Atoms in Rotating Lattices and Cavity QED
Measurements of Atomic Dynamics
byBrandon March Peden
M.S., Mathematics, Western Washington University, 2004
B.S., Physics, Western Washington University, 2002

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy in Physics
Department of Physics
2010This thesis entitled:
Atoms in Rotating Lattices and Cavity QED Measurements of Atomic Dynamics
written by Brandon March Peden
has been approved for the Department of Physics

Murray Holland

Ana Maria Rey

Date

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

iii
Peden, Brandon March (Ph.D., Physics)
Atoms in Rotating Lattices and Cavity QED Measurements of Atomic Dynamics
Thesis directed by Prof. Murray Holland

The first part of this thesis describes the behavior of ultracold atoms confinedin a rotating optical lattice. We consider gases of strongly-interacting bosons and noninteracting fermions in ring and square lattice geometries. We derive modified Hubbard
models to describe the gas and use the quasi-angular momentum (QAM) representation
to label the eigenstates. Exact level crossings between states of different QAM are
predicted. We identify signatures of these transitions in the momentum distribution,
indicating that these states should be distinguishable in time-of-flight experiments.
The second part of this thesis describes a scheme for nondestructively probing
the dynamics of atoms in optical lattices by coupling to the modes of an optical resonator. The cavity fields set up both the optical lattice and the probe field so that noexternal interrogation fields are necessary. The probe is weak so that the atoms can becontinuously monitored without affecting the atomic motion. This scheme is applied toa measurement of Bloch oscillations; SNR’s as high as 104 are predicted.
In the third part of this thesis, we study the dynamics of atoms in a tilted latticenear the Mott-insulator regime. The dynamics involve the creation and annihilationof dipoles, states generated from the unit-filled states by introducing defects where an
atom has hopped exactly one site to the left. We describe how these states can beexperimentally distinguished by coupling the atoms to the modes of an optical cavity.
Finally, we describe a quantum non-demolition measurement of the cavity photon
number using Raman interferometry of atoms coupled to the cavity modes. Using
Bayesian inference, we show that there is a measurement protocol for which the cavity
photon number can be determined with just a few measurements.

Dedication

To Amanda, Ashley, Dylan, and Sean.
To Chuck, Marilyn, and Marty.

Acknowledgements

Special thanks go to Brad Johnson, Murray Holland, and Dominic Meiser asadvisors/mentors: they meant well. Special thanks go to Boris Nowak and David Tieri:
without them putting up with me telling them wrong things all the time about masterequations and quantum Monte Carlo simulations, I might never have learned how to use
those methods. Special thanks go to Ron Pepino, Rajiv Bhat, and Brian Seaman: theoffice was a spectacularly fun place to work (I mean do crosswords, watch online videos,and play flash games). And special thanks go to Seth Rittenhouse and Rob Wild for the
excellent variety of random discussions about physics, baseball, probability, magnetic
healing, egg-nog recipes, puzzles and riddles, natural selection, books—most had overa glass of beer.
Here are even more people who helped me understand physics: Alan Bristow,
Marilu Chiofalo, Jinx Cooper, Jami Kinnunen, Meret Krämer, Ed Meyer, Ana Maria
Rey, Chester Rubbo, Evan Salim, and Jochen Wachter.
These are the people—besides those already mentioned—who are responsible (toblame?) for who I am: the Clans Anderson, Bostwick, and Peden; Mike Burghart,
Ariel Coleman, Katy Davis, Matt Fuller, William Funk, Jason Ho, Josh Kaffer, KevinLangstaff, Patrick Leonard, Brian Long, Alex Mazurkewycz, Molly Painter, Tanya
Rivers, Zev Rivers, Lizzie Rothwell, Tim Salazar, Ben Schielke, Jesse Sherman, Ali
Smith, Rick Wallace, James Wells, Jason Whittington, Glenn Willis; the cat, Stewart;
the Clan Holland; the CU hiking club VIPs; the WWU math and physics departments.

vi
Special acknowledgements go out to Cheez-its, cheese, Charles Darwin, India paleales, the Southern Sun, community-supported agriculture, Salman Rushdie, scotch, theLucksmiths, InfiniteRegress Collective Mix Tapes, GarageBand, the Mariners (especially, Ichiro!, Felix, and Jack Z), FactCheck.org, Alton Brown, Google reader, Bon
Iver, Radiohead, Built to Spill, Braid, Xtc, Death Cab For Cutie, The Daily Show, USS
Mariner, FanGraphs, J. R. R. Tolkien, Lookout Landing, babies, xkcd, Dinosaur Comics,
Ernest Hemingway, Dresden Codak, Indexed, MLB.tv, PhD Comics, Pitchfork Music,
Daytrotter, La Blogotheque Take-Away Shows, Wins Above Replacement, Michael Pollan, the Beaver Inn, Richard Dawkins, National Public Radio, Wait Wait Don’t Tell
Me, New York Times crosswords, Craig Calcaterra of Shysterball, Dave Cameron of
USSM, Joe Posnanski of SI, television, Bellingham, Budapest, Boulder, beer-brewing
and cheese-making, jam-canning and bread-baking, Accustomed to Failure, and My
Little Brother.
Plus everyone else.

Contents

Chapter
1 Introduction

1

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2 Optical Lattices and Optical Resonators
2.1

2.2

7

Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1.1

Band theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.2

Hubbard models . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Optical Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.2.1

Resonant modes . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.2.2

Cavity field dynamics and atom-field coupling . . . . . . . . . . .

30

3 Bose and Fermi Gases in Rotating Optical Lattices

32

3.1

Bloch Theory for Systems With Discrete Rotational Symmetry . . . . .

33

3.2

Single Particle on a Rotating Ring Lattice . . . . . . . . . . . . . . . . .

35

3.2.1

Effects of rotation . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.3

Many-Body Description of Quasi-Angular Momentum . . . . . . . . . .

41

3.4

Bose and Fermi Gases in Rotating Optical Lattices . . . . . . . . . . . .

45

3.4.1

Strongly-interacting bosons . . . . . . . . . . . . . . . . . . . . .

47

3.4.2

Non-interacting fermions . . . . . . . . . . . . . . . . . . . . . . .

53

viii
3.5

Experimental Signatures of Quasi-Angular Momentum . . . . . . . . . .

58

3.6

Conclusion

67. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Mott-Insulator to Superfluid Phase Diagrams
4.1

69

MI-SF Phase Diagram for Strongly-Interacting Bosons . . . . . . . . . .

70

4.1.1

Comparison between two-state and mean-field approximations .

71

4.2

Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.3

Conclusion

75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Non-Destructive Cavity QED Probe of Atomic Dynamics in an Optical Lattice 77
5.1

5.2

5.3

Model and Detection Scheme . . . . . . . . . . . . . . . . . . . . . . . .

78

5.1.1

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.1.2

Eliminating higher-order correlations . . . . . . . . . . . . . . . .

83

5.1.3

Detection scheme and signal-to-noise ratio . . . . . . . . . . . . .

88

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.2.1

System dynamics: Bloch oscillations . . . . . . . . . . . . . . . .

93

5.2.2

Signal and SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6 Cavity QED Measurements of Many-Body Dynamics Near the Mott-Insulator
Regime
6.1

6.2

102

Atomic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.1.1

Dipole Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.1.2

Symmetric Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 107

6.1.3

Validity of the approximations . . . . . . . . . . . . . . . . . . . 111

Detection Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2.1

Dipole-subspace detection . . . . . . . . . . . . . . . . . . . . . . 118

6.2.2

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

ix
7 Quantum Non-Demolition Measurement of the Photon Number in a Cavity

122

7.1

Ramsey Interferometry in the Bragg Regime . . . . . . . . . . . . . . . . 123

7.2

Bayesian Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2.1

Measurement protocols . . . . . . . . . . . . . . . . . . . . . . . 128

7.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.4

Experimental Considerations . . . . . . . . . . . . . . . . . . . . . . . . 133

7.5

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8 Conclusion

137

Bibliography

141

Appendix
A Numerical Implementation

150

A.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.1.1 Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.1.2 Hamiltonian decomposition in momentum space . . . . . . . . . 154
A.1.3 Hard-core boson approximation . . . . . . . . . . . . . . . . . . . 156
A.1.4 Jordan-Wigner fermions . . . . . . . . . . . . . . . . . . . . . . . 158
A.2 Tilted Lattice Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.2.1 Dipole subspace Hamiltonian . . . . . . . . . . . . . . . . . . . . 161
A.2.2 Symmetric Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 163
B Atoms Interacting with Quantized Fields
B.1 Hamiltonian of the free electromagnetic field

165. . . . . . . . . . . . . . . 167

B.2 Optical dipole potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
B.3 Adiabatic elimination of high-lying excited states . . . . . . . . . . . . . 170

Figures

Figure2.1

Dispersion relation for free particles . . . . . . . . . . . . . . . . . . . .

11

2.2

The Bloch function for quasi-momentum equal to zero . . . . . . . . . .

13

2.3

Lowest band dispersion relation . . . . . . . . . . . . . . . . . . . . . . .

15

2.4

Examples of Wannier functions for sinusoidal lattices . . . . . . . . . . .

17

2.5

Tight-binding tunneling energies . . . . . . . . . . . . . . . . . . . . . .

19

2.6

The Mott-insulator to superfluid phase diagram . . . . . . . . . . . . . .

21

2.7

Schematic of a linear optical cavity . . . . . . . . . . . . . . . . . . . . .

23

2.8

Schematic of the spectrum of a linear optical cavity . . . . . . . . . . . .

24

2.9

Longitudinal envelope of a Hermite-Gaussian mode . . . . . . . . . . . .

26

2.10 Transverse envelope of the (00) Hermite-Gaussian mode . . . . . . . . .

27

2.11 Cross-sections of sample Hermite-Gaussian modes . . . . . . . . . . . . .

29

3.1

Equivalence between a ring lattice and a 1D lattice with periodic BCs .

36

3.2

Exact energy-level crossings due to rotation . . . . . . . . . . . . . . . .

39

3.3

The m = 0 Bloch function at ω = 0 . . . . . . . . . . . . . . . . . . . . .

40

3.4

Illustration of the rotation operator in the many-body formalism . . . .

43

3.5

Ground state of three strongly-interacting bosons . . . . . . . . . . . . .

52

3.6

Single-particle energy spectrum in a 4 × 4 square lattice . . . . . . . . .

57

3.7

Quasi-angular momentum as a function of rotation for 1 to 16 particles

59

xi
3.8

Quasi-angular momentum as a function of rotation for 1 to 16 particles

60

3.9

Quasi-angular momentum as a function of rotation for 1 to 16 particles

61

3.10 Fourier transform of single-particle quasi-angular momentum states . . .

64

3.11 Momentum distributions computed via imaginary time propagation . . .

66

4.1

Bose-Hubbard phase diagram for strongly-interacting bosons . . . . . .

72

4.2

Comparison between the MI to SF phase diagrams . . . . . . . . . . . .

72

4.3

MI-SF phase diagram for an infinite one-dimensional lattice . . . . . . .

75

5.1

Schematic of the coupled atom-cavity system . . . . . . . . . . . . . . .

80

5.2

Schematic of the balanced heterodyne detection scheme . . . . . . . . .

88

5.3

Example of system dynamics for V0 = −3ER

. . . . . . . . . . . . . . .

96

5.4

Examples of the (a) signal power and (b) signal power spectrum . . . .

97

5.5

Signal-to-noise ratio as a function of the initial wave-packet width . . .

98

5.6

Signal-to-noise ratio as a function of lattice depth . . . . . . . . . . . . .

99

6.1

Schematic of the atomic system . . . . . . . . . . . . . . . . . . . . . . . 103

6.2

Overlaps S0

6.3

Total population PD of the dipole subspaces . . . . . . . . . . . . . . . . 112

6.4

Total population PD of the dipole subspaces . . . . . . . . . . . . . . . . 112

6.5

Total population PS in the symmetric states . . . . . . . . . . . . . . . . 113

6.6

Population in the symmetric states as a function of time . . . . . . . . . 114

6.7

Hubbard model coupling parameters fi,j (q0 ) . . . . . . . . . . . . . . . . 117

6.8Time sequence for |hn̂(q0 )i|2 . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.1

Forward probability distribution P (e|n) . . . . . . . . . . . . . . . . . . 129

7.2

Forward probability distribution P (e|n) . . . . . . . . . . . . . . . . . . 130

7.3

A three-measurement Bayesian sequence starting with a coherent state . 131

7.4

Three 200-step Bayesian measurement sequences . . . . . . . . . . . . . 132(M )in the M -dipole subspace . . . . . . . . . . . . . . . . . . 110

xii
7.5

A three-measurement Bayesian sequence starting with a coherent state . 133

7.6

Three 16-step Bayesian measurement sequences . . . . . . . . . . . . . . 134

7.7

Reconstruction of the initial coherent state of mean 3

7.8

Forward distribution for various initial momentum distribution widths . 135. . . . . . . . . . 134

Chapter 1

Introduction

1.1

BackgroundExperimental and theoretical studies of quantum gases of ultracold atoms havebeen successful in replicating the behavior of a wide class of condensed matter systems.
In particular, Bose-Einstein condensation (BEC) [4, 16, 32] and the degenerate Fermi
gas [33] have both been realized, and there have been extensive studies of superfluidity,
vortex formation [135], and the BCS-BEC crossover [48, 90]. More recently, a gas ofultracold polar K-Rb molecules has been created near quantum degeneracy [92]. There
are also proposals for simulating quantum Hall physics in rotating BEC’s [5, 38, 100,130]. However, experimentally reaching the parameter regimes necessary to observe thisbehavior is difficult [117].
The recognition that systems comprised of ultracold atoms trapped in opticallattices could be used to cleanly realize many-particle systems from condensed mattersystems [64] has motivated a concentrated effort to study these systems. These systems
are highly tunable: lattice spacing and depth can be varied by tuning the frequency
and intensity of the lasers, and interactions between atoms can be tuned via a Feshbachresonance; in addition, these systems are very clean, i.e. they lack defects that typicallyshow up in condensed matter systems [15]. By varying the lattice depth, the existenceof the Mott-insulator to superfluid transition in a system of interacting bosons trappedin a lattice has been verified [47]. Fundamental concepts from condensed-matter physics

2such as Fermi surfaces and interaction-induced inter-band transitions have been investigated [68]. Experiments observing novel phases of ultracold matter in atomic mixturesof Bose and Fermi gases in optical lattices have also been performed [2, 60, 97, 96].
On the theoretical side, proposals for a range class of condensed matter and highenergy models abound. By imprinting an effective magnetic field on two-dimensional
systems by either rotating the lattice [9] or driving atomic transitions with a set oflasers [65], phenomena such as the Hofstadter butterfly and quantum Hall physics canbe observed. Other recent studies have made direct connections between fractional
quantum Hall effect (FQHE) physics and strongly-interacting bosons in optical lattices [10, 52, 99, 125]. Similarly, there have been proposals for realizing systems exhibiting non-Abelian symmetries [98, 114], Kondo-type physics [41], superradiant exciton
states [134], and supersolid states [81].
Concurrently, experimental activity in quantum optics and cavity quantum electrodynamics (QED) has provided access to interesting regimes of the light-matter interaction [89]. The strong-coupling regime—where interaction effects dominate overdissipative effects—has been achieved, and both vacuum-Rabi oscillations and mechanical light forces can be monitored in cavity QED setups where the mean photon-number
is at all times less than one [57]. By coupling atoms to a cavity mode, efficient conversion of collective atomic excitations into a single photon points the way towards
high-efficiency single-photon sources [123]. Kerr non-linearities and dispersive opticalbistability due to the displacements of atoms in the cavity have been observed [51].
Photon blockade, where a single photon within a cavity blocks further photons fromentering the cavity, leading to photon anti-bunching, has been observed [13].
All along, there has been a productive marriage of the two fields of ultracold
atoms in optical lattices and cavity QED. Using the long-range interactions between
atoms induced by the field-atom coupling in an optical cavity, the Dicke quantum phasetransition has been demonstrated [6]. In addition, cooling of a single trapped ion in

3
an optical cavity has been achieved by way of cavity sideband cooling [72]. There are
numerous theoretical studies of novel models where the atomic and light-field degrees
of freedom both play integral parts, such as self-organization of atomic states due tothe back-action of light on atoms in optical resonators [70] and phase transitions tosuperradiant-like phases driven by competition between atom-atom and atom-field interactions [122].
In the great majority of the atomic experiments listed above, destructive measurement schemes such as time-of-flight absorption imaging and Bragg spectroscopy are
employed to probe atomic states and dynamics. This allows for high resolution images
and a strong signal using only a single measurement. In addition, be preparing several experimental copies with identical initial conditions and varying the duration of
the experiment, these schemes provide access to dynamical information. Thus, thesemeasurements are sensitive to a large number of atomic observables in experimental
setups. On the theoretical side, linear response theory has been applied to current
flow in the presence of a “potential gradient” in order to observe FQHE physics [10],noise correlation analysis has been applied to bosons in a rotating ring lattice [106], andBragg spectroscopy has been mentioned as a probe for vortex states [129]. The drawback in these schemes is that the atomic sample is destroyed in the process of making
a measurement.
The advantage of cavity QED setups is that they come equipped with a natural
measurement device, i.e. the light-field out-coupled through the cavity mirrors. In one
proposed detection scheme, a weak probe beam is scattered off of atoms trapped in an
optical lattice into a cavity mode, and signatures of many-body states such as Mott
insulators and superfluids appear in the out-coupled fields [86]. In another, atoms ina lattice interact with two counter-propagating ring-cavity modes, and atomic number
statistics can be inferred from the behavior of the cavity fields [27]. Bloch oscillationsof atoms in a lattice can also be monitored via the transmission of light through a

4cavity [128].
Related techniques have been applied to nondestructive optical measurements ofRabi oscillations in gases of Cs atoms [131], of the Cs clock transition pseudo-spin [26],and of nonlinear dynamics in cold gases [124]. In addition, state preparation suchas atomic spin squeezing via measurements on out-coupled cavity fields has been proposed [85, 87, 93] and measured [73, 115]. It has also been demonstrated that the motionof individual atoms in an optical cavity can be tracked by the transmission of a probe
field [58].
Quantum non-demolition measurements of the photon number in microwave cavities by way of Ramsey interferometry of Rydberg atoms sent through the cavity havebeen performed in Paris. Through these experiments, fundamental concepts such as
quantum jumps [46] and non-demolition photon counting have been explored [50]. In
addition, this setup was used to perform quantum process tomography to reconstruct
the master equation describing dissipative processes for photons trapped in microwave
cavities [17].
One final example of an area in which ultracold atomic gases and cavity QED
have made great strides is in the area of precision measurement. For instance, thenewest generation of optical lattice clocks and ion clocks are becoming competitive withthe time and frequency standards of the standard fountain atomic clocks [78, 95, 109].
In addition, there is hope for new technologies that promise to increase the stability ofthese clocks by two orders of magnitude [84]. Finally, by referencing against a highly
stable optical cavity, precision measurements have been performed to test variation of
fundamental constants such as the fine structure constant [14, 40].

1.2

Overview
Throughout these examples, light-matter interactions have been exploited to ex-

plore and measure the quantum behavior of both atomic and electromagnetic systems.

