cancellation with gravity
compensation in 7 Li is therefore about 80 mHz.

The coil-winding processThis section provides a detailed description of the production process for a pair of coils. The wire
chosen for the coils in this experiment is 5 mm × 5 mm square copper with a 2 mm diameter roundhollow core and fiberglass insulation, manufactured by MWS, Inc.. The wire is rather stiff, and inorder to maintain enough tension to wind it tightly, it should be wound around a jig which can berotated by a lathe. The advantages of larger wire cross-section are mainly current handling andcooling capacity. The greater the tension the lathe can apply, the fewer imperfections there will bein the coil. In order to wind coils using a lathe, one needs:• A low-speed, high-torque gear option for the lathe (10 rpm is a reasonable speed).
• A rig for holding and applying tension to the wire spool, and possibly a wire guide or straightener to remove kinks before winding.
32• A form around which to wind the coil. This form should be spiral-shaped, with a pitch equal
to the wire diameter, and should be made of aluminum in order to withstand the significant
forces experienced during winding (figure 2.11a).
• A dowel to attach the form to the lathe chuck. This dowel should have one round or hexagonalend, in order to fit into the chuck, and one end that can transfer torque to the winding form,
e.g. D-shaped, and should also be made of aluminum.
• A pair of plates to keep the wire in the plane of the form during winding and to secure thewire during drying. These plates should be made of clear, scratch-resistant acrylic; clear
acrylic enables observation the winding process, and the scratch-resistant surface is smoothenough that epoxy does not adhere to it very strongly (figure 2.11b).
• Epoxy to bind the layers of the coils. NM-252 (Duralco) is a non-magnetic, high temperature
two-part epoxy that works well for this purpose.
The plate design used for these coils is pictured in figure 2.11a; two of the same plates canbe used to make the form for a bilayer coil. For the other coil in the pair, the spirals must havethe opposite handedness and so require inverted plates. In order to wind bilayer coils with tighttolerances, the following procedure was developed:

1. Unspool enough wire to form one of the layers, including the lead, and make a 90◦ bend at
the part closest to the spool. It helps to have a form around which to bend the wire.
2. Place one of the guide plates and one of the coil forms on the dowel; place the 90◦ bend of
the wire on the starting point of the form, and clamp in place with the second guide plate.
The excess wire that will eventually form the second layer should be pointing along the axisof the lathe, where it can rotate harmlessly in place.
3. Wind the coil, making sure to fully coat the inside surface of the wire with epoxy before eachlayer is wound. After completing the coil, continue winding until enough excess wire has beenwound in order to form a lead. While continuing to maintain tension on the spool, clamp theguide pates tightly in place with C-clamps and wait until the epoxy has dried (figure 2.11c).
4. Remove the outer guide plate, add the second coil form to the dowel, bend the wire so that
it is perpendicular to the lathe axis and replace the outer guide plate. Cut the first layer freefrom the spool.
5. Wind the second layer as the first, except that wire should be fed from the other side and thewinding direction of the lathe should be reversed (figure 2.11d).
Once the epoxy on the coils has dried completely, the guide plates and coil forms can becompletely removed and the leads bent into position (figures 2.11e and f). Three separate coils
are bonded with epoxy in order to create a multi-layer coil package. The coils are further secured
with epoxy to a ceramic washer that has threaded holes for brass rods. These rods attach to an
33

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 2.11: (a) Render of the winding form for bilayer coils. The form is an Archimedian spiralwith a pitch equal to the wire diameter. A second form with opposite helicty must be made forthe opposing coil. (b) Render of a completed bilayer coil with winding form and acrylic guideplates. (c) Photograph of a half-completed, left-handed bi-layer coil at the end of winding the firstlayer and (d) the beginning of winding the second layer of a right-handed coil. (e) The leads arebent around a form with the appropriate radius. (f) Three completed top coils and three completed
bottom coils. (g) A view of the top three coils, assembled into a single package with ceramic washer
and aluminum mounting bracket. (h) A view of the top coils mounted in the bucket window on
the main chamber.

34

aluminum bracket which can be screwed to the top of the bucket windows in order to clamp thecoil package against the inner surface of the bucket window (figures 2.11g and h).

Design and characterization of the main coils
In order to be able to generate very high fields with low curvature, the top and bottom coils are
made of three bilayer segments each (as in figure 2.10). The segments closest to the atoms haveeight windings, with the more distant bilayers containing seven and six windings each. This design
leaves space for the leads, and also varies the ratios of |B|′′ /|B| produced by each coil. Following
the procedure outlined in the previous two sections, relative angles of +70◦ and −15◦ are chosen forthe second and third bilayers relative to the first. A full thick-wire simulation further reveals thatthe trap frequency is limited by quartic terms to about 500 mHz, which is significantly higher than
the theoretical limit of 80 mHz derived in the context of gravity compensation. While such perfectcurvature cancellation is not necessary for any of the lattice simulations discussed in this thesis, it
may prove useful for future projects and it would be interesting to characterize the curvature of
the coils in situ.

Figure 2.12: Simulated cuts through |B| in the XY plane at 750 G, with the curvature in the
X and Y directions completely zeroed by varying the relative currents in the layers (only quartic
terms remain). The field is plotted in temperature units (µB |B|/kB ), with 0.5 Hz and 1Hz trapping
potentials plotted for comparison.
35

When the transverse overlap of the quadrupole zero and the bias field saddle point are measured
in situ by finding the equilibrium position of the atoms in each field configuration, they are foundto be offset by only 150 microns, thanks to the bilayer design and careful construction. On theother hand, the overlap along the z-axis is not nearly so good. In the plane of the quadrupolezero, the atoms experience a force upwards in a 750 G bias field which is about twice the force ofgravity. It is suspected that the origin of this gradient, which vertically offsets the field centers, is adifference in the ferromagnetic content of the two bucket windows in which the coils reside, rather
than due to a problem with the coils themselves. The top and bottom layers are connected inseries, so the currents are identical, and further there exist no winding errors or shorts which could
possibly give rise to a vertical gradient while simultaneously preserving the cylindrical symmetry
necessary to create good transverse overlap. Regardless, it is possible to simply servo out thisgradient by applying a small quadrupole field with layer 3– which was necessary anyway to cancel
gravity. When all three layers are water-cooled in parallel at a pressure of 200 psi, they equillibrate
at 5◦ C above the temperature of the cooling water in 10 s when supporting a series current of 500
A. In Helmholtz configuration, the temperature coefficient of the field is measured to be +2mG/◦ C
via field-sensitive Zeeman transitions within the range 14◦ C to 27◦ C.

2.2.4

Computer Control and Data Collection

The control signals for the experiment consist of computer-generated analog voltages and TTL
signals, synced by a variable-timebase clock with a minimum resolution of 2 µs. The DACs are
National Instruments PXI-6733s, while the TTL outputs are NI PXIe-6535s. Details about theclock, experimental sequence software environment (Cicero) and output card software controller(Atticus) can be found in the PhD thesis of Aviv Keshet, who designed and coded the first versions.
4

The TTL channels are all optocoupled for safety and peace of mind with respect to grounding;the analog channels are isolated using Analog Devies AD215 analog isolators when connecting tohigh voltage equipment such as power supplies, or when the analog inputs of equipment are notdifferential already. The same computer control system synchronizes data collection from various
cameras and ADCs, which after the completion of each experimental run or “sequence” are written
4
The software is
Cicero-Word-Generator

open-source

andavailable

36

fordownloadathttps://github.com/akeshet/into a database (MariaDB) along with all the control parameters of the sequence and basic preanalysis of the data. Results are live-plotted and can be used for closed-loop feedback via the
database, which is available via standard SQL queries. A MATLAB library has been developedthat provides an object-oriented environment for analysis of absorption images of cold atomic
clouds, BECs, and lattice superfluids in various trapping geometries and in time of flight (TOF).
Data is recorded by ADCs and CCD cameras. The experiment currently contains two fixedposition cameras and one auxiliary camera. On the axis with the re-entrant viewport (“imagingaxis”) is an imaging system with variable magnification (×1/3 through×20) which images the atoms
onto a low-noise EMCCD supplied by Princeton Instruments. The vertical axis is imaged by a PI
CoolSnap CCD. The auxiliary camera is a PixelFly which can be moved about the chamber toassist with beam alignment procedures (there are imaging systems along each axis of the chamber,including mounting hardware for the PixelFly). The ADC channels are provided by an NI PXI-5105
card.

2.3

Laser Cooling

Many of the laser cooling techniques employed in this experiment are quite standard in the field
and will not be reviewed here, except to provide some context for design choices, and to report
operational details and characterizations of their performance. There is an incentive to improvephase space density as much as possible during laser cooling, because evaporation is a lossy coolingmechanism, and furthermore the lossiness (efficiency) of evaporation can be improved by startingwith higher phase space density. Gains made at any stage of the laser cooling process therefore
result in faster evaporation and larger degenerate clouds.
On the bright side, magneto-optic trapping often results in clouds cooled to below the Doppler
limit (in fact, the very first MOT exhibited this surprising result, much to the consternation of theexperimenters). The explanation for this effect is polarization gradient cooling which takes placenear the zero-field center of a standard type-I (J → J +1) red-detuned MOT. Reaching the Doppler
limit requires a detuning of Γ/2, which reduces capture velocity and decreases achievable densities
due to enhanced light-rescattering. Additionally, the density can be increased by using a hollow
repumping beam so that atoms in the center of the MOT are shelved in a dark state (F = 1),37

allowing the density to build up without rescattering [48].
Unfortunately, neither of these techniques provide much of an advantage in 7 Li . It has beenfound that polarization gradient cooling of the red-detuned σ + − σ − type, which usually enhances
the cooling power of a MOT, does not function effectively in lighter alkalis on account of theirdegenerate hyperfine manifolds. The essence of the argument is that these schemes rely on alevel of detuning-enhanced selectivity between the possible F2 transitions which is foiled by thetight spacing of the F’ hyperfine levels in 7 Li . Furthermore, the combination of indistinguishable
hyperfine levels in the excited state and branching ratios that determine the hyperfine fate of atoms
that undergo spontaneous emission means that atoms are approximately half as likely to end up in
the F1 manifold as in the F2. This situation requires that there be nearly as much F1 light as F2.

Atomic Beam Source
The oven is a stainless steel cup, connected to a nozzle by a 90◦ elbow and operated at 485◦ C
by means of a regulated band heater (Omega Engineering). Whereas effusive flow from a pinhole
nozzle would normally create a distribtuion with a solid-angle flux density proportional to cos(θ),
the nozzle used in this experiment is a special micro-channel plate which admits only particles with()
atan v⊥ /v∥ < atan(d⊥ /d∥ ) ≡ θ0 for length and width d∥ and d⊥ of the micro-channels (see figure2.13). For a micro-channels with aspect ratios of 10, θ0 ≈ 5◦ and this pre-collimation of the beam
is estimated to save 99% of the flux of atoms, as the atoms which are prevented from leaving theoven have velocities that would cause them to be lost before reaching the chamber (only atoms
within a solid angle of θ < 0.37◦ reach the chamber, given the length and diameter of the Zeeman
slower).
The effect of this nozzle can be observed by measuring the absoprtion of a laser beam propagating through the effusive flux as a function of laser frequency ωL . In order to see why, consider
distribution of atoms after the nozzle. An atom passes through a nozzle of area A0 in a time dt if
it can be found in the volume A0 vz dt before the nozzle. The probability that a randomly selectedparticle leaves the oven is
A0 vz dt
· p(v)d3 v
V0(2.11)

for oven volume V0 and where p(v) is the 3D Maxwell-Boltzmann distribution. The total number
38

(a)

(b)

Micro-channel Nozzle = 525°C, Cup = 485°C
Micro-channel Nozzle = 540°C, Cup = 500°C
Micro-channel Nozzle = 550°C, Cup = 510°C
Micro-channel Nozzle = 560°C, Cup = 520°C
Effusive Nozzle = 525°C, Cup = 485°C

0.25

Doppler-sensitive
probe beam

5.4

vrms = 1000 m/s

Optical Density

0.2
0.15
0.1
0.05
0

-3

-2

-1
0
1
Frequency (GHz)2

Figure 2.13: (a) A schematic representation of the oven filled with Lithium vapor. An effusive nozzle
produces a density distribution with an angular dependence of cos θ. The micro-channel nozzle
limits the maximum emitted angle to θ0 ≈ 5◦ . (b) A near-resonant laser passing perpendicularly
through the atomic beam produces a Doppler-sensitive spectrum which reveals the distribution oftransverse velocities in the atomic beam. The thermal atoms are randomly distributed between
the two hyperfine ground states, leading to relative absorbtion strengths of 3/8 and 5/8 on theF1 and F2 lines, due to the multiplicity of the Zeeman sublevels. These two peaks can be seen
distinctly in the spectra taken with the micro-channel nozzle, due to the significant decrease in
Doppler-broadening. In the spectrum taken with the ordinary effusive nozzle, the two peaks must
be distinguished by fitting (dashed gray line). The spectra have been shifted so that the F2transition is at zero detuning.
dN of such particles, recast in spherical coordinates d3 v = v 2 dvdΩ, isdN = n0 A0 v 3 cos θ · dt · p(v)dvdΩ .

(2.12)

Here n0 is the number density, which is a function of temperature via the Clausius-Clapeyron
relationn0 (T ) = p∞ ·e−L/kB T
kB T(2.13)

with p∞ = 2.341 × 104 Pa and L = 1.108 × 10−19 J for 7 Li [49]. The solid-angle flux of particles
dN in velocity group dv emitted into the solid angle dΩ in a time dt is
dNj(θ, ϕ, v) == n0 A0 v 3 cos θdtdΩdv39(m
2πkB T

)3/2

2

e

mv
− 2k
T
B

.

(2.14)

In order to calculate the local density of this atomic beam, consider that particles emitted into dΩpass through an area dA = r2 dΩ at a distance r from the nozzle, and that these particles have
moved by a distance vdt in the time dt and so are located within a volume d(Av)dt. Thusdn(r, θ, ϕ, v) =

dN
dN
1 1
=
=
j(θ, ϕ, v)dv ,dVd(Av)dt
v r2(2.15)

which can be integrated along the path of a probe beam to calculate its absorption.

Photons propagating through an atomic vapor, assumed for now to have a single electronictransition with resonant cross-section σ0 ≡ 6πλ̄2 , are absorbed with a frequency dependent crosssection
σ(ω) = σ0 ·(Γ/2)2.
(ω − ω0 )2 + (Γ/2)2(2.16)

If a particle is moving at velocity v, the Doppler shift is added to the detuning so that σ(ω−k·v) for
photon wavevector k. Thus the total optical density ρOD experienced by a laser beam propagating
perpendicular to the flux of the oven is an integral over all atomic velocities and along the path lof the laser beam

∫ ∞
ρOD =

∫
drσ(ω − k · v(r, v))dn(r, v) .

dv
0(2.17)l

In the limit that kB T ≫ mΓ2 , atomic lineshapes can be approximated by delta functions and thisline integral reduces ton0 σ0 ΓA0
ρOD =
4z0 k

√

)
(
(
)
ω − ω0 2 1
kB T
m
exp −
2πm
2kB T
k
sin2 θ0(2.18)

at a distance z0 from the nozzle. The resulting intensity I/I0 = − exp(ρOD ) can be used to fit the
spectroscopic signals plotted in figure 2.13b in order to obtain θ0 and A0 , from whence it is possible√
to calculate the effusion rate R = n0 A0 sin2 θ0 kB T /2πm. Given T = 485◦ C, n0 can be calculated
from (2.18), leading to A0 = 11.2 mm2 , θ0 = 5.4 and finally R = 2.1 × 1015 /s, with 6.5 × 1012 /sreaching the chamber.
40

Differential Pumping and Beam Shutter
After leaving the oven, the atoms next pass through another collimating nozzle, or cold cup, whose
purpose is to protect the subsequent differential pumping tubes from atomic vapor deposition.
During the design phase of this experiment, it was estimated that the melting point of Lithium
is so high that the cup would not require active cooling. No significant deposition has ever been
observed on the cup’s surface, which casts doubt on the verity of this assumption, but in any
case the use of a micro-channel nozzle renders the discussion moot. Two stages of differential
pumping, the second of which includes the Zeeman slower tube, separate the oven chamber, whichis measured to have a pressure of 5 × 108 Torr, from the main chamber, whose pressure is measured
to be 2 × 10−11 Torr. A solenoid-operated beam shutter, in the form of a copper paddle attached
to a vacuum bellows, is located between the two differential pumping tubes. The oven chamber,beam shutter chamber and main chamber are each continuously pumped by 60 L/min ion pumps(Varian StarCell).

Zeeman Slower
The experiment uses a spin-flip Zeeman slower, which consists of decreasing-field and increasingfield sections pointing in opposite directions, joined at a zero-crossing. The advantage of thisdesign is minimized fringing field in the chamber: although the magnitude of the magnetic fieldat the terminus of the slower is more than 400 Gauss, the fringing field due to the slower at theposition of the atoms 10 cm away is less than 1 Gauss, which can be easily compensated with a
low-current shim coil. The frequency of the slowing light is determined by kvcap + µB Bf inal : thesum of the Doppler shift corresponding to the capture velocity of the MOT vcap , and the Zeeman
shift experienced by the atoms at the end of the slower. It is necessary that Bf inal ̸= 0, so that theslowing light can be sufficiently detuned from resonance at zero field that it does not interfere withthe operation of the MOT, through which it necessarily passes. In this experiment, the detuningof the slower is chosen to be 400 MHz ≈ 68Γ, by which point the scattering rate has been reduced
from resonance by a factor of 5 × 105 .
The slower performance saturates with a power of 40 mW, with 6% of the power in the sidebands(3% on the repumping transition at +803.4 MHz, and 3% wasted). The sidebands are created with41

Scheme{
MOT{
GMTransition
D2, F1
D2, F2
D1, F1
D1, F2

Detuning (Γ)
−5
−7
+5
+5

Intensity (mW/cm2 )
2.3
3.9
2.0
68.2

Table 2.1: Parameters of the cooling beams.

a home-built resonant EOM operating as a phase modulator. The slowing is optimized either
by observing velocity distribution of the atomic beam with Doppler-sensitive spectroscopy, or bysimply observing the initial loading rate of the MOT (< 500 ms), which is linear in time and hasa slope that depends on the slowing rate. The loading rate is observed to improve by about 20%
when a small magnetic field (< 10 G) is added in the 30 cm of differential pumping preceding
the slower, providing a quantization axis along which the atoms can be optically pumped into the|F, mF ⟩ = |2, 2⟩ state, which has a cycling transition on the D2 line. This trick ensures that theatoms are prepared in the |2, 2⟩ state before they enter the Zeeman slower.

Magneto-optic Trap

The MOT is a standard type-I configuration operating on a J → J+1 transition (in this case the D2line). As the cooling and trapping properties of MOTs are described in many references [46, 50], thiswork will briefly describe only the operation and characterization of the particular implementation
used in this experiment. The MOT is created from three pairs of counter-propagating, circularly
polarized beams. In order to maximize capture velocity, the detunings of the F2 and F1 light are
made as red as possible given the available power, such that the scattering rate on each transition
remains high enough to capture the Zeeman-slowed atoms. In order to maximize the capture
volume, the beams are also made as large as the chamber will permit, with a 1/e2 radius of
2.5cm. The beam parameters are summarized in table 2.3. The strength of the quadrupole field is
Bz′ = 15 G/cm, for ẑ along the axis of symmetry of the coils. With these parameters, the MOT iscapable of loading 5 × 109 atoms in 3 seconds.
42

|3⟩
δ2 ′
F =2

δ1

D1
|2⟩
|1⟩

F=2

F=1

Figure 2.14: The Gray Molasses cooling level scheme.
Compressed MOT
While the parameters of the MOT are chosen to maximize atom number and loading rate, they
do not simultaneously maximize phase space density. Rather than compromise capture velocity or
re-scattering by adjusting the MOT parameters during the loading, the parameters of the MOT
are dynamically adjusted after the loading stage is completed. As the CMOT will be followed bya sub-Doppler cooling stage with limited beam size (7 mm, compared to 2.5 cm) the parametersof the CMOT are empirically tuned to prioritize maxiumum density. The detuning of the coolinglight is brought close to resonance, in order to maximize compression, while simultaneously theintensity of the repumping light is lowered to zero in order to limit the scattering rate. As a result,the cloud is compressed by slightly more than a factor of two and its temperature is lowered by
30%. Somewhat surprisingly, it was found that increasing the strength of the quadrupole field did
not significantly affect the density or temperature of the cloud.