5
In this thesis, we continue this tradition by investigating the behavior of gases of ultracold atoms in optical lattices and proposing methods for detecting atomic states anddynamics in these systems.
In chapter 2, we review the basic physics of optical lattices and optical resonators.
We derive the second-order atomic Stark shift that gives rise to an optical lattice potential. We review the basic physics of band theory and the Bose-Hubbard model. Finally,
we discuss the properties of optical cavities and describe the mathematical formalism
used to treat cavity-field dynamics.
In chapter 3, motivated by the search for quantum Hall physics in atomic systems,
we investigate the ground-state behavior of ultracold Bose and Fermi gases in rotating
optical lattices. Using the quasi-angular momentum representation, we use exact diagonalization and imaginary-time propagation techniques to explore the atomic behavior.
Exact energy-level crossings in the spectrum indicate the possible existence of quantum
phase transitions between states of different quasi-angular momentum. The momentum
distributions differ significantly for states of different symmetry, indicating that there
may be signatures of these transitions in time-of-flight experiments.
In chapter 4, we demonstrate the utility of the many-body formalism developedin chapter 3 by computing approximations to the Bose-Hubbard phase diagram forsuperlattices.
In chapter 5, we propose a scheme for nondestructively probing the dynamics of agas of ultracold atoms confined in an optical lattice and subjected to an external potential by coupling the atoms to the counter-propagating modes of an optical ring-cavity.
The light field out-coupled through the mirrors is used as the measurement device. This
scheme accomplishes three main goals. The cavity field is used to simultaneously set
up both the lattice field and the probe field, so that no extra interrogation fields arenecessary. The probe is weak so that the atoms can be continuously monitored without
affecting their motion. The signal-to-noise (SNR) ratio is large enough to be readily

6experimentally detected. Integration of the signal in time and across the atomic cloudyields measurements of dynamical atomic observables such as the center-of-mass oscillation frequency with a large SNR in a single experimental run. This comes at the priceof losing information about single atoms and real-time dynamics, but the atomic sample
is not destroyed during the measurement. We apply this scheme to a measurement of
the Bloch oscillation frequency of atoms trapped in an optical lattice and subjected to a
constant, external force such as gravity. We simulate the atomic dynamics and compute
the SNR of a heterodyne signal at the cavity output. The maximum predicted SNR is104 .
In chapter 6, we investigate the dynamics of atoms in tilted optical lattices near
the Mott-insulator regime. When the Bose-Hubbard interaction strength U is equal tothe energy difference between adjacent lattice wells, the atoms can hop in a restricted
way about the lattice. The system can be described by way of a “dipole” Hamiltonian,where the allowed atomic states consist of adjacent two-particle/zero-particle sites. In
order to detect the dynamics in this system, we employ the scheme from chapter 5,
relaxing the condition that the cavity field acts as the lattice potential. This detection
scheme is insensitive to dipole states near the unit-filled (i.e. Mott-insulator) state, butthere is a large signal for the state with the maximum possible number of dipoles.
In chapter 7, we propose a quantum non-demolition measurement of the cavity
photon number via Ramsey interferometry of atoms allowed to interact with the cavity
field in the Bragg regime. We employ quantum Bayesian inference to project the cavity
field onto a number state after a certain number of repeated measurements. This method
can be used for both quantum state preparation and quantum state tomography of thecavity field.
Finally, we conclude with a summary and outlook for future work in chapter 8.

Chapter 2

Optical Lattices and Optical Resonators

Much of the work in this thesis relies on two related light-matter systems. Firstand foremost, we consider optical lattices, which are external, sinusoidal lattice potentials for atoms made up of one or more laser beams. This system is paradigmaticof systems with a discrete translational symmetry, and it gives rise to physical phenomena from condensed matter physics such as band theory, many-body models suchas the Bose-Hubbard Hamiltonian, and quantum phase transitions such as the superfluid/insulator transition. The other system is that of atoms interacting with the quantized, resonant modes of an optical cavity. This system is highly flexible: it can be used
to generate external optical lattices for the atoms, to measure the dynamics of atoms
trapped in cavities, and it can be used to enter the strongly-interacting regime where
single photons can affect the behavior of atoms.
In this chapter, we explain the basic physics of optical lattices and resonators
including the general mathematical framework by which we describe these systems andtheir interactions with ultracold atoms. The discussion of optical lattices relies on thematerial in references [12, 67, 80, 107]. The discussion of optical resonators, including
general characteristics and the Gaussian resonator modes, relies on the material inreferences [121, 133]. The mathematical description of cavity field dynamics relies onmaterial contained in references [77, 88].

8

2.1

Optical Lattices
Two-level atoms experience a position-dependent Stark shift when subjected to

an off-resonant laser field. This shift gives rise to an external potential. In appendix
B, we perform a rigorous calculation where high-lying excited states are adiabaticallyslaved to two low-lying states. Here, we make a plausibility argument through the useof second-order perturbation theory.
Given two internal states |ei and |gi of an atom, the interaction Hamiltonian
between the atom and a single quantized mode of the electromagnetic field is given by
ĤI =

~Ω +
~Ω − † ∗σ̂ âu(x̂) +σ̂ â u (x̂),
2
2

(2.1)

σ̂ + = |eihg| = (σ̂ − )† ,

(2.2)

where

and â is the annihilation operator for the field mode with mode function u(x). We havemade both the dipole and rotating wave approximations in writing this Hamiltonian.
In the optical domain, these are very good approximations. Typical wavelengths for
optical fields are in the hundreds of nanometers, two orders of magnitude larger than
the size the atoms. In addition, the rotating-wave approximation requires that|Ω|  |ω + ωeg |,(2.3)

where ω is the frequency of the field mode, and ωeg is the atomic transition frequency.
This is easily satisfied in the optical domain where ω/2π ≈ 1015 Hz.
When the detuning δ = ω − ωeg between the atomic transition frequency and field
frequency satisfies|δ|  |Ω| ,(2.4)

we can compute the second-order correction to the ground state energy as
Eg(2) =

|he|ĤI |gi|2~|Ω|2 †=â â|u(x̂)|2 .
~δ
4δ

(2.5)

9
This expression takes the form of an external potential, the strength of which depends
on the field intensity.
If the electromagnetic field is in a coherent state with a large average value,α, then the intensity fluctuations ∆â† â/â† â are on the order of 1/|α|, which can bemade extremely small. In this case, we can neglect the effects that field fluctuations
and atom-field correlations have on the atoms. We absorb the field intensity into aneffective coupling constant, and the atoms see a potential,
V (x̂) = ~g0 |u(x̂)|2 .

(2.6)

If the field mode is a standing-wave, then|u(x̂)|2 ∝ cos2 (k · x̂),

(2.7)

and we can see that the atoms experience a sinusoidal lattice potential of lattice constant,2π/|k|.
This simple model leads to a wealth of physical phenomena. At the single-particle
level, the discrete translational symmetry of the system gives rise to band theory andBloch functions. When the lattice depth is large enough, the system enters the tightbinding regime which is characterized completely by an atomic tunneling rate between
adjacent lattice wells. At the many-body level, a gas of ultracold atoms trapped in an
optical lattice gives rise to clean realizations of models from condensed matter physics
such as the Bose-Hubbard model [64].

2.1.1Band theory

The concept of band physics can be approached from two directions. The particle
can be treated as nearly free, in which case the introduction of a weak periodic potential
breaks the continuous translational symmetry of the free problem. States with the same
symmetry properties mix with each other under the perturbation, giving rise to gaps

10in the free-particle dispersion relation. That is, the perturbation gives rise to bands ofallowed energies separated by gaps on the order of the perturbing potential. On theother hand, the particle can be treated as being tightly bound to the sites on a periodic
lattice, where the energy associated with the particle being localized to specific sites is
identical across the lattice. Allowing the particle to hop between adjacent sites at a rate
J adds a width to the originally flat energy spectrum, giving rise to a band of allowed
energies of width, 2J. These two ways of viewing the physics of an atom moving in
the presence of an external periodic potential are complementary, as one gives rise to atheory about band-gaps, and the other gives rise to a theory about bandwidths.
Here, we introduce band physics using the concept of nearly-free particles. Later,when discussing the Bose-Hubbard model, we use the language of the tight-binding
model. Moving between these languages requires the concepts of Bloch and Wannier functions, which we move on to now. In most of what follows, we consider onedimensional systems. The generalization to two and three dimensional systems is relatively straightforward.
A small periodic potential V (x) of period d couples two free-particle wave func√√
tions ψ1 (x) = eik1 x / V and ψ2 (x) = eik2 x / V only if k1 and k2 differ by 2π/d, as
shown by the calculation,Z ∞e−ik2 xeik1 xdx √ V (x) √ =
V
V
−∞

Z ∞

e−ik2 (x−d)
eik1 (x−d)√
V (x − d) √
V
V
−∞
Z ∞
−ik
x
2
eik1 xe= e−i(k1 −k2 )ddx √ V (x) √ .
V
V
−∞
dx

(2.8)

Due to the phase multiplying the integral, this matrix element must be zero unless k1 and√k2 differ by a multiple of 2π/d. In particular, the two wave functions ψ1 (x) = eiπx/d / V√and ψ2 (x) = e−iπx/d / V at the Brillouin zone boundaries (q = ±π/d) are coupled toeach other. The potential acts as a perturbation that breaks the degeneracy between

11

HbL

8

E Hunits of ER L

E Hunits of ER L

HaL

6
4
2
0
-3

8
6
4
2
0

-2

-1

0

1

2

3

-1

-0.5

q Hunits of ΠdL

0

0.5

1

q Hunits of ΠdL

Figure 2.1: Dispersion relation for free particles (blue, solid) and particles moving in a
one-dimensional sinusoidal lattice (red, dashed) plotted in (a) extended zone scheme and(b) reduced zone scheme. Free space dispersion relations are plotted versus momentum,
and lattice dispersion relations are plotted versus quasi-momentum. Energy is in unitsof the recoil energy, ER = ~2 π 2 /2md2 , where d is the lattice spacing. The lattice depth
is V0 = 5ER .

these two states, creating new eigenfunctions,
1  iπx/d
ψ± (x) = √e± e−iπx/d ,
2V

(2.9)

with corresponding energies,
E± =~2 π 2+ hψ± |V̂ |ψ± i.
2m d2(2.10)

This process opens up a gap,
∆ = |hψ− |V̂ |ψ− i − hψ+ |V̂ |ψ+ i|,(2.11)

at the Brillouin zone boundaries. In a reduced zone scheme, this energy spectrum
becomes the familiar one-dimensional Bloch band diagram (see figure 2.1).
The introduction of a periodic potential breaks the continuous translational symmetry of the system, and therefore the wave functions are no longer momentum eigenstates. Instead, the eigenfunctions of the Hamiltonian are eigenfunctions of a discrete
translation operator, Td , that translates the entire system by one lattice spacing,ψ(x − d) = Td ψ(x) = e−iqd ψ(x),(2.12)

12and can be labeled by q, the quasi-momentum. A quasi-momentum eigenstate of aperiodic Hamiltonian can be written as the product of a plane wave and a function
periodic in the lattice:ψq(n) (x) = eiqx uq(n) (x),(n)u(n)
q (x − d) = uq (x),(2.13)(2.14)

where n is the band index, and q takes on the values,
−

π
π
≤q≤ .
d
d

(2.15)

This is the essence of Bloch’s theorem, and these eigenfunctions are called Bloch functions.
When these functions are expanded in the atomic momentum basis, the only
momenta contributing to the sum are those that differ from k by a reciprocal lattice
vector, Gj = 2πj/d. The expansion is then given by
ψq(n) (x) =∞i(q+2πj/d)x
X
dk
(n) e
√ eikx ψq(n) (k) =
√
cj
,
2π
2π
−∞
j=−∞

Z ∞

(n)

where the expansion coefficients cj
(n)
cj =

(2.16)are given by

Z ∞
dx
−∞

e−i(q+2πj/d)x (n)√ψq (x).
2π(2.17)

Figure 2.2(a) is an example of a Bloch function in a sinusoidal lattice, and figure 2.2(b)is its Fourier transform.

2.1.1.1

Lowering the symmetry

Since this will be an important concept in the chapter describing gases of ultracold
atoms trapped in rotating optical lattices, we note that the arguments made above canbe used again when an extra periodic potential is added to a system already having a
discrete translational symmetry. If we introduce an extra potential U (x) satisfying
U (x − nd) = U (x),(2.18)13

HaL

HbL 14

0.20

ÈΨ 2

ÈΨ 2

0.15
0.10
0.05
0.00

0

1

2
x Hunits of dL

3

4

12
10
8
6
4
2
0
-2

-1

0

1

2

k Hunits of 2ΠdL

Figure 2.2: The Bloch function for quasi-momentum equal to zero in the first band for
an infinite sinusoidal lattice of lattice depth V0 = −10ER . (a) The real space wave
function is concentrated in the lattice sites. (b) The wave function in momentum space
exhibits peaks at reciprocal lattice vectors. The envelope of these peaks is given by aGaussian for large lattice depths, consistent with a lattice of simple harmonic oscillator
wells.

14
where n ≥ 2 is an integer, and d is the lattice spacing, then Bloch states hx|qi = ψq (x)and hx|q 0 i = ψq0 (x) will mix with each other if q and q 0 differ by 2πm/nd. We canshow this using a calculation entirely analogous to equation 2.8. Eigenstates of the new
Hamiltonian will therefore be linear combinations of these Bloch states. The potential
increases the lattice periodicity and, accordingly, decreases the size of the first Brillouinzone to 2π/nd.
In figure 2.3, the lowest band (red, dashed) of a one-dimensional lattice withan arbitrary periodic potential is plotted in a reduced zone scheme. Introducing an
infinitesimal potential whose periodicity is three times the lattice spacing d, the first
Brillouin zone is reduced to the region −π/3d ≤ q ≤ π/3d. Equation (2.8) showsthat the states lying on a vertical line in figure 2.3 mix with each other under the
addition of the new potential—i.e., hq 0 |Û |qi is non-zero for these states. In particular,
the degenerate states at q = 0 mix, breaking the degeneracy and opening up a gap at
q = 0 (blue, solid). The single band splits into three bands due to the reduction of thesymmetry by U (x).
When the addition of a potential reduces the symmetry of a system as just described, the original eigenstates of the system mix according to their symmetry. As
a consequence, gaps open up where there are degeneracies between states of the same
symmetry, creating a band structure.

2.1.1.2

Wannier states

Bloch functions can be thought of as the replacements for momentum eigenfunctions when there is no continuous translational symmetry. Indeed, in the infinite latticecase, the Bloch function representation diagonalizes the atomic momentum operator restricted to a single band. It is a natural question to ask whether there is an alternative
representation where the functions instead diagonalize the position operator restricted
to a single band. This type of representation is a natural one when we are investigating

15

E Hunits of tL

2
1
0
-1
-2
-1.0

-0.5

0.0

0.5

1.0

q Hunits of Π3dL

Figure 2.3: Lowest band dispersion relation (red, dashed) for a one-dimensional tightbinding model plotted in a reduced zone scheme that anticipates the lowering of thesymmetry via an additional potential with periodicity 3d. After the introduction of thispotential, states with the same quasi-momentum q in the reduced zone mix with eachother, opening up gaps anywhere that there is a degeneracy. This results in a modified
dispersion relation (blue, solid).

16
the physics of atoms hopping between sites of a lattice. The Bloch representation can
obscure this physics since the Bloch functions are delocalized across the entire lattice.
Such a representation exists, and the resulting functions are called Wannier functions. They take the form,
1 X iqxj (n)(n)
wj (x) = √e ψq (x),N q(2.19)

where xj is the position of the center of lattice well j. It is easy to show that these
Wannier functions are just translations of each other, that is,(n)wj (x) = w(n) (x − xj ),(2.20)(n)and that they are orthogonal. In addition, wj (x) is localized to the lattice site j. Itshould be noted here that this localization condition depends on a choice of relative
phases between the different Bloch states. For the most part, this is an issue when performing numerical calculations, and we discuss this problem more carefully in appendix
A.
In figures 2.4(a) and 2.4(b), we have plotted Wannier functions localized at xj =3d for lattices with depth −ER and −15ER , respectively; ER is the recoil energy ER =~2 k 2 /2m. As the lattice depth is increased, the Wannier functions get more localized to
the lattice sites. The Wannier functions approach Gaussians as the lattice depth goesto infinity.

2.1.2

Hubbard models

The Wannier function representation is the natural one in systems where thelattice depth is relatively large. In this case, the atoms are well-described as being
localized to individual lattice sites with tunneling allowed only between adjacent sites.
This is known as tight-binding, and is it in this regime that models such as the BoseHubbard model can be derived. In order to investigate this, we consider the single-

17

HbL 1.2
1.0

HaL 0.5

Èw 2

0.4

0.8

0.3

0.6

0.2

0.4

0.1

0.2

0.0

0.0
0

1

2

3

4

x Hunits of dL

5

6

7

0

1

2

3

4

5

6

7

x Hunits of dL

Figure 2.4: Examples of Wannier functions for sinusoidal lattices. Superimposed onthese plots is the lattice potential in arbitrary units. (a) Wannier function for a latticeof depth V0 = −ER , localized to x = xj = 3d. (b) Wannier function for a lattice of
depth V0 = −15ER , localized to x = 3d.

18
particle Hamiltonian,H(x) = −~2 d2+ V0 cos2 (kL x) ,2M dx2(2.21)

which corresponds to the second-quantized, bosonic Hamiltonian,
Z
Ĥ0 =



~2 d2
2
dx Ψ̂ (x) −+ V0 cos (kL x) Ψ̂(x).
2M dx2†(2.22)

If the atoms interact via a short-range, s-wave interaction, we include a term
Û =

~g
2

Z

dx Ψ̂† (x)Ψ̂† (x)Ψ̂(x)Ψ̂(x),(2.23)

where
g=

4π~as
M

(2.24)

with as the s-wave scattering length.
We expand the field operators in the Wannier states asΨ̂(x) =

X(n)w(n) (x − xj ) b̂j ,(2.25)

n,j

in which case the Hamiltonian can be written as
Ĥ = −XX

X X(mnpr) (m)† (n)† (p) (r)b̂i b̂j b̂k b̂l ,Uijkl(2.26)

dx w(n)∗ (x − xj )H(x)w(n) (x − xk ),(2.27)

dx w(m)∗ (x − xi )w(n)∗ (x − xj )w(p) (x − xk )w(r) (x − xl ).

(2.28)

n0 ,n

j,k

1
(n) (n)† (n)b̂k +2

Jjk b̂ji,j,k,l l,m,n,p

where
(n)

Z

Jjk = −
(mnpr)
Uijkl
= ~g
(n)

Z

(n)

Clearly, Jjk = Jj−k+1,1 .
(1)

(1)

(1)

In figure 2.5, we have plotted J = J2,1 and J3,1 /J2,1 = J2 /J as a function
(1)

(1)

of lattice depth. At about V0 = 5ER , J1,1 /J2,1 = 0.05, and we can neglect all but
nearest-neighbor hopping. This defines the tight-binding regime. Without numericaljustification, we also assume that the dominant interaction term comes only from the(1111)

on-site, single-band interaction U = U1111 . This approximation come about because

19
HbL 0.25

0.15

ÈJ2 È Hunits of JL

ÈJÈ Hunits of ER L

HaL 0.20

0.10
0.05

0.20
0.15
0.10
0.05
0.00

0.00
0

5

10

15

20

0

5

V0 Hunits of ER L

10

15

20

V0 Hunits of ER L

Figure 2.5: Tight-binding tunneling energies for nearest neighbor and next-to-nearest
neighbor hopping, plotted as a function of lattice depth. (a) Nearest-neighbor hopping
parameter J has a maximum of about 0.2ER . This quickly decays as a function of lattice
depth. (b) Next-to-nearest neighbor hopping rate as a function of V0 . At V0 = 5ER ,the ratio |J2 /J| is about 0.05. This is the cut-off for neglecting all but nearest neighbor
hopping, which defines the tight-binding regime.

the Wannier functions are highly peaked about the centers of the lattice wells for large
lattice depths. In this case, the Wannier functions between different sites do not overlap
to a great extent, indicating that the other interaction parameters are small.
We have therefore arrived at the single-band Bose-Hubbard model,Ĥ =UX † †b̂j b̂j b̂j b̂j ,2(2.29)

dx w(n)∗ (x − xj )H(x)w(n) (x − xj ).