Gray Molasses
Due to the temperature dependent scattering length of 7 Li (see section 2.1), it is necessary to coolwell below the Doppler limit in order to ensure the thermalization rates required for evaporative
cooling. A good choice for alkalai atoms is a class of Λ-enhanced sub-Doppler schemes known as
Gray Molasses (GM), a form of Electromagnetically Induced Transparency (EIT) cooling [51] (distinguished here from EIT-sideband cooling, which requires a trapping potential [52]) that combines
the cooling of a blue-detuned of σ + -σ − cooling on a J=1→J’=1 transition with the momentum43

2P

2S

1/2

|2, −2⟩|2, −1⟩|2, 0⟩

1/2|2, −2⟩|1, −1⟩|1, 1⟩|2, −1⟩|2, 0⟩|2, 1⟩|2, 1⟩|2, 2⟩|2, 2⟩|1, 1⟩

Figure 2.15: A dark state pumping scheme in which atoms collect in the |2, 2⟩ ground state. As
this is the D1 transition, there is no F’ = 3 level in the excited state. Not pictured is spontaneousemission from the excited state, which takes atoms from the excited state to F = 1 or F = 2 in
the ground state with ∆mF = ±1, 0.

dependent dark state mechanism of Velocity-selective Coherent Population Trapping (VSCPT) [53].
While the original GM proposal called for a 1D, single-transition molasses in 87 Rb on the D2 line[54], subsequent iterations have expanded the scheme to other atoms and three dimensions [55,56], and recently also simultaneous J→J transitions (i.e. on F=1 and F=2 transitions) [57]. In thelatter case, the motional dark state is the result of the three-level Λ-system depicted in figure 2.14.
While the relative detuning δ1 − δ2 must equal zero in order for the dark state to correspond tozero atomic velocity, the overall detuning ∆ must be positive in order for the molasses to function.

Because of the indistinguishability of the hyperfine levels within the 2 P3/2 in 7 Li , only the D1line contains a suitable transition, as the 2 P1/2 hyperfine levels are separated by nearly 16Γ. The
performance of the scheme is sensitive to the relative detuning δ1 − δ2 at the level of Γ/10, but theoverall detuning may be anywhere from 4Γ to 6Γ blue of the F′ = 2 excited state. The cooling rateof the molasses is improved by higher optical intensity, so the GM beams are 7 mm in diameter,
which is approximately the size of the atomic cloud after the CMOT. The beam parameters areagain given in table 2.3. This scheme achieves a final temperature of about 100 µK in 4 ms with
14 mW per cooling beam, without any loss of density.
44

Dark State Pumping

In order to prepare the atoms in the |2, 2⟩ state (see section 2.4.1), an mF pumping technique isneeded. In the presence of a magnetic field in order to set the quantization axis, σ + light will tend
to increase the angular momentum of the atoms over many scattering events so that they gather
in the stretched state. If a D2 transition is used, as in the MOT, this pumping has the undesirable
side effect of causing the atoms which collect in the stretched state to scatter many photons on
the |2, 2⟩g ↔ |3, 3⟩e cycling transition. This leads to significant heating, negating the hard-won
gains of laser cooling. The entire problem can be side-stepped by using a scheme in which thestretched state is also a dark state. The D1 transition affords just such a possibility. The strategy
is to illuminate the cloud with σ + light on the D1 line in the presence of a weak bias field (≈ 5G), so that atoms will scatter until they reach the |2, 2⟩ state and subsequently go dark (both F2and F1 light are necessary, as spontaneous emission from the excited state can result in an atom
populating either F=2 or F=1).

One can characterize the effectiveness of this scheme by measuring the acceleration of the cloud
due to photon recoils, e.g. by imaging the cloud from the side after some pumping time. If
the stretched state were perfectly dark, one would observe that the center of mass of the cloud
would simply acquire a momentum equal to the photon recoil times the average number of photons
necessary to pump an atom into the dark state. A dark state that is imperfect or “leaky” will,
however, continue to scatter and accelerate. The final velocity of the cloud can be determinedfrom the position of the cloud’s center of mass after a time of flight following the pumping pulse,
assuming this time of flight is significantly longer than the pumping pulse. The measured velocity√
normalized by the recoil velocity vR = h̄2 k 2 /m2 gives the number of photons scattered. In thescheme implemented here, these data suggest the scattering rate asymptotes to about 3 photons/ms(see figure 2.16), which, while not perfect, is at least suppression by more than forty times of theinitial scattering rate of 125 photons/ms. This may be due to imperfect alignment of the pumping
light with the bias field, or to imperfect σ + polarization, but in either case the scheme pumps
nearly all of the atoms into the dark state within half a millisecond with minimal excess heating.
45

Number of Photons Scattered

14
12

ns/ms
3 photo

10
8
6
4
2
0

0

.2

.4

.6

.8
1
1.2
Pumping Time (ms)

1.4

1.6

1.8

2

Figure 2.16: Suppression of scattering from the dark state. The dashed line is a linear fit to the dataat the ten longest times, whose slope shows the asymptotic behavior of the scattering cross-section
as the atoms are pumped into |2, 2⟩.

2.4

Evaporation

Even using a sub-Doppler cooling scheme, the phase space density achieved by laser cooling in
7 Li is only 5 × 10− 5, which is still five orders of magnitude lower than ρcrit (see section 1.3.1).

Although the first condensate of 7 Li was formed by solo evaporative cooling of 7 Li in a magnetic
trap, to this author’s knowledge all successive 7 Li condensates have been made by evaporating a
mixture of 6 Li and 7 Li . The mixture has the advantage that the inter-species scattering length isabout a factor of two larger than the scattering length of 7 Li with itself. This section describes thedevelopment of a forced evaporative cooling scheme for 7 Li alone, using both magnetic and opticaltraps in succession.

2.4.1

Magnetic Trap Evaporation

In principle, it would be ideal to transfer the laser-cooled atoms directly to an optical dipole trap
(ODT), in which the scattering length can be adjusted to a large and positive value via a Feshbachresonance, so that evaporation can proceed with maximum efficiency. The light mass of 7 Li ,however, requires a significant investment in optical power: a 10 W optical dipole trap beam witha 1/e2 waist of 100 µm gives a trap depth for 7 Li of about 100 µK, which is on the order of thelowest achievable temperature via laser cooling. A crossed ODT made from such beams would,
46

1500

1/ 2 ,3/ 2

1000

Zeeman Shift (MHz)

500
2,2

0
2,-2

-500
-1000

1,1
-1/2

-1500

-1/2 , 3

-2000
-2500

, - 3/2

/2

0

200

400

600

800

d
c
b
a

1000

B (G)

Figure 2.17: Hyperfine structure of 7 Li in a magnetic field. The low-field states, dominated byspin-orbit coupling, are labeled with the quantum numbers |F, mF ⟩, where F=I+S, while the highfield states are labled with the quantum numbers |S, I⟩ and are indexed (a,b...) in ascending order.
however, have terrible overlap with the spacial distribution of the laser-cooled cloud, which is a
few millimeters in diameter. Just from overlap considerations, the captured number of particles
would be about 105 , which is far too low: evaporation should end with about 105 atoms in thecondensate. The ODT beams could be made larger in order to increase spatial overlap, but thatwould require higher optical power, which is expensive and difficult to fiber-couple.

Trappable states
Instead, a short step of evaporation in a highly compressed magnetic trap can boost the density andlower the temperature enough to efficiently stitch together laser cooling and ODT evaporation. 7 Li
has three magnetically trappable states at low fields (figure 2.17). Near |B| = 0 the |1, −1⟩ and |2, 1⟩
states both have slopes of half a Bohr magneton, whereas |2, 2⟩ is a stretched state and therefore
always has the slope of the full Bohr magneton. In the |1, −1⟩ state the magnetic moment goes
through a minimum and changes sign at around 160 G, limiting the achievabe trap strength; boththe |1, −1⟩ and |2, 1⟩ states are susceptible to spin-changing collisions which lead to losses, which
as a stretched state the |2, 2⟩ is not, and in any case the greatest trap strength is achieved using
the highest magnetic moment (in principle, dipolar relaxation is still available as a loss channel,
47but the rate is too low to measure in bulk).
The tightest possible confinement is achieved by use of a quadrupole trapping potential |Btrap |2 =(B ′ )2 · (x2 /4 + y 2 /4 + z 2 ). The use of a quadrupole trapping potential has one caveat, which is amagnetic field zero at its center, where passing atoms can be spin-flipped to an untrapped state.
This “Majorana hole” introduces a single-particle loss rate which is proportional to the peak densityof the cloud, and thus increases with the phase space density. In this experiment, the atoms are
kept away from this region by a repulsive potential created by a 10 W, 532 nm beam with a waistof about 23 µm, which is on for the duration of the magnetic trap evaporation.
Adiabatic compression
The effects of adiabatic compression in a quadrupole trap can be derived from simple scalingarguments. It is possible to define a characteristic length x̃ = kB T /(µB B ′ ) and also a characteristic
momentum p̃2 /(2m) = kB T for temperature T , atomic mass m and magnetic field trapping gradientB ′ . Adiabatic changes in this trap strength imply x̃p̃ = const such that T ∼ (B ′ )2/3 in such a trap.
Thus the density scales as n ∼ 1/x̃3 ∼ (B ′ /T )3 ∼ B ′ . Most importantly, these two results can be
combined to show that the two-body elastic collision rate nσv scales as (B ′ )4/3 . This collision rate
determines the equilibration rate of thermal distributions and is critical for efficient evaporative
cooling.
RF-induced evaporative cooling
The principle behind evaporative cooling is the removal of energy from a trapped gas by removing
the most energetic particles: that is, the particles in the high-energy tail of the Maxwell-Boltzmann
distribution. In a magnetic trap, it is straightforward to achieve this culling by using RF transitions
between trapped and untrapped Zeman sublevels [58]. The most energetic atoms explore highest
parts of the trapping potential, and thus experience the largest Zeeman shifts, so it is possible tochoose a radio frequency such that only these atoms are spin-flipped. The cutoff energy can be
described by multiples of the mean kinetic energy of the atoms η ≡ Ecut /(1/2mv̄ 2 ).
The hyperfine structure of 7 Li presents two options for this spin flip: the atoms can be transferred to the lower half of the F = 2 manifold (often referred to as “mF evaporation”), or directly to
the |1, 1⟩ state (referred to as “hyperfine evaporation”). The two options further are characterized48

by diffferent frequency requirements. Whereas mF evaporation requires RF between 100 MHz and
1 MHz, hyperfine evaporation requires RF between 1 GHz and Ehfs /h = 803.5 MHz. It was foundthat mF evaporation interferes with other low-frequency lab equipment, such as the Doppler-free
spectroscopy locking scheme for the diode lasers; and since the high-|B| spin flips performed laterin the experiment require a 100 MHz - 1 GHz amplifier and antenna anyway, hyperfine evaporation
was selected.
A trapped gas of particles with contact interactions reaches thermal equilibrium through collisions–in two dimensions and higher, elastic two-particle collisions can redistribute kinetic energy andmomentum. However, collisions also present an opportunity for particle loss and heating, whether
by changing the internal state of the particles or via a chemical reaction that changes the spatial component of the two-body wavefunction. A sample of atoms in a single hyperfine state (a“spin-polarized” sample) may decay to a mixture of states, subject to the constraints of angularmomentum conservation. These states may be untrapped, leading directly to loss, or in the presence of the trapping magnetic field may convert enough Zeeman energy to kinetic energy that theyescape the trapping potential (see figure 2.17). At ultracold temperatures, two-particle chemical
reactions are ruled out by the need to both conserve the center of mass momentum of the twoparticle system and to find a place to put the energy liberated by the chemical bond. When threeparticles collide, however, there is a chance for two of them to form a molecule while the third
carries away the binding energy (which can be hundreds of recoils), which results in the loss of all
three particles. As both the two- and three-body loss processes scale with density (as n and n2 ,respectively), they will tend to heat a gas trapped in a non-uniform potential by removing particles
with below-average kinetic energy from the coldest and densest part of the trap.
Due to ever-present single particle losses caused by background gas collisions, it is desirable
to evaporate as quickly as possible. The constraint is the thermalization rate of the trapped gas,
proportional to the two-body collision rate nσv for collisional cross section σ and average velocity
v. Therefore it is desirable to compress the cloud as much as possible. On the other hand, therate of three-particle collisions is Γ3 n2 , for three-body loss coefficient Γ3 . The ratio of two-to-threeparticle collision rates is then σv/Γ3 n. Optimization of forced evaporative cooling is thus equivalent
to finding a route to degeneracy in which the densisty of the atomic sample is controlled in such a
way as to keep the two-body collision rate high and to simultaneously straddle the constraints of49

single- and three-body losses.
Optimization of Evaporative Cooling
When the scattering length is small, as is the case in 7 Li , the key to fast and efficient evaporationis compression. There are really two aspects to this: the maximum achievable gradient, andgradient at which the atoms can be caught after laser cooling. These two endpoints determine the′′compression factor Bmax
/Bcapture≡ α. The maximum achievable density is therefore α · ncapture

and the two-body collision rate receives a boost α4/3 , via the scaling arguments introduced in theprevious section. The coils whose design was introduced in section 2.2.3 are capable of compressing
the laser-cooled cloud by a factor of 5, which boosts the elastic collision rate by a factor of about
8.5.
Compression optimizes the starting conditions, but it still remains to determine the RF(t) andB ′ (t) that maximize cooling efficiency. Various optimization techniques exist, many of which arereviewed in [58]. First of all, the most efficient path is the one which optimizes the efficiency
at each point, so that the frequency ramp can be tuned experimentally by breaking it into a
number of linear segments and optimizing each separately. It is further possible to assume thatthe distribution of kinetic energies amongst the atoms follows a (truncated) Maxwell-Boltzmann
distribution at all times, in which case the problem of optimizing evaporation amounts to solving
a pair of coupled differential equations for Ṅ /N and Ṫ /T . An alternative approach, originallyintroduced by Foot and Wu [59], is to run a Monte-Carlo simulation of the full dynamics of thetrapped gas, including kinetics, losses, and collisions. This method has the advantages that it
makes for relatively straightforward incorporation of various sorts of losses and energy-dependent
scattering, and it does not assume anything about the distribution of kinetic energies of the atoms.
A version of this approach is used to optimize the evaporation procedure in this work. It does,
however, assume classical statistics and non-hydrodynamic densities (both of which are valid forthe regimes modeled in this work). The DSMC algorithm, which simulates the dynamics of N
particles with N/p representative particles, can be summarized as follows:

1. Initialize N/p particles (results converge for N/p > 105 ).
2. Advance a time dt.
50

(mK)

(m)

100

1

3

el

50
0

0

1

10 9
10 8
0

1

2

(cm-3)

3

0

0

3

10

10 12

10 -3

10 11

0

1

2

0

3

1

2

3

el

0

1

2

3

1

2

5

3

0

1

2

3

B',f

1000

900

125

0
0

in

6.5

0

250

/

3

8

-1

10 -5
0

2

1000

PSD

10 13

1

2000

0.25

n

(G/cm)

2
in

0.5(s -1)

100(s -1)

2

10 -4

N

10 10

10 -3

10
0

rrms

10 -2

0.5

1

1.5

time (s)

2

2.5

(MHz)

T

1000

800
3

Figure 2.18: Output of the DSMC evaporation optimizer when the evaporation time has been set to
2.5 s. 5 × 109 7 Li atoms at 100 µK are brought to just below degeneracy with an efficiency of γ=2.2
in an optical quadrupole trap. The strength of the quadrupole field B ′ is reported as the transverse
gradient in G/cm (the Z-gradient is twice as large). Γin is the loss rate per particle due to the
combined effects of background gas collisions, dipolar relaxation and three-body recombination. T
is defined via the average kinetic energy of the particles, although the particle velocities are not
restricted to follow a Maxwell-Boltzmann distribution. n is the peak density, rrms is the root-meansquare cloud radius in the transverse direction, and η is the potential energy of the spin-flipped
atoms normalized by kB T .

51

(a)

(b)

(c)

(d)

Figure 2.19: Cuts along an axis of the optically-plugged quadrupole trap during different stages of
evaporation. (a) Atoms are captured in a mode-matched magnetic trap. (b) The atomic cloud isadiabatically compressed by ramping up the magnetic potential. (c) RF spin flips remove the mostenergetic atoms from the edges of the trap. (d) The trap is adiabatically decompressed while RFevaporation continues, in order to manage density-dependent losses.

3. Update positions and velocities of atoms moving in the conservative trapping potential bysolving Newton’s equations, e.g. using symplectic integration such as the Beeman algorithm.
4. Bin the atoms into 3D cells.
5. Compute two-particle collisions using Bird’s constant-time NTC algorithm [60] and update
particle velocities.
6. Compute density-dependent particle losses in cells.
7. Redraw the grid of 3D cells and adjust particle weight p to remain at a constant number ofsimulation particles and return to step 2.