(2.30)

Xjb̂†j b̂j −

X
hj,ki

J b̂†j b̂k +

j

where
Z
=

This model has been very well-studied in both the condensed-matter and atomic physics
communities. There is a quantum phase transition that occurs in the (µ/U, J/U )plane—where µ is the chemical potential—between superfluid and insulator phases,
known as the Mott-insulator to superfluid transition [39]. We briefly outline the derivation of this transition in a mean-field approximation.
As described in references [111, 127], we introduce an order parameter ψ = hb̂i iwhich characterizes the superfluid phase of the system. Interpreting this quantity as an20order parameter is a posteriori justified by the calculations performed below and a priori
justified by the expectation of an insulator-superfluid transition: in an insulator statehb̂i i = 0, and in a superfluid state hb̂i i =6 0. Assuming that the quantum fluctuationsb̂i − hb̂i i = bi − ψ(2.31)

are small, we write the Hamiltonian in terms of the order parameter as UX
Ĥj = −zJ (b̂†j + b̂j )ψ − ψ 2 +n̂j (n̂j − 1),
2(2.32)jwhere n̂j = b̂†j b̂j , z is the number of nearest-neighbor lattice sites for site j, and the full
Hamiltonian is a sum over the Ĥj . We now have a set of non-interacting Hamiltonians
that describe each lattice site independently, treating the atoms on all other lattice sites
in a mean-field approximation.
Near the superfluid-insulator transition, the order parameter is small, so we treat
H (1) = −zψJ(b̂† + b̂) + zJψ 2(2.33)

as a perturbation of the on-site Hamiltonian
H (0) = −µn̂ +Un̂(n̂ − 1),
2(2.34)

where µ is the chemical potential. We work in the grand canonical ensemble.
The first-order and second-order corrections to the unperturbed energies
(0) = −µn + U n(n − 1)/2 yield an approximate ground-state energy given byg ≈ −µn +




U
n
n+1
n(n − 1) + zJ + z 2 J 2+
ψ22−µ + U (n − 1) µ − U n= A0 + A2 ψ 2 .

(2.35)(2.36)In order to determine the true ground state and the phase boundaries, we minimize gwith respect to ψ. If A2 > 0, then ψ = 0 minimizes g ; if A2 < 0, then some ψ 6= 0
minimizes g . Thus, setting A2 equal to zero determines the boundary between insulator

21
Μ Hunits of UL

4
3
2
1
0
0.00

0.05

0.10

0.15

0.20

J Hunits of UL

Figure 2.6: The Mott-insulator to superfluid phase diagram as computed in one dimension using mean-field theory. Inside the rounded lobes are insulator ground states ofinteger-filling. Outside these so-called Mott lobes, the system is a superfluid.

and superfluid phases. Solving for µ, this yields
1p1µ = (U (2n − 1) − zJ) ±(zJ)2 − 2zJU (1 + 2n) + U 2 .
22(2.37)We plot these phase boundaries in figure 2.6 for a one-dimensional lattice, i.e.
z = 2. The phase diagram consists of a sequence of lobes. The parameter A2 is positive
within these lobes, so that ψ = 0, and the ground state is an insulator with a fixed
number of particles on each site. Outside of these lobes, A2 < 0, so the ground state isa superfluid.
The Bose-Hubbard model is a jumping-off point for the simulation of a varietyof condensed-matter systems in gases of ultracold atoms trapped in optical lattices. In
this thesis, we investigate two in particular. In chapter 3, atoms trapped in a twodimensional rotating lattice are investigated with an eye to emulating quantum Hall
physics in optical lattices. In chapter 6, a linear shift is added to the lattice, and the
dynamics of atoms near the Mott-insulator regime are investigated.

2.2

Optical Cavities
Our interest in marrying the fields of atomic gases in optical lattices and cavity

quantum electrodynamics (QED) is two-fold. First, the resonant modes in optical cav-

22
ities are isolated from each other, allowing access to the regime where atoms interact
with single electromagnetic field modes. This not only simplifies the mathematical description of the light-matter interaction, but it also allows investigation of fundamental
models from quantum optics such as the Jaynes-Cummings model. In addition, there isan enhancement of the atom-field interaction strength due to the presence of the cavity,
allowing access to the strong-coupling regime of cavity QED, where single photons havean effect on the atomic dynamics. Second, these systems come equipped with an inherent measurement device via detection of the light-field out-coupled through the cavity
mirrors. This can be exploited for schemes for non-destructive measurement of atomic
dynamics in optical lattices. This is discussed in detail in chapters 3 and 6. Here, we
discuss the basic characteristics of optical cavities, the resonant field modes that they
support, and the mathematical treatment of cavity field dynamics in terms of a masterequation.
We consider the simple case of a linear resonator geometry consisting of two
spherical mirrors with equal radii of curvature (see figure 2.7). A laser pumps a resonant cavity mode through the cavity mirrors. The geometry of the resonant mode isdetermined by the geometry of the cavity. These mode functions can be made from linear combinations of free-space Hermite-Gaussian or Laguerre-Gaussian functions, which
are solutions to the paraxial wave equation. These modes are defined by a beam waist
w0 , a resonant frequency ω, and the length L of the cavity.
The presence of the mirrors enforce approximately box-like boundary conditions,in which case the spectral response of the cavity consists of a set of discrete peaks
separated in frequency by the free spectral range ωfsr , as illustrated in figure 2.8. Since
the mirrors are not perfectly reflecting, the modes couple weakly to electromagnetic
field modes outside the cavity. This means that there is a finite decay time 1/κ forphotons in the cavity, and the resonant peaks acquire a linewidth κ. To the extent thatthe linewidths of the cavity and of the in-coupled laser are much smaller than the free

23

R

w0

Η

Κ

Figure 2.7: Schematic of a linear optical cavity. A laser couples with strength η tothe cavity modes through a mirror of reflectivity R. A standing-wave pattern is set up
within the cavity. Photons leak out the cavity through the mirrors at a rate κ.

spectral range, these modes can be populated independently.
In place of the parameters, ω, ωfsr , and κ, cavity properties are often expressedin terms of the quality factor Q and the finesse F. These are defined as
ω
,
κ
ωfsr
F=
.
κ
Q=

(2.38)(2.39)In order to get an idea as to what the state-of-the-art is for cavities, we quote the quality
factor and finesse for an optical cavity used as the reference to stabilize an optical latticeclock [78] and for a microwave cavity used in quantum-non-demolition measurements[46]. For both cavities, Q ≈ 1010 − 109 and F ≈ 105 . Of course, these two cavities
operate in wildly different regimes: the optical cavity operates at 4.2 × 1014 Hz, and themicrowave cavity operates at 5.1 × 1010 Hz. The lifetime of a photon in the microwave
cavity is about 0.1s, whereas in the optical cavity, the lifetime is about 10−4 s.
We make one final note: if the reflectivity R of the mirror is large enough, thenthe finesse takes the approximate form,
√
π R
F≈
.
1−R

(2.40)

This means that the ratio of free spectral range to cavity linewidth is determined solely
by the properties of the mirror and not by any other aspects of the cavity design such

24

Ωfsr
Κ

...

...

Figure 2.8: Schematic of the spectrum of a linear optical cavity. The discrete spectral
peaks are separated by the free spectral range ωfsr and acquire a width κ due to thefinite coupling to modes outside the cavity.

as the length. In other words, the isolation of the resonant modes from each other isdetermined solely by the reflectivity of the mirror.
In the following, we examine the structure of the resonant modes more carefully.

2.2.1

Resonant modes

Electromagnetic fields in free space follow the Helmholtz equation,∇2 + k 2 E(r) = 0,(2.41)

where E(r, t) = E(r)e−iωt , and ω = ck. Due to the mirror geometry discussed above,
we will be concerned with beams that primarily propagate along the axis ẑ of the cavity,in which case we defineE(x, y, z) = f (x, y, z)e−ikz .

(2.42)The Helmholtz equation becomes
∂2f
∂2f
∂2f
∂f
+
+
− 2ik
= 0.
2
2
2
∂x
∂y
∂z
∂z

(2.43)

Having removed the “fast” spatial dependence along ẑ, we make the paraxial approximation,
∂2f
∂2f
∂2f
∂f

,
, 2k
.
∂z 2
∂x2
∂y 2
∂z

(2.44)

25
This approximation only holds as long as the plane waves that have appreciable overlap
with f have wave vectors that lie within an angle θ0 ≈ π/6 of the z-axis [121].
To solve the paraxial wave equation, we separate variables as
fnm (x, y, z) = fn (x, z)fm (y, z),(2.45)

where both fn (x, z) and fm (y, z) satisfy the equation,
∂2f
∂f
− 2ik
= 0.
∂x2
∂z

(2.46)

A set of solutions to this equation are given by√ !
 1/4 r
1 ei(n+1/2)ψ(z)
2x
x2
x2
2
Hn
−
, (2.47)
exp −ik
fn (x, z) =
π
2n n!
w(z)
w(z)
2R(z) w2 (z)where !1/2z 2,w(z) = w0 1 +z0 z 2 
0R(z) = z 1 +
,
z
 
z
ψ(z) = arctan,z0z0 = kw02 /2,(2.48)(2.49)(2.50)(2.51)and Hn is a Hermite polynomial. We reconstruct the full solution as
fnm (x, y, z; k) = fn (x, z)fm (y, z)e−ikz
 r

2 1/2
1
ei(n+m+1)ψ(z) e−ikz=
Lπ2n+m n!m!
w(z)
√ !
√ !


2x
2y
(x + y)2 (x + y)2
× Hn−, (2.52)
Hm
exp −ik
w(z)
w(z)
2R(z)
w2 (z)
called Hermite-Gaussian modes. They satisfy the orthogonality relation,
Z ∞δnn0 δmm0 δkk0 =

Z ∞
dz

−∞

Z ∞
dx

−∞

−∞

dy fn∗0 m0 (x, y, z; k 0 )fnm (x, y, z; k).

(2.53)

From equation 2.52, we can see that the function w(z) determines the width ofa Gaussian function that falls off away from the z-axis. This is a minimum at z = 0

26
HaL 0.0004

HbL

Èu+00 20.00030.00020.0001
0
-2

-1

0

1

2

z Hunits of ΛL0-2 ´104

2 ´104z Hunits of ΛL

Figure 2.9: Longitudinal cross-section and envelope of a Hermite-Gaussian mode asa function of z with w0 = 57λ, L ≈ 5 × 104 λ, and x = y = 0. (a) The mode isapproximately a cosine function near z = 0. (b) The envelope of the mode function
decreases away from z = 0.

and takes on the value w0 . For this reason, w0 is called the beam-waist. The width of
the Gaussian increases as |z| is increased from zero, indicating that the mode spreads
out away from z = 0. This is a diffraction effect that arises because the electric field isbeing confined to some finite region in space.
We construct standing wave modes from equation 2.52 as(x + y)2
kz + k− (n + m + 1)ψ(z) ,
2R(z)
r
√ !
√ !
2
2
1
2x
2y
1
e−(x+y) /w (z)Hnunm (x, y, z) =Hm,
Lπ 2n+m−2 n!m!
w(z)
w(z)
w(z)

u±
nm (x, y, z; k) = unm (x, y, z)



cos
sin

(2.54)(2.55)

where unm determines the envelope and transverse behavior of the mode function. In
2figure 2.9, we plot the longitudinal behavior of the standing-wave mode |u+k00 (x, y, z; k)|for x = y = 0. For this particular choice of parameters, explained below, the envelope|u00 (0, 0, z)|2 is roughly constant over hundreds of wavelengths, meaning that the mode
function is well-described by a cosine function in this region. The envelope eventually
appreciably decreases far enough away from z = 0. In figure 2.10, we plot the transverse
2
behavior of |u+k00 (x, y, z; k)| . The width w(z) of the Gaussian decay of the mode func-tion away from the z-axis increases as |z| is increased. Finally, in figure 2.11, we plot

27

HaL 0.0004

wHzL Hunits of ΛL

HbL 140

Èu+00 20.00030.00020.00010-150 -100 -50

050

x Hunits of ΛL

100

150

120
100
80
60
40
20
0
-2 ´1040

2 ´104z Hunits of ΛL

Figure 2.10: Transverse behavior of the (00) Hermite-Gaussian mode with w0 = 57λand L ≈ 5 × 104 λ. (a) The mode at z = 0 falls off as a Gaussian away from the z-axis
with width given by the beam waist w0 . (b) The effective width w(z) of the Gaussian
increases from w0 as |z| is increased.

28
transverse cross-sections of three Hermite-Gaussian modes. These modes have markedly
different transverse structures but essentially the same longitudinal behavior. We now
turn to the exact formulas for the resonant frequencies.

2.2.1.1

Resonant frequencies and mode parameters

The total phase shift that the Hermite-Guassian modes undergo as they traverse
the cavity is given by
φ(z2 − z1 ) = k(z2 − z1 ) − (n + m + 1)(ψ(z2 ) − ψ(z1 ))= kL − (n + m + 1)(arctan(z2 /z0 ) − arctan(z1 /z0 )).

(2.56)The parameters z2 and z1 are the positions of the right-hand and left-hand mirrors,
respectively. We take them to be z1 = −L/2 and z2 = L/2. The resonance condition
for a standing-wave cavity mode states that the total round-trip phase shift must be an
integral multiple of 2π, leading to


 
 
1
z2
z1
ωqnm = ωfsr q + (n + m + 1)
arctan− arctanπz0z0(2.57)

where ωfsr = πc/L, and q is an integer. Defining
g =1−

L
,
R

(2.58)

where R is the radius of curvature for the mirrors, we can show thatcos−1 (g 2 ) = arctan(z2 /z0 ) − arctan(z1 /z0 ),g=

4π 2 (w0 /λ)4 − (L/λ)2.
4π 2 (w0 /λ)4 + (L/λ)2(2.59)(2.60)

This implies that the resonant frequencies are given byωqnm = ωfsr (q + (n + m + 1)π −1 cos−1 (g 2 )).
These are also the resonant frequencies for the Laguerre-Gaussian beams.

(2.61)

29

HaL

HbL

150

y Hunits of ΛL

100
50
0
-50
-100
-150

HcL

150

HdL

y Hunits of ΛL

100
50
0
-50
-100
-150
-150 -100 -50

050

x Hunits of ΛL

100

150

-150 -100 -50

0

50

100

150

x Hunits of ΛL

Figure 2.11: Cross-sections of sample Hermite-Gaussian modes with√ w0 = 57λ andL ≈ 5 × 104 λ. (a) The (00)-mode at z = 0. (b) The (00)-mode at z = 2z0 ≈ 104 λ. (c)The (20)-mode at z = 0. (d) The (23)-mode at z = 0.

30
2.2.2

Cavity field dynamics and atom-field coupling

We now turn to the quantum description of cavity field dynamics in a cavity. The
Hamiltonian for a set of resonant cavity modes being pumped by an in-coupled laser of
frequency ωP is given byĤ =X~ (ωk − ωP ) â†k,λ âk,λ +

k,λ


X 
∗
~ ηk,λâk,λ + ηk,λ â†k,λ ,(2.62)k,λwhere k = ω/c is the wave-number of the resonator mode corresponding to âk,λ , andλ indexes its polarization. The parameter ηk,λ measures the coupling strength of thein-coupled laser to the resonant mode (k, λ). It is a function of the mirror reflectivity
and the pumping laser intensity. The decay of photons through the mirrors is governedby a Liouvillian,L̂ρ̂ = −X ~κk,λ  †âk,λ âk,λ ρ̂ + ρ̂â†k,λ âk,λ − 2âk,λ ρ̂â†k,λ ,2(2.63)k,λwhere ρ̂ is the reduced density matrix of the cavity field, and κk,λ is the rate at whichphotons in mode (k, λ) decay. The density matrix follows the equation of motion,i~dρ̂= [Ĥ, ρ̂] + iL̂ρ̂.
dt(2.64)

It is apparent from this equation of motion that each mode acts independently. For thisreason, we treat only a single mode â in what follows. This will not be the case whenatoms are allowed to interact with the field.
The field amplitude within the cavity is proportional to hâi = α. This quantity’sequation of motion is given by
i

dα 
κ
= ω − ωP − i
α + η.
dt
2(2.65)

This equation is solved by
α(t) = α(0)e−i(ω−ωP )t e−κt/2 + (1 − e−i(ω−ωP )t e−κt/2 )−η,ω − ωP − iκ/2(2.66)

31
which describes some transient phase evolution e−i(ω−ωP )t of α and a steady state value,α(t → ∞) =−η.
ω − ωP − iκ/2

(2.67)

We turn to the interaction of these modes with a gas of atoms. As we havementioned, we will consider the special cases of a single-mode linear cavity and a twomode, degenerate ring cavity. In addition, we will assume that the cavity field is far
enough detuned from the atomic transition that high-lying excited states of the atoms
can be adiabatically slaved to the lower lying states, as described in appendix B. In
this case, we can write the interaction asZ
ĤI = ~gdz Ψ̂† (z) â†k âk + â†−k â−k + â†k â−k e−i2kz + â†−k âk ei2kz Ψ̂(z)(2.68)

for the ring cavity and†ĤI = ~g â â
for the linear cavity.

Zdz Ψ̂† (z) cos2 (kz)Ψ̂(z)(2.69)

Chapter 3

Bose and Fermi Gases in Rotating Optical Lattices

The results in this chapter were published in the Physical Review A as
Phys. Rev. A 74, 063606 (2006) [11] and in the Journal of Physics B:
Atomic, Molecular, and Optical Physics as J. Phys. B: At. Mol. Phys. 40,3725 (2007) [101].
One of the current thrusts of research in cold quantum gases trapped in optical lattices is demonstrating and investigating strongly-correlated physics. In particular, there
have been a number of theoretical efforts aimed at observing quantum Hall physics insystems comprised of ultracold atomic gases in optical lattices [10, 65]. A promising lineof research takes advantage of a mapping between the Hamiltonian of electrons confinedin two dimensions in the presence of a constant transverse magnetic field and that ofa rotating atomic gas. This has led to various theoretical studies predicting fractional
quantum Hall effect (FQHE) behavior in rotating BEC’s [5, 38, 100, 130]. However,experimentally reaching the parameter regimes necessary to observe this behavior isdifficult [117]. Optical lattices offer a solution to this problem by tightly confining
atoms in the lattice wells, thereby enhancing correlations.
Motivated by this, preliminary exact diagonalization studies of bosonic atoms inrotating lattices were carried out using a Bose-Hubbard model modified by a rotation
term [9]. This work identified quantum phase transitions between states of different
vorticity as the rotation rate was increased. It was then extended to explicit studies ofquantum Hall physics in rotating lattices [10].

33
Further work is required for a physical understanding of the transitions. In thischapter, we present analytical and numerical results for strongly-interacting bosons andnon-interacting fermions in rotating optical lattices. These results are based on the notion of quasi-angular momentum, which is a quantum number for systems with a discrete
rotational symmetry. Quasi-angular momentum is analogous to quasi-momentum forperiodic translationally invariant systems and has been previously used in the context
of rotating ring lattices [19, 20, 126] and carbon nanotubes [105] to label eigenstates.
A formalism for this quantum number for second-quantized systems is presented here
and applied to gases of bosons and fermions in rotating optical lattices. We identify
transitions between states of different symmetry for the ground state of a rotating system. A possible avenue for experimentally detecting these results via the momentum
distribution of the ground state is presented.
The chapter is structured as follows. In section 3.1, we draw the analogy between
systems with a discrete translational symmetry and systems with a discrete rotational
symmetry, thereby generating a Bloch theory for the latter and introducing the notionof quasi-angular momentum. The theory of one particle on a rotating ring lattice is used
to illustrate these ideas in section 3.2. This is extended to a many-body description insection 3.3, where a many-particle, tight-binding model is diagonalized. Specific investigations of quantum gases of strongly-interacting bosons and non-interacting fermionsin ring and square lattice geometries are carried out in section 3.4. In section 3.5, signatures of the quasi-angular momentum in the momentum distribution of a state are
identified. Finally, section 3.6 summarizes the main results of this chapter.