By using this simulator, it is possible to optimize the path the evaporated cloud traces throughphase space, given certain constraints (e.g. on the total time). That is the approach followed
here: the optimizer adjusts the RF and trap compression ramps in a piece-wise fashion in order to reach a target PSD within a given time. For example, it finds an evaporation procedurethat increases the PSD by more than four orders of magnitude in 2.5 s with an efficiency ofγ ≡ log(ρi /ρf )/ log(Ni /Nf ) = 2.2, which is remarkably good for an atom with such a small collisional cross-section. The results of the simulation are found to be highly consistent with theexperiment; in fact, the number of atoms with which the target PSD is reached in the experimentis slightly higher than predicted, suggesting that the rates for one or another loss processes areover-estimated (but the optimal path turns out not to be terribly sensitive to collisional loss rates,
as long as adiabatic decompression maintains the density at a level where τloss per atom is much
shorter than remaining evaporation time).
52

2.4.2

Optical Dipole Trap Evaporation

The difficulty posed by the attractive interactions the |2, 2⟩ state must be addressed before BEC
can be achieved. The lowest two hyperfine states of 7 Li (|1, 1⟩ and |1, 0⟩ at low field, a and b at highfield, as in figure 2.17) have Feshbach resonances around 740 G and 940 G respectively; further|1, 1⟩ is the absolute spin ground state, so no two-body losses can occur. It is therefore in either the|1, 1⟩ or |1, 0⟩ state, at a magnetic field where the scattering length is about 150a0 , that the final
evaporation to BEC takes place. Both choices of hyperfine state are incompatible with magnetic
trapping, so the atoms are transferred to an optical dipole trap (ODT), which uses the conservative
potential arising from the AC-Stark shift to trap the atoms in a focused beam of light.
Optical Dipole Trapping
The conservative (induced dipole potential) and non-conservative (scattering) aspects of the lightforce can be expressed as
)
(
Γ
3πc2
Γ
+
I(r) ≡ s(ω)I(r) , andUdip (r) = − 32ω0 ω0 − ω ω0 + ω)2( ) (3πc2 ω 3ΓΓ
Γsc (r) =+I(r)ω0 − ω ω0 + ω
2h̄ω03 ω0(2.19)(2.20)

for laser frequency ω, transition frequency ω0 , natural linewidth Γ, and intensity distribution
I(r) [61]. The detuning and optical power should be chosen such the light can provide enough trap
depth to contain atoms with kinetic energy kB Tevap , while simultaneously keeping the characteristic
scattering time 2π/Γsc long compared to the duration of the experiment. In the context of small
detuning δ = ω − ω0 , the term that depends on 1/(ω + ω0 ) becomes negligible, leading to
Udip
δ
≈ .
h̄Γsc
Γ

(2.21)

Formulation in this rotating wave approximation makes it clear that the strategy should be to detune
as far as possible, given constraints on optical power. However, for extremely large detunings, thisapproximation significantly underestimates both the induced dipole potential and scattering rate.
Without inclusion of the counter-rotating term, trapping forces due 1064 nm light on 7 Li , which
has an S→P transition at 671 nm, would be underestimated by about 40%.
53

The ODT itself takes the form of a Gaussian beam propagating in the ẑ direction, described by
2−2 r 2I(r, z) = I0

w0e
1 + (z/zR )2(2.22)

where w0 is the 1/e2 beam waist and zR = πw02 /λ is the Rayleigh length. I0 = 2P/πw02 is the peakintensity for a beam with total optical power P . An atom located near the focus of such a beam
will experience an approximately harmonic dipole potential
1
U (r, z) ≈ m(ωr2 r2 + ωz2 z 2 )2√where ωr =(2.23)√s(ω) √s(ω) √
8P/πm and ωz = w0 zR 4P/πm. As the Rayleigh length scales like 1/w02 , thew02

longitudinal confinement provided by a single beam is much weaker than the transverse confinement(unless the beam is focused to a wavelength-sized spot). In order to achieve more symmetricconfinement of the atomic cloud, and therefore higher densities, this apparatus uses two beams
that cross at 45◦ . With beam parameters of λ =1064 nm, w0 =100 µm, and P =5 W, the ODTbeams each achieve a radial trapping frequency for 7 Li of 2 kHz and a depth sI0 /kB of 70 µK.
Due to poor spatial overlap, the ODT can only capture 10% of the atoms prepared by evaporation in the magnetic trap. After loading the atomic cloud into the ODT, an RF sweep causes a
hyperfine transition |2, 2⟩ → |1, 1⟩ in 10 ms with close to 100% fidelity. A gradient of 10 G/cm isapplied in order to clear atoms out of the wings of the ODT. The cleanup step may be followedby another sweep into the |1, 0⟩ state, depending on whether an a- or b-condensate is to be made.
The main coils then ramp the bias field to Bevap .

Evaporative Cooling to Degeneracy
At this point, the atom number with wich BEC is reached can be adjusted lower by ramping themagnetic field towards the Feshbach resonance for some time, in order to induce three-body loss. In
this way, BEC numbers between 1×104 and 5×105 can be reached via the same evaporative coolingparameters, with fluctuations of ±3% and without loss of condensate fraction. After adjusting thestarting number, evaporation in the |1, 1⟩ or |1, 0⟩ state proceeds quickly because a scattering length
can be selected that affords the best ratio of elastic to inelastic collisions (about 150a0 , at 730 G or
54

870 G, depending on the hyperfine state). The depth of both ODTs are lowered exponentially over
2 s with a time constant of 500 ms in the presence of a magnetic field gradient equal to 8 times theforce of gravity, which helps to spill the light Lithium atoms from the trap.
Limits of evaporative cooling
After crossing the BEC transition, the collisional density of the sample leads to fast thermalization
times and ought to result in extremely efficient evaporation. Further cooling, however, tends
to be hampered by the fact that the in situ size of the thermal cloud remains determined by
the temperature, while the size of the condensate is determined by the condensate number andinteractions (in the Thomas-Fermi limit). (In reality, there is some interplay between the size ofthe condensate and the size of the thermal cloud, because the presence of the BEC modifies thetrapping potential experienced by the thermal atoms, at least at the center of the trap.) In order
to avoid cutting into the condensate while evaporating, which would tend to heat the sample, theη of the evaporation may have to become prohibitively high, leading evaporation to stall. Thiscan to some extent be limited if interactions can be tuned via a Feshbach resonance, so that the
BEC radius can be made smaller, at the price of increasing the density and exposing the sample tothree-body losses, but is typically only a consideration at very high condensate fractions (> 95%).

2.5

Summary of Cooling Results

The excellent performance of the cooling sequence is contingent upon two accomplishments: (1) the
final temperature achieved by laser cooling is low enough that the compression ratio of the magnetic
trap is high and the temperature-dependence of the scattering length does not play a role, and (2)adiabatic compression kick-starts the evaporative cooling process, followed by decompression inorder to keep the time constant of three-body loss above a second. As a result, BECs with > 105
atoms and high condensate fraction (> 95% is too high to fit reliably) can be made in the |1, 1⟩ or|1, 0⟩ state in about 8 seconds.

55

Cooling Step
MOT
CMOT
GM
DSP
MT (Capture)
MT (Compressed)
MT (Evaporation)
ODT (Capture)
ODT (Evaporation)

Duration (ms)
3000
4
4
0.6
100
250
2500
500
2000

N
2 × 109
2 × 109
1 × 109
1 × 109
1 × 109
1 × 109
5 × 107
5 × 106
5 × 105n0 (cm−3 )1 × 1011
2 × 1011
2 × 1011
2 × 1011
2 × 1011
5 × 1012
2 × 1013
4 × 1013
1 × 1013

T (µK)
1200
700
100
100
120
410
8
35
0.65

ρ0
1 × 10−6
1 × 10−6
5 × 10−5
5 × 10−5
4 × 10−5
4 × 10−5
0.25
0.05
5.5

Table 2.2: The duration of each cooling step, as well as the atom number, peak density, temperature,and phase space density it achieves.

56

Chapter 3

A Platform for Simulations
Atoms in Optical Lattices

57

3.1

An Atom in an Optical Lattice

In this thesis, the role of the optical lattice is to create localized Fock states which will form thesites of a lattice spin model. Its initialization is therefore the first leg of the adiabatic connection
between the superfluid ground state and the ground state of said spin model. Atoms in an opticallattice are analogs of electrons in a solid crystal, where the lattice sites correspond to potential wellsof nuclei. As in a solid material, the periodic external potential changes a particle’s effective mass,modifying relationship between the its energy and its momentum. Derivation of this single-particle
band structure in that context is readily available in textbooks [62, 63], but bears repeating here
because of its central importance in the discussion to follow.

Figure 3.1: Render of a standing wave of light, formed by retro-reflecting a Gaussian beam. The
potential experienced by the atoms is in the region of the focus between two lenses, where thelight is most intense and the longitudinal variation of the transverse beam profile is simultaneously
minimized. The lattice spacing is exaggerated for clarity.

3.1.1

Optical Lattice Potentials

The simplest optical lattice– and the one used throughout this work– is formed by retro-reflecting
a laser beam, Eei(kz−ωt) , upon itself. Here k = 2π/λ is the wavevector and ω = 2πc/λ is the opticalfrequency. The resulting standing wave is of the form

V (r) cos2 (kz)

(3.1)

where V (r) = 4 · s(ω) · I(r) is the depth of the lattice and carries units of the recoil energy ER . One
factor of two comes from retro-reflection, and the other from interference, while s is a wavelengthdependent parameter which relates potential energy to intensity, defined as in (2.19). The resulting
optical potential is a stack of “pancakes” along the ẑ axis, with a transverse envelope arising from58

the curvature of the Gaussian beams, as in (2.22), and is illustrated in figure 3.1. To form apotential akin to an array of tubes, one must overlap an orthogonal, non-interfering lattice beamwith the first. The non-interference condition may be realized either by orthogonal polarizations,or by sufficiently detuning the frequency of the orthogonal lattice beams so that the interference
between them creates beat frequencies that are higher than any inter-band transitions present in
the lattice. Adding a third mutually orthogonal lattice beam results in a cubic array of point traps.

3.1.2

Wavefunctions in a Lattice: The Momentum Basis

The behavior of an atom in the lattice is governed by the one-dimensional Schrödinger equation()
p̂2− V (x) ψ(x) = Eψ(x)
2m

(3.2)

where the momentum operator is p̂ = −ih̄∂/∂x and the potential energy V (x) = Vlat (x) + Vtrap (x)has short-range structure due to the optical lattice and long-range variation due to the Gaussian
curvature of the lattice beams and optical traps. Making the approximation that Vtrap does notvary on the scale of the periodicity of the lattice, one may invoke Bloch’s theorem in order to(n)write the eigenstate ψq(n)

as a plane wave with quasimomentum q times a scalar function uq (x) =(n)uq (x + π/k) that has the same periodicity as the lattice:(n)
iqx/h̄
ϕ(n).
q = uq (x)e(3.3)

Here n labels the eigenvalues and in the context of a lattice potential is called the “band index.”Plugging in this trial wavefunction gives
(

)
1
2
(n)
−
(p̂ + q) − V (x) u(n)q (x) = Euq (x)
2m
59

(3.4)

n=1

n=2

3

| (x)|2

2

1

0

-1

0

-1

1

0

sites

1

sites

Figure 3.2: The square norms of the Bloch functions in the ground and first excited bands (n=1,2)
of a lattice with a depth of 10 ER . The solid lines depict the Bloch functions at q=0, while thedashed lines are at q=h̄k.

(n)where the common phase factor has been dropped. Both uq (x) and V (x) have the same periodicity
π/k, so their Fouirer decompositions are sums of plane waves with wavevector 2k:

V (x) =∑

Vr ei2rkx , and

(3.5)

r

∑ (n,q)u(n)cl ei2klx .
q (x) =

(3.6)

lThus the Bloch functions are superpositions of plane waves with wavevectors q/h̄ + 2kl, with q
defined between −h̄k and h̄k. This range of quasimomenta is known as the first Brillouin zone.
Given the Fourier decomposition (3.5) of the Bloch functions, the potential energy term of the
Schrödinger equation can be re-expressed as

V (x)u(n)q (x) =

∑∑
r

=

(n,q)

l′

∑∑
r

′

Vr ei2(r+l )kx cl′

l

60

(n,q)

Vr ei2lkx cl−r

(3.7)

V0=5E R50

V0=15ER

V0=30ER

40

E/E R

30

20

10

0
-1

-0.5

0

0.5

1 -1

-0.5

0

q/hk

0.5

1 -1

-0.5

q/hk

0

0.5

1

q/hk

Figure 3.3: The band structure is plotted for several lattice depths as a function of quasimomentum,
ranging over the first Brillouin zone.

while the kinetic energy becomes∑ (2h̄k + q)2 (n,q)
1(p̂ + q)2 u(n)cl ei2lkx .
q (x) =
2m
2m

(3.8)

l

The optical lattice potential (3.1) is itself a sum of counter-propagating plane waves

V (x) = V cos2 (kx) =)V ( i2kxe+ e−i2kx + 2
4

(3.9)

so that the only non-zero coefficients Vr are V0 = V /2 and V±1 = V /4. Finally, the Schrödinger
equation can be recast in the form
∑

(n,q)

Hll′ cl′(n,q)= Eq(n) cl

l′61(3.10)

Bloch Bands in 3D

Bloch Bands in 1D

25

2

2,

1,

n=

20
,2

1,1
n=

n=2

n=1

E/ER

15

3
n=

10

5

0

,1,1

n=1

0

5

10

15

20

25

30 0

5

10

V0 /ER

15

20

25

30

V0 /ER

Figure 3.4: The first three bands of the lattice, plotted in 1D and 3D as a function of the lattice
depth in recoils. In 3D, one quanta of excitation is added per band. While the bands are fully
separated at all non-zero lattice depths in 1D, the ground and first excited band do not fully
separate in 3D until about 2.3 ER .

with the matrix elements(2l + q/h̄k)2 Erec + V /2 if l = l′Hll′ =

if |l − l′ | = 1

V /4






0

(3.11)

else

This matrix, truncated at some sufficiently large |l| ≫ n, can be numerically diagonalized inorder to obtain the eigenenergies and eigenvectors (higher energy waves and do not contributesignificantly to the lowest energy eigenvalues or their associated eigenvectors). The eigenenergies,
calculated as a function quasimomentum, are arranged in continuous bands with periodic boundary(n,q)

conditions (figure 3.3), and the eigenvectors define the coefficients cl(n)

to reconstruct the Bloch waves ψqwhich can in turn be used(figure 3.2). At zero lattice depth, the band structure simply

resembles the dispersion relation of a free particle (modulo h̄k), and the Bloch functions are simply
single-frequency plane waves. As the depth of the lattice is increased, a gap opens at the edgesof the band structure (q = ±h̄k), and the width of the bands decrease to zero, starting with theground band. At very large lattice depths, each lattice site is effectively an isolated harmonic62

oscillator and so the lower bands become entirely flat and the gaps approach h̄ω.
Due to the separability of the Hamiltonian, the Bloch wavefunctions in multiple dimensions are
simply products of the independent wavefunctions in each dimension. The energies are therefore thesums of the eigenenergies associated with those Bloch waves. It is then evident the band structurechanges both quantitatively and qualitatively, depending on the dimensionality of the lattice (figure3.4). In a single dimension the bands are always gapped for V > 0, but in three dimensions the
(0)

(1)

first excited band has a minimum energy 2 × E0 + E0

which is lower than the maximum energyof the ground state 3 × E±h̄k . In fact, the bands do not fully separate until approximately 2.3 Erec .

3.1.3

Wavefunctions in a Lattice: The Position Basis

The Bloch wavefunctions are momentum eigenstates and are thus completely delocalized in positionspace. If one wishes to know the wavefunction of an atom localized on the ith site of a lattice, onemust construct a coherent superposition of all Bloch waves that maximizes the probability to findthe atom on site i, which is to say one must find the phase factors by which to multiply the Blochwaves so that they interfere constructively on the site [64]. This can be accomplished by taking
1 ∑ −iqx (n)w(n) (x − xi ) = √eϕq (x)
M q(n)where ϕq(3.12)

are the (normalized) Bloch wavefunctions and M is the number of quasimomenta in therange of the sum. These so-called Wannier wavefunctions, while not eigenstates of the Hamiltonian,are extremely useful for describing the state of an atom localized by a deep lattice or by interactions
with atoms on neighboring lattice sites. In a 3D cubic lattice, the Wannier functions are simply
products of the 1D Wannier functions w(n) (x) = w(nx ) (x)w(ny ) (y)w(nz ) (z). In very deep lattices,
the on-site potential of the lattice approaches a harmonic trap, i.e. V0 sin2 (kx) ≈ V0 (kx)2 forkx ≪ 1, so that in low-energy bands the Wannier functions approach the solutions to the quantum
harmonic oscillator. For example, in the ground state,
√
w

(0)

(x) ≈

4 mω0 −mω0 x2 /(2h̄)

πh̄
63

e

(3.13)

-3

-2

-1

0.1 ER

1.0 ER

10 E R

100 ER

0

1

2

3 -3

-2

-1

0

1

2

3

Sites

Sites

Figure 3.5: Square norms of the Wannier functions on a lattice site (solid blue lines) are plottedin arbitrary units for a selection of lattice depths. The corresponding ground state harmonic
wavefunctions (dashed red lines) are plotted for comparison. At lower lattice depths, the Wannierfunctions feature off-site revivals that are not captured in the harmonic approximation.

for a trap frequency ω0 =√2V0 k 2 /m. At lower lattice depths, however, one notes significant

revivals of the wavefunction away from the central peak, suggesting an enhanced probability todetect the atom on a neighboring lattice site (figure 3.5). The shape of wings of the wavefunction
will play a key role in the description of atomic motion and interactions within the Hubbard model.

3.2

Many Atoms in an Optical Lattice

If the atomic interactions become comparable to the width of the ground band, they will also
play a role in determining the configuration of atoms in the ground state. Following the approachoriginally suggested by Jaksch et al. [65, 66], this section will introduce the Bose-Hubbard model
for interacting bosonic atoms in the lowest Bloch band of an optical lattice, and will subsequently
explore the effect these interactions have on the ground state.
64

Figure 3.6: A schematic illustration of the matrix elements of the Hubbard model.

3.2.1

The Bose-Hubbard Model

The single-component Hubbard model describes particles which live on a lattice, may hop betweensites, and interact only when they meet on a site, resulting in the Hamiltonian
∫
H=

[
(
)
]
g
−h̄2
+ Vlat (x) + Vtrap (x) ψ̂(x) + ψ̂ † (x)ψ̂ † (x)ψ̂(x)ψ̂(x) ,d3 r ψ̂ † (x) −2m
2(3.14)

where g = 4πh̄2 a/m is the contact energy for atomic scattering length a. The field operators ψ̂ † (x)and ψ̂(x) are responsible for the creation and annihilation of an atom at position x and can be
expanded in the Wannier basis asψ̂(x) =∑∑n(n)w(n) (x − xi )âi.

(3.15)i

Assuming that the interaction energies and the kinetic energy of the system due to the temperature
T are much less than energy gap between the ground and first excited band, the sum over n canbe dropped as atoms exist only in the ground band. The Hubbard Hamiltonian then reduces toH=−∑tij (â†i âj + h.c.) +

⟨i,j⟩

∑
1∑
Ui n̂i (n̂i − 1) +µi n̂i2i(3.16)iwith the number operator n̂ = â†i âi counting the atoms on site i.

Matrix elements of the Bose-Hubbard model
The coefficient of the kinetic energy term is
∫
t=−

(
)
h̄ 2
dx w ∗ (x − xi ) −∇ + Vlat (x) w(x − xj )2m
65

(3.17)

for a particle hopping from site i to site j. It has also been assumed that the off-site interactions are
small compared to all the other energies in the system, so that the interaction energy only involves
site i and is equal to∫
U =gdx |w(x)|4 .

(3.18)

This assumption is well-justified for all but the lowest lattice depths: by V = 2ER , the contributionof nearest-neighbor sites to the interaction energy is already suppressed by 102 compared to the onsite interactions. Lastly, the term µ, often called the chemical potential, characterizes the potential
energy due to the external confinement and is equal to∫µi =dx |w(x − xi )|2 Vtrap (x − xi ) .

(3.19)

Approximations to the matrix elements

To see how the matrix elements scale with lattice depth, it is useful to find their analytic approximations in the limit of high lattice depth. In the case of U , one may replace the Wannier functions
with harmonic oscillator wavefunctions (3.13), leading to
√
U≈

8
ka
π

(

V0
ER

)3/4
ER .

(3.20)The harmonic oscillator wavefunctions are further a little narrower than the Wannier functions(see figure 3.5), leading to a slight overestimate of the interaction energy. On the other hand, thetunneling matrix element t can, in the context of the tight-binding model, be inferred from theband structure calculation as t = 14 (E (±h̄k) − E (0) ) and can therefore be approximated [67] as()
4 −2√V0 /ER V0 3/4t≈ √ eER .
ER
π(3.21)

These considerations show that U/t scales exponentially with the lattice depth, which means thatan extremely wide range of Hubbard parameters are available within a reasonable range of lattice
depths (figure 3.7).
66

100

(a)

Exact Result
Approximate Result(b)

Exact Result
Approximate Result(c)

Exact Result
Approximate Result

10-1

t/ER

10-2

10-3

10-4

10-5
1
0.9
0.8
0.7

U/ER

0.6
0.5
0.4
0.3
0.2
0.1
0
104

10

3

U/t

102

101

100

10-1

10-2

0

5

10

15

20

25

30

35

40

45

50

V0 /E R

Figure 3.7: The matrix elements of the Hubbard model are plotted as a function of the lattice
depth in recoils. In each case, the approximate analytical result is plotted (dashed black line) as
a comparison to the exact numerical result calculated from the Wannier functions(solid blue line).
The 1D tunneling energy t shown in (a) does not depend on the choice of atomic species and isa general result for a cubic lattice. In (b), the 3D on-site interaction energy scales linearly withthe scattering length, which in the F=1 manifold of 7 Li can be tuned arbitrarily with Feshbachresonances. For the plot, a typical scattering length of 100 a0 is chosen. The same scattering length
is chosen in (c), which shows how U/(6t) can be tuned over many orders of magnitude simply byvarying the lattice depth.
67

Spin-flipped Atoms (a.u.)