3.1

Bloch Theory for Systems With Discrete Rotational Symmetry
The concepts discussed in section 2.1 hold for systems with symmetries other than

discrete translational symmetry. We can see this by investigating the fact that a moving
N -site linear lattice with periodic boundary conditions is analogous to a rotating N -site34ring lattice. We can make this analogy explicit by considering the Hamiltonian of amoving, one-dimensional, sinusoidal N -site lattice in the co-moving frame, given by
H(x) = −~2 ∂ 2~ ∂+ V0 cos2 (qx) − v,
2M ∂x2i ∂x(3.1)

where v is the velocity of the lattice. In this expression, we have used the fact that ina moving frame,
Ĥ = Ĥ0 − p̂ · v,

(3.2)

where Ĥ0 is the time-independent Hamiltonian in the non-moving frame, and v is thevelocity of the moving frame with respect to the lab frame [69].
The lattice wave-number q can be rewritten as q = π/d, where d is the latticespacing. If periodic boundary conditions,ψ(x + N d) = ψ(x),(3.3)

are included, then the Hamiltonian describes the system illustrated in figure 3.1(a) forthe explicit case of an 8-site lattice. On the other hand, the Hamiltonian of a rotating,sinusoidal N -site ring lattice (figure 3.1(b)) is given in the rotating frame by


φ
~ ∂
~2 1 ∂ 2
2+ V0 cos N−Ω,H(φ) = −2M R2 ∂φ2
2
i ∂φ

(3.4)

where Ω is the rotation frequency, and R is the radius of the ring. The inclusion of theterm −ΩLz has the effect of moving to a frame co-rotating with the lattice [69]. The
two Hamiltonians, equations (3.1) and (3.4), are mathematically identical if we perform
a transformation x = φN d/2π and identify N d/2π with R and v/R with Ω.
Since the Hamiltonians are exactly identical, all of the properties of one-dimensional
systems with a discrete translational invariance carry over for ring systems with a discrete rotational invariance. The analogy can be carried further, starting with the twodimensional free-space solution in polar coordinates,ψj (φ, ρ) = eijφ Rj (ρ),

(3.5)

35
where Rj (ρ) is a radial function, irrelevant for our discussion, and j is an integer. In
the presence of a potential that breaks the rotational symmetry, the eigenstates arelinear combinations of these free-space solutions. If this potential has a discrete N -foldrotational symmetry, its eigenstates can be expanded in the free space solutions as
ψm (φ, ρ) =∞X

i(N j+m)φ
(m) e

aj

j=−∞

√

Nπ

Rj (ρ).

(3.6)

It is evident from this expansion that∞i(N j+m)(φ−2π/N )
X
2π
(m) e
√
ψm (φ −, ρ) =Rj (ρ) = e−i2πm/N ψm (φ, ρ).
aj
N
Nπ

(3.7)

j=−∞

and that therefore ψm (φ, ρ) is an eigenvector with eigenvalue e−i2πm/N of the discrete
rotation operator R2π/N that rotates the system by the angle 2π/N ; R2π/N takes theplace of the discrete translation operator, Td . The analogy is complete when we notethat the eigenstates are linear combinations of angular momentum eigenstates, in whichcase we call the number ~m the quasi-angular momentum of the state, ψm (φ, ρ). For
simplicity, we drop the factor of ~ and call the value m the quasi-angular momentum.

3.2

Single Particle on a Rotating Ring Lattice
As an application of the discussions in the previous section, we examine the prob-

lem of a single particle trapped on a rotating ring lattice. This problem is analyticallysolvable in terms of Mathieu functions, and we can examine the effects of rotation on
both the spectrum and the eigenstates.
The Hamiltonian for a single particle trapped on a rotating ring lattice is given
by equation (3.4). The N lattice sites are situated at


2πnj
,
(ρj , φj ) = R,
N

(3.8)

where j = 1, . . . , N . To simplify the problem, we make a transformation φ → 2φ/N
(φ ∈ [0, N π]) and scale energies by the “recoil energy”,
 2
~2
N
~2  π 2
ER =
=
.
2m 2R
2m d

(3.9)

36

`
Td

1

d
2

3

4

5

6

7

8

1

1
2

8
`
R2 Π8
Φ

3

7

4

6
5

Figure 3.1: Equivalence between a ring lattice and a one-dimensional lattice with periodic boundary conditions. The translation operator T̂d that translates through one
lattice site maps onto the rotation operator R̂2π/8 that rotates the lattice through one
lattice site.

37
In this case, the Schrödinger equation is given byEψ(φ) = ER

∂2
N ∂
v0
v0
− 2 +i ω+cos(2φ) +
∂φ
2 ∂φ
2
2


ψ(φ),

(3.10)

where ω = ~Ω/ER , v0 = V0 /ER , and ψ(φ − N π) = ψ(φ).
In order to take into account the effect of rotation, we “imprint” an angularmomentum on the wave function that coincides with the rotation rate, i.e.
ψ(φ) = eiφωN/4 ϕ(φ).

(3.11)In this case, the Schrödinger equation becomesEϕ(φ) = ER

∂2
v0
v0
N2
− 2+
cos(2φ) +
− ω2
∂φ
2
2
16


ϕ(φ).

(3.12)

Defining
Ev0 ω 2 N 2
−
+
,
ER
2
16
v0
t= ,
4

a=

(3.13)(3.14)

this equation may be re-written as
0=∂2
ϕ(φ) + (a − 2t cos(2φ))ϕ(φ).
∂φ2(3.15)

This is exactly Mathieu’s equation. See appendix A for a discussion of the solutions of
this equation.

3.2.1

Effects of rotation(n)

We know that the eigenfunctions, ψm (φ), of equation (3.15) should satisfy
(n)
(n)
ψm
(φ − π) = e−i2πm/N ψm(φ).

(3.16)(n)ψm(φ) = eiφωN/4 ϕ(n)q (φ).

(3.17)In this case,

38
Thus,
(n)
(n)
(n)
e−i2πm/N ψm(φ) = ψm
(φ − π) = e−iπ(q+ωN/4) ψm(φ).

(3.18)The quantity q is then a function of the quasi-angular momentum number m and therotation rate ω, given by
q → qm (ω) = mod2


2m ωN
−
+ 1 − 1,N
4

(3.19)

in which case


(n)
(n)
(n)
ψm
(φ) = eiφωN/4 Cqm (ω) (φ) + i Sign(q)Sqm (ω) (φ) ,(3.20)

where C and S are Mathieu cosine and sine functions.
We can make two interesting observations about this solution. First, apart fromthe integer values of qm (ω), we may write the energy of one of the quasi-angular momentum states as
v0 ω 2 N 2
(n)
−
.
Em
(ω) = ER akn,qm (ω) +2
16(3.21)(1)

Let’s consider the ground state of the system. At ω = 0, this is clearly the state ψ0 (φ)
sinceq0 (0) = 0 =⇒ akn,qm (ω) = a0 ,(3.22)

which is the smallest Mathieu characteristic value. However, when, say, ω = N82 , wecan see from equation (3.19) that2|q0 (ω)| = mod2 − + 1 − 1 ≥ 0,N

(3.23)|q1 (ω)| = |mod2 (1) − 1| = 0.

(3.24)The m = 1 quasi-angular momentum state is now the ground state. It is clear fromequation (3.19) that this behavior generalizes, and as ω increases, the ground state
cycles through the different quasi-angular momentum states. Since these states do not
mix with each other, this is signaled by exact energy level crossings in the spectrum as39
E Hunits of ER L

1
0
-1
-2
-3
-2

-1

0

1

2

W Hunits of ER ÑL

Figure 3.2: Exact energy-level crossings due to rotation for a four-site sinusoidal latticewith periodic boundary conditions. At zero velocity, the zero quasi-momentum state is
the ground state. At some finite positive velocity, the k = π/2d state is energeticallyfavorable and becomes the ground state. The ground state eventually cycles throughall quasi-momenta.

a function of rotation. This is illustrated in figure 3.2, where the m = −2, −1, 0, and1 states in the first band for a four-site lattice are plotted as a function of the rotation
speed.
This behavior is general, and we can understand it by invoking first-order perturbation theory. In a stationary one-dimensional ring lattice, the zero quasi-angularmomentum Bloch function is always the ground state. When the lattice is rotating,the energetically favored state is one that rotates along with the lattice. In order toinvestigate this phenomenon, we consider the Hamiltonian,Ĥ = Ĥ0 − ΩL̂z ,(3.25)

where Ĥ0 is the time-independent Hamiltonian in the non-moving frame. The groundstate of Ĥ is the ground state in the lab frame but written in rotating frame coordinates.
We consider an eigenstate of Ĥ0 , ψk (x). In first-order perturbation theory, theenergy Ek of an eigenstate ψk (x) of Ĥ0 is shifted by the rotation term to(0)

Ek = hk|Ĥ0 − ΩL̂z |ki = Ek − Ωhk|L̂z |ki.

(3.26)

At zero rotation rate, the ground state is a zero quasi-angular momentum state, ψ0 (x).
However, the energy of a state that has a positive average angular momentum, hk|L̂z |ki >

40
0.20

ÈΨ 2

0.15
0.10
0.05
0.00
0

2

4

6

8

10

x Hunits of dΠL

Figure 3.3: The m = 0 Bloch function at ω = 0 (black, solid) and ω = 1 (blue, dashed).
The Bloch function is squeezed tighter within the wells for larger rotation rates.

0, falls below that of ψ0 (x) for large enough Ω. This change is signaled by an exact energy level crossing in the energy spectrum between two quasi-angular momentum states(see figure 3.2). As the rotation rate is increased, the effect of the rotation term is toreorder the energies of the quasi-momentum states.
The second observation we can make is about the Bloch function solutions themselves. In figure 3.3 are plotted the densities for the m = 0 state in the first band fortwo different rotation speeds. The Bloch function with the larger rotation speed, ω = 1,
is more localized to the lattice sites. In fact, we can observe that the ground state isalways the state that is least-localized to the lattice sites. This effect may be understoodin the following way. A particle that is moving slower than the lattice will be pulled
along, effectively “piling up” the wave function against the side of the lattice well. This
simultaneously decreases the width of the wave packet within the well and increases the
kinetic energy. A particle at the same speed as the lattice will not feel this effect, andits wave function will not be “squeezed” by the lattice wells. This is the physical basisof both the wave function for m = 0 being more localized for higher rotation rates andfor the exact energy level crossings in the spectrum as a function of rotation.

41

3.3

Many-Body Description of Quasi-Angular Momentum
The symmetry considerations of the previous sections necessarily hold for many-

particle systems where the underlying single-particle Hamiltonian has a discrete rotational symmetry. The eigenstates of the many-particle Hamiltonian can then be chosen
to be quasi-angular momentum eigenstates. Alternatively, it is sometimes possible tore-cast the symmetry properties of the eigenstates in terms of the creation and annihilation operators in a second quantized formalism. In this section, we examine thespecial case of a Hubbard Hamiltonian for non-interacting fermions, which for instancedescribes fermions in a one-dimensional optical lattice in the tight-binding regime.
The Hamiltonian takes the simple form,
Ĥ =

N
X

â†j âj −

j=1

N 
X


J ∗ â†j+1 âj + J â†j âj+1 ,(3.27)

j=1

where âN +1 = â1 in order to satisfy periodic boundary conditions, andhâi , â†ji

= δij ,

(3.28)[âi , âj ]+ = 0,(3.29)+

so that the particles are fermions. This Hamiltonian can be diagonalized using operator
methods. To this end, we define new fermionic operators by means ofN

1 X −i2πmk/N
dˆm = √
e
âk ,
N k=1

(3.30)N −1

1 X −i2πmk/N ˆâk = √
e
dm .
N m=0

(3.31)

Under this canonical transformation, the on-site energy term becomes
Ĥ0 =

N
X

â†j âj = 

j=1

N
−1
X

dˆ†m dˆm ,(3.32)


2πm
−φ ,
N

(3.33)

m=0

and the hopping term becomesĤ1 = −2t

N
−1
X
m=0

dˆ†m dˆm cos

42
where t and φ are defined by the relation J = teiφ . The eigenstates of this Hamiltonian
are then|n0 , . . . , nN −1 i = (d†0 )n0 · · · (d†N −1 )nN −1 |0i,

(3.34)

where |0i is the vacuum, nm can take on the values 0 or 1, and the corresponding
eigenvalues are
En0 ,...,nN −1 =

N
−1
X


nm

m=0


2πm
 − 2t cos(
− φ) .
N(3.35)

As we can see from its solutions, this Hamiltonian is exactly the many-body generalization of the tight-binding model for atoms hopping around a one-dimensional lattice.
From the expansion of dˆ†m in the â†k ’s we can see that d†m adds one particle tothe system, populating each site with equal probability but differing phase. We canconclude that particles created by dˆ†m carry quasi-angular momentum. To make thisstatement exact, we define a discrete rotation operator R̂ by
R̂â†k R̂−1 = â†k+1 ,(3.36a)R̂|0i = |0i.

(3.36b)This operator commutes with the Hamiltonian, and acting it on dˆm yields
NN
1 X i2πmk/N † −1
1 X −i2πm/N i2πm(k+1)/N †R̂dˆ†m R̂−1 = √R̂âk R̂ = √
e
e
e
âk+1
N k=1
N k=1

= e−i2πm/N dˆ†m .

(3.37)

Therefore,R̂(d†1 )n1 · · · (d†N )nN |0i = exp

−i2π

N
−1
X
m=0

mnm
N

!
(d†1 )n1 · · · (d†N )nN |0i.

(3.38)

This state carries quasi-angular momentum,m = modN

N
X

!
mnm

.

(3.39)

m=1

A pictorial representation of this discussion is shown in figure 3.4 for the specific caseof one particle in an eight-site ring lattice.

43

HaL

HbL

2

2

1

1

3

3

`
R2 Π8
8

8
4

4

7

7
5

5
6

ÈΨ\

6

`
R2 Π8 ÈΨ\Figure 3.4: Illustration of the rotation operator in the many-body formalism. (a) The
operator dˆ†1 adds one particle to the system, populating every site with equal probability
but a phase that differs by 2π/8 between lattice sites. The state that is illustrated is|ψi = dˆ†1 |0i; arrows indicate phase. (b) When this state is rotated by one lattice site, theresulting state is R̂2π/8 |ψi. Subtracting the phases at a site before and after rotation,
we get φ = −2π/8. The picture then illustrates the fact that |ψi is an eigenstate ofR̂2π/8 with eigenvalue e−i2π/8 .

44Armed with an understanding of the eigenstates of this system, it is straightforward to compute physical quantities such as the energy. The ground state of the systemis particularly easy to find, as we simply populate the lowest energy levels to create a“Fermi sea”.
If there is an additional potential that reduces the symmetry of the problem,
quasi-angular momentum can still be exploited to simplify the problem. The dˆm operators act as a natural basis in which to describe problems that have some rotational
symmetry. For instance, we can consider an additional potential that is some arbitrary
function of the creation and annihilation operators on an N -site lattice,
V̂ =

N
X

Vjk dˆ†j dˆk ,(3.40)

j,k=1∗ . We suppose for simplicity that this potential is only 4-fold symmetric,
with Vjk = Vkj

i.e. it is symmetric under rotation by N/4 sites, effected by the operator R̂N/4 . This
limits the possible terms dˆ†j dˆk that can appear, which we can see by computing
R̂N/4 V̂ R̂−N/4 =N
X

Vjk R̂N/4 dˆ†j R̂−N/4 R̂N/4 dˆk R̂−N/4 =

j,k=1

N
X

Vjk eiπ2(j−k)/4 dˆ†j dˆk . (3.41)

j,k=1

In order for the potential to be invariant under this four-fold rotation, we must havethat eiπ2(j−k)/4 = 1, in which case mod4 (k − j) = 0. Any terms involving dˆ†j dˆk where
this equation is not satisfied are zero. We define
k = 4l1 + m1 ,(3.42a)j = 4l2 + m2 ,(3.42b)li = 1, . . . , N/4,

(3.42c)mi = 0, 1, 2, 3,(3.42d)which allows us to write the potential as
V̂ =

N/4
3
X
X

(m)
Vl1 ,l2 dˆ†4l1 +m dˆ4l2 +m ,(3.43)

m=0 l1 ,l2 =1(m)

Vl1 ,l2 = V4l1 +m,4l1 +m .

(3.44)

45
This re-writing automatically sorts the operators appearing in V̂ according to their fourfold symmetry, which is the one relevant to the new problem. The full Hamiltonian Ĥ+V̂
is then block-diagonal in the four-fold quasi-angular momentum, m, making numericalsolutions to this problem more tractable. This scheme can be used to facilitate thesolution for a system of atoms in a rotating two-dimensional lattice.

3.4

Bose and Fermi Gases in Rotating Optical Lattices
Interacting bosons in one dimension enter the Tonks-Girardeau regime when in-teractions get very strong [45, 75]. The gas takes on many of the characteristics of a gasof non-interacting fermions; for instance, the number density in position space is thesame in both cases. It differs in certain aspects, however; for instance, the momentum
distribution is more highly peaked about a central value for strongly-interacting bosons
than it is for fermions [61]. It then becomes particularly interesting to compare the casesof strongly-interacting bosons and non-interacting fermions in rotating lattices. In thissection, gases of non-interacting fermions and hard-core bosons in rotating optical lattices are investigated. The many-particle formalism for quasi-angular momentum states
discussed previously is used to monitor the symmetry of the ground state as a functionof rotation. The second-quantized Hamiltonians described below are generalizations of
those found in reference [37].
The Hamiltonian for interacting bosons or fermions in a rotating, two-dimensional,
square, optical lattice is
Ĥ =

XZ

d x Ψ̂†σ (x)
2

σ

+

X Z

d2 x

Z



~2 2
−
∇ + VL (x) − ΩLz Ψ̂σ (x)2MdD x0 Ψ̂†λ (x)Ψ̂†λ0 (x0 )Vλλ0 ,µµ0 (x, x0 )Ψ̂µ0 (x)Ψ̂µ (x0 ),(3.45)

λλ0 ,µµ0

where
VL (x) = V0 cos2 (kL x) + V0 cos2 (kL y),(3.46)

46
Vλλ0 ,µµ0 (x, x0 ) is a two-particle interaction that is possibly spin-dependent, λ, µ, and σ
are spin indices, and Ψ̂σ (x), Ψ̂†σ (x) are field operators satisfying the (anti-)commutation
relations,hiΨ̂σ (x), Ψ̂†σ0 (x0 ) = δσσ0 δ(x − x0 ),(3.47)hiΨ̂σ (x), Ψ̂σ0 (x0 ) = 0.

(3.48)±±The two-particle interaction Hamiltonian, Vλλ0 ,µµ0 (x, x0 ), is chosen to be an s-wave
contact-interaction.
A Hubbard Hamiltonian can be derived in a lowest-band, tight-binding approximation by expanding the field operators in the lowest-band Wannier functions, w(x−xj )[64]. A basis better suited in the presence of rotation is given by the modified Wannierfunctions,M
Wj (x) = exp −i
~

Z x

!
A(x0 ) · dx0w(x − xj ),(3.49)

xj

where A(x0 ) = Ω × x0 . This modification captures the effect of rotation at low rotation
speeds [11]. However, as we have seen, at higher rotation speeds, the density of the
Wannier functions changes along with the phase, and in this case the approximationneeds to be further modified.
The result for spinless bosons is [10]
X



1
2
J + M Ω A1
ĤB = −
e−iφij â†i âj + eiφij â†j âi
2
hi,ji

X

1
UX † †+
 − M Ω2 rj2 + A2 â†j âj +âj âj âj âj .
22
j(3.50)j

A similar derivation for spin-1/2 fermions yields
X 



1
2
ĤF = −J + M Ω A1
e−iφij â†i,σ âj,σ + eiφij â†j,σ âi,σ2hi,ji,σX †1UX † †
22+
 − M Ω rj + A2 âj,σ âj,σ +âj,↑ âj,↓ âj,↓ âj,↑ .
2
2
j,σ

j

(3.51)

47
For both cases, i, j are site indices with hi, ji indicating a sum over only nearest neighbors, and ri is the radial position of the i’th site. The phase φij is given by the expression,
Z
M xi
MΩ
φij =
A(x0 ) · dx0 =(xi yj − xj yi ).
(3.52)~ xj~
The parameters , J, and U are the on-site zero-point energy, the hopping parameter, and the on-site interaction energy, respectively, as computed in chapter 2. The
parameters A1 and A2 arise due to the phase factor in equation (3.49) and are given byZ
A1 = dx w∗ (x − xi )(x − xi )2 w(x − xj ),(3.53)
Z
A2 = 2 dx w∗ (x − xi )(x − xi )2 w(x − xi ),(3.54)
where w(x − xi ) are one-dimensional Wannier functions. All of these parameters canbe numerically evaluated for a lattice of specific shape, period, and depth [10]. If aharmonic trap of trapping frequency ΩT is included, we replace every instance of Ω2with Ω2 − Ω2T .
3.4.1

Strongly-interacting bosons

In this section, we describe hard-core bosons in a rotating ring lattice. The
quasi-angular momentum of an eigenstate for a one-dimensional system is then definedand monitored as a function of rotation speed. The analytic results derived here areconsistent with a previous numerical treatment of a two-dimensional system [11].
In the limit of very strong interactions, U → ∞, a gas of bosons enters the hardcore boson regime. In an optical lattice, this regime can be characterized by using a
number basis where the occupation number of each site is either l or l + 1, with l an
integer. We define new operators b̂j that satisfy the relations,hb̂j , b̂†jib̂j , b̂ji

= 1,

(3.55a)= 0,(3.55b)hib̂i6=j , b̂†j = 0.

(3.55c)

h

+

−

−

48
and
â†j âj = b̂†j b̂j + l,

(3.56a)â†j âj±1 = (l + 1)b̂†j b̂j±1 .

(3.56b)

The particles described by these operators have boson-like properties except for
the fact that no more than one particle can occupy a single site, which is enforced byequation (3.55a). In terms of these new operators, the Hamiltonian for hard-core bosons
on a ring lattice is given by
N

N

j=1

j=1

X †
X
1
Ĥ = U N l(l − 1) + U lb̂j b̂j +(ω)b̂†j b̂j2− J(l + 1)NX

e−iπωN/4 b̂†j b̂j+1 + H.c.,(3.57)

j=1

where
(ω) =  −

N 2ω2
.
16

(3.58)

The first two terms respectively represent the energy of a particle on site j interactingwith the l particles already there and the interaction energy of the l particles on site
j summed over all sites. For practical purposes, U = 100J, where J is the tunnelingenergy, is large enough to enter this regime [11].
The mixed commutation/anti-commutation relations of equation (3.55) make thisHamiltonian hard to interpret, since we have neither bosons nor fermions. However, for
one-dimensional problems of this sort, we can perform a second transformation of theannihilation operators that transforms this Hamiltonian into a purely fermionic one.
This is called the Jordan-Wigner (JW) transformation.
The essence of this transformation is to include a factor of ±1 in the definitionof b̂j , where the sign depends on the number of particles to the “left” of the site j. It
is standard to first map this Hamiltonian onto an N -spin- 21 Hamiltonian [66, 116], but
we prefer to work strictly with the creation and annihilation operators.

49
The JW transformation is given by defining new annihilation operators,ĉj = b̂j eiπ

Pj−1 †
i=1 b̂i b̂i,(3.59)

These operators are fermionic, as we show in appendix A. The Hamiltonian can thenbe written as
N

N

j=1

j=1

X †
X
1
Ĥ = U N l(l − 1) + U lĉj ĉj +(ω)ĉ†j ĉj2− J(l + 1)N−1X

e−iπωN/4 ĉ†j ĉj+1 + H.c.

j=1

− J(l + 1)e−iπωN/4 ĉ†N ĉ1 (−1)N̂ +1 + H.c.

(3.60)

This Hamiltonian is manifestly not strictly quadratic in the ĉ operators since N̂ appears
in an exponential. However, as we will soon see, the fact that N̂ is conserved allows us
to diagonalize the Hamiltonian in even- and odd-number particle subspaces separately.
Before we move on, we note that for two-dimensional lattices, the remaining phasefactors are non-trivial, and therefore the JW transformation is not useful in this case.

3.4.1.1

Diagonalization

The Hamiltonian just derived has a form identical to that of equation (3.27)
except for the factor of (−1)N̂ +1 . This factor is a constant within a constant particlenumber subspace. For this reason, we replace this factor with (−1)n+1 , where n is thenumber of particles, and diagonalize Ĥ as if it were subspace-independent, resulting in
two separate sets of fermionic operators. At the end of the derivation, we describe how
the normal picture of fermionic excitations is modified by this procedure.
We modify the definition of the operators form section 3.3, defining two sets of50
fermionic operators asN

1 X −iπ2mj/N
dˆ(o)
e
ĉj ,
m = √
N j=1

(3.61)N

1 X −iπ(2m−1)j/N
e
ĉj .
dˆ(e)
m = √
N j=1

(3.62)The Hamiltonian in terms of the dˆm operators is given by
N −1

X
1
ˆ(σ)
Ĥ = U N l(l − 1) +Em dˆ(σ)†
m dm ,
2
m=0


2
2
πfm πNN ω− 2(l + 1)t cos−
ω ,Em = U l +  −
16
N
4


 2m − 1 if n = even
fm =
,

 2m
if n = odd(3.63a)(3.63b)(3.63c)where n is the number of particles, and σ = e for n even and σ = o for n odd.
The eigenstates of the system are given by
 
n0

n

ˆ(e)† N −1 |0i if PN −1 nj = even
 dˆ(e)†
·
·
·
d
0
m=0
N −1
|n0 , . . . , nN −1 i =
,




n
n
0
P
N
−1

N −1
 dˆ(o)†
ˆ(o)†
·
·
·
d
n
=
odd
|0i
if
0
m=0 j
N −1

(3.64)and the corresponding eigenenergies are
1
En0 ,...,nN = U N l(l − 1)
2



N
−1
X
N 2ω2
πfm πN−
nm U l +  −− 2(l + 1)t cos
−
ω
.
16
N
4

(3.65)

m=0

We note that states that mix the two operator definitions are not eigenstates, e.g.
(e)† (o)†|ψi = dˆ0 dˆ1 |0i(3.66)

is not an eigenstate of Ĥ.
In order to complete the diagonalization procedure, we need to interpret dˆ(σ)†
as an operator that creates a particle in the quasi-angular momentum state m. As in
the previous section, we can do this by first defining a rotation operator. Due to the†many phases eiπĉk ĉk in this problem, it is not a priori obvious that the rotation operator

51
as defined in section 3.3 is a convenient choice for the definition of the quasi-angularmomentum. For this reason, we consider the Hamiltonian in terms of the JW fermions,
equation (3.60), rather than the original. DefiningR̂ĉ†j6=N R̂−1 = ĉ†j+1 ,(3.67a)R̂ĉ†N R̂−1 = (−1)n+1 ĉ†1 ,R|0i = |0i,

(3.67b)(3.67c)we have thatN −1

1 X iπfm k/N † −11R̂dˆ†m R̂−1 = √R̂ĉk R̂ + √ eiπfm N/N R̂ĉ†N R̂−1
e
N k=1
N
N −1

=e

−iπfm /N

1 X iπfm (k+1)/N †
√
e
ĉk+1
N k=1

1
+ e−iπfm /N eiπfm /N √ eiπfm (−1)n+1 ĉ†1 .
N(3.68)

Noting that eiπfm = (−1)n+1 , this becomesN

1 X iπfm k/N †1R̂dˆ†m R̂−1 = e−iπfm /N √eĉk + e−iπfm /N √ eiπfm /N ĉ†1
N k=2
N
= e−iπfm /N dˆ†m .

(3.69)

Thus,
N −1
 n0
π X
†
†
nN −1
R̂ d0
nm fm
· · · (dN −1 )|0i = exp −iN!
(d†0 )n0 · · · (d†N −1 )nN −1 |0i. (3.70)

m=0

The quasi-angular momentum of this state is defined to be

P


N −1
n
 modNmn−
if n = even
m
m=0
2
m=
.



 modN PN −1 mnm
if n = odd
m=0
3.4.1.2(3.71)

Results

We are now in a position to use this formalism to investigate the ground state of
the system. The ground state is determined by populating the lowest energy levels inHbL -0.08

0.02

-0.10

E Hunits of ER L

0.04

E Hunits of ER L

52
HaL

0.00
-0.02
-0.04
-4

-2

0

2

4

q Hunits of ΠdL

-0.12
-0.14
-0.16
-4

-2

0

2

4

q Hunits of ΠdL

Figure 3.5: Ground state of three strongly-interacting bosons in a rotating 8-site ringlattice. Solid curves are guides for the eye. (a) At Ω = 0, the three lowest singleparticle energies carry quasi-angular momentum m = −1, 0, and 1. The three-particle
ground state therefore has quasi-angular momentum m = mod8 (−1 + 0 + 1) = 0. (c) AtΩ = 0.14ER /~, the three lowest single-particle energies carry quasi-angular momentum
m = 0, 1, and 2. The three-particle ground state therefore has quasi-angular momentum
m = mod8 (0 + 1 + 2) = 3.

the way standard for fermions. Equation (3.63b) is the single-particle energy spectrum
Em . The qualitative behavior of the system does not depend on the filling l and wetherefore set l = 0; in addition, the zero-point energy  has no physical effect, so we set
it to zero.
As ω is increased, the minimum value of the cosine moves to the right, as illustrated in figure 3.5. The ground state is then determined by populating the lowest
energy dˆ†m dˆm eigenstates. In figures 3.5(b) and 3.5(c), this process is illustrated for thecase of three particles in an eight-site ring.
For an odd number of particles, n = 2p + 1, if the minimum of the cosine functionin Em is near m = m0 , we populate the states m0 , m0 ±1, . . . , m0 ±p. The quasi-angularmomentum of the state is thenpXm = modN (m0 + l) = modN ((2p + 1)m0 + 0) = modN (m0 n).

(3.72)

l=−p

For an even number of particles, n = 2p, when the minimum of the cosine function is
between m0 and m0 + 1, we populate the states m0 − p + 1, m0 − p + 2, . . . , m0 + p.

53
The quasi-angular momentum of the state is then


p

X
n
n
m = modN (m0 + l) −  = modN 2pm0 + p −= modN (m0 n). (3.73)22l=−p+1Thus, as rotation is increased from ω = 0, the quasi-angular momentum cycles throughthe values,m = modN (jn) :j = 0, 1, 2 . . .

(3.74)

We make one final note for the case of bosons. If there is an infinitesimal potential,V̂ , that breaks the symmetry of the original Hamiltonian, Ĥ0 , down to a four-fold rotational symmetry, then the N -fold quasi-angular momentum is no longer a good quantum
number. We treat the problem perturbatively. Tiny band gaps in the single-particle
energy spectrum, equation (3.63b), open up at the new Brillouin zone boundaries. The
eigenstates of Ĥ0 are independent of Ω. Therefore, to first order in V̂ , the energy level
shifts are independent of Ω, and the eigenstates remain unchanged. The effect of rotation is then exactly the same as in the non-perturbed case, and the quasi-angularmomenta have the same behavior as a function of rotation. However, since the old
quasi-angular momentum is no longer a good quantum number, we have to takem → mod4 (m).

(3.75)

These results are consistent with the numerical treatments in references [9, 11].

3.4.2

Non-interacting fermionsMoving beyond the bosonic systems previously studied, we consider non-interacting
fermions in rotating optical lattices. Because of the JW mapping, this system shouldhave properties similar to those of the bosonic case. In addition, by considering fermions,
we move closer to the system originally displaying quantum Hall physics, that of a twodimensional gas of electrons moving in the presence of a constant transverse magnetic
field.

54
3.4.2.1

Ring lattice

Here, we describe non-interacting fermions in ring lattices and compare to theboson case. The quasi-angular momentum of the ground state changes as rotation isincreased. The allowed values of this quantity for fermions differ from those of the bosoncase.
For single-species, non-interacting fermions, we drop the spin index, and the
Hamiltonian for a ring lattice is given by

N 
N
X
X
N 2ω2 †Ĥ =−ĉj ĉj − J
e−iπωN/4 ĉ†j+1 ĉj + H.c.
16
j=1

(3.76)

j=1

With the identifications,
→−

N 2ω2
,
16

(3.77a)J → teiπωN/4 ,(3.77b)this Hamiltonian is identical to the one discussed in section 3.3, and it can be written
in the form
Ĥ =

N
−1
X

Em dˆ†m dˆm ,(3.78a)m=0

Em =  −N 2ω2− 2t cos16

2πm ωN
−N
4


,(3.78b)N

1 X i2πmk/N
dˆm = √
e
ĉk .
N k=1

(3.78c)

The calculation of the quasi-angular momentum of an odd particle-number eigenstate is identical to that of equation (3.72). However, for an even particle-number
eigenstate, the n/2 in equation (3.73) does not appear, and we have


 

p
X
1
n . (3.79)m = modN (m0 + l) = modN (2pm0 + p) = modNm0 +2
l=−p+1Thus, the quasi-angular momentum assumes the values,
modN (nj)
if n = oddm=, modN (n(j + 1/2)) if n = even(3.80)

55
as ω is increased.
As expected, the single-particle cases for both bosons and fermions are identical.
It turns out that all cases of odd numbers of particles also coincide. However, there isan interesting distinction between the two cases when the number of particles in thesystem is even (compare equations (3.73) and (3.79)). For instance, a system of two
fermions in a four-site lattice cycles between quasi-angular momenta 1 and 3 whereasa system of two bosons cycles between values 0 and 2. This is one way in which noninteracting fermions and strongly-interacting bosons in one dimension differ despite the
JW transformation that maps between the two cases.

3.4.2.2

Two-dimensional square latticeDifferences between fermions and bosons remain when considering two-dimensional
systems. Here we describe non-interacting fermions in a two-dimensional 2P ×2P squarelattice, which consists of P square ring lattices, as illustrated in figure ?? for P = 2.
The Hamiltonian is given by


X
1
2
Ĥ = −J + M Ω A1
e−iφij ĉ†i ĉj + eiφij ĉ†j ĉi
2
hi,ji

X
 †
1
2
2
+
 − M Ω rj + A2 ĉj ĉj .
2(3.81)j

We numerically diagonalize this Hamiltonian for one particle in a 16-site lattice. We thendetermine the quasi-angular momentum of the ground state as a function of rotation rate
for different particle numbers by filling up the lowest-energy states in the way standard
for fermions. Since the states are sorted according to their quasi-angular momentum, we
can directly compute the quasi-angular momentum of the eigenstate. We illustrate thisprocess for the case of n particles in a sixteen site lattice. In figure 3.6(a) is plotted thesingle-particle energy spectrum for the Hamiltonian in the absence of hopping between(2,1)
the rings (Jˆm = 0) and Ω = 0. When the hopping is “turned on” (figure 3.6(b)), levelcrossings become avoided crossings due to the mixing of states with the same four-fold

56
quasi-angular momentum. In figure 3.6(c), the case for zero hopping between rings isplotted for Ω = 0.05ER /~, whereas in figure 3.6(d), the hopping has turned on, and the
avoided crossing behavior is again observed.
In figure 3.7 are plotted the quasi-angular momenta of the ground state as afunction of rotation speed for non-interacting fermions in a sixteen-site lattice. Similar
to the ring lattice case, the results for odd numbers of particles are identical to thosefor hard-core bosons, but differ for even numbers of particles. There are however otherinteresting properties that do not appear in the ring lattice case. The most obvious
such property in 3.7 is the lack of particle-hole symmetry. The rate at which transitions
between quasi-angular momenta occur as a function of rotation speed is much higher forsmall particle numbers. This is easy to understand once we see that holes have a negative
effective mass, meaning that they move in the opposite direction as particles and alsoseek regions of higher potential. Due to the effective inverse-parabolic potential in thethird term of equation (3.81)—a result of the centrifugal barrier—particles are pushed
to the outer ring of the lattice whereas holes are pushed to the inner ring. Transitions
between quasi-angular momenta occur faster as a function of rotation speed for theouter lattice rings as compared to the inner lattice rings, since the tangential speedof particles is greater at larger radii. Thus, for small particle-numbers (1, 2, 3) where
the particles are mostly confined to the outer ring, the transitions occur faster than forlarge particle numbers (9, 10, 11) where the outer ring is filled, and the dynamics occurs
mainly on the inner ring due to Pauli exclusion.
This argument is further verified in figure 3.8, where a strong trapping potentialof trap frequency ΩT = 2ER /~ reverses the situation. Since Ω < ΩT in this series ofplots, the particles are first attracted to the center of the lattice, so that the dynamics
for small numbers of particles occur on the inner ring, and thus the transitions between
quasi-angular momentum states occur further apart in rotation rate. Finally, in figure
3.9 where we include a trapping potential with ΩT = Ω so that the lattice is flat at

57

-2

-1

0

1

2

E Hunits of ER L

HaL 0.06
0.04
0.02
0.
-0.02
-0.04
-0.06
-0.08-2

-1

0

1

2

E Hunits of ER L

HcL 0.06
0.04
0.02
0.
-0.02
-0.04
-0.06
-0.08
-2

-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

HbL

HdL

-1

0
m

1

2

m

Figure 3.6: Single-particle energy spectrum in a 4 × 4 square lattice for (a) zero rotationand zero hopping between inner and outer rings, (b) zero rotation and non-zero hopping,(c) Ω = 0.05ER /~ and zero hopping, and (d) Ω = 0.05ER /~ and non-zero hopping
between inner and outer rings. The dashed and solid line are aids to the eye thatindicate the band. In (a) and (c), the red dashed line is the spectrum for the inner
4-site lattice.

58
all rotation speeds, the system exhibits particle-hole symmetry up to the fact thatparticles and holes travel in opposite azimuthal directions. This can be seen by for
instance comparing figures 3.9(a) and 3.9(e) where the plots are identical up to the factthat they cycle through quasi-angular momenta in opposite directions as a function ofrotation speed.
The case of ΩT = Ω is particularly important for quantum Hall physics, as this is
the condition necessary to reach the degenerate lowest Landau level. In this regime, phenomena such as the Hofstadter butterfly energy spectrum and the quantized resistancecharacteristics of quantum Hall systems arise [10].

3.5Experimental Signatures of Quasi-Angular Momentum
While the quasi-angular momentum m is a good quantum number and a useful

tool in investigating the symmetry properties of the ground state, it is not a quantitythat can be directly measured in experiments. Instead, experiments are routinely performed in which time-of-flight, far-field, density images of the gas contain information
about the momentum distribution before expansion. As a first step towards detecting
quasi-angular momentum states, we look for a signature of quasi-angular momentum inthe momentum distribution of a state.
The momentum distribution of the many-body state, |ψi, is given byn(k) = hψ|Ψ̂† (k)Ψ̂(k)|ψi,(3.82)

where Ψ̂(k) is the Fourier transform of the field operator Ψ̂(x). Recalling that we expand
the field operators asΨ̂(x) =

Xjâj Wj (x) =Xjâj e−iφij w(x − xj ),(3.83)

59

HaL 3

HbL

HcL

HdL

HfL

HgL

HhL

HjL

HkL

HlL

HnL

HoL

HpL

m

2
1
0

HeL 3

m

2
1
0

HiL 3

m

2
1
0

HmL3

m

2
1
0
-1

-0.5

0.

0.5

W Hunits of ERÑL

1.

-1

-0.5

0.

0.5

W Hunits of ERÑL

1.

-1

-0.5

0.

0.5

W Hunits of ERÑL

1.

-1

-0.5

0.

0.5

W Hunits of ERÑL

Figure 3.7: Quasi-angular momentum as a function of rotation for 1 - 16 particles ina sixteen-site lattice in the absence of a trapping potential. The number of particles is(a) 1, (b) 2, (c) 3, etc.

1.

60

HaL 3

HbL

HcL

HdL

HfL

HgL

HhL

HjL

HkL

HlL

HnL

HoL

HpL

m

2
1
0

HeL 3

m

2
1
0

HiL 3

m

2
1
0

HmL3

m

2
1
0
-1

-0.5

0.

0.5

W Hunits of ERÑL

1.

-1

-0.5

0.

0.5

W Hunits of ERÑL

1.

-1

-0.5

0.

0.5

W Hunits of ERÑL

1.

-1

-0.5

0.

0.5

W Hunits of ERÑL

Figure 3.8: Quasi-angular momentum as a function of rotation for 1 - 16 particles ina sixteen-site lattice in the presence of a strong trapping potential, ~ΩT = 2ER . The
number of particles is (a) 1, (b) 2, (c) 3, etc.

1.