Lattice Depth

-4

-3

-2

-1

0

1

2

3

4

-25

-20

-15

Quasimomentum

-10

-5

0

5

RF detuning (kHz)

Figure 3.8: Comparisons of time-of-flight images (which probe quasimomentum) and lattice interaction spectroscopy (which probes site occupancy) reveal the differing nature of the ground states
at low and high lattice depths (upper panels and lower panels). Images of the ballistically expanded
cloud reveal characteristic superfluid peaks at ±h̄k momenta at low lattice depths, √whereas at highlattice depths this coherence disappears. (The horizontal axis has units of ±h̄k/ 2 because thelattice is viewed at a 45◦ angle.) RF interaction spectroscopy reveals that the excitation spectrumof the MI contains a sharp peak at δ = 0 and −U/h. At low lattice depth, this structure subsides
into a broadened peak centered around −U/h̄ · n̄, due to mean-field shifts.

3.2.2

Ground State of the Hubbard ModelIn the Hubbard model, competition between t, which lowers the kinetic energy of the system by
delocalizing the particles, and U , the potential energy arising from interactions, sets up an interplay
between two phases with very different characteristics. When t/U → ∞, the Hamiltonian describes
a lattice superfluid, while when t/U → 0 the Hamiltonian describes a Mott Insulator (MI). In theformer, site occupancy is described by a coherent state with number fluctuations and well-definedphase. The ideal superfluid state can, in the absence of external confinement, be written as(|ΨSF ⟩ =

1 ∑ †
√
âi
M i=1
M

68

)N
|0⟩

(3.22)

for M lattice sites containing N bosons, which approaches|ΨSF ⟩ =

M
∏

(√
exp

i=1

N †
â
M i

)
|0⟩i

(3.23)

for sufficiently large N and M . In 3D at U/t = 0, all N bosons will be fully delocalized over the(0)lattice, occupying the Bloch state ψ0 (x). In contrast to the superfluid, the site occupancy of theMott Insulator is rather described by a Fock state|ΨMI ⟩ =∏|n̄⟩i(3.24)iwith the average filling factor n̄ = N/M . The MI phase is an insulator (i.e. the atoms are localized
to a single lattice site) because of the interactions, which penalize delocalization. It is possible toview the interaction portion of the Hamiltonian as a number-squeezing term which suppress number
fluctuations on a lattice site.
These two different phases have important consequences for both the spectra of energy and of
quasimomentum in the lattice system. The superfluid phase exists at a well-defined quasimomentum and is phase-coherent, which is readily observed by projecting the quasimomentum onto real
momentum by ballistic expansion. In contrast, the MI exists at a well-defined occupation number,which splits the lowest Bloch band into many “Hubbard bands” and produces an energy spectrum
that can be probed e.g. via lattice depth modulation (see section 3.3.1). The uncertainty relation
between phase and number suggests that the characteristic momentum spectrum of the superfluid
phase should be entirely washed out for the MI phase, and vice versa for the energy excitation
spectrum.

Quantum phase transition
The critical value of t/U that describes transition between the superfluid and the MI can becalculated in the context of the mean field approximation√ni , and(3.25)⟨â†i âj ⟩ = ⟨â†i ⟩âj + â†i ⟨âj ⟩ − ⟨â†i ⟩⟨âj ⟩ = (âj + â†i )ψ − ψ 2 .

(3.26)

ψ = ⟨âi ⟩ = ⟨â†i ⟩ =

69

3
3.5

n=3
3

/U

2

1.5

2.5

Superfluid

n=2

2

1

0.5

0

Site Occupancy

2.5

1.5

n=1

1

0.05

0.1

0.15

0.2

0.25

zt/U

Figure 3.9: The expectation value of the site occupancy n obtained from a mean field approximation
to the 3D Bose-Hubbard model is plotted vs the chemical potential µ and tunneling t (both inunits of the interaction energy U ), as calculated for nmax = 7. The white dotted lines show theperturbative approximation to the phase boundary between the Mott insulator and superfluid.
Within the Mott insulating lobes, n is an integer, while in the superfluid region n is a real number,which suggests one should expect the number fluctuations commensurate with a superfluid phase.

Substitution into (3.16) and scaling by the number of sites gives an effective Hamiltonain
Hef f = −ztψ∑∑U∑(âi + â†i ) + ztψ 2 Ns +n̂i (n̂i − 1) − µ
n̂i
2
i

i

U
= n̂i (n̂i − 1) − µn̂i − (âi + â†i )ψ + ψ 2
2i(3.27)

where U and µ have both been rescaled by zt, the product of the tunneling coefficient t and thecoordination number z of the lattice. An advantage of this mean field Hamiltonian is that it iscompletely local, depending only on the occupancy of each site i, and can thus be diagonalized inorder to determine the energy spectrum and wavefunctions.

As a first step, one can gain some intuition into the phase diagram by treating the tunneling as70

a perturbation. Following Oosten et al. [68], the ground state energy for g particles is thenEg(0) =Ug(g − 1) − µg ,2(3.28)

for U (g − 1) < µ < U g. The first order of perturbation theory produces no non-zero matrix
elements. To second order,(2)

EGS = ψ 2=

∑ |⟨g|V |n⟩|2(0)(0)
n̸=g Eg − Eng
g+1+
U (g − 1) − µ µ − U g(3.29)The mean field ground state energy is thus the sum

Eg (ψ) = E0 (g, U, µ) + E2 (g, U, µ)ψ 2 + ...

(3.30)

minimized as a function of ψ. When a2 (g, U, µ) > 0, ψ = 0, while when a2 (g, U, µ) < 0, ψ ̸= 0.
Thus the boundary between phases is at a2 (g, U, µ) = 0. Rescaling zt and µ by U and solving forµ gives11√(c)µ± = [(2g − 1) − t] ±1 − 2t(2g + 1) + t2
2
2

(3.31)

which outlines the Mott lobes for each g, which are plotted in figure 3.9.
Returning to the local effective Hamiltonian, one may calculate the density per site by direct
diagonalization, after truncating the Hamiltonian at nmax sufficiently higher than the highest site
occupancy expected in the system. The eigenvectors give the amplitude of the Fock states |0...n⟩on each site. This method thus provides the expectation value of the number operator within the
Mott insulating lobes, where it is integer-valued, and also in the superfluid regions without, where
it is not (see figure 3.9).
One may use this phase diagram in order to quantitatively predict the ground state of bosons
in an optical lattice. In general the ground state depends not only on the lattice depth V0 , but also
on the external confinement V (r). Making the local density approximation, one may consider thechemical potential on each site to be µ − V (r). Along a single dimension, a 3D optical lattice withharmonic confinement samples a line along the phase diagram, from µ = 0 to µ = µmax . If the
71

2.5
20 ER
15 ER
12 ER

2

n

1.5

1

0.5

0
-25

-20

-15

-10

-5

0

5

10

15

20

25

site

Figure 3.10: A cut of the expectation values of the number operator n along a lattice axis areplotted for lattices of various depths. The number of atoms is 40,000, the lattice beam waists are
100 microns and the scattering length is 100a0 . The site occupancy develops a kink as each plateaucrosses the Mott insulating transition.

lattice depth is uniform, this line is vertical, although in the case that the confinement arises fromthe curvature of the lattice beams themselves, the depth of the lattice is greatest where V (x) = 0
and so tunneling increases as a µ−V (x) decreases. Thus at sufficiently high filling, the ground stateof interacting bosons in an optical lattice is said to have a “wedding cake” structure, with a plateau
of sites occupied by n atoms in the center, surrounded by a plateau of occupancy n − 1, and soon until n = 0, with thin shells of superfluid in between. These superfluid shells are regions where
tunneling is not suppressed by interactions, and have expectation values of the number operator
between n and n − 1 (see figure 3.10).

Super-exchange

The ground state of the MI at infinite lattice depth (infinite U/t) is a product state of integersite occupancy, as in equation (3.24). At finite but small U/t, one may again treat tunneling as a72

perturbation on top of the MI ground state at infinite lattice depth, to find that|Ψ⟩(1) ≈ |ΨMI ⟩ +t ∑ †âi âj |ΨMI ⟩U(3.32)⟨i,j⟩

to lowest order in t/U , which suggests that the ground state fundamentally contains a fraction ofparticle-hole excitations proportional to (t/U )2 . These coherently admixed doublons are responsible
for restoring finite visibility of superfluid peaks above the MI transition [69] and have been directly
observed in quantum gas microscope experiments [70], and will be important considerations forthe lifetime of an n=2 MI, where the particle-hole excitations create triplons that are subject tothree-body loss (section 4.3.3). Their presence, however, already suggests it is worth considering
second-order tunneling process which involves an atom tunneling to a virtual excited state (which
does not conserve energy) and tunneling back: in analogy to the way an off-resonant two-photon
process with one-photon Rabi frequency ω and detuning ∆ occurs at a rate Ω2 /∆, tunnelingadmixes particle-hole pairs at a rate J ≡ t2 /U . The off-site interaction energy arising from thissecond-order tunneling process is about an order of magnitude larger than the “real,” zeroth-order
off-site interaction energy computed directly from the overlap of the Wannier functions (3.18), andresults in oscillations in the population of a double-well system with two particles [23].

Figure 3.11: Schematic illustrating two different sorts of super-exchange processes. Two particles
which could not tunnel separately, due to conservation of the interaction energy, are allowed totunnel as a pair. Two neighboring particles (red and blue) can also switch places, which does not
modify the overall-density distribution but nonetheless shifts the energy of the state by the exchange
coupling energy Jex . In contrast to bound-pair tunneling, this latter process is only non-trivial insystems with at least two distinguishable components.

The presence of super-exchange interactions in a MI gives rise to all sorts of fascinating phenomena, such as repulsively bound pairs of atoms [71] and distillation of vacancies [72]. One especiallyrelevant feature it brings to the current work is the idea of partitioning the Hilbert space of thelattice system into a density (or “charge”) sector, which is frozen, and a spin sector in which all
the dynamics take place. Because the Mott insulator (under appropriate filling conditions) has aninteger number of particles per site, it is an ideal platform upon which to implement a quantum73

simulation of a lattice model in which the quantum information is carried by internal states ofatoms: the density distribution remains in quasi-equilibrium, but “spin” information can be exchanged between sites via super-exchange. The resulting interaction energy breaks the degeneracyof the different pseudospin configurations in the lowest of the Hubbard bands (which themselves
exist in the ground state of the lowest Bloch band), giving rise to a spinful ground state that will beshown in section 4.1.1 to have features reminiscent of magnetic ordering in the Heisenberg model.

3.3

Lifetimes and Coherences

A variety of heating mechanisms can limit the fidelity of the single-component ground state. Indeed,
any non-adiabatic variation of the parameters of the Hamiltonian will lead to excitations of somekind– imperfections in the adiabaticity of the lattice ramp will limit the initial fidelity, and technicalsources of noise in the control beams and fields will limit the coherence lifetime of the model. The
ground state of atoms in an optical lattice is protected by an energy gap to the first excited state at
all depths in 1D and at all depths above 2.3 ER in 3D. While care is taken to ramp adiabatically withrespect both to this bandgap and to the over-all confinement, dynamic processes in the lattice cannevertheless lead to a decay of the ground state occupation. This section will provide an overview ofthe decoherence mechanisms present in an optical lattice experiment, and provide some insight on
how to diagnose various sources of heating– a process which is not always straightforward. Whilethe effects of intensity and pointing noise on inter-band excitations have been carefully characterized
in fermionic Lithium [73], the effects of position noise of the lattice sites will also turn out to be an
important consideration. Further, there are intra-band excitations possible in bosons which are notin fermions (due to the Pauli exclusion principle), although the heating rates for these excitations
turn out to be rather low.

3.3.1

Inter-band Heating Mechanisms

Transitions from the ground band to excited bands of the optical lattice can be driven by fluctuationsin the lattice potential at frequencies on the order of the bandgap wbg . If the occupation is mostly
in the ground band, these transitions are mostly single-particle physics, and are relatively simple
to evaluate in the limit of harmonic oscillator lattice sites (i.e. deep lattices) where most lattice74

physics takes place.

Intensity fluctuations
By considering the fractional fluctuations of the trapping frequency ωT as a perturbation, Gehm etal. [74] showed that the expected rate of transitions out of the ground band of a harmonic oscillatorisΓ0→2 =πωT2
Sk (2ωT )8(3.33)

where Sk (ω) is the one-sided power spectrum of the perturbation ϵ(t) = δωT /ωT in units of (1/s)−1 ,∫∞normalized so that 0 dω Sk (ω) = ϵ2RMS . The rate of excitation into the first excited band is
identically zero: the perturbation is of the form ϵ2 x2 , which is even and therefore connects only
states with the same parity. It is interesting to note that such parametric heating is only sensitive
to noise at the frequency corresponding to Eg→e /h̄ (which in the harmonic oscillator approximation
is simply the trap frequency times the difference in band indices). The excitation rate as a functionof the lattice depth is thus sensitive to the shape of the noise. For example, if the noise is white– i.e.
it has a power spectral density1 (PSD) that is constant as a function of frequency– the excitation
rate will increase as ωT2 ∝ V0 . If, however, the PSD falls off as 1/ω 2 , the excitation rate will be aconstant function of the lattice depth.
It is therefore important to characterize the PSDs of the various sources of intensity noise in a
lattice. White noise is the signature of a Poissonian process, or of many uncorrelated noise sources(as a consequence of the central limit theorem), and in electronics is often the result of JohnsonNyquist noise in resistors– which is especially likely to manifest in analog circuits as noise on the
input offset of the input amplifier of a circuit (or wherever the signal is smallest). Alternatively,a white noise profile might be the result of shot noise in the measurement of an optical signal. It
could also be present in the light itself as a result of stimulated Brillouin scattering (SBS), whichis essentially white noise at frequencies lower than 1 GHz. Brownian noise (S ∝ 1/ω 2 ), on theother hand, is the result of a random walk, and is the likely noise profile of vibrations (neglecting
resonances), which can couple into optical intensity via pointing noise at fiber-coupling inputs or
1

The power spectral density used here is not the spectral density of the optical power, but rather the squaremodulus of the Fourier transform of the lattice depth V (t)– which itself is proportional to the optical power in thelattice beam.

75

-80

Relative intensity noise in dBc/Hz

-90

(b)

-100

Relative intensity noise in dBc/Hz

(a)

Azur

-110
-120

Nufern-130-140

Mephisto

-150
-160
-170

-60
-80
-100

Free running

-120
-140

Noise eater active-160

2 x Photodiode Shot Noise
Photodiode Shot Noise

-180

-180
10 0

10 1

10 2

10 3

Frequency in kHz

10 -3

10 4

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

10 5

Frequency in kHz

Figure 3.12: (a) Relative intensity noise at full output power of two models of 50 W 1064 nm
fiber amplifier (Azur Light Systems and Nufern), seeded by the same ND:YAG laser (Coherent
Mephisto). (b) Relative intensity noise of the Azur fiber amplifier with and without an EOM-basednoise-canceling scheme, after passing through a single-mode fiber. One photodiode is used for
feedback, and a second independent photodiode is used for the measurement, so that the measured
noise floor is the sum of two uncorrelated white noise processes.
via polarization fluctuations in fibers. It is also the expected intensity noise profile of a laser, absent
excess electronic noise in the control circuitry, due to the lowpass filtering effect of the gain cavity.
Significant effort has been made to characterize– and to some extent mitigate– intensity noise in
the lattice lasers. Two different 50 W, 1064 nm fiber amplifiers have been used to generate latticelight. The first is a customized Nufern NuAmp with optimized gain fiber and shortened output fiber(in order to reduce stimulated Brillouin scattering, which is a limiting source of noise in the factorymodels), and the second is a stock model Azur Light Systems amplifier. The relative intensity noise(RIN) of both are plotted in figure 3.12a, along with the RIN of the Coherent Mephisto seed, where
the RIN in dBc/Hz is defined as 10 log10 S(f ), if S(f ) is the PSD of the fractional noise in 1/Hz,
i.e.
S(f ) = 2π

1
T

∫ T
0

V (t) − V̄ iωt
e dt
V̄

2

(3.34)

for a photodiode signal V (t). For both fiber amplifiers, two regimes are apparent: at frequencies
below 100 kHz, noise produced by the fiber amplifiers themselves dominates (from a combinationof power supply noise and vibrations), while above 100kHz the limit is the RIN of the seed laser.
The plateau of noise between 100 kHz and 1 MHz is likely due to the shot noise of the internal
photodiode which the Mephisto laser uses to regulate its own output power (most units contain76

a “noise eater” which suppresses endemic intensity oscillations at a few hundred kilohertz), after
which the noise power falls off with a slope of -30 dB/Hz.
This performance can be readily exceeded by the use of an additional noise-canceling scheme.
Figure 3.12b shows the results of such a scheme, which uses an EOM between crossed polarizers
as an intensity modulator to achieve shot noise limited performance in the region of the bandgap
frequencies, from 1 kHz to 1MHz. The remaining noise at low frequencies comes from a combinationof amplifier noise and pointing noise (turned into intensity noise by fiber-coupling). Addressing thislow frequency noise directly with the EOM scheme is difficult because the amplitude of fluctuations
grows larger as 1/f , which subsequently requires servoing very high EOM control voltages (> |±15|V), which are incompatible with high bandwidth, and/or operating the EOM polarization rotator
at a setpoint of 45◦ to increase the sensitivity to the control voltage, which sacrifices optical power.
A better approach is to use an AOM to take care of low frequency noise (which is well within the
bandwidth of typical AOM feedback systems). Intensity noise is not currently a limiting source ofheating (see next section), but such additional noise-canceling schemes will likely play an importantrole in future upgrades.

Characterizing sensitivity to intensity noise

A model-free way to measure the local slope of the sensitivity to noise is to add noise and measure
the response of the system. It is usually easier to add noise than it is to take it away, so that
if there are many uncorrelated heating processes, this method can help to demonstrate that acertain heating process is not limiting (or at least is not the only limit) before making a serious
technological investment in noise-canceling. In order to test the atoms’ sensitivity to intensity noisein the lattice beams, it is straightforward to introduce broadband white noise into a lattice beamand measure the resulting coherence time as a function of the amplitude of the noise. The results(figure 3.13) indicate that the coherence lifetime of the 3D Mott Insulating phase was limited by aheating mechanism other than intensity noise at the time of the measurement.
77

(a)

(b) 12080 MHz RF

LO

+

IF

RF

X40

AOM

Mixer

PID

Coherence Time (ms)100White Noise

Photodiodes
To Spectrum
Analyzer

80
60
40
20

To Atoms

0
-120

-115

-110

-105

-100

-95

Noise Power Spectral Density at 2

-90

-85

(dBc)
bg

Figure 3.13: (a) Diagram of the scheme to measure the effect of added intensity noise. Whitenoise (0-1 MHz) is summed with the output of one of the lattice feedback controllers. The summed
signal controls the amplitude of the RF powering the AOM, which controls the intensity of the light.
Two separate photodiodes supply the feedback signal and a monitor signal, which can be used tomeasure the magnitude of the added noise. Parametric heating is driven by noise at frequencies ofapproximately twice the bandgap ωbg , which is too high in 7 Li for the added noise to be attenuatedby the feedback controllers. (b) Coherence lifetime of a Mott Insulator at 35 ER as a function of
intensity noise in one of the three lattice arms, measured by adiabatically ramping back to the
BEC and probing the condensate fraction. The dashed line is a fit to a model in which the loss
rate depends linearly on the PSD of the intensity noise, as in (3.33), and another independent loss
rate.