61

HaL 3

HbL

HcL

HdL

HfL

HgL

HhL

HjL

HkL

HlL

HnL

HoL

HpL

m

2
1
0

HeL 3

m

2
1
0

HiL 3

m

2
1
0

HmL3

m

2
1
0
-1

-0.5

0.

0.5

W Hunits of ERÑL

1.

-1

-0.5

0.

0.5

W Hunits of ERÑL

1.

-1

-0.5

0.

0.5

W Hunits of ERÑL

1.

-1

-0.5

0.

0.5

W Hunits of ERÑL

Figure 3.9: Quasi-angular momentum as a function of rotation for 1 - 16 particles ina sixteen-site lattice in the presence of a trapping potential equal in strength to theeffective centrifugal potential: ΩT = Ω. The number of particles is (a) 1, (b) 2, (c) 3,
etc.

1.

62
we haven(k) = hψ|Φ† (k)Φ(k)|ψiZZ0ik·x Xe−ik·x X†
2 e∗= hψ| d x √âl Wl (x) d2 x0 √âl0 Wl0 (x0 )|ψi
V l
V
l0
X
=
hψ|â†l âl0 |ψiWl∗ (k)Wl0 (k).

(3.84)

ll0

Defining
kl = k + ΩM
xl × ẑ,~(3.85)

the modified Wannier function can be re-written as
Wl (k) = e−ik·xl w(kl ),Z
e−ikl ·(x−xl )√w(x − xl ).
w(kl ) = d2 xV

(3.86)(3.87)

Thus, the momentum distribution takes the form,n(k) =

Xw∗ (kl )w(kl0 )eik·(xl −xl0 ) hψ|â†l âl0 |ψi.

(3.88)

l,l0This distribution is written in the momentum coordinates, k, corresponding to the rotating frame coordinates, x. Since the number density rotates in the lab frame coordinates,
xL , the momentum distribution also rotates in the lab frame momentum coordinates,
kL . However, far-field pictures of the gas will be snapshots of the momentum distribution at the moment the trap and lattice are turned off, provided the switch-off time isfast enough. Thus, the momentum distributions presented here are accurate representations of what will be measured in time-of-flight measurements, although they may berotated relative to each other.
Since quasi-angular momentum is a reflection of the symmetry of the system, considering single-particle states should be sufficient to capture the basics of how this quantum number affects the momentum distribution. We therefore concentrate on singleparticle states only. In this case, n(k) is merely the square of the Fourier transform of
the wave function.

63
In order to develop an understanding of the momentum distribution, we first neglect the Wannier functions, which neglects the envelope of the momentum distribution.
In this case, n(k) is the Fourier transform of a sum of weighted delta functions. On afour site-lattice, the wave function is of the form
ψ(x) ∝

4X

ei2πmj/4 δ(x − xj ),(3.89)

j=1

where xj are the four corners of a square centered at x = 0. In figure 3.10 are plotted
the momentum distributions for m = 0, 1, 2, and 3. The m = 0 state is peaked
at k = 2πn/d, with n an integer and d the lattice spacing, whereas the other statesvanish at these points. The spacing between peaks for both m = 0 and m = 2 is√
∆k = 2π/d, whereas for m = 1, 3 this spacing is ∆k = 2π/ 2d. Since the lattice spacing
is known, there are clear measurable distinctions between different generic quasi-angularmomentum states. Note that m = 1 and m = 3 are identical as they represent similarstates with counter-propagating current patterns.
Including the Wannier functions contributes to the momentum distribution anpoverall envelope of approximate width V0 /ER /4, where V0 is the depth of the lattice.
The modification k → kl , equation (3.85), moves the peak of this envelope away fromk = 0 as rotation is increased, though the center of the envelope is always located at
k = 0. This is consistent with higher momenta being accessed for higher rotation rates.
At this point, it is important to note that locating k = 0 in the distribution isnecessary for distinguishing the m = 0 and m = 2 states. This can be done easily due
to the overall envelope in momentum space that is centered at k = 0.
There are two additional considerations that come into play for larger lattices,
both system-size effects. An overall envelope in position space determines the widthof the peaks in momentum space. This envelope determines the size of the system,
and if the width is roughly L, then the width of the peaks in momentum space scales
as 1/L. If the lattice spacing is increased by a factor of n, then the peak-spacing in64

ky Hunits of ΠdL

HaL 4

HbL

2
0
-2
-4

ky Hunits of ΠdL

HcL 4

HdL

2
0
-2
-4
4

2

0

-2

kx Hunits of ΠdL

-4

4

2

0

-2

-4

kx Hunits of ΠdL

Figure 3.10: Fourier transform of single-particle quasi-angular momentum states on foursites, neglecting an overall envelope caused by the width of the Wannier functions. The
quasi-angular momentum is (a) m = 0, (b) m = 1, 3, (c) m = 2, and (d) m = 3, andthe momentum distributions are plotted over a range 4π/d ≤ kx , ky ≤ 4π/d; the first
Brillouin zone is given by π/d ≤ kx , ky ≤ π/d.

65
momentum space is decreased be a factor of 1/n. The general peak structure is not
modified by these considerations. We can conclude that the distinctions between quasiangular momentum states are sustained for more physical systems, which we turn to
now.
The ground state momentum distribution for a one-hundred-site lattice in thepresence of a harmonic trap of frequency ΩT = 0.15ER /~ is computed via imaginary
time propagation. The trap ensures that the number density falls off to zero outside
the central 36 sites, eliminating the edge effects of the hard-wall boundary at the edgesof the lattice.
The results are summarized in figure 3.11, where we have plotted the momentum
distributions for the m = 0, 1, 2, and 3 states at rotation speeds of 0, 0.1ER /~, and
0.145ER ~, increasing to the right. We can again make the observation that the m = 0
state is the only one that is non-zero at exactly the reciprocal lattice vectors, i.e. k =2πn2πmd k̂x + d k̂y .

The peak-spacing is smaller for the m = 1 state than for the m = 2

state, distinguishing them from each other. However, the exact peak-spacing is unclear
due to complex interference effects between sites at different radii.
These differences are washed out at higher rotation speeds. The m = 1, m = 2,
and m = 3 states have very similar structures for large rotation. The peaks overlap sothat it is difficult to resolve them; the momentum distributions all appear to be rings
centered at reciprocal lattice vectors. They are distinguishable via the radius of thering, as the radius is smallest for m = 1 and largest for m = 3, but as this is in orderof increasing energy, the larger radius may just be an artifact of the larger width in
the number density envelope at higher energies. Indeed, figures 3.11(a), 3.11(b), and3.11(c) are consistent with this conclusion, as the momentum peaks are narrowing due
to the number density spreading out in real space. These results are consistent withthose calculated via direct diagonalization of the Hubbard models previously discussed.

66

ky Hunits of ΠdL

HaL 4

HbL

HcL

HeL

HfL

HhL

HiL

HkL

HlL

2
0
-2
-4

ky Hunits of ΠdL

HdL 4
2
0
-2
-4

ky Hunits of ΠdL

HgL 4
2
0
-2
-4

ky Hunits of ΠdL

HjL 4
2
0
-2
-4
4

2

0

-2

kx Hunits of ΠdL

-4

4

2

0

-2

kx Hunits of ΠdL

-4

4

2

0

-2

-4

kx Hunits of ΠdL

Figure 3.11: Momentum distributions computed via imaginary time propagation for
one particle in a 100-site lattice in the presence of a trap of frequency ΩT = 0.15ER /~.
Plotted are m = 0 ((a), (b), and (c)), m = 1 ((d), (e), and (f)), m = 2 ((g), (h), and(i)), and m = 3 ((j), (k), and (l)) states for rotation speeds of Ω = 0 ((a), (d), (g), and(j)), Ω = 0.1ER /~ ((b), (e), (h), and (k)), and Ω = 0.145ER /~ ((c), (f), (i), and (l)). At
zero rotation, the distinctions between different quasi-angular momenta are sustained.
As rotation is increased, it becomes harder to distinguish the m = 1 and m = 2 states.
Due to the centrifugal term, higher rotation increases the width of the envelope of thenumber density, decreasing the width of the peaks in momentum space.

67

3.6

Conclusion
By analogy with quasi-momentum in translationally invariant periodic systems,

the notion of quasi-angular momentum has been introduced to label the eigenstates ofa Hamiltonian that has a discrete rotational symmetry. We have shown that quasiangular momentum is useful in analyzing the ground-state properties of quantum gasesof bosons or fermions in rotating optical lattices. In particular, monitoring the quasiangular momentum of the ground state as a function of rotation allows us to identify
transitions between different circulation values.
We have also presented a possible avenue by which the quasi-angular momentumof a state can be experimentally determined. We identify characteristics in the momentum distribution distinguishing between different quasi-angular momentum states
at low rotation speeds, such as the existence of a peak at reciprocal lattice vectors foran m = 0 state only, the peak-spacing, and the overall structure of the momentum
distribution.
There are still open questions as to how the momentum distributions will change
when the lattice size or number of particles is increased. The effects of statistics will
be of fundamental importance for many-particle systems; there is a question of how
well the peaks can be resolved for larger lattices, since peak-spacing decreases with increasing lattice size; and interference effects between adjacent sites and between rings
complicates our treatment of the exact spacing between peaks in the momentum distribution. However, there will always be signatures of the quasi-angular momentum
in the momentum distribution, as it is connected with the phase information of theground state, which in turn influences the momenta of the system. In addition, a zeroquasi-angular momentum state can be distinguished from a non-zero one, which allows
the experimenter to verify that vorticity has entered the system.
The approximations used to describe strongly-interacting bosons can be used to68analyze many different ground-state features of the system. In particular, we can use
these results to generate Mott-insulator to superfluid phase diagrams for both flat andsuperlattices, indicating changes in dynamical behavior.

Chapter 4

Mott-Insulator to Superfluid Phase Diagrams

One of the first successes in the AMO physics subfield of simulation of many-bodymodels using ultracold, neutral atoms trapped in optical lattices was the verificationof the superfluid (SF) to Mott-insulator (MI) transition in the Bose-Hubbard model[39, 47]. The experiment set off an explosion of research investigating the phase diagramof various models related to the Bose-Hubbard Hamiltonian, including the introductionof superlattices and the effects of the presence of both fermionic and bosonic particles.
Unanswered questions from condensed matter systems, including the nature of quantum
criticality and the behavior of certain dynamical structure factors of one-dimensional
systems, might be resolved in ultracold, atomic systems. A complete understanding of
the phase transitions that occur in the simplest of many-body systems, such as thosemodeled by the Bose-Hubbard Hamiltonian, is therefore essential.
As a final application of the formalism developed and discussed in the previous
chapter, we compute the phase diagrams for strongly-interacting bosons in a two-state(i.e. fermionized) approximation in the presence of a superlattice. This captures theexact nature of the diagram for small J/U . The behavior of the Mott-insulator lobes asa function of the energy difference between adjacent wells was useful in an early version
of atomtronics [118]. We compare the phase diagram in this approximation to thatgenerated via the mean field calculation in section 2.1.2.

70

4.1

MI-SF Phase Diagram for Strongly-Interacting Bosons
In this section, by diagonalizing a one-dimensional, fermionized Bose-Hubbard

Hamiltonian, we generate the phase for strongly-interacting bosons trapped in an opticallattice. The phase boundaries in the (µ/U, J/U )-plane are the contours of zero energy
in the single-particle spectrum.
We previously derived the fermionized Hamiltonian for strongly-interacting bosons
in a rotating ring lattice, equation 3.60. Taking ω = 0 and eliminating the periodic
boundary conditions, we have the Hamiltonian for bosons on a one-dimensional lattice,Ĥ (l) = U N

N

N

j=1

j=1

X †
X
l(l − 1)− µN l + (U l − µ)ĉj ĉj +(ω)ĉ†j ĉj2− J(l + 1)N−1Xĉ†j ĉj+1 + H.c.

(4.1)

j=1

where U is the on-site, two-particle interaction strength, J is the tunneling energy, and lis the filling fraction, i.e. the number of particles per lattice site. In this approximation,
the fermionic operator ĉ†j adds one particle to site j, raising the number of particles
from l to l + 1. We have introduced the chemical potential, µ, so that we can work in
the grand-canonical ensemble. We perform the canonical transformation,
r


N
2 X
πmk
sin
dˆm ,
ĉk =
N +1
N +1

(4.2)

m=1

which diagonalizes the Hamiltonian, yielding


N 
X
πm
l(l − 1)(l)Ĥ =U l − µ − 2(l + 1)J cosdˆ†m dˆm − µN l + U N
.
N +1
2

(4.3)

m=1

Fixing l, we can see that the ground state is given by populating those eigenstatesof the dˆ†m dˆm operators for which the energy U l−µ−2(l+1)J cos(πm/(N +1)) is negative.
Therefore, every time there is an m for which this energy is zero, the particle number
changes by one. This argument implies that the boundaries between phases of different
particle number are truly given byJµ= l − 2(l + 1) cos
U
U



πm
N +1


.

(4.4)

71
The phase diagram with these phase boundaries is plotted in figure 4.1(a) forthe case of an eight-site lattice. The overlaps of the phase boundaries completes thestory. Each time there is an intersection between boundaries, a different particle-number
becomes the global ground state of the system. As the number of sites goes to infinity,
the number of phase boundaries goes to infinity, and the area outside the empty triangles
in the figure becomes a continuum of particle-number states. This implies that outside
the empty triangular regions, the ground state is a state of uncertain site-number, ora superfluid. Within the empty triangles, the ground state is a number state. We canthen interpret these empty spaces as Mott lobes in the Mott-insulator to superfluid
phase diagram, with the triangular regions the Mott insulator states and outside theseregions superfluid.
This approximate phase diagram has been derived in a slightly modified form in[24] using a semiclassical approach.

4.1.1

Comparison between two-state and mean-field approximations

In figure 4.2, we plot the MI-to-SF phase diagram as computed in the two-state
approximation and as computed via the mean-field calculation from section 2.1.2. The
two methods agree in the strong-interaction regime, i.e. for small J/U , but significantly
differ from each other for larger values of the tunneling parameter. The most visible
difference is the behavior of the phase boundaries near the tip of the Mott lobe. In the
two-state approximation, this tip is pointed, whereas in the mean-field approximation,
the tip is rounded.
The true character of the phase diagram may lie somewhere in between. In
reference [39], it is noted that the phase transition that occurs at the tip of the Mott
lobe is of a markedly different character than the one that occurs away from this tip.
This is because the transition at the tip occurs at a fixed density—or rather, at a fixed
integer filling fraction—whereas the transition that occurs at the phase boundaries away

HaL

4

Μ Hunits of UL

72

3

HbL

2
1
0
0.00

0.05

0.10

0.15

0.20

0.00

J Hunits of UL

0.05

0.10

0.15

0.20

J Hunits of UL

Figure 4.1: Bose-Hubbard phase diagram for strongly-interacting bosons in a onedimensional lattice. In the large-N limit, for large enough J/U , the strongly-interacting
approximation breaks down, and the tips of the Mott lobes become rounded. (a) Phaseboundaries for an eight-site lattice. Each white region is a region where the groundstate is a state of exact particle-number. (b) As the number of sites increases, the region
outside the triangular spaces becomes a continuum of different particle-number groundstates. This region is then considered to be superfluid.

Μ Hunits of UL

4
3
2
1
0
0.00

0.05

0.10

0.15

0.20

J Hunits of UL

Figure 4.2: Comparison between the MI to SF phase diagrams computed using a twostate approximation and a mean-field approximation.

73
from the lobe tip involves adding or subtracting a particle from the system.

4.2

Superlattices
A direct generalization of the procedure above is to introduce an extra periodic

potential with periodicity larger than the original lattice. The simplest way to implement this is to add an energy to every M ’th site, leaving the rest of the lattice flat. The
ring and one-dimensional lattices should be identical in the large-N limit, so for ease of
future calculation, we re-include periodic boundary conditions in this section.
This extra periodic potential takes the form,
N/M

V̂ =

X

ĉ†M j ĉM j .

(4.5)

j=1

In terms of the operators that diagonalize the Hamiltonian for the flat lattice, this is
N/M

V̂ =

=

N −1
N −1
1 X −i2πnM j/N ˆ† 1 X i2πmM j/N ˆ
√
e
dn √
e
dm
N n=0
N m=0
j=1

X

N
 X ˆ† ˆdm dn δmodN/M (m−n),0 .
M

(4.6)

m,n=1

TakingN
q1 + m1 ,MNn=
q2 + m2 ,MN
mj = 0, . . . ,
− 1,
M
m=

qj = 0, . . . , M − 1,

(4.7)
(4.8)
(4.9)
(4.10)

We can re-write the full transformed Hamiltonian asl(l − 1)2 
N/M −1 M −1 
X X
2π N
+
U l − µ − 2(l + 1)J cos
q+mdˆ†(N/M )q+m dˆ(N/M )q+m
N M

Ĥ (l) = −µN l + U N

m=0

+


M

q=0

N/M −1 M −1
X X
m=0 q1 ,q2 =0dˆ†(N/M )q1 +m dˆ(N/M )q2 +m .

(4.11)

74
This Hamiltonian can be diagonalized separately at every m. In order to do this,we define the column vector,

d(m) = dˆm , dˆN/M +m , dˆ2N/M +m , . . . , dˆ(M −2)N/M +m , dˆ(M −1)N/M +m .

(4.12)The Hamiltonian can then be rewritten as
Ĥ(l)

N/M −1Xl(l − 1)d(m)† h̄(l,m) d(m) ,= −µN l + U N+
2(4.13)

m=0

where h̄(l,m) is an M × M matrix with elements,(l,m)
hq0 q = δq0 ,q




2π
2π

q+
m
.
U l − µ − 2(l + 1)J cos+
M
NM(4.14)(l,m)

Diagonalizing h̄(l,m) yields eigenvectors, ψ (l,m;n) , with corresponding eigenvalues λn.

The canonical transformation,fˆn(l,m) =MX(l,m;n)∗ ˆd(N/m)q+m ,ψj(4.15)

q=1

results in the manifestly diagonal Hamiltonian,Ĥ(l)N/M −1 M
X X
l(l − 1)λ(l,m)= −µN l + U N+
fˆn(l,m)† fˆn(l,m) .
n2(4.16)

m=0 n=1(l,m)

The eigenvalues, λn, can be used to determine the phase boundaries in a modified

MI-SF phase diagram.
As an example, we consider the case of M = 3. We first take the large-N limit
where the matrix elements of h̄(l,m) take the form,(l,m)
hq0 q = δq0 ,q
U l − µ − 2(l + 1)J cos

2π
q
3




+ .
3

(4.17)The eigenvalues of this matrix are
(l)

λ± =
(l)


1
1p 2
+ lU − µ − (l + 1)J ± + 2(l + 1)J + 9(l + 1)2 J 2 ,222

λ0 = lU − µ + (l + 1)J.

(4.18)(4.19)

75
2.0
Μ Hunits of UL

1.5
1.0
0.5
0.0
-0.5
0.00

0.05

0.10

0.15

0.20

0.25

J Hunits of UL

Figure 4.3: MI-SF phase diagram for an infinite one-dimensional lattice. The dashedblack curve is the phase diagram for the flat lattice. The red curve is the phase diagram
for a lattice with every third site shifted down in energy by  = −5J. The blue curve isthe phase diagram for a lattice with every third site shifted up in energy by  = 5J.

We consider the special case where the on-site energy shift is proportional to the hopping
rate,  = αJ, in which case these reduce to(l)

λ± = lU − µ +pJ
α − (l + 1) ± α + 2(l + 1)α + 9(l + 1)2 .
2(4.20)An argument similar to that in the previous section yields the boundaries between
the MI and SF phases. As we can see in figure 4.3, the Mott lobes are sheared upwards
when every third lattice site is shifted up in energy relative to the rest of the lattice,whereas the Mott lobes are sheared downwards when every third lattice site is shifted
down.