Position fluctuations
Fluctuations in the position of a harmonic trapping potential can be evaluated in a similar way tothe intensity fluctuations discussed in the previous section, leading to an excitation rate ofΓ0→1 =πmωT3 Sx (ωT )h̄(3.35)

where Sx (ω) is the PSD of the trap position fluctuations in m2 (1/s)−1 . In this case, the perturbation
ϵx is odd and connects only states with opposite parity, leading to transitions from the zeroth tothe first exited band. For position noise, the scaling of the excitation rate is one power of ω stronger
than for intensity noise, so that even if the noise is Brownian (which one would expect it to be forunattenuated vibrations, for example), excitation rates should still increase as a function of trap
frequency.
78

Figure 3.14: Interferometrically measured position noise of a lattice site. The noise floor is determined by twice the noise in the signal if one of the arms of the interferometer is blocked (blueline). The stability of the reference arm is characterized by using two very short interferometers,
rather than one short and one long (red line). The resulting position variation in a √lattice site(yellow line) is calculated from the photodiode signal V as ∆z = (V /Vmax − 1/2)/( 2k). The
contribution of the frequency fluctuations of the laser (dash-dot purpule line) are calculated fromthe manufacturer-specified phase noise of the Innolight Mephisto√laser used for the experiment.
The ultimate noise floor of the measurement, around 5 × 10−13 m/ Hz, comes from the shot noiseof the light.
There are fundamentally two different sorts of noise which can cause the position of a latticesite to fluctuate: noise in the position of the retro-reflection mirror, and noise in the frequency ofthe lattice light. The retro-reflecting surface defines a node of the standing wave and the number ofnodes between this surface and a point l away is fixed by the wavelength to 2l/λ. Thus fluctuationsin the longitudinal position of the retro mirror directly couple to fluctuations in the positions of
the lattice sites. However, a change in the frequency of the lattice light ωL by δ moves a latticesite by approximately l ωδL . For typical ωL of 100 THz and lattice path lengths of 1 m, a frequency
sweep of δ = 100 MHz moves a lattice site by 1 µm. The effect of frequency noise can therefore be
decreased by shortening the “lever arm” of the retro-reflection path length.
Position fluctuations can be evaluated with the aid of a Michaelson-Morely interferometer. A
non-polarizing beamsplitter cube is introduced into the lattice path near the launch point. Half ofthe light travels through the chamber and is retro-reflected as usual (path length ∼1 ); the other79half is retro-reflected immediately (path length ∼1 mm). The two beams recombine at the cube
and the resulting intensity is measured by a photodiode. The phase of the interference signal is ameasure of the arm length difference, modulo λ/2. Assuming the short arm is a good reference,
this path length difference is a measure of the fluctuations in the position of a lattice site. It canbe further assumed that relative motions of the beamsplitter cube and retro-reflection mirror arecomparable but uncorrelated, so that the position fluctuations of a lattice site in the inertial frame√of an atom are 1/ 2 smaller than what is measured by the interferometer. In order to make acareful measurement, the interferometer must be locked to one of the interference fringes at half
the contrast, so that the response of the interference signal is locally a linear function of the path
length difference. Because of the path length difference of 1 m between the reference and thelattice arm, this can be accomplished by feeding back on the frequency of the laser with a slow
servo (bandwidth < 1 hZ). The noise floor of this measurement is determined by the intensity noisein the light, which produces fluctuations of the interference signal which are uncorrelated with path
length difference, so it is important to use a light source with low intensity noise.
The results of this measurement can be seen in in figure 3.14. Although photon shot noise
prevented a precise determination of the spectral power density of the position noise between 100
kHz and 500 kHz, which are the relevant trap frequencies for 7 Li , it can be determined that thePSD decreases as 1/ω 2 , ignoring low frequency acoustic resonances, which are unlikely to occur
above 10kHz. This scaling comes as no surprise, as absent any unusual driving or resonances,the position of some object ought indeed to be the result of Brownian motion. Extrapolating tohigher frequencies, a lattice of 16 ER has an on-site trap frequency of ωT = 2π × 200 kHz andshould suffer from position-noise-driven transitions to the first excited band at a rate of about 1
s−1 in one dimension. Since these data were taken, compressed air vibration isolation has been
added to the legs of the table which supports the apparatus. This isolation system is expected toreduce the vibration noise power at frequencies above 100 Hz by at least 20 dB. While repeating
the interferometric measurement of the lattice position noise would be a direct confirmation thatthe system works as intended, the subsequently measured excitation rate of the atomic sample(see figure 3.15) is consistent with these expectations, and Mott insulator coherence lifetimes (asmeasured by adiabatic return to the superfluid phase) increased by a little more than a factor of
10.
80Fluctuations in the external trapping potential
Any noise present in external potentials (the ODT, or external confinement of the transverse lattice
beams) can also couple to the position and depth of the optical lattice sites. Treating all potentials
as harmonic, for fractional perturbations in the eternal trap frequency ϵext , the total trap frequency
can be written as
2
2
2
ωtot
= ωsite+ ωext(1 + ϵext )2 .

(3.36)

Similarly, perturbations in the external trap position lead to a fluctuating position of the trap
minimum satisfying)∂ ( 2 22
ωsite x + ωext
(x − ϵext )2 = 0 .
∂x(3.37)

In both cases, it turns out that the effect of the external perturbations are reduced by a factor(ωext /ωsite )2 < 10−3 for a wide range of lattice parameters, including those used in this experiment.
The small size of this coefficient means that intensity or pointing noise in externally confining beams
are unlikely to contribute to excitation rates in a lattice.

Choice of atomic species
The physics of the Hubbard model is controlled by the dimensionless ratio t/U . In section 3.2.1 it√was shown that this quantity scales as e−2 V0 /ER , which means that, ignoring for the present any
variation in the s-wave scattering length among different atomic species, access to the same partsof the Bose-Hubbard phase diagram is possible with atoms of any mass at the same lattice depth
V0 , measured in units of ER . At the same time, the on-site trap frequency of an optical lattice is√ωsite = 2V0 h̄k 2 /m, which means that that ωsite ∼ 1/m for a given V0 , depending on the choiceof atom. The inter-band excitation rate at a given lattice depth due to either intensity noise or
position noise therefore scales as 1/m2 · S(ω), which leads to higher heating rates for lighter atoms
if S(ω) ∝ const and no advantage if S(ω) ∝ 1/ω 2 .
On the other hand, if one wishes to observe super-exchange physics, the relevant comparison is
the super-exchange time τSE = 1/h̄ · t2 /U ∝ (t/U )2 · U ∝ const · 1/m. As already discussed, thecoherence time τC ∝ m2 /S(ω), so that τSE /τC ∝ m/S(ω). It is therefore critically important tounderstand the functional form of S(ω), as lighter atoms may either be advantaged or disadvantaged81

with respect to single-particle band excitations, when it comes to observing super-exchange physics!

3.3.2

Intra-band Heating Mechanisms

Excitations within the first Bloch band may either be created directly by perturbations of the
Hamiltonian, or via inter-band excitations which then relax to the lowest band via collisions. The
former process creates particle-hole pairs whose energy scales with U , while the latter eventuallysees the bandgap energy h̄ωbg distributed over many doublons in the ground band.
Intensity fluctuations
Just as inter-band excitations can be caused by various sorts of noise at the frequencies of the
bandgaps, intra-band excitations can be caused by perturbations of the Hamiltonian at frequencies
within the bandwidth. The elementary excitation is the particle-hole pair, the creation of which
has been studied by Pichler et al. [75], who find that, as t/U → 0 deep in an optical lattice, white
noise on the lattice depth drives creation of particle-hole pairs as

ΓPH =

1
S0 V0 t2 z (⟨n⟩ + 1)h̄2

(3.38)

for a PSD S0 of the fractional noise in the unitless lattice depth V0 . The particle-hole pairs are
created with opposite quasimomenta and travel on top of the Mott insulator, but in sufficiently
deep lattices have energy U regardless of quasimomentum, due to the flatness of the band. Thisrate scales with E2R ∝ 1/m2 for white noise, but is vanishingly small for 7 Li in a 15 ER lattice– lessthan 10 mHz for 7 Li in a 15 ER lattice created by beams with intensity noise at the level of 100
dBc/Hz.

Thermalization of band excitations
The band excitations discussed up to this point have been treated as single-particle phenomena,
adding an entropy per particle of kB /n · ln n to the Mott-insulating ground state by creating randomly located holes. If the first or second excited band atoms can collide with ground-band atoms,
however, it is possible for each band excitation to further cause multiple intra-band excitations
within the ground band. Müller et al. found that atoms in the |nx , ny , nz ⟩ = |1, 0, 0⟩ state relax
82

% of sites with doublons

10
9

15 ER

8

20 ER
25 ER

7

30 ER

6

5%/s

5
4
3
2
1
0

0

200

400

600

800

1000

1200

1400

hold time (ms)Figure 3.15: Doubly occupied lattice sites recorded as a function of time in 3D lattices of various
depths, starting with 30,000 singly-occupied sites (the maximum number of doubly occupied sites
would be 50%). The dashed line is a fit to the data for 20 ER for t > 200 ms. The initial fractionof doublons is due to loss of sensitivity of the measurement technique at very low atom number,and does not change when the total atom number is varied within a range of ±30%.

to the ground band via collisions at a rate which is nearly constant in the range of lattice depths
between 15 ER and 45 ER , when the excited state population starts at 80% [76]. Starting with alattice entirely in the |000⟩ state, it is possible to directly monitor the number of doubly-occupied
sites as a function of hold time in the lattice (using RF transitions and interaction blockades todistinguish between sites with differing occupancies, as in section 4.2.1, rather than the bandmapping technique used by Müller et al.). This rate will serve as an ultimate limit on the lifetimeof the many-body ground state in the optical lattice, and so is of great relevance for adiabatic state
preparation. It is found that the rate of doublon creation is about 5% of the originally occupied
sites per second, and does not depend on lattice depth (see figure 3.15). These data rule out direct
creation of such excitations within the lowest Bloch band, which should occur at a rate that scales[ √ ]as V0 exp −4 V0 , but are consistent with a mechanism that involves inter-band excitations driven
by position noise of the lattice sites followed by relaxation. It is possible that further steps toprovide vibration isolation for the retro-mirrors in the lattice setup will improve the lifetime of the
n=1 MI beyond several seconds.
83

3.4

Implementation

Both the ODT and lattice light are supplied by a 1064 nm laser, whose detuning from the S to P
transition in 7 Li is nearly 340 THz, leading to a scattering rate which is below 10 mHz for a 3D, 35
ER lattice. The next sections discuss how this light is generated and controlled in order to create
an optical lattice.

Beam generation and delivery
The 1064 nm optical trapping and lattice light is provided by an Azur Light Systems 50 W fiber
amplifier, seeded by a Coherent Mephisto ND:YAG laser with a linewidth specified at 1 kHz (forits contribution to the position noise of the lattice sites, see figure 3.14). The light is divided
by a dynamically variable beamsplitter into a beam path supplying the ODT light and a beam
path supplying the lattice light. During initial transfer of the cold atomic cloud into the ODT,
the beamsplitter is set to maximize the amount of light reaching the ODTs, while during lattice
operation, the opposite is the case. This setup is pictured in figure 3.16, and is capable of delivering
either 11 W of power to each ODT beam or 8 W of power to each lattice beam. The crossover
between the two regimes of operation takes place during the ODT evaporation step, when the
power in the ODT is decreased to near-zero. Each of the two ODT beams and three lattice beams
is coupled through an AOM (Gooch and Housego AODF 4090-6) to provide intensity control
and frequency offsets, and then through a polarization-maintaining photonic crystal fiber (NKT
Photonics LMA-PM-15). The large mode area of the fiber is important in order to minimize thepotential for stimulated Brillouin scattering (SBS).
The fibers are connectorized with a PCF-SMA connector by AlphaNov. The connectors include
an epoxy-free mode-stripper to prevent uncoupled light from propagating into the fiber cladding, inorder to improve power handling. It was found that mode-strippers which contain epoxy tend to failwhen the power of the uncoupled light reaches 6 W. As the in-coupled power of each lattice beam
is about 10 W, this failure mode necessitates the use of an interlock mechanism which monitors
the coupling efficiency of each fiber and switches off the laser beam when the coupling drops below
50%. The epoxy-free mode-strippers display no such failure mechanisms, and making an interlock
is unnecessary. In addition, each mode stripper has been supplied with a homemade water-cooling
84

AOM

ALS 50 1064 IR
VariableAOM

1 2 -Waveplate

LegendFaraday Isolator

AOM

ISO

AOM

Dielectric Mirror
1 2 -Waveplate

AOM

Polarizing Beamsplitter
Cube
APC PM Photonic
Crystal Fiber

To OD
T

1

To OD
T

2

ISO
To Latt

ice 1

ISO
To Latt

ice 2

ISO
To Latt

ice 3

Figure 3.16: Schematic of the IR path. Not pictured are beam-shaping telescopes, focusing lenses,
folding mirrors and polarization clean-up optics.

jacket, which makes the coupling efficiency virtually independent of the input power.
As each of the lattice beams is retro-reflected after passing through the chamber in order tocreate a standing wave potential, it is extremely important to provide adequate directional isolation
along the lattice beam paths, both to protect the lasers from the retro-reflected power and the atoms
from tertiary back-reflections. This goal is accomplished in two stages: a Faraday isolator with 55
dB of directional isolation is placed between each AOM and lattice fiber input, and a PBS andquarter-wave plate pair are placed after each lattice fiber output. The latter technique provides
only about 40 dB of directional isolation, but mostly serves to protect the lattice beams fromreflections of the retro-reflected beam from the fiber tip. These tertiary reflections are further
minimized by the use of an angle-cleaved fiber. The cube-and-waveplate isolation technique also
has the dual advantage that it does not suffer from the sort of thermal lensing typical of the TGG
crystal inside a Faraday isolator, and can be easily un-done by the addition of a half-wave plate to
the lattice beam path, allowing for easily alignment of the retro-reflected beam by back-couplingat low power. The fact that the lattice beams form a σ + -σ − pair proves to be unimportant as1064 nm is far enough detuned from any atomic transition in 7 Li that vector AC-Stark shifts are
negligible, and the interference terms between different lattice beams are suppressed enough byrelative detunings of 3 to 5 MHz (not commensurate with any inter-band transitions) that further
isolation by orthogonal polarizations does not provide any benefit. It is worth noting that using the85

positive and negative orders of AOMs to create beams with detunings of +δ and −δ is not sufficient
to decouple the beams. In general, AOMs suffer from a one part in a hundred contamination of
the positive order by the negative order and vice versa, due to photons scattered by the reflected
acoustic wave, so that interference between the two beams can still cause transitions within theband structure.

Figure 3.17: Cartoon of a lattice beam with cat-eye configured retro-reflection mirror. The forwardpropagating beam (gray) is delivered by an optical fiber and focused onto the atoms. If the angleof the retro-mirror changes, the retro-reflected beam (blue) still passes through the atoms. Not
pictured are folding mirrors, beam shaping optics and polarizing elements.
The lattice and ODT beams are all alignable via 45◦ mirrors with two-axis piezoelectric control(Newport Picomotor 8821). The minimum angular step-size of these devices is 0.7 µrad; they areplaced before lenses which focus the beams onto the atoms (f = 300 mm) so that the position of
the beams is adjustable with a minimum resolution of 200 nm. The ODT beams are shaped so thatthe beam waists at the atoms have a 1/e2 radius of 100 µm, while the lattice beams have waistsof 125 µm, which is the maximum size at which the intensity is high enough to reach a lattice
depth of V0 = 40ER with the available power. The lattice beams are additionally retro-reflectedby a gimbal-mounted mirror at the focus of an f = 50 mm lens in a cat-eye configuration. The
cat-eye lens and lattice focusing lens form an imaging system which images the waist of the beam(where the atoms are) onto the mirror surface; this geometry guarantees that the beam passes backthrough the atoms regardless of the angle of the retro-reflection mirror. Similarly, the beam waist
in the chamber is the image of the fiber tip, so the retro-reflected light is also guaranteed to hit thefiber–although not necessarily with the correct angle for efficient fiber-coupling. This arrangement
makes the lattice alignment quite insensitive to the angle of the retro mirror.

Beam control and feedback
In addition to providing the frequency offset between the beams which is necessary to suppress
interference, AOMs also provide intensity control, which can be used in a feedback loop to both86

set the total intensity and cancel intensity noise. The frequency response of an AOM has two
major components– a lowpass filtering behavior with characteristic frequency of about 3 MHz, anda constant time delay introduced by the propagation time of the sound wave in the AOM crystal,
which is about 1 mm / (1000 m/s) = 1 µs, as in figure 3.18. The additional phase shift accrued
due to this time delay is quite significant, and has the effect of limiting the bandwidth of the AOM
which is usable for feedback, as will be illustrated. The delay can in principle be reduced slightly byfocusing the beam tightly and moving it closer to the transducer, but such a strategy is ultimately
limited by thermal lensing in the AOM crystal for the high powers required by Lithium.

Gain (dB)

0
-10
-20
-30
10 3

10 4

10 5

10 4

10 5

10 6

10 7

10 8

10 6

10 7

10 8

45

Phase Shift

0
-45
-90
-135
-180
-225
10 3

f (Hz)

Figure 3.18: Bode plot of the transfer function of an AOM with a typical bandwidth of 3 MHz(indicated by vertical dashed dashed line). The blue lines show the gain and phase shift of an AOM
that behaves only as a first-order lowpass filter. The red line shows the total phase shift, including
the extra phase shift accrued due to 1 µs of time delay.

Closed loop feedback is a fascinating topic with many implications for the work in this thesis
and broad application in atomic physics generally, which will not be discussed in detail;2 it suffices
here to say that the open loop gain of a linear feedback circuit should cross 0 dB where the phaseshift of the total loop, including the servo device, is less than 180◦ . This condition is necessary inorder to maintain the stability of the circuit when the loop is closed, which means the PI-corner ofa simple feedback controller should be set to wherever the total phase shift is −45◦ . The controller
2

A nice introduction to feedback can be found in [77].

87gain can then be increased until the phase margin is too small to avoid amplification of noise
beyond the loop bandwidth. In the presence of a time delay, however, the PI-corner must be set
to significantly lower frequency (in the typical example of figure 3.18, at about 100kHz). Not only
does this limit the low frequency noise-canceling gain, but it also allows any output noise of thecontroller at frequencies above the feedback bandwidth to be written into the light, where the AOM
gain is still responding proportionally. Note that a feedback system driving a servo with simple
first-order lowpassing behavior and no time delay would not have this problem, as noise above thefeedback bandwidth would never reach the light. In heavier atoms, frequencies between 100kHz
and 3 MHz should be relatively unimportant; in Lithium, however, this range corresponds precisely
to the inter-band transitions that can be driven by intensity noise. The problem of excess output
noise can be addressed by introducing a low-pass filter between the feedback controller and theAOM driver at a frequency corresponding to the time-delay-limited bandwidth, but this solution
is not ideal, as it limits the feedback bandwidth even more. Further efforts to build a feedback
controller with lower output noise therefore remain a priority.

Alignment procedures
In order to ensure that spurious forces experienced by the atoms are minimized and to provide for
adiabatic hand-off between the ODT and the lattice, the ODT beams must be aligned with thesaddle point of the Feshbach field, and the lattice beams must be aligned with the crossed ODT.
To that end, the following procedure has been developed:

1. After making a BEC in the crossed ODT, ramp down one of the ODT beams adiabatically.
The BEC will then be confined by the combined potentials of the remaining ODTl and thehorizontal curvature of the magnetic field (this only works if the field is confining in the XY
plane and anti-confining in Z). Draw a line perpendicular to the ODT which passes throughthe center of the cloud (see figure 3.19a).
2. Repeat with the other ODT beam. Move the ODTs to that position where the two center-lines
cross, and confirm that the untrapped cloud experiences no horizontal forces.
3. Load a BEC in the ODT beams. Snap off both ODT beams and pulse on one of the threelattice beams at high power for a short time (at most 1/4 ofthe radial trap period in the latticebeam). After some time of flight, image the cloud along the beam axis. Its displacement fromthe trap center will be a dispersive function of the distance between the ODT and latticebeam centers, which close to perfect alignment displays linear sensitivity (see figure 3.19b).
88

T1
OD

OD
T2

(b)
Position in Time of Flight(a)

Beam Offset

Figure 3.19: (a) Composite of the two-step alignment procedure to align the ODT beams with themagnetic field saddle point. The thin solid circles represent equipotential lines of the magnetic fieldin the XY-plane. (b) The position of the cloud in time of flight, after being kicked by a latticebeam which was offset along one axis from the ODT position.
The cloud displacement also scales linearly with the time of flight, which allows this method
to magnify small offsets that would be difficult to resolve in situ.
4. Adjust the lattice beam until the kicked cloud remains at the same position in time of flight.
Repeat with the other two lattice beams.
5. Align the retro-reflected beams by maximizing the back-coupled efficiency through the latticefibers.
This method gives a lattice depth within a few percent of the expected value, given the measured
beam power and waist size, suggesting that the alignment is procedure is highly successful. A temperature stabilized enclosure surrounding the experiment (±0.1◦ C) ensures the alignment procedure
only has to be repeated every few weeks.
Lattice calibration
The noise-driven processes discussed in section 3.3 can also be used intentionally to calibrate thelattice depth. When the lattice depth is modulated at a single frequency ωmod , peaks will beobserved in the heating rate when ωmod = U/h̄, as discussed in section 3.3.1, but also at ωmod =ω1→3 (see figure ). The lattice depth can then be backed out of the band structure calculation ofsection 3.2.1, with a typical precision of 1%.

89

Atom Number in ODT

2

10 5

1.5

1

0.5

0
300

320

340

360

380

400

420

440

460

480

500

Lattice Modulation Frequency (kHz)

Figure 3.20: A near-degenerate thermal cloud is loaded into a lattice whose depth is modulated by
3% peak-to-peak for 30 ms. The number of atoms remaining is a function of modulation frequency,
where the highest frequency response is associated with transitions between the ground and second
excited bands at the center of the lattice (dashed line). The width of the feature is dominated bytransitions between other bands, which take place at lower frequencies, and transitions at positions
far from the trap center where the lattice is weaker. In a lattice made from 125 micron beams, thelattice depth varies by only 1% over ±20 sites.

90

Chapter 4

Towards Simulations of Spin-1
Heisenberg Chains

91

4.1

Spin Models

One of the goals of the experimental platform described in this thesis is to provide a simple, tuneable
analog system for Heisenberg models. These models share, as a defining feature, competition
between nearest-neighbor exchange interactions along different axes or between these exchangeinteractions and other local terms. They provide insight into the behaviors of real materials, buteven in the simplified context of toy models these behaviors can be difficult to compute, especiallyfor large numbers of strongly interacting particles. The resulting systems contain not only states
whose long range magnetic order is significantly influenced by the quantum nature of the spins
themselves, such as xy-antiferromagnets, but also states displaying symmetry-protected topologicalorder (SPT order) with robust, non-local correlations, such as the Haldane phase. This chapter
describes progress towards simulating the dynamics and ground states of a spin-1 model that
contains both of these features.
In 2006, Duan et al. showed that by matching up the matrix elements of a magnetic Hamiltonian and the matrix elements of the perturbative atomic Hubbard model (including terms up tosecond order in t/U ), it is possible to define a mapping between the two [78]. This mapping shows
how the physical pseudospin configuration within the lowest Bloch band of an optical lattice isanalogous to idealized quantum spins on a lattice. The model with a single fermion per site, corresponding to spin-1/2, has been studied extensively in fermionic regimes; recent successes include
observation of 3D and 2D Néel ordering in fermions [11, 79], measurements of spin correlations in
2D spin-imbalanced systems [80] and spin-charge correlations in the presence of hole doping [16].
The bosonic system, however, which features the anisotropic exchange couplings and integer-spin
models, remains unexplored.

4.1.1

Effective Hamiltonians and the Two-component Bose-Hubbard Model

While this chapter will focus mostly on implementing a spin-1 model, the spin-1/2 case provides a
straightforward context in which to introduce the mapping and elucidates a few important differences between bosonic and fermionic models, while also providing a useful point of comparison forthe spin-1 model. This section therefore begins by reviewing the spin-1/2 mapping, and then goeson to suggest how it can be extended to spin-1 [81].
92Spin-1/2

One of the simplest descriptions of a spin-1/2 system is the anisotropic Heisenberg model in an
external magnetic field, which has the Hamiltonian

Hspin =

∑(

)
∑
Siz
Jz Siz Sjz + Jxy (Six Sjx + Siy Sjy ) − hz

⟨i,j⟩

(4.1)

i

where Jz and Jxy are the longitudinal and transverse exchange couplings, respectively, and hz isthe magnetic field. To see how this model can be implemented with cold atoms, consider the
Bose-Hubbard model (3.16), which, when modified to describe a lattice of atoms with two possible
internal states, readsH=∑ (⟨i,j⟩,σ|) 1∑∑
Uσσ niσ (niσ − 1) +tσ a†iσ ajσ + h.c. +U↑↓ ni↑ ni↓
2
i,σ
i
{z
}
{z
} |

(4.2)

H0

V

for σ = [↑, ↓]. The use of pseudospin indices is intended to recall the mapping, first introduced
in section 1.2.2, between spins and hyperfine states of atoms. With two different hyperfine states,
a composite spin-n/2 model can be constructed out of n atoms per site. The MI ground state of
H0 provides plateaus of integer occupancy n, which constitutes a platform for any dynamics orordering that may occur when spins are added to the system. In the limit of small t/U , as insection 3.2.2, one may treat the tunneling portion of the Hamiltonian perturbatively. With one
atom per site (spin-1/2), the model has large manifold of degenerate ground states, which are all
the possible configurations of pseudospins in the lattice. At finite t, however, the perturbation V
will lift this degeneracy. As in the single-component MI, there are no first-order corrections to theenergies. Following Complement BI of [82], the matrix elements of the effective Hamiltonian for
perturbation V , to second order, are given by1∑⟨m| Heff |n⟩ ≈ En δnm + ⟨m| V |n⟩ +⟨m| V |k⟩
2
k

(

1
1
+
Em − Ek
En − Ek)⟨k| V |n⟩ ,(4.3)

which can then be compared to the matrix elements of ⟨m| Hspin |n⟩, where the states ⟨m| and |n⟩are the different positional configurations of nearest-neighbor spins/hyperfine states. This process93

leads to four equations which can be solved simultaneously to find the coefficients of the spin
Hamiltonian in terms of the coefficients of the effective atomic Hamiltonian:Jz =t2↑ + t2 ↓
2U↑↓

t2↑

t2↓

t↑ t↓
−
−
, Jxy = ±, and hz = z
U↑↑ U↓↓
U↑↓

(

t2↓
U↓↓

−

t2↑
U↑↑

)
,

(4.4)

where z is again the coordination number of the lattice, and the −(+) sign in Jxy is for bosons(fermions), due to the parity of the wavefunction with respect to exchange of particles in each case.
The term hz is a magnetic field in the ẑ direction in the spin model (not the real magnetic field B),
which in this cold atoms system is fixed by the net magnetization M = N↑ − N↓ and is therefore
identically zero when the system contains the same number of a atoms and b atoms. There is also a
total constant energy offset equal to ϵ = −(t2↑ + t2↓ )/(2U↑↓ ) − t2↑ /U↑↑ − t2↓ /U↓↓ , which can be dropped
from the Hamiltonian without loss of generality.
For fermions, U↑↑ and U↓↓ are infinite, due to the Pauli exclusion principle, so Jz = Jxy , withpositive U↑↓ favoring anti-ferromagnetic order in both z and xy. The system therefore implements
an isotropic spin-1/2 Heisenberg model, which can be switched between ferromagnetic and antiferromagnetic order by switching the sign of U↑↓ via, for example, a Feshbach resonance. The story
is quite different for bosons, however: U↑↑ ̸= U↓↓ in general, leading to competition between theJz and Jxy terms and the reintroduction of the bias field hz . At the same time, the overall sign of
Jxy is switched, so that negative U↑↓ favors anti-ferromagnetic coupling in the transverse direction.
Varying the three interactions relative to one another therefore is expected to realize a wide varietyof phases. One can gain some insight into the different ground state configurations by considering
the spin Hamiltonian acting on trial states that are the classical magnetic ground states, as has
been done by Altman et al. [81], but it turns out that applying this mean-field picture to the spin-1
configuration reveals only a part of the story.
Spin-1
With two atoms per site and two hyperfine states a and b, the three possible configurations,√|+1⟩ = |aa⟩, |0⟩ = (|ab⟩ + |ba⟩)/ 2 and |−1⟩ = |bb⟩, form a spin triplet manifold on each lattice√site.1 These are unique to bosons; fermions would instead form the spin singlet (|ab⟩ − |ba⟩)/ 2.
1
Hereafter, for compactness, these three states will be referred to as |aa⟩, |ab⟩, and |bb⟩, using the Fock basis
notation.

94

|bbi
ωbb→ab√1 (|abi + |bai)2|aai

ωaa→abB

Figure 4.1: The three different combinations of two hyperfine states on a lattice site have interactionenergies Uaa , Uab , and Ubb which can be tuned via Feshbach resonances. At a given field B,
the splittings will in general be unequal, giving rise to an interaction blockade: the two possible
transitions can be individually addressed by an RF drive with frequency ω = ωZeeman + ∆U/h̄.
As in the spin-1/2 system, the nearest neighbor interactions arise from super-exchange. Becauseof Feshbach resonances, Uaa ̸= Ubb ̸= Uab in general:they are functions of the applied magnetic
field B so that one can tune the configurational energies of different distributions of spins in thelattice (Fig. 4.1). The appearance of a spin triplet is already quite suggestive of the presence ofan analog to a spin-1 model. It is helpful to first zero the energy of the |0⟩ state by subtracting
an overall energy offset U↑↓ , and then to subtract the energy due to a fictitious magnetic fieldhz = (U↑↑ − U↓↓ )/2, which affects only the |±1⟩ states and sets their energies to ±D. As before,
this fictitious magnetic field has no effect within a subspace of fixed magnetization, determinedby the hyperfine state populations. One may then proceed as in the spin-1/2 case by finding thematrix elements of the effective atomic Hamiltonian (see figure 4.2) and collecting terms with thesame spin operators, which results in

H=J∑

Si · Sj + D

⟨i,j⟩

∑

(Siz )2 + O(δ 2 ) .

(4.5)

i

The correction terms of order δ 2 come from the normalized difference between the intra-spin interactions δ = (U↑↑ − U↓↓ )/(U↑↑ + U↓↓ ), and affect only states of the form |±1⟩i |0⟩i+1 . The terms oforder δ have meanwhile been removed by the previous introduction of the fictitious magnetic field,
which can now be written as hz = −zJ · 3δ/2, and which provides, as usual, just a constant energy
95Diagonal

Off-Diagonal

Figure 4.2: Pictorial representation of the calculation of two example matrix elements of the effective
Hamiltonian– one diagonal and one off-diagonal– to second order in t/U . The tunneling is presumed
to be uniform, i.e. there is no state-dependent lattice potential.
offset while constrained to a subspace of constant magnetization. The coefficients themselves takethe forms
J = −4U↑↑ − U↓↓t2
, and D =
− U↑↓
U
2

(4.6)

for an effective U which is the harmonic interactions of the two intra-spin interactionsU≡

2.
1/U↑↑ + 1/U ↓↓(4.7)

While the spin exchange term in the approximate Hamiltonian (4.5) is again simply the unadorned
Heisenberg model, a number of differences from the spin-1/2 case are immediately clear. In thebosonic spin-1/2 system, the model has an exchange coupling anisotropy which depends on the
difference in intra-state interactions, while the exchange coupling in the spin-1 model is isotropic
regardless of the interactions. On the other hand, in the spin-1/2 system, the ground state isdegenerate at t/U = 0, whereas in the spin-1 system the states already have different energies,
which leads to the local spin-squeezing term (Siz )2 in the Hamiltonian.2 Further, the exchangeinteraction in the spin-1/2 model depends only on U↑↓ , while the exchange interaction in the spin-1
model does not depend on U↑↓ , but only on U↑↑ and U↓↓ , and additionally picks up a factor of four
due to having two bosons per site. Finally, the mapping to spin-1/2 is exact, while the mapping tospin-1 is exact only up to first order in δ, and the form of the corrections may be important when2
The spin-squeezing term is sometimes referred to in condensed matter literature as single-ion anisotropy, uniaxial
large ion anisotropy, or easy-axis anisotropy.

96considering the symmetries of the Hamiltonian.

Ground state phase diagram

This Hamiltonian describes the competition between an exchange term Si · Sj , which favors correlations between sites, and a spin-squeezing term |Siz |2 , which favors only local (on-site) pairing.
When D is large and positive, the |0⟩ state is strongly favored everywhere, leading to a groundstate, in the limit D → ∞, which can be written as
|Ψ⟩ =

∏

|0⟩i ,(4.8)

iwhere |0⟩ indicates the spin zero state of two atoms (and not a zero-occupancy site). This productstate of doublons, or Spin Mott, is highly reminiscent of the MI ground state (3.24), and in thecontext of magnetic ordering behaves like a Z-paramagnet. When D is large and negative, however,the spins are constrained to be |±1⟩ on each site, and any residual exchange coupling leads to Isingtype interactions. Finally, the Hamiltonian has a special point defined byU↑↑ + U↓↓ = 2U↑↓ ,(4.9)

which is the point at which D = 0 and the super-exchange process |1⟩i |−1⟩i+1 ⇔ |0⟩i |0⟩i+1 becomes
resonant. At this degeneracy point (DGP), what remains is an isotropic Heisenberg Hamiltonian,and the ground state then depends on the total spin, the dimensionality of the lattice, and on thelattice coordination number z. Further, whenD/|J| approaches 1 from the positive side, one might
expect competition between the terms of the Hamiltonian to lead to a phase transition between
the uncorrelated state (4.8) and a state that displays some sort of exchange-mediated ordering.
In fact, spin-1 chains of this sort have been the topic of significant study in the context ofcondensed matter systems ever since Haldane’s 1983 prediction that the ground state of 1D chainsof integer spins is gapped, contrary to the gapless spectrum of spin-wave excitations on an XYantiferromagnetic phase that a mean field phase diagram would suggest [12]. Importantly, Afflek
97

et al. showed that the larger class of 1D spin Hamiltonians

H=

∑[
]
Si · Si+1 − β(Si · Si+1 )2(4.10)

iexhibit the same symmetries as the Heisenberg Hamiltonian, and furthermore that 4.10 has anexactly solvable ground state for the case of β = 1/3 (the Heisenberg Hamiltonian is recovered
for the case of β = 0) [83]. The construction of this so-called valence bond state arises fromthe realization that each spin-1 site can be viewed as the symmetric combination of two spin-1/2
particles on the same site (in the case of the model implemented in this work, this configuration
exists in real space, but in the original theoretical work it was simply a construction). These
constituent spin-1/2 particles can form a spin singlet with neighboring spin-1/2 particles |↑⟩i |↓⟩i+1 −|↓⟩i |↑⟩i+1 . The symmetric combination of the two spin-1/2 particles on each site restores the total
spin on each site to S=1. This construction makes the origin of the gap very clear (the excitations
are spin singlets, not spin waves), but also brings to light another interesting feature of the model:
in an open chain, there will be un-paired spin-1/2 particles at the ends, which gives rise to a fourfold degenerate ground state. In their second paper on the topic, Afflek et al. showed that thesefour ground states consist of a spin triplet and a spin singlet, the latter of which is a fundamentally
non-local state [84, 85]. Several condensed matter groups measured the presence of a Haldane
gap in crystalline materials (see [86] for a partial list), but it is only quite recently that a fuller
understanding of the Haldane ground state has been developed. The ground state of the isotropic
spin-1 Heisenberg model is one of the simplest examples of a Symmetry Protected Topological(SPT) phase [87, 88]. These phases feature order which is protected by some symmetry of the
Hamiltonian, in contrast to intrinsic topological order, which is robust even in the presence ofany local perturbations. It was originally proposed that the Haldane phase in spin-1 chains wasprotected by inversion symmetry [89]. More recently, Gu and Wen pointed out that violating any
one of inversion, translation or time-reversal symmetry will lead to a decay of the string-orderedcorrelations that characterize the Haldane phase [90].
The model Hamiltonian (4.5) features a competition between the Heisenberg exchange term
and a spin squeezing term, leading to a first-order phase transition between the spin-squeezed
paramagnetic phase and the Haldane phase. Sakai and Takahashi showed that the Haldane phase98

Figure 4.3: The valence bond ground state. Symmeterized spin-1 lattice sites are represented by thedashed ovals. The solid lines represent the “valence bonds” completing the off-site anti-symmetricspin singlets |↑↓⟩ − |↓↑⟩. Illustration based on [84].
persists in quasi-1D– that is, in a lattice of chains with weak inter-chain coupling [91]. Pollman
et al. argue, however, that the Haldane phase really only exists for odd-integer spins in 1D (andnot just integer spins, as Haldane originally conjectured), and correspondingly in the quasi-1D
situation, the number of coupled tubes must be odd (e.g. no spin ladders) [13]. Similarly, Gong etal. argue that the Haldane phase persists in the presence of weakly anisotropic spin coupling [92].
A simulator of the sort described in this work, with tunable parameters and access to a selectionof ground states in the phase diagram, ought to add substantially to the data which have beengleaned from condensed matter systems.
Néel Order

0.032

/J

0.016

XY-AFM

J

Z-AFM

0.008

Haldane

0
-1

SM

0

1

2

D/J

Figure 4.4: Cartoon of the phase diagram of the quasi-1D spin-1 chain with antiferromagnetic
exchange interactions, as a function of the spin-squeezing parameter D/J and the inter-chain coupling strength J⊥ /J. In isolated chains, the system features only three phases: the Spin Mott, theHaldane phase and the Ising-type z-antiferromagnet. As weak inter-chain coupling is re-introduced,
the system gains another phase that competes with, and eventually overwhelms, the Haldane phase.
This phase is the quantum xy-antiferromagnet (XY-AFM). While the point on the D/J axis where
the Haldane and SM phases touch is a first order phase transition, the SM to XY-AFM transition
is a mean field transition. This figure is based on work by Wierschem and Sengupta [86, 93] andSakai and Takahashi [91].
994.1.2Ingredients for an atomic spin-1 model

The ability to implement a spin-1 model with neutral cold atoms in optical lattices thus has anumber of specific requirements, especially if one is interested in accessing the (arguably moreinteresting) anti-ferromagnetic exchange couplings. First, one needs a two-component system whichforms a spin triplet on each site, which implies a bosonic atom with two hyperfine states. The two
hyperfine states should be stable against spin-changing collisions, which strongly favors stretchedor hyperfine ground states. There should be some way to vary the interactions between these
two states, in a regime which includes (1) all on-site interactions large enough to be in the MI
at reasonable lattice depth and (2) differences between the interaction energies of comparable
magnitude, ideally the same at one point. The exchange energy J must be positive to achieveanti-ferromagnetic coupling in bosons, for which it is a necessary (but not sufficient) condition
that one or both of U↑↓ and U↓↓ be negative. The magnitude of the rate J/h̄ should also exceed
the decoherence rate in the lattice, so that many super-exchange events may take place before
the introduction of excitations that break up the spin chain and limit the correlation lengths ofmagnetically ordered states. And finally, there should be some way to prepare one of the groundstates in the spinful phase diagram from a ground state in the single-component system.
Given these requirements, 7 Li presents a number of unique advantages compared to other atomic
systems, but also several challenges. The two lowest hyperfine states have a unique spectrum of
Feshbach resonances that includes a double-resonance in the b state (see figure 4.9b). This particular
structure provides for a unique opportunity to scan Ubb over a wide range while Uaa > Uab remain
almost constant, which provides for the variation of D/J between ±∞, as will be discussed ingreater detail in section 4.3.2. In the same region of magnetic field, all the relevant scattering
lengths are negative, providing for anti-ferromagnetic interactions. While the integer-occupancy
density distribution of the MI with this configuration is the actually most excited Hubbard band,
it is meta-stable in a 3D optical lattice for the simple reason that atoms which would tunnel toneighboring sites have no way to dispose of the potential energy they would lose; the lifetime of
the attractive MI is therefore not detectably shorter than the lifetime of the normal repulsive MI,
as has been observed in two-component MIs in this experiment and previously in single-component
137 Cs [94].