4.3

Conclusion
We have generated the Mott-insulator to superfluid phase diagram for strongly-interacting bosons trapped in a one-dimensional lattice. This computation is exact forsmall enough J/U , since in this regime, the bosons are well-described via the JordanWigner transformation that maps strongly-interacting bosons onto non-interacting fermions.
One significant difference between this computation and the phase diagram as computed
via mean-field theory is the character of the tip of the Mott lobe, where the diagram in

76
the two-state approximation has pointed tips.
The hard-core boson approximation breaks down for large enough J/U , whichmeans that it does not correctly capture the behavior of the phase boundaries near thetip of the Mott lobe. On the other hand, the mean-field computation assumes thatfluctuations in the order parameter hci i are small near the phase transition, which isalso a bad approximation. The truth probably lies somewhere in between these two
extremes.

Chapter 5

Non-Destructive Cavity QED Probe of Atomic Dynamics
in an Optical Lattice

The bulk of this chapter was published in the Physical Review A as Phys. Rev. A
80, 043803 (2009) [102].
In this chapter, we propose a method for optically probing atomic gases in opticallattices subject to an external potential. The method is in situ and non-destructively
measures properties of the atomic motion via weak-coupling to the existing lattice fields.
The technique satisfies three main goals. The probe is weak so that the atoms can becontinuously monitored without affecting their dynamics; the existing lattice fields are
employed as the probe, so that no additional, external interrogation fields are necessary;and the signal-to-noise ratio (SNR) is large enough for experimental detection. In a ringcavity, two counter-propagating running-wave modes interact with a gas of ultracold
atoms and simultaneously set up both a conservative, external lattice potential for theatoms and a weak, quantum optical probe of the atomic center-of-mass dynamics. The
probe field leaks out of the cavity and is detected with a balanced heterodyne schemeat the cavity output.
This method is in a sense dual to strong measurement schemes such as timeof-flight absorption imaging and Bragg spectroscopy. In these schemes, laser light iseither absorbed by or scattered off of the atomic cloud. This allows for high resolution
images and a strong signal using only a single measurement, but the atomic sample isdestroyed in the process. In contrast, in our measurement scheme, the probe field is very

78
weak so that a continuous measurement is made without affecting the atomic dynamics.
Integration of the signal in time and across the atomic cloud yields measurements ofdynamical properties of the atoms with a measurable SNR in a single experimental run
at the price of losing information about individual atoms and real-time dynamics.
The procedure is similar in spirit to recent proposals for optical detection of
many-body atomic states [27, 86, 128] and occupies a place in AMO physics sharedby proposals for state preparation in cavity QED systems [85, 87, 93]. In addition,
experiments have already been carried out for nondestructive probing of various systems[26, 58, 124, 131].
We here provide a test of the technique for the conceptually simple motion of noninteracting atoms in an optical lattice driven by a constant force, which leads to Blochoscillations [8]. This choice is motivated by the fact that Bloch oscillations can be viewed
as a probe for investigating quantum gases in optical lattices. These oscillations may
be used in the measurement of fundamental constants [28], to provide levels of precision
up to δg/g ≈ 10−7 in the measurement of the acceleration of gravity [3, 23, 36, 62, 108],and to measure Casimir forces on small length scales [132]. When interactions are
significant, damping of Bloch oscillations provides information on correlation-induced
relaxation processes [18, 34, 42, 82, 94]. Finally, this investigation is a starting point forother measurement schemes, such as periodically driven lattices acting as a spectroscopic
probe of atomic motion [62].

5.1

Model and Detection Scheme
We briefly outline the physical basis of the model and approximations used. This

includes a discussion of obstacles in the way of satisfying the goals outlined above,
avoiding these problems, and the conditions required for the method to work.
To set up a conservative lattice potential, many photons must be present in thecavity field so that fluctuations can be neglected, and this necessitates strong pumping

79
from the in-coupled lasers. On the other hand, the probe field amplitude must be smallenough so that it does not affect the atomic dynamics, requiring weak pumping. In
addition, the probe and lattice fields are coupled to each other through the scattering ofphotons off of the atoms. This acts as an extra source for probe dynamics. The probe
field is then not a direct measure of atomic dynamics and can act back on the atoms,
altering the properties we are attempting to measure.
We can circumvent these problems by first choosing the relative phase on thein-coupled lasers so that only one of two standing-wave modes in the cavity is pumped.
Strong pumping and the properties of a bad cavity—where the fields are at all times
in steady state—ensure that the pumped mode acts as a lattice potential. The otherstanding-wave mode is not pumped. Any field leaking out of the cavity from this modearises solely because of events occurring in the cavity, and it can therefore act as a probe
for system dynamics.
Ensuring that the non-pumped mode acts as a probe of atomic dynamics requiresthat two conditions be met. The probe must have as its source only the motion of theatoms. This means that any probe dynamics due to the effective coupling to the latticefield must be small compared to that induced by the motion of the atoms. The backaction of this mode on the atoms must also be negligible. Any atomic motion inducedby coupling to the probe field must be small compared to the motion induced by bothlattice and external potentials.
Two conditions are also required for the pumped field to act as a conservative
lattice potential. The back-action of the atoms on the lattice field must be negligible,
meaning that deviations from the mean field amplitude caused by coupling to the atoms
is small compared to the mean field amplitude itself. In addition, any atomic motion
induced by fluctuations away from the mean lattice field must be small compared tothat induced by the external potential, since this is the interesting dynamics.
The scheme is realized in the setup illustrated in figure 5.1. We consider Na

80Η

ei kC zΚ

e-i kC zΚΗ

Figure 5.1: Schematic of the coupled atom-cavity system. In-coupled lasers set up twocounter-propagating fields within the cavity. The atoms interact with the cavity fields
via the optical dipole potential. Photons from the cavity beams exit the cavity throughthe mirrors at a rate κ.

ground-state atoms interacting with two counter-propagating running-wave cavity modes
in a ring resonator setup. The two modes of the cavity have frequency ωC and wave vectors ±kC ẑ. The two cavity modes are coherently pumped at a detuning ∆p = ωL − ωC
where ωL is the frequency of the pumping lasers. Photon decay through the cavity mirrors is treated within the Born-Markov approximation. We treat the atom-cavity-fieldinteraction in rotating wave and dipole approximations. The cavity-modes are fardetuned from atomic transitions. This same setup has been used in a cooling scheme
for atoms [49].

5.1.1

Model

The effective Hamiltonian for the coupled atom-cavity system is given by



X 
~2 d2
†
†
Ĥ = dz Ψ̂ (z) −
+
V
(z)
Ψ̂(z)
+
~ηâ
+
~ηâ
−
~∆
â
â
ext
p k k
k
k
2m dz 2
k=±kC
Z



+ dz Ψ̂† (z) ~g0 â†kC e−ikC z + â†−kC eikC z âkC eikC z + â−kC e−ikC z Ψ̂(z). (5.1)
Z†

Here, Ψ̂ is the atomic field operator, and âk is the annihilation operator for the cavity
mode, k. The parameter, m, is the the mass of the atom, g0 is the two-photon atom-

81
cavity coupling, and η is the strength of the cavity field pumping, taken to be real. Dueto the far detuning of the cavity fields from the atomic transition, excited states of theatom have been adiabatically eliminated, and the atoms couple to the field intensity.
The three terms in equation (5.1) are respectively the atomic kinetic energy and externalpotential, the bare cavity mode Hamiltonian, and the atom-cavity interaction. Cavity
losses through the cavity mirrors are treated via a master equation with Liouvillian,L̂ρ̂ = −~κ X  †âk âk ρ̂ + ρ̂â†k âk − 2âk ρ̂â†k ,
2

(5.2)

k=±kC

where ρ̂ is the reduced density matrix for the atom-cavity system, and κ is the cavity
linewidth.
We perform a canonical transformation of the cavity mode operators to symmetricand anti-symmetric modes,b̂± =âkC ± â−kC
√
.
2

(5.3)

The bare cavity Hamiltonian in terms of these operators is given by
ĤC =

√





2~η b̂+ + b̂†+ − ~∆p b̂†+ b̂+ + b̂†− b̂− .

(5.4)

The symmetric mode, b̂+ , is pumped by the in-coupled lasers whereas the anti-symmetricmode, b̂− , is not. The b̂+ mode has a mode function proportional to cos (kC z) and sets
up the lattice potential, as follows.
The equation of motion for the symmetric field amplitude isi√dhb̂+ i  κ= −i − ∆p hb̂+ i + 2η + 2g0 hb̂+ Ĉi + ig0 hb̂− Ŝ2 i,
dt
2

(5.5)

where
Z
Ĉ =
Z
Ŝ2 =

dz cos2 (kC z) Ψ̂† (z)Ψ̂(z),(5.6)

dz sin (2kC z) Ψ̂† (z)Ψ̂(z).

(5.7)

82
We perform another transformation to a fluctuation operator, dˆ+ = b̂+ − β, leaving
the anti-symmetric mode unchanged, dˆ− = b̂− . The mean steady-state amplitude, β, isgiven by√β = hb̂+ i =2η

∆p − 2g0 hĈi0 + iκ/2

,

(5.8)

where hĈi0 is hĈi evaluated at t = 0.
The equations of motion for both dˆ+ and dˆ− are given byidhdˆ− i κ ˆ= −∆p − i
hd− i + 2g0 hdˆ− Ŝi − ig0 hdˆ+ Ŝ2 i − ig0 βhŜ2 i,
dt
2

(5.9)

and
i



dhdˆ+ i κ ˆ= −∆p − i
hd+ i + 2g0 hdˆ+ Ĉi + ig0 hdˆ− Ŝ2 i + 2g0 β hĈi − hĈi0 ,dt
2(5.10)

where
Z
Ŝ =

dz sin2 (kC z)Ψ̂† (z)Ψ̂(z).

(5.11)

Finally, the equation of motion for the atomic field operator is given bydΨ̂(z)
=
i~
dt



~2 d2
−
+ Vlat (z) + Vext (z) Ψ̂(z)2m dz 2

+ i~g0 β ∗ dˆ− − β dˆ†− sin(2kC z)Ψ̂(z) + 2~g0 β dˆ†+ + β ∗ dˆ+ cos2 (kC z) Ψ̂(z)+ 2~g0 dˆ†− dˆ− sin2 (kC z)Ψ̂(z) + 2~g0 dˆ†+ dˆ+ cos2 (kC z) Ψ̂(z)+ i~g0 dˆ†+ dˆ− − dˆ†− dˆ+ sin(2kC z)Ψ̂(z),(5.12)

where Vlat (z) = V0 cos2 (kC z) is a conservative lattice potential of depth, V0 = 2~g0 |β|2 .
This depth is proportional to the number of photons |β|2 in the symmetric mode.
Since the coupling strength ~g0 will be relatively small due to the large detuning between
cavity fields and atomic resonances, the number of photons in the symmetric mode will
have to be relatively large. Using the parameters below, we arrive at a coupling strengthof about 10−4 ER , and assuming that we want lattice depths of at least a couple ER ,|β|2 must be on the order of 104 –105 . In addition, in order to be able to neglect field

83
fluctuations relative to the mean field, we must have that |β|  |β|2 . This is clearlysatisfied for |β|2 ∼ 104 –105 .
At the beginning of this section, we outlined the conditions that need to be met
in order for the cavity fields to act as both an external, lattice potential and a weakprobe for the atoms. We can now put a more precise spin on these conditions.
First, in equations (5.9) and (5.10), the third term represents an effective coupling
to the other mode mediated by the atoms, i.e. coherent scattering of photons off of theatoms between the modes. In order for dˆ− to act as an atomic probe, it needs to be
decoupled from dˆ+ , so we need to be able to neglect this term. Examination of theseequations indicates that the symmetric fluctuation field amplitude hdˆ+ i must be muchsmaller than the steady-state symmetric mode amplitude β. Equivalently, the numberof photons in the fluctuation field must be much smaller than the number of photons in
the mean field. Using the scaling arguments described below, this condition translates
into satisfying the relation|g0 β|Na|g0 |Na
 |β| =⇒
 1.
κ
κ(5.13)

Secondly, we can interpret the terms in lines two, three, and four of equation(5.12) as extra effective lattice potentials that arise due to the presence of photons in
the anti-symmetric mode and in the symmetric fluctuation field. For instance, the firstterm in line three is a sine-squared lattice whose depth is 2g0 dˆ†− dˆ− . In this case, it isnecessary that the number of photons in the anti-symmetric mode is much smaller than
the number of photons in the symmetric mean field. Equation (5.13) again guarantees
this condition.

5.1.2

Eliminating higher-order correlations

We make the preceding discussion more rigorous by computing the equations of
motion for higher-order moments in order to generate scaling relations for correlations

84
such as hdˆ− Ŝi. By comparing these scales, we can eliminate many of the offending terms
in equations (5.9), (5.10), and (5.12).
Upon examining equations (5.9) and (5.10), it is apparent that in order for dˆ− toact as a probe field, we must be able to neglect the higher-order atom-field correlations
such as hdˆ+ Ŝ2 i. In addition, we must be able to neglect all but the first term in equation(5.12) in order that the lattice field act as an external, conservative potential.
To ensure that these conditions are satisfied, we examine the equations of motion for operators such as hdˆ+ Ĉi. Specifically, we compute the equation of motion for
hdˆ± Ψ̂† Ψ̂i, given byi~

dhdˆ± Ψ̂† Ψ̂i~κ= hdˆ± [Ψ̂† Ψ̂, Ĥ]i + h[dˆ± , Ĥ]Ψ̂† Ψ̂i − i hdˆ± Ψ̂† Ψ̂i.
dt
2

(5.14)

The first term is small and can be neglected. For this reason, we drop it in the following
discussion and return to it at the end. For dˆ− and dˆ+ separately, the equations of
motion becomedhdˆ− Ψ̂† Ψ̂i
i~
=
dt



~κ
−~∆p − ihdˆ− Ψ̂† Ψ̂i − i~g0 βhŜ2 Ψ̂† Ψ̂i2+ 2~g0 hdˆ− Ŝ Ψ̂† Ψ̂i − i~g0 hdˆ+ Ŝ2 Ψ̂† Ψ̂i,

(5.15)

~κ−~∆p − ihdˆ+ Ψ̂† Ψ̂i + 2~g0 hdˆ+ Ĉ Ψ̂† Ψ̂i2+ 2~g0 β hĈ Ψ̂† Ψ̂i − hĈi0 hΨ̂† Ψ̂i + i~g0 hdˆ− Ŝ2 Ψ̂† Ψ̂i.

(5.16)

anddhdˆ+ Ψ̂† Ψ̂i
i~
=
dt



For the purpose of making a scaling argument, we drop all terms involving the
product of a field operator with four atomic operators. Scaling relations for thesequantities can be derived by adiabatically slaving them to the lower-order correlation
functions. This is done by setting the time derivatives to zero, resulting in
4g0 β(hĈ Ψ̂† Ψ̂i − hĈi0 hΨ̂† Ψ̂i),κ
2g0 β
hdˆ− Ψ̂† Ψ̂i = −hŜ2 Ψ̂† Ψ̂i,κhdˆ+ Ψ̂† Ψ̂i = −i(5.17)(5.18)

85
implying that
2g0 β(hĈ Ŝ2 i − hĈi0 hŜ2 i),κg0 βhdˆ− Ŝi = −hŜ2 Ŝi.
κhdˆ+ Ŝ2 i = −i

(5.19)(5.20)

Plugging these back into the equations of motion for hdˆ± i, we haveidhdˆ− i κ ˆ
g0
= −∆p − i
hd− i − 2 g0 β(hĈ Ŝ2 i − hĈi0 hŜ2 i − hŜ2 Ŝi) − ig0 βhŜ2 i, (5.21)
dt
2
κ

and
i

κ
g0 β
dhdˆ+ i
= −i hdˆ+ i − 4ig0
(hĈ 2 i − hĈi0 hĈi)
dt
2
κ
g0 β
hŜ2 Ŝ2 i + 2g0 β(hĈi − hĈi0 ).
− ig0κ

(5.22)The terms with four atomic operators scale as
g0|g0 Na |g0 βhĈihŜ2 i ∼|g0 βhŜ2 i|
κκ
g0|g0 Na |g0 β(hCˆ2 i − hĈihĈ0 i) ∼|g0 βhĈ − Ĉ0 i|κκ
|g0 Na |g0
g0 βhŜ2 Ŝ2 i ∼|g0 βhŜ2 i|.
κκ(5.23)

As long as |g0 Na |/κ  1, these higher-order correlations can be neglected.
The scaling relations for dˆ± are then
2g0 βhŜ2 i,κ
4g0 β
hdˆ+ i = −i(hĈi − hĈi0 ).
κhdˆ− i =(5.24)(5.25)

We can use these relations to determine the approximate size of the non-conservative
terms in equation 5.12. Recalling that V0 = 2~g0 |β|2 , we have2

4g
2~g0 dˆ†− dˆ− ≈ V0 20 |hŜ2 i|2 ∼ V0
κ



g0 Na
κ

2
,


g0 Na 2
,
κ


8g02
g0 Na 2† ˆ† ˆˆˆi~g0 (d+ d− − d− d+ ) = −V0 2 hŜ2 i(hĈi − hĈi0 ) ∼ V0
.
κ
κ
2

16g
2~g0 dˆ†+ dˆ+ ≈ V0 2 0 |hĈi − hĈi0 |2 ∼ V0
κ(5.26)

(5.27)(5.28)

86
Each of these coupling constants is down from V0 by at least a factor of (g0 Na /κ)2 ; we
can neglect them. The last terms involve products of β with the fluctuation operatorsdˆ± , which come in scaling as V0 (g0 Na /κ) rather than V0 (g0 Na /κ)2 . It turns out thatthese are zero to first order, however:
∗

4g0 β
4g0 β
β ∗ dˆ+ + β dˆ†+ ∼ −iβ ∗(hĈi − hĈi0 ) + βi
(hĈi − hĈi0 ) = 0,
κ
κ
2g0 β
2g0 β ∗β ∗ dˆ− − β dˆ†− ∼ β ∗hŜ2 i − βhŜ2 i = 0.
κκ(5.29)(5.30)

Thus, we can also neglect these terms.
Finally, we return to the terms, hdˆ± [Ψ̂† Ψ̂, Ĥ]i. The equations of motions for theatomic number density isdΨ̂† Ψ̂~2[Ψ̂ Ψ̂, Ĥ] = i
=−
dt
2m
†

!
∂ 2 Ψ̂(z) ∂ 2 Ψ̂† (z)Ψ̂ (z)
−
Ψ̂(z)
∂z 2
∂z 2
†

(5.31)

To first approximation, this is essentially hdˆ± i times the expectation value of the atomic
kinetic energy. The kinetic energy is approximately the square of the group velocity of
the atomic wave-packet. In the tight-binding limit (V0 ≥ 5ER ), the energy of thesingle-particle wave-packet is given by~ = −~2J cos(dq),(5.32)

where d is the lattice spacing, q is the (continuous) quasi-momentum, and J is the
hopping energy. The group velocity isvg =∂= 2Jd sin(dq).
∂q(5.33)The maximum kinetic energy is then112
T0 = mvg,0= m(2Jd)2 = π 2 ER
2
2



~J
ER

2
.