The magnitudes of the scattering lengths are also important, as they fix the depth of
100

40
aaa

20

aab
abb

a/a 0

0

I

-20

II

III

-40
-60
-80
50

55

60

65

70

75

80

B (G)

Figure 4.5: Scattering lengths of 39 K around the low field bb resonance. Regions II and III exhibit
degeneracy points for an n=2 system with a and b atoms. Feshbach resonance data for 39 K is takenfrom [95].

the lattice at which the experiment must be performed, to which the parameter J is exponentiallysensitive: the MI transition happens at fixed t/U , so J ∼ U · (t/U )2 ∼ U · const. Fortunately, themagnitudes of the scattering lengths in the lowest two hyperfine states of 7 Li are all at least 50a0 .
The super-exchange rate is also benefited by the light mass of Li through U ∼ m−1 . Lastly, at thehigh bias field where the Feshbach resonances are found, the Zeeman detunings of the two states
also have similar slopes, which grants a degree of magnetic field insensitivity to the RF-transitions
necessary for initialization and readout (µrel ≈ 0.02 µB ). On the other hand, the Feshbach resonancethat allows for the tuning of D/J is rather narrow, so that precision control of the bias field is stillrequired (to better than one part in 105 ). Furthermore, 7 Li ’s light mass and fast tunneling areaccompanied by high three-body loss rates, especially near Feshbach resonances. How these effects
balance each other will be the subject of section 4.3.3.
It is worth considering what other atomic systems might afford an opportunity to explore thismodel, and what portion of the parameter space would be available to them. To this author’s
knowledge, the sort of Feshbach resonance structure that affords both anti-ferromagnetic coupling
and the ability to scan D/|J| through the region containing the phases depicted by figure 4.4 exists
elsewhere in bosonic alkalis only in the lowest two hyperfine states of 39 K (figure 4.5). The relevant
scattering lengths in this region are, however, more than a factor of two smaller than those in101

7 Li , which decreases the maximum achievable super-exchange coupling by the same factor.

The

heavier mass of 39 K reduces the super-exchange even further. On the other hand, the relevant
Feshbach resonance structure exists at substantially lower magnetic field and the slope of thetuneable interaction (again bb) is shallower, requiring much less dynamic range for magnetic field
feedback. Ultimately, the most important concern would be the two- and three-body loss rates
near the bb Feshbach resonance, for which published results are not yet available. While it ispossible that J/(hΓloss ) would be promising, a definite determination requires further study. One
might also attempt such a simulation in atoms which have no Feshbach resonance, rather using
spin-dependent lattice to vary Uab . These atoms would universally have U > 0 for all possible
interactions if they can form a stable spin mixture (e.g. 87 Rb), and so in order to achieve antiferromagnetic exchange coupling, one could reverse the sign of super-exchange by tilting the systemwith an optical gradient so that J = t2 /U → t2 /(U +∆)+t2 /(U −∆) = 2t2 U/(U 2 −∆2 ) [23]. In thecase of 87 Rb, Uaa ≈ Ubb ≈ .986 Uab , and beacuse the spin-dependent lattice reduces Uab but leaves
the intra-species interactions unchanged, D/|J| can actually access the entire range of parameters
necessary to realize the phase diagram of figure 4.4.

Attempts have also been made to model spin-1 Heisenberg spin chains using trapped ions[96]. The JQI group has recently implemented a spin-1 simulator with short chains of ions (upto n=4) that models a version of the anisotropic Heisenberg Hamiltonian with spin squeezing andlong-range interactions (rather than nearest-neighbor). The addition of long range interactions
frustrates the XY-order that might otherwise arise in 1D in the presence of anisotropic exchange
coupling, although it is predicted that the Haldane phase persists if the coupling is weak enough[97]. However, the reduced dimensionality of the system prevents adiabatic preparation of the
Haldane phase from the large-D phase (Spin Mott): by definition, a state with symmetry protected
topological order is not adiabatically transformable into a topologically trivial state– that is, aproduct state like the large-D phase. It remains an open question, at least experimentally, whether
an xy-antiferromagnetic phase can be adiabatically transformed into the Haldane phase in thepresence of suitable symmetry-breaking perturbations– a question which future work on this 7 Li-based quantum simulator may help to answer.
102

4.2

Interactions in 7 Li

In order to find out the appropriate magnetic fields which implement various parts of the phase
diagram of the anti-ferromagnetic spin-1 Heisenberg chain, it is necessary to know the on-site
interactions in 7 Li to better than the scale of J, which ranges from a few hundred Hertz to akilohertz, depending on dimensionality. Before the work discussed in this thesis, the Feshbachresonances in 7 Li had not been characterized so precisely, and what is more, the crucial ab resonance
had never been measured at all. This following section is a discussion of direct measurements of
the on-site interaction energies Uaa , Uab and Ubb , which constitute the direct inputs to the manybody Hamiltonian 4.5, by applying a technique which has not previously been used to characterize
Feshbach resonances [36]. The use of an optical lattice is crucial for stabilizing the density of theground state two-component mixture at a variety of combinations of scattering lengths, including
when all or some of these scattering lengths are negative or have vastly different magnitudes.

4.2.1

Interactions on a Lattice Site

Section 3.2.1 derived the on-site interaction energy U from the overlap integral of two Wannierfunctions, which is valid in the limit of U ≪ ωbg , so that higher bands may be neglected. While thislimit is satisfied for the scattering lengths used in the lattice under normal experimental conditions,Feshabch resonance spectroscopy involves some measurements at relatively high scattering length(up to 250 a0 ).

The Busch modelBusch et al. showed that there is an exact analytic solution for the energies in the case of two
atoms with contact interactions occupying a spherical harmonic trap [98]. The Hamiltonian canbe diagonalized when the interactions are expressed in terms of the regularized delta-function
δreg (r) ≡ δ (3) (r)(∂/∂r)r, where r is the relative coordinate of motion. The modified eigenenergies
obtained from this process, plotted in figure 4.6, can be expressed by
1 Γ(−E/2 + 1/4)a= √.
2 Γ(−E/2 + 3/4)
103

(4.11)

10
8
6
4
2
0
-2
-4
-6
-8
-8

-6

-4

-2

0

2

4

6

8

Figure 4.6: Exact solution to the energy of two interacting atoms in a spherical harmonic trap withtrap frequency ω, as a function of the scattering length a. The dashed lines are the asymptotic
energies reached as a → ±∞ at a resonance.
This result beautifully captures several salient aspects of the system of two confined, interacting
particles. First, the energy shift experienced due to attractive (repulsuve) interactions saturates at+(−)h̄ω, which is to say the magnitude of an effective scattering length a saturates at the harmonic√
oscillator length losc ≡ h̄/(mω). Second, passing through a resonance from the +∞ side promotes
the atoms from oscillator level n to level n + 1, and vice versa for passage in the opposite direction.
Finally, the calculation illustrates the formation of a bound molecular state of the two atoms in
the lowest state for a > 0, shifted upwards by the ground state trap energy 1/2h̄ω.

Interaction Blockades
Much of the work that follows will make use of the fact that interactions in a many-body system
split the degeneracy in the excited state, shifting the energy levels of an otherwise a harmonicallyspaced system by the difference in the interaction energies in each state. These shifts, or interaction
blockades, have been observed in a variety of quantum systems, including neutral atoms in thecontext of RF spectroscopy of MI shells [99] and inter-band transitions in a optical lattice [100],and in Rydberg atoms, where the so-called “dipole blockade” has been used to implement a varietyof engineered interactions and quantum gates [21, 101, 102]. In an n particle system, this situation104

is further characterized by a collective enhancement of the Rabi frequency, e.g. by√

n for thetransition between the ground state and the state with one quantum of excitation. In the simplest
case of two particles, each in either states |a⟩ or |b⟩, this effect makes the transition from state to
the superposition state |ab⟩ + |ba⟩ addressable independently of the transition to the fully excitedstate |bb⟩ (see figure 4.7). Such interaction blockades will play an important role not only inspectroscopically characterizing the interactions of 7 Li and the distributions of occupancies in the
optical lattice, but also in coherent control of the hyperfine states in an occupancy-dependent way.

Non-Degenerate

DegenerateState Occupany(2-atom Basis)

1.0

aa
ab + ba
bb

0.8
0.6
0.4
0.2

State Occupancy(1-atom Basis)

0
2.0

a
b

1.5
1.0
0.5
0

0

2

4

6

8

10 0

time (a.u.)

2

4

6

8

10

time (a.u.)

Figure 4.7: Rabi oscillations of two interacting bosons. The left panels depict the case where theinteractions in all three possible configurations are degenerate, whereas the right panels depict thecase where the drive is resonant only with the |bb⟩ → |ab⟩ transition. The upper panels show the
expectation value of the state occupancy in the basis of the two-atom states, while the bottom
panels show the total detection probabilities√ for atoms in each hyperfine state. The non-degeneratecase exhibits half the contrast but a 2 enhancement of the Rabi frequency compared to thedegenerate case, which is typical of an interaction blockade effect.

4.2.2

Interaction Spectroscopy

In order to measure the differential interaction energy of two confined atoms as a function ofmagnetic field, the system is prepared in an n = 2 Mott insulator in the b hyperfine state. The
number of atoms is 1 × 105 and the lattice depth is 35 ER in each dimension. The central n = 2
plateau, approximately 3 × 104 sites, is surrounded by an n = 1 shell containing a similar number
105

Atom Number (a.u.)

Magnetic Field

n=1

n=2

847.0 G
847.4 G
847.7 G
-6

-2
0
-4
RF Detuning (kHz)848.1 G2

848.3 G
848.5 G849.5 G
849.9 G
850.9 G
861.0 G
868.1 G
871.1 G
874.6 G
877.7 G
881.5 G

-80

-60

-40

-20

0

20

40

60

80

RF detuning (kHz)

Figure 4.8: Fits to representative spectra of the transition between b and a, taken between the two
bb Feshbach resonances. The spectra have been plotted so that the Zeeman-shifted peaks of then = 1 transitions overlap. The inset shows an example spectrum in which each point is an averageof four measurements and the fit is a sum of two Gaussians. The crossing of the frequencies of thetwo peaks corresponds to Uab − Ubb = 0.
of sites. The RF drive is pulsed for 2.9 ms and monitor the number of atoms in the other hyperfine
state as a function of drive frequency. At the frequency corresponding to the transition of a bareatom, a peak is observed coming from the atoms on n = 1 sites. A second peak is observed comingfrom the n = 2 atoms, which is shifted by the difference in interaction energy between the initialstate (|aa⟩ or |bb⟩) and the final state (|ab⟩). The pulse length corresponds to a π-pulse for the√n = 2 sites (so that the signal is maximized), which have a Rabi frequency 2 greater than thatof the n = 1 sites, due to bosonic enhancement. The interaction blockade that arises from unequalinteraction energies in the three states means that one may drive the system selectively between|aa⟩ and |ab⟩ (or |bb⟩ and |ab⟩). Thus when the atoms are probed absorptively after an RF pulse
with light that is resonant only for a (or b), a single flipped atom per n = 2 site is measured (seethe inset of figure 4.8).
The frequency of the n = 1 peak corresponds to the Zeeman shift and thus to the magnitudeof the applied magnetic field (the hyperfine constant and nuclear g-factor for 7 Li are taken from[103]). The frequency shift of the n = 2 peak, which may be positive or negative, is a direct measureof the differential two-body interactions. Using this technique, RF spectra are obtained at many106

selected bias fields from which the two-body interaction splittings Ubb − Uab and Uab − Uaa (Fig.
4.8) can be derived. The technique works equally well for attractive and repulsive interactions, solong as the system remains in the Mott insulating state.
The precision to which the differential interaction energies on a site must be determined is set
by the super-exchange rate, which at Mott insulator depths for 7 Li in a 1064 nm optical lattice
ranges from hundreds of hertz to several kilohertz, depending on lattice depth and dimensionality.
Here the differential interaction energies are measured as a function of magnetic field to a precisionof about 100 Hertz, limited only by the stability of the magnetic field (about one part in 105 at
103 G) and the sensitivity to magnetic fields of the RF-transition between the two hyperfine states(approximately 33 kHz/G at high field). Each n=2 spin flip data point includes an independentcalibration of the magnetic field via the n=1 spin flip. Each of these measurements includes 100
points, sampled 3x each. At 10s/point, this is about an hour of data-taking during which themagnetic field can drift. However, the data used to determine the interaction shift at a certain
field are taken out-of-order, and so the field drift is expected only to broaden the resonances, notshift their positions. Indeed, the n=1 resonances approach Gaussians with standard deviation of
about .5-.7kHz. Therefore, an upper bound on the field noise is a Gaussian distribution with awidth of 0.7kHz/(33kHz/Gauss)=20mG. This value is very much an upper bound, as (1) it does
not take into account the intrinsic width of the spin-flip resonances due to finite probe time and (2)the dense sampling of the resonance means drift should not significantly affect the positions of theresonances, but rather their widths. As the interaction energies range over many tens of kilohertz,
the error bars are too small to see on a full scale plot (Fig. 4.9a). The technique of RF interaction
spectroscopy is particularly well-suited to atoms with characteristically large interaction energies,
such as Li, because the wide separation between singlon and doublon spin-flip resonances permits
the use of high Rabi frequencies, which maximizes signal size and decreases sensitivity to magnetic
field noise.
Amplitude modulation spectroscopy
While RF spectroscopy in a lattice is a powerful and precise tool for characterizing differential
interactions, another technique is necessary to measure the absolute interaction energy (i.e. Uaa
or Ubb ), for these interactions determine the lattice depth for the transition to the Mott insulator
107Interaction energy (kHz)

50

(a)

Uab - U aa
Ubb - U ab

0

-50
400

(b)

aaa

a/a 0

200

aab
abb

0
-200
-400

650

700

750

800

850

900

950

Feshbach field (G)Figure 4.9: (a) The relative interactions Uab − Uaa and Ubb − Uab are plotted as a function ofmagnetic field, measured via RF interaction spectroscopy in 35, 35, 35 ER optical lattice. (b) The
scattering lengths of the aa and bb interactions are plotted as a function of magnetic field, measured
using lattice AM and shown in units of the Bohr radius a0 . Also shown as a bold dashed line isthe ab scattering length, obtained using simultaneous hyperbolic fits to the RF spectroscopy andlattice AM data sets. The fits are shown as solid black lines. Dotted vertical lines indicate theposition of a resonance.
in each hyperfine state. In previous studies, lattice amplitude modulation (AM) has been used
to drive singlon-to-doublon conversion in the lowest Hubbard band, and the resonant frequencyof this process has been associated with the on-site interaction energy in both repulsive [105] andattractive [94] single-component bosonic systems. Here the same technique is employed to map theon-site intra-species interactions of 7 Li across a broad range of magnetic fields. The lattice depth
is modulated by 30% peak-to-peak along the shallow dimension of a 35, 35, 20 ER optical lattice
and the entropy added to the system is found by adiabatically ramping back to the BEC from theMott insulator and measuring the recondensed fraction. While the system must remain in the Mott
insulator in regions of magnetic field where the scattering length is negative in order to preventthe collapse of the atomic cloud, data taken above but close to the transition for either attractive
or repulsive interactions display a bias towards higher frequency [105]. While this systematic
bias prevents the collection of data for very small scattering lengths, limiting the accuracy of ourdetermination of the background scattering length in each channel, the locations of the resonances
themselves are not significantly affected.
108

Channelabg /a0∆ (G)Bres (G)RF data only

Bres (G)Combined fit

Bres (G)Previous works

aa-25.8(1.2)-135.9(6.9)

737.58(10)738.29(15)738.2(2)[34]
737.8(2)[104]736.97(7)[33]

ab
bb
bb

-29.8(1.3)-23.0(1.4)-23.0(1.4)-90.5(4.0)-14.9(0.9)-172.7(10.0)

794.64(07)845.42(01)893.34(12)794.59(12)845.45(02)893.84(18)844.9(8)[34]
893.7(4)[34]

Table 4.1: Parameters of the Feshbach resonances in the lowest two hyperfine states of 7 Li, determined with a simultaneous fit to RF interaction and lattice AM spectroscopy data (except where
specified otherwise). The reported errors in our measurements are 1σ statistical uncertainties in
the fit parameters. The AM data slightly bias the resonance positions derived from combinedfits towards higher magnetic fields, possibly due to unaccounted-for systematics. As the RF data
are influenced only by two-body, on-site effects rather than by many-body physics, the resonancepositions are likely captured more accurately by the RF data alone.
Extraction of the Feshbach resonance parameters
A simultaneous fit to both the RF differential interaction spectroscopy data and the lattice AM
spectroscopy data determines the inter-species interaction energy Uab , and extracts the parametersof the Feshbach resonances (see Fig. 4.9b). Feshbach resonances are dispersive features and can beparamatarized as hyperbolae(
as = abg

1−

)

∑

∆(i)

i

B − Bres(4.12)(i)(i)

with a background scattering length abg , width ∆(i) , and resonance location Bres [35, 106]. In orderto provide a useful comparison with existing literature, the scattering lengths are calculated using(3.20):
14πh̄2 1=asm U∫
|ψ(r)|4 d3 r

(4.13)

where U is the measured on-site interaction energy and ψ(r) is the calculated Wannier wavefunction
on a site, given calibration of the lattice depth by inter-band parametric excitation. This approximation for ψ systematically biases the scattering lengths towards lower values because interactions
modify the actual two-particle wavefunction by admixing higher bands, but the correction is not√
significant as long as |as |/aHO ≪ V 1/4 / 2π where aHO is the harmonic oscillator length on a site
and V is the lattice depth in recoil units [107].3 The complete Feshbach resonance spectrum for3
While the moderate scattering lengths considered here remain small enough that this approximation does notmeaningfully affect the parameters of the hyperbolic fits, a more precise method would be to use the Busch solutions

109

the lowest two hyperfine states of 7 Li is plotted in (Fig. 4.9b) and the parameters of the resonances
can be found in Table 4.1.
This technique confirms the presence of a single resonance in the lowest hyperfine state, whose
position is in good agreement with the most recent measurements made using RF-spectroscopy ofmolecular binding energies [34], although previous measurements made using the modification ofin-trap condensate size due to the mean field energy of interactions have reported the resonanceat slightly lower magnetic fields [33, 104, 108]. There is also a single resonance between the aand b states, previously unmeasured, whose parameters are particularly relevant for studies of twocomponent systems. Of interest is also the double resonance in the b state, studied previously
using three-body atom loss [34, 109] and also RF-spectroscopy of molecular binding energies [34].
Compared to previous techniques, RF interaction spectroscopy allows for the exploration of theinter-species resonance. The method is as precise as RF spectroscopy of molecular binding energies,
but does not require knowledge of the molecular potential in order to extract the scattering lengths.
(a)ΔU (kHz)

50
0
- 50

Residuals (kHz)(b) 1.0
0.5
0.0
- 0.5
650

700

750

800

850

900

950

B (G)

Figure 4.10: (a) Fit to a coupled channel calculation of the interaction energy differences Uaa − Uab(blue) and Ubb − Uab (gold). (b) The residuals of the fits. Figure reproduced with permission from[110].

Fit to a coupled channel calculation
The interaction energy differences measured via RF-spectroscopy can also be used as the inputs
to a coupled channel calculation, which allows for an especially precise determination of the scattering lengths at all magnetic fields, which does not suffer from the approximation inherent in theto the energies (4.11), as will be done in the next section.

110

hyperbolic parameterization, or the shortcomings of the lattice AM spectroscopy (especially at low
lattice depths) [110]. By fitting scattering lengths obtained from modifying the Busch solution to
the two-particle energies (4.11) so that the lowest two harmonic oscillator levels are separated by
the same bandgap as the lowest two Bloch bands of the sinusoidal lattice potential, the resonancepositions of table 4.2 are obtained. The fits are plotted in figure 4.10.
Channel
aa
ab
bb
bb

Bres (G)737.883(31)795.307(160)
845.499(14)893.989(195)

Table 4.2: Resonance positions obtained from a fit of the RF spectroscopy data to a coupled channel
calculation. The errors represent 2σ statistical uncertainty in the fit parameters.

4.3

Considerations for Spin-1 Experiments

The results of the interaction spectroscopy reveal that there is indeed a point near 848 G where
the degeneracy condition for the spin-1 model is fulfilled, i.e. where D = 0. This point is notjust a degeneracy point for super-exchange, however: it is also the point at which the interaction
blockade on a single lattice site vanishes, and it is possible to rotate the total spin on a lattice sitearound the entire spin-1 Bloch sphere. Figure 4.11 shows an experimental realization of this effect,
which is both a neat demonstration of interaction blockade physics and verification that such a
degeneracy point exists in the Feshbach resonance spectrum of 7 Li .

4.3.1

Coherent Control and State Preparation

As much as RF interaction spectroscopy in a lattice enables precise measurements of scattering
lengths, it is also a tool for state preparation and diagnostics. Starting with an n = 2 Mott insulator in a single hyperfine state, one can prepare the fully paired, S z = 0 state |ab⟩ ≡ |0⟩ on every
doubly-occupied lattice site by means of a π-pulse or a Landau-Zener sweep. Figure 4.11b demonstrates coherent preparation of this fully paired state, or Spin Mott state, which has a large gap
and is a promising starting point for adiabatic state preparation, in analogy to the band insulator
in fermions [25]. Full spin-sensitive diagnostics of doubly occupied sites can be realized using tran111

2
1
0

(b)

2
1

0

1

2
Time (ms)

30Atoms/site detected in state a(a)Figure 4.11: Rabi oscillations of doublons (a) exactly at and (b) far away from the magnetic field
at which the interactions are degenerate, as in (4.9). The measured atom number is normalized tothe total number of n = 2 sites. The system is initially prepared in state b and the total number ofatoms in state a is detected after applying a resonant RF drive. Decaying sinusoidal fits determinethe√ Rabi frequencies to be 1.68(4) kHz and 2.30(1) kHz respectively, consistent with the expected
2 ratio in Rabi frequency between resonant three-level and two-level systems. The time constantfor the decoherence away from degeneracy is 10.2 ms. The oscillations in (a) seem to decay fasterbecause this data was taken at the closest magnetic field to the degeneracy point which is permitted
by the present resolution limit of our magnetic field setpoint, so that ultimately beating between
two nearly-equal but off-resonant Rabi frequencies decreases the signal contrast.

sitions to a third hyperfine state, i.e. the c state, by selectively measuring the number of doublonsin different configurations of hyperfine states at a magnetic field where the on-site interactions canbe distinguished by RF transitions.

4.3.2

Magnetic Field Stability

The magnetic field is the parameter which tunes the scattering lengths, and through them, the ratio
D/J that controls the phase of the system. Controlling both short term fluctuations (“noise”) andlong term drift (“reproducability”) of the magnetic field is therefore critical for quantum simulation
experiments. The Feshbach resonances of 7 Li begin above 700 G, so experiments will naturally
involve high bias fields. These two requirements– detection of small variations and very highbiases– require a stabilization system with large dynamic range.
112

(a) -2.5

(b) 5
U -U
ab

-2.6

U -U
bb

aa

4

ab

3
2

R

12.5 E
15 E

R

R

1

-2.8

D/|J|

U (kHz)

-2.7

10 E

-2.9

0
-1
-2

-3

-3

-3.1

-4

-3.2
847.44847.45847.46847.47847.48847.49

847.5

B(G)

-5
847.44847.45847.46847.47847.48847.49

847.5

B (G)

Figure 4.12: (a) Relative interactions plotted in the vicinity of the degeneracy point DGP, for alattice depth of 12.5 ER . (b) D/J plotted for several values of the lattice depth in the strongcoupling direction (the transverse lattices are set to 35 ER ) for the same region of magnetic field.
The values are calculated by using the Feshbach resonance parameters reported in table 4.1.
Requirements for tuning D/J
The width of the magnetic field region where |D/J| < 1 for the two-component 7 Li system issharply dependent upon the depth of the optical lattice V0 along the strong-coupling axis, mostly
through the value of J. The depth V0 in turn depends on the location of the MI transition for
doublons with the minimum Uon-site , because the system must be in the integer-occupancy regime
in order for the mapping to the Heisenberg model to remain valid. At a reasonable lattice depthof 10 ER the width of the region in which |D/J| < 1 is about 50 mG, but gets smaller very quickly
at higher lattice depths (figure 4.12)b.
In order to make access to the regime in which D ≈ 0 accessible, the 20 mG of field noise
detected via the interaction spectroscopy measurements requires some further improvement. Either
the magnetic field stability can be improved, as will be discussed in the next section, or modifications
can be made to the lattice setup. In 1D, where the coupling between chains should be zero, thetransverse lattices can be made as deep as the available laser power allows, which increases U as
3/4

V0

and in turn allows for a decrease of the lattice depth along the chain such that t∥ /U ∼ const.

If the transverse coupling is to be finite, the wavelength of the transverse lattice can be decreased,
in which case U ∼ k 2/3 and again the depth along the chain can be decreased while still remaining
in the MI. Both techniques would effectively broaden the range of magnetic field values for which113

|D/J| < 1.
Field control and feedback
If it were possible to measure the magnetic field at the position of the atoms, one could simply
feed back on the bias current. As vacuum-compatible sensors with the required dynamic range(or bandwidth) have not yet– to the knowledge of this author– been invented, it is necessary tostitch together measurements from several sorts of sensors. To this effect, the large magnitude of
the applied bias magnetic field presents both benefits and challenges. Because the relevant atomic√
effects depend on the magnitude of the magnetic field |B| = Bx2 + By2 + Bz2 , experiments will beinsensitive to changes in background field along the two axes perpendicular to the bias field, as
their contributions to |B| will be suppressed by Bbias . Another advantage of high field is that therelative magnetic moments of the states at high field are only tens of kilohertz per Gauss, compared
to megahertz per Gauss at zero field, so that in a spinful system magnetic field fluctuations mostly
just rescale the total energy. In order to stabilize the bias field, a practical approach is to measure
the applied current in a way that is insensitive to background magnetic fields. This task can beaccomplished with Hall or fluxgate sensors that effectively measure the integral of the magnetic
field along a closed loop; these can be placed around the conductor supplying the magnetic fieldcoils. The approach of stabilizing the current, rather than the field itself, relies on the geometricstability of the coils with respect to the position of the atomic sample, which in turn depends on
the thermal stability of the coil package and any hysteretic effects in the mounting hardware. In
Helmholtz configuration, the bias field generated by a certain current varies by only +2 mG/◦ C
for variations in the coil temperature (see section 2.2.3), and if any hysteresis in the field exists,
its effects are too small to detect given other contributions to the noise, making the stabilizationof current sufficient for this apparatus’ needs. The remaining 20 mG of field noise comes rather
from a combination of setpoint drift (due to the DAC used to control the fields, see section 2.2.4)
and input offset drift of the error amplifier in the commercial feedback controller. Both issues arerather easily addressable via electronics upgrades.
In addition to stabilizing the applied bias field, fluctuations in the background field must be
addressed as well. These can be as large as ±10 mG, but are in general due to far-away sources
and are quite uniform throughout the lab (the exception being 60 Hz fields, which are due mostly114

to power supply transformers in lab equipment). The slow fluctuations in the component of the
background field parallel to the applied bias can be measured by a field sensor such as fluxgate orHall probe. The applied bias field presents an additional challenge, however, as one must be able
to differentiate it from the background. In addition, near the chamber, the magnitude of the bias
field is hundreds of Gauss, which will tend to overwhelm the dynamic range of any sensor.
The bias field falls off as 1/r3 , so one might consider placing the sensor many meters from thechamber and subtracting a scaled bias field control signal from the measurement. This method has
the disadvantage that the sensor must be far from the chamber. While the background fluctuations
are presumed to be uniform in space, the distance between the sensor and the chamber prevent
it from being used in a feedback configuration, where the compensation coils surround both thechamber and the sensor. Alternatively, one might place the sensor close to the chamber and winda pair of compensation coils around it that cancel the fringing field of the bias. The advantage ofsensor proximity can, however, be achieved with minimal additional complexity by recognizing thatthe fringing field of the bias coils is perpendicular to the coil axis everywhere on the surface of adouble cone centered on the atoms and passing through the coils. This trick actually works better
than one might expect: the sensor can be aligned by modulating the bias field and minimizing the
response of the sensor. A small fluxgate sensor package (Bartington 690) with an active area of
about 4 mm2 can be placed within 20 cm of the coils such that sensitivity to the bias field is alreadysuppressed by five orders of magnitude. Thus a large set of coils enclosing both the sensor and thechamber can be used in a feedback configuration to cancel ambient field drifts down to the 0.5 mGlevel, with a bandwidth of about 100 Hz.

4.3.3

Decoherence Mechanisms

In order to have any hope of seeing coherent dynamics or ground states in this system, it must
be the case that the super-exchange rate is much greater than the decoherence rate. In addition
to the inter- and intra-band excitations (see section 3.3), the system is also susceptible to twoand three-body losses. Although state a is the hyperfine ground state, and therefore stable against
spin-changing collisions, sites with two b atoms can decay to an a atom and a c atom, releasingtens of megahertz of Zeeman energy and ejecting the atoms from the lattice. Furthermore, sites
with three particles can be lost due to three-atom recombination. Although the system is in the
115

104
bb lifetime (measured)h/J (calculated at DGP)
1D MI Transition

(ms)

103

102

101

100

10-1

0

5

10

15

20

25

30

35

V0 /ER

Figure 4.13: Measured lifetimes of bb sites and super-exchange period h̄/J at a range of lattice
depths, at the degeneracy point (B ≈ 847.5 G). The lattice depth Vcrit at which the 1D MI
transition takes place for the weakest interaction in the system (Uaa ) is indicated with a graydotted line [111]. Because the MI lobes are rather more peaked in 1D than in 3D, the presenceof external confinement means the superfluid phase is not really suppressed throughout the lattice
until a depth of more than twice Vcrit is reached.

n=2 MI state, there is a certain endemic three-body loss due to the finite admixture of tripley
occupied sites (see section 3.2.2), which in this context can be viewed as the intermediate step ofsuper-exchange. These two loss mechanisms together form a fundamental limit to the lifetime ofthe system. Because the degeneracy point is very close to a bb resonance, the loss rate of bb sites
should bound the total loss rate. Figure 4.13 shows the exponential decay constants measured foratom number in a lattice of bb sites at the degeneracy point, for a range of lattice depths. For
comparison, the rate associated with the super-exchange parameter J/h̄ is also plotted, based on
the Feshbach resonance parameters reported in 4.1. The loss rate scales like t2 for lattice depths
below 20 ER , suggesting that below lifetimes of 1 s, the dominant loss process is three-body loss via
coherently admixed triplons. Above lifetimes of 1 s, the lifetime is likely limited by noise-induced
tunneling, resulting in band excitations and ultimately three-body loss. The three-body loss rate
for b-state 7 Li atoms at this magnetic field is unfortunately rather high [109]. For a range of lattice
depths above the MI transition, however, the super-exchange rate is about 20 times higher than116

the loss rate, suggesting that it will still be possible to observe effects arising from some tens ofsuper-exchange events before the system decoheres completely. Encouragingly, the loss rate is likely
to depend strongly on the local environment: ab doublons are less likely to decay if the neighboringsites contain aa or ab doublons than if the neighboring sites contain bb doublons.

4.3.4

First Experiments

In light of these moderately encouraging results, some preliminary experiments can be performed
which begin to explore the phase diagram of the spin-1 Heisenberg chain. The first step is tocreate a single-component MI in the b state with a core of doubly-occupied sites (given available
lattice power, about 600 chains of 25 atoms each can be created, but the length can be increased
by compensating external confinement along the tube axes with a repulsive potential, i.e. from ablue-detuned beam). Interaction blockades then permit RF preparation of the Spin Mott state in a
deep lattice (J ≈ 0), in a region of magnetic field where this state is the ground state (D/|J| ≫ 1).
It would then be possible to adiabatically sweep D/|J| in presence of weak inter-chain coupling inorder to produce the xy-antiferromagnet. This sweep towards D/|J| ≈ 1 requires both lowering
the magnetic bias field and lowering the lattice depth along the chains, which in principle can beaccomplished in either order: lowering the magnetic bias field towards the DGP mostly decreases
D, while lowering the lattice depth mostly increases |J|. Changing the field first will likely produce
better results, as the magnetic field can be stabilized near the DGP while the sites are effectively
isolated and the lifetime of the system is extremely long.
The formation of an xy-antiferromagnet from the Spin Mott phase should only require as manysuper-exchange events as the correlation length of the magnetically ordered state, which is anticipated to be a few sites. Thus, a second advantage of the Spin Mott phase as a starting point for
adiabatic quantum simulation of this portion of the Heisenberg model phase diagram is that thephase transition does not require any long-range transport, which would be slow. Measurement of
the correlation length would require techniques such as Bragg scattering [79, 112], noise correlations
in time-of-flight [113, 114], or spin-resolved single-site imaging [115], but it may be possible to initially detect the presence of a phase transition simply by tracking the spin populations. Interaction
blockades can again be used to distinguish aa, ab and bb sites by flipping in an occupancy-dependent
way one atom per site into a third state which can be imaged separately, e.g. the c state. In a117

perfect Spin Mott, the population of doubly occupied sites should be exclusively ab, while in an
xy-antiferromagnet the distribution of spins ought to be half ab sites and a quarter each of aa andbb (this is in contrast to a thermal distribution of spins, which ought to be one third of each state).
It may even be possible to detect whether the transition is truly adiabatic by reversing the ramp
of D/J, and measuring whether the distribution of spins respond accordingly. On the far side of
the phase diagram, the z-antiferromagnet can be distinguished by the spin population being halfeach of aa and bb, with no ab sites at all.
Distinguishing the Haldane phase, however, is not nearly so straightforward. The spins are
predicted to minimize the string order parameter


⟨
σ
Ostr
≡

lim

|j−k|→0Sjσ exp iπ∑Slσ  Skσ⟩(4.14)

j≤l<kwhich, as opposed to involving only products of nearest neighbors, is an inherently non-local measure [13, 116]. The sum in the exponent means that the order parameter is minimized only when
every site containing a |+1⟩ spin is separated from a site containing a |−1⟩ spin by some numberof |0⟩ sites (or none at all). Thus spin populations do not give an easily distinguishable pictureof the state, and from the point of view of Bragg scattering, the Haldane state would look likethe disappearance of magnetic ordering. In order to measure the long-range spatial correlations
that string-ordering entails, one would really need the single-site resolution of a quantum gas microscope. An alternate approach to distinguishing the Haldane phase from the anti-ferromagnet
would be to probe the system’s response to a perturbation that creates spin waves, e.g. a magnetic
field gradient. While the xy-antiferromagnet supports spin-waves, the Haldane phase does not,which leads to two different excitation spectra. The effect of these spin waves could be observed byreturning (as adiabatically as possible) to the Spin Mott and observing the fidelity of the S z = 0
state.

118

Chapter 5

Conclusion and Outlook

119This thesis has reviewd the design and construction of a general purpose, fast bosonic quantum
simulator using 7 Li . Because dynamics such as tunneling and super-exchange scale with therecoil energy, lighter atoms provide a significant advantage with respect to avoiding fundamental
decoherence mechanisms like background gas collisions and off-resonant photon scattering, whose
effects are independent of mass. Most technical sources of heating in the lattice, however, also
scale with the mass (or with some higher power of the mass)– so that light atoms are a better
choice if and only if such technical sources of noise can be mastered. In this respect light atoms
present a dual challenge: not only are they more sensitive to noise, but the noise to which they aresensitive is at higher frequencies, so that techniques for mitigating it (i.e. feedback) can prove to
be more technically demanding. Notwithstanding, the ratio of super-exchange rate to decoherence
rate achieved by the 7 Li quantum simulator at the time of this writing is large enough to enable
experiments in many regimes. In addition, an experimental duty cycle of under ten seconds androbust, repeatable cooling performance should prove useful for densely sampling large parameter
spaces in order to reveal the ground state phase diagrams of spin models and other Hamiltonians.
Aside from being the lightest bosonic alkali atom, 7 Li ’s interactions are eminently tunable, due
to its Feshbach resonance spectrum. This thesis has discussed experiments in which the interactions
between all three of the combinations of the lowest two hyperfine states were measured by exploiting
interaction blockades in tightly-confining 3D optical lattice. The results of these measurements are
the most precise achieved to date for 7 Li by any method, and also constitute the first measurementof an inter-state resonance in 7 Li . It has also been shown how the same interaction blockade
effects may be used for coherent preparation of non-trivial combinations of spin states on a lattice
site, and for spin- and number-sensitive readout of those combinations. Using these techniques inexperiments requires the application of very large yet very stable magnetic fields. The requisite
coils and feedback architecture have been described and characterized. As future experiments may
further require the same high bias fields with extremely low curvature in order to achieve low
external confinement along one or more axes, a proscription for how to create such fields withelectromagnetic coils has been developed.
The particular case of the quasi-1D spin-1 Heisenberg model has been shown to provide an
excellent proving ground for the unique strengths of the 7 Li platform. Using the interaction spectroscopy data, a magnetic field has been found at which the combination of interactions in an n=2120

Mott insulator of 7 Li implements the isotropic spin-1 Heisenberg model. Further, the surrounding
magnetic field regions have been identified with differently ordered magnetic ground states, including z- and xy-antiferromagnets and a Spin Mott/paramagnet state, affording ample opportunities
to study quantum phase transitions and to adiabatically prepare those states from a BEC. Whilesome concerns about three-body losses remain, this system nonetheless provides a unique opportunity to explore magnetic ordering in bosons and perhaps study some aspects of symmetry-protected
topological phases. If initial experiments monitoring population dynamics across the phase transitions seem promising, there may be sufficient arguments to upgrade the apparatus with single-site
imaging, inspired by excellent results already achieved in fermionic Lithium [11, 115].
Of course, the 7 Li platform can simulate many other models as well. Another promising avenue
is the spin-1/2 system implemented by an n = 1 Mott Insulator, which promises a similarly richarray of magnetically ordered ground states and phase transitions between them, and should provide
an interesting bosonic counterpart to recent successes in simulations of fermionic systems. Whileplatforms using fermionic atoms can only implement isotropic exchange interactions, bosons canprobe anisotropic models as well, which considerably expands their phase space. Like the spin-1
system, the spin-1/2 system should provide opportunities to study spin dynamics and spin diffusion
at high temperature, and adiabatic preparation of magnetically ordered ground states at lower
temperatures. These and other experiments will hopefully contribute to the understanding of thetruly fascinating quantum materials that exist in nature, and further to the use of adiabatic analog
quantum computers as general tools for problem-solving.

121

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