(5.34)The kinetic energy is about ten times the tunneling rate squared in units of the recoil
energy. We can approximate the tunneling rate as~J = ER e−V0 /4ER ,(5.35)

87
which indicates that the kinetic energy is on the order of ten recoils at most.
We can make the approximation,~J 2 ˆ ††2
ˆhd± [Ψ̂ Ψ̂, Ĥ]i ≈ π ER
hd± Ψ̂ Ψ̂i ≤ π 2 ER hdˆ± Ψ̂† Ψ̂i.
ER

(5.36)

This term is a dispersive shift, and since
π 2 ER  ~κ,(5.37)

we may neglect it. Finally, ensuring that we can neglect all of these terms compared tothe external potential requires explicit knowledge of the form of Vext and will therefore
be left for the next section.
Thus, in the correlation hierarchy, each consecutive order scales as |g0 |Na /κ times
the previous order. For instance, hdˆ+ Ψ̂† Ψ̂i scales as |g0 |Na /κ times hΨ̂† Ψ̂i. Therefore
requiring |g0 |Na /κ  1 allows us to neglect all terms involving correlations between
three or more operators. We reiterate finally that this condition is equivalent with thephysical requirement that the numbers of photons in dˆ± are small compared with thenumber of photons in the symmetric mean field, since|hdˆ± i| ∼|g0 β|Na |β|.
κ(5.38)

With these approximations in hand, we solve equations (5.9) and (5.10) by adiabatically slaving the fields to the atomic motion, yielding.
hdˆ− i =−ig0 βhŜ2 i,∆p + iκ/2(5.39)

hdˆ+ i =2g0 β(hĈi − hĈi0 ).
∆p + iκ/2(5.40)The equation of motion for the atomic field operator is


~2 d2
dΨ̂(z)i~= −+ Vlat (z) + Vext (z) Ψ̂(z),dt
2m dz 2

(5.41)and the probe field amplitude is given byhdˆ− (t)i =−ig0 βhŜ2 i.
∆p + iκ/2(5.42)

88a` kcaS`b-

a` -kca

`
b+

5050

5050
a`

LO

Figure 5.2: Schematic of the balanced heterodyne detection scheme. The out-coupledcavity beams, â±kC , are combined to form symmetric (b̂+ ) and anti-symmetric (b̂− )modes. The antisymmetric mode beats against a strong local oscillator (LO), âLO andphotodetectors count the number of photons in the quadratures of b̂− . The differenceof these counts is the signal.

Since the atoms are independent of each other and couple symmetrically to the lightfield, we can in equation (5.42) make the replacement,hŜ2 i → Na hψ(t)| sin(2kC ẑ)|ψ(t)i,(5.43)

where |ψ(t)i is a single-particle state satisfying equation (5.41) with Ψ̂(z) → |ψ(t)i.

5.1.3

Detection scheme and signal-to-noise ratio

Through equation (5.42), hdˆ− i provides a measure of the atomic dynamics inside
the cavity. In reference [85], two schemes for detection of atomic motion using the outcoupled cavity fields were presented. We here review the superior case, where balanced
heterodyne detection of dˆ− is performed. Briefly, the signal is combined with a strong
local oscillator (LO) at a 50/50 beam-splitter, and the two output signals are measured
with photo-detectors and subtracted from each other. This differencing cancels the large
mean-field signal created by the LO. The noise is dominated by the LO noise and isconveniently cancelled when computing their signal-to-noise ratio. The general setup is
illustrated in figure 5.2.

89
In what follows, we compute the exact expressions for the signal and the signalto-noise ratio for this heterodyne detection scheme. We use a basic quantum Langevinapproach [44] combined with beam-splitter theory [76] to exactly calculate the signal
power spectrum and signal variance, and the signal-to-noise ratio is computed as theratio.
According to input-output theory [44] this field is proportional todˆout =√κ dˆ− + dˆin .

(5.44)

Input-output theory relates the fields inside and outside a cavity using a quantum
Langevin approach, writing the output (i.e. signal) field in terms of the input field andthe field out-coupled from the cavity. Equation (5.44) has a simple interpretation as√
the superposition of two electromagnetic fields, κ dˆ− and dˆin , and can be viewed asa boundary condition at the cavity output. In this case, the input field state is the√vacuum, which results in hdˆout i = κ hdˆ− i.
The output field is combined at a 50/50 beam-splitter with a strong local oscillator. The output fields at the beam-splitter are given by





ˆ
1  −i 1   dout  b̂2 = √ 
.
2
1 −i
b̂1
âLO

(5.45)The intensities of these two fields are measured with photodetectors. These signals are
subtracted from each other, mixed with an oscillating signal cos(ωt) and integrated intime. The final signal is proportional toZ T
N̂ (ω, T ) =0
dt b̂†2 (t)b̂2 (t) − b̂†1 (t)b̂1 (t) cos(ωt).

(5.46)

Defining
αLO = hâLO i = |αLO |e−iωLO t eiφLO ,(5.47)âLO = αLO + v̂,(5.48)

90
we can write∗ ˆ∗ ˆb̂†2 b̂2 − b̂†1 b̂1 = i(αLOdout − αLO dˆ†out ) + i(v̂ † dˆout − dˆ†out v̂) ≈ i(αLOd − αLO dˆ† ),where we have taken dˆ =√(5.49)κ dˆ− for convenience, and we have assumed that the local

oscillator has a very large amplitude. Finally,
Z T
N̂ (ω, T ) =ˆ − |αLO |e−iωLO t eiφLO dˆ† (t) cos(ωt)
dt i |αLO |eiωLO t e−iφLO d(t)0iT |αLO |e−iφLO ˆ
iT |αLO |eiφLO ˆ=d(ωLO + ω) +d(ωLO − ω) + H.c.
22

(5.50)

where we have defined
1
ˆ
d(ω)
=
T

Z T

ˆ
dt eiωt d(t).

(5.51)

0

The signal variance is given byEDh∆N̂ 2 (ω, T )i = (N̂ (ω, T ) − hN̂ (ω, T )i)2 .

(5.52)We define the signal fluctuation operator,ˆˆ − hd(α)i,ˆ∆d(α)= d(α)(5.53)and make the assumption that the signal is at all times a coherent state, in which caseˆ∆d(α)
is in the vacuum, and therefore only anti-normal-ordered correlation functions
involving these operators are non-zero. In addition, we assume thatˆˆ 0 )i = δ(t − t0 ),h∆d(t)∆d(t(5.54)

which is approximately valid when the interrogation time T is much larger than 1/κ.
We are left with4
T 2 |α2LO |ˆ LO + ω)∆dˆ† (ωLO + ω)i + h∆d(ωˆ LO + ω)∆dˆ† (ωLO − ω)ih∆N̂ 2 (ω, T )i = h∆d(ω+ (ω → −ω).

(5.55)

91
Noting that
Z
Z T
1 T
0 0
† 0
iαt 1
ˆ
ˆ
ˆ
h∆d(α)∆d (α )i =dt e
dt0 e−iα t h∆d(t)∆dˆ† (t0 )i
T 0
T 0
Z
Z
1 T
1 T 0 −iα0 t0=dt eiαtdt e
δ(t − t0 )
T 0
T 0

 Z
1 1 T
i(α−α0 )t
,
dt e
=
T T 0

(5.56)

we can see that only the diagonal terms contribute here—with a factor of 1/T —so thath∆N̂ 2 (ω, T )i =|αLO |2 T.
2(5.57)We are finally in a position to compute the signal-to-noise ratio. We define thesignal quadrature operator,iq̂− (α, φLO ) = √ (e−iφLO dˆ− (α) − eiφLO dˆ†− (α)).
2(5.58)

in which case our signal is given by√√
T |αLO | κ
T |αLO | κ√√hN̂ (ω, T )i =hq̂− (ωLO + ω, φLO )i +hq̂− (ωLO − ω, φLO )i. (5.59)
2
2
The signal-to-noise is given by the ratio of the signal power to signal variance, which is
SNR =|hN̂ (ω, T )i|2h∆N̂ 2 (ω, T )i

= κT |hq̂− (ωLO − ω, φLO )i + (ω → −ω)|2 .

(5.60)

If we are looking for a particular dynamical frequency ω0 of the atomic motion, we setω ≈ ωLO − ω0 . For the system investigated below, we find that when the field has alarge amplitude at ω0 , it does not have a large amplitude at 2ωLO − ω0 . Thus, we may
drop the first term, leaving
SNR = κT |hq̂− (ω, φLO )i|2 .

(5.61)

By varying the phase φLO on the local oscillator, we can maximize this signal-to-noise
ratio.

92

5.2

Results
In this section, we consider the motion of atoms confined in the optical lattice inthe presence of gravity,
Vext (z) = mgz,

(5.62)

and use the scheme outlined in the previous section to probe the motion of the atoms.
Gravity measurements are important for instance for optical lattice clocks [79]. For thisreason, we treat the specific system of a gas of 87 Sr atoms, although the method certainly
applies to many species of atoms. The parameters for the coupled atom-cavity system
are chosen to reflect current experimental conditions. They are λC = 2π/kC = 813 nm,~κ = 100ER , ∆p = ωp − ω = 0, ~g0 = 10−4 ER , and Na = 104 , implying derived
2 /2m is the recoil energy
parameters of ER ≈ 2π 4 kHz ~ and mgd ≈ 0.25ER ; ER = ~2 kCof the lattice, and d = π/kC is the lattice spacing.
We have to ensure that the back-action of both dˆ− and dˆ+ on the atoms isstill negligible. Specifically, the coupling strengths in equation (5.12) must be small
compared to the characteristic coupling strength of Vext , ~ωB = mgd. These conditions
are met if |V0 |(g0 Na /κ)2  ~ωB . This inequality is satisfied for the parameters above.
While these conditions are necessary, the exact criteria for being able to neglect the
back-action of the fields on the atoms requires more careful numerical study. We do not
perform these investigations here.
Within this setup, we envision an experiment in which the atoms are initiallyloaded into a harmonic trap. A vertical one-dimensional optical lattice is slowly ramped
on so that the atoms are in the ground state of the combined potential of trap and lattice
for a non-interacting gas. The trap is switched off over a time that is long compared tothe time-scale of vibrational dynamics in order to prevent inter-band transitions during
the switch-off. The gas is then allowed to evolve under gravity. In the presence of such a
constant force, the atoms undergo Bloch oscillations. This dynamics is briefly reviewed

93
in the following discussion.

5.2.1

System dynamics: Bloch oscillations

The central result of the theory describing Bloch oscillations is the acceleration
theorem, based on a semi-classical equation of motion [8], which states that the average
quasi-momentum of a wave-packet restricted to the first band increases linearly in time
until it reaches the Brillouin zone (BZ) boundary, at which point it is Bragg-reflected.
Explicitly, this is~hqi(t) = ~hqi(0) + mgt,

(5.63)

where the quasi-momentum, q, is restricted to the range, −kC ≤ q ≤ kC . Since the groupvelocity of the atomic wave-packet is given by the derivative of the dispersion relation[8], the periodic nature of the quasi-momentum implies that the atomic momentum
oscillates at a frequency, ωB = mgd/~. These Bloch oscillations will persist as long as
there is negligible Landau-Zener tunneling to higher bands. Each time the wave-packet
reaches the BZ boundary, a fraction of population is transferred to the second band,
given by [91]π 2 ∆2
PLZ = exp −8 mgdER
,(5.64)

where ∆ is the band-gap at the boundary. When ∆2 < 4mgdER , the population transfer
is appreciable, and vibrational dynamics within lattice sites significantly complicate thebehavior of the atoms. For this reason, we restrict our attention to lattice depths greaterthan 3ER , where PLZ is at most 10−5 for our choice of parameters.
In order to understand how Bloch oscillations are reflected in the time-dependenceof the probe field, we carefully consider equation (5.42). The operator, sin(2kC ẑ), isperiodic in space with period d and has odd parity, implying that it connects two Bloch
(n)

(n0 )

states, |ψq i and |ψq0 i, only if the quasi-momenta are equal, q = q 0 , and the bands

94
satisfy n − n0 = odd. Expanding the atomic wave function in the Bloch states as
|ψi =

X

(n)
c(n)
q |ψq i,(5.65)

n,q

we can approximate the matrix element in equation (5.42) ashψ| sin(2kC ẑ)|ψi ≈X

ρ(1,2)
q,q + h.c.,(5.66)q(1,2)

where ρq,q(1)∗ (2)cq is the coherence between bands one and two.

= cq

This assumes aninitial state confined to the first band in the case of negligible coupling to bands threeand higher. Using equation (5.41), we can derive an approximate equation of motion
for the coherence; it is
(1,2)

dρq,q
i
dt
(1)

where ρq(1)
= ∆(1,2) (q)ρ(1,2)
q,q + ωB ρq ,(5.67)

is the population of the q-quasi-momentum state in the first band, and
(2)

(1)

~∆(1,2) (q) = Eq − Eqis the energy difference between the q-quasi-momentum Blochstates in the first two bands. Since ∆(1,2) (q)  ωB , the coherence follows the first-band
(1,2)

(1)

population adiabatically. In this approximation, ρq,q = −ρq ωB /∆(1,2) (q).
Combining this expression for a wave-packet that is narrow in quasi-momentum
with equation (5.63), equation (5.42) becomes approximatelyhdˆ− (t)i ≈ig0 βNaωB.
(1,2)∆p + iκ/2 ∆(mgt)(5.68)

This expression implies that the probe field amplitude is largest when the atomic wavepacket is centered at the BZ boundary since ∆(1,2) (mgt) is smallest at this point.
Equations (5.41) and (5.42) are numerically integrated for an initial state that is
a Gaussian of spatial width, σ, projected into the first band. This approximates theground state of the combined potential of lattice and harmonic trap for a non-interactinggas. An example of the system dynamics is illustrated in figure 5.3, where V0 = −3ER ,and σ = 2d. A vertical slice through figure 5.3(a) is the wave function density in the95first band plotted versus quasi-momentum at an instant in time. The center of thiswave-packet moves linearly in time and is reflected at the BZ boundary (q = kC ), as
in equation (5.63). Bloch oscillations are illustrated in figure 5.3(b), where the atomic
momentum oscillates in time. Finally, the response of the probe field to this dynamics
is illustrated in figure 5.3(c). As predicted above, the probe field intensity peaks when
the atomic wave-packet reaches the BZ boundary.

5.2.2

Signal and SNR

As described in section 5.1, the probe field is combined at the cavity output witha strong local oscillator, and the resulting signal is proportional to a quadrature of theprobe field, equation (5.58). An example of such a signal is plotted in figure 5.4. Thereis a clear peak at the Bloch oscillation frequency in the signal power spectrum, but there
are also several harmonics present. In calculating the SNR, equation (5.61), we place anotch-filter about ωB and count only the total number of photons out-coupled from thequadrature at this frequency.
There are a few properties of the system that can affect the SNR. First, the widthof the initial wave-packet has an effect. It is easiest to see why this is so by taking as
the initial state a Wannier function, which is a coherent superposition of Bloch statesin a single band, populated equally. According to equation (5.63), the wave-packet iscontinuously reaching the BZ boundary, and the oscillation in the signal is washed out.
Second, when the lattice is too deep, the first two bands are essentially flat, in which(1,2)

case ∆qdoes not change with quasi-momentum, eliminating the oscillations in thesignal according to equation (5.68).
The temperature of the atomic gas can also significantly influence the SNR. For a
finite temperature, the pure state in equation (5.43) is replaced with a density matrix.
The incoherent sum over different states of the atoms will alter the expectation value,
possibly reducing it. In addition, the temperature and chemical potential of the gas de-

96

HaL

1.0

q Hunits of kCL

0.5

0.0

-0.5

`
XPat\ Hunits of Ñ kCL

HbL`
ÈXd -\ 2

HcL

0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3

0.025
0.020
0.015
0.010
0.005
0.000

0

20

40
60
80
t Hunits of ÑERL

100

120

Figure 5.3: Example of system dynamics for V0 = −3ER and initial state a Gaussian ofwidth σ = 2d projected into the first band. (a) Atomic density in the first band plottedversus quasi-momentum. White corresponds to zero population, black to maximal population. Population in the second band is at most 0.001Na . (b) Expectation value of
atomic momentum illustrating Bloch oscillations. (c) Number of photons in the probe
field.

97`ΚÈXq- \ 2HaL

HbL 4104

10

3104

7.5
5

2104

2.5

1104

0
0

100

200

t Hunits of ÑER L

300

400

0
0.0

0.25

0.5

0.75

1.0

Ω Hunits of ER ÑL

Figure 5.4: Examples of the (a) signal power and (b) signal power spectrum, computedwith parameters, V0 = −3ER , ~g0 = 10−4 ER , Na = 104 , and ~κ = 100ER . The signal
displays a clear oscillation at the Bloch frequency, ωB = 0.25ER /~.

SNR

98
104
103
102
101
100
10-1
10-2
10-3

-3 ER -4 ER -5 ER -10 ER -15 ER

0

2

4

6

8

10

12

14

Σ Hunits of dL

Figure 5.5: Signal-to-noise ratio as a function of the initial wave-packet width for aninterrogation time of 1s and for lattice depths of V0 = −3, −4, −5, −10, −15ER . For
σ < 2d, the SNR is reduced due to parts of the wave-packet constantly moving past theBrillouin zone boundary, where the signal peaks. The SNR saturates near σ = 2d.

termine the relative populations of the various Bloch states, and appreciable populationin higher bands can destroy Bloch oscillations. As long as the temperature is smaller(1,2)

than the bandgap, kB T  ~∆kC , this effect will be small.
Finally, the replacement, equation (5.43), can not be made when the atoms are
not independent, e.g. if they interact with each other. In this case it is possible thatthe scaling of the SNR with the number of atoms will be reduced from Na2 . A proper
treatment of thermal and interaction effects is necessary for exact results. Here, we
neglect both interactions and finite temperature for the case of 87 Sr atoms, in whichcase equation (5.43) is valid.
Equation (5.12) is numerically integrated for a time t = 400~/ER . The resulting
wave function is used to compute the probe field amplitude, equation (5.42), which is
Fourier-transformed and squared, yielding the power spectrum. The SNR is computed
and scaled up linearly to an interrogation time of 1s, which assumes that coherence timeof the Bloch oscillations is longer than 1s.
The results are plotted in figures 5.5 and 5.6. The SNR rises from zero for small
wave-packet widths and saturates near σ = 2d. The decrease in SNR for σ < 2d is a
result of the fact that the wave-packet is wide in quasi-momentum, which means that

99

104

SNR

103
102
101
100
10-1
10-2
5

10

15

20

25

30

35

V0 Hunits of ER L

Figure 5.6: Signal-to-noise ratio as a function of lattice depth for σ = 2d and aninterrogation time of 1s. For |V0 | < 3ER , significant Landau-Zener tunneling to thesecond band destroys Bloch oscillations. For |V0 | > 3ER , the SNR decreases due to theincreasing flatness of the lowest band.

100
a significant portion of the wave-packet is at the BZ boundary for all times. We get amaximum when the lattice depth is relatively small, |V0 | ≈ 3ER , and the SNR decreases
with increasing lattice depth.

5.3

Conclusion
We have described a general cavity QED system in which properties of atomic

dynamics can be probed in situ and non-destructively. One cavity field is strong enoughto act as a conservative lattice potential for the atoms, and the other cavity field is weakso that it acts as a non-destructive probe of atomic motion. This technique is applied tothe detection of Bloch oscillations. Balanced heterodyne detection of the probe field at
the cavity output combined with integration in time and across the atomic cloud allows
for SNRs as high as 104 .
Examining equations (5.42) and (5.61), we can see that the SNR can be increased
by either decreasing the cavity linewidth, κ, at fixed lattice depth and atom-cavity
coupling or increasing the coupling constant, g0 , at fixed V0 and κ. The linewidth canbe decreased as long as the system remains in the bad cavity limit. However, a linewidthof κ = 100ER is already very small from an experimental standpoint, so decreasing it
beyond this level is a technological challenge. On the other hand, g0 can be varied
merely by varying the detuning between the cavity fields and atomic transitions. In
addition, the SNR scales with the square of the number of atoms, so increasing Na
beyond the 104 level assumed in this paper is also desirable. This can all be done tothe extent that the conditions outlined in section 5.1 and section 5.2 are still met.
This scheme can be extended for use in detection of various atomic properties,
and the measurement of Bloch oscillations itself can be viewed as a general DC probe foratomic dynamics and states. For instance, Bloch oscillations may be used for measurement of fundamental constants [28] and for Casimir forces [132]. Varying the detuning
between two lattice beams gives rise to an effective acceleration of the lattice [91], and101band physics may be probed by varying the Bloch oscillation frequency in such a setup.
Breakdown of Bloch oscillations are a signature of many-body effects in an atomic gas[42], and this is signaled by a reduction in SNR compared to the non-interacting case.
Two generalizations of this measurement technique are readily realizable. We may
implement a time-periodic force whose tunable driving frequency can be a spectroscopic
probe of atomic dynamics. The simplest examples of this include shaking the lattice[62] and modulating the amplitude of the lattice [1]. Another important extension of
the method involves measuring higher-order correlation functions of the out-coupledprobe field. Since one cavity field operator couples to two atomic field operators (see for
instance equation (5.9)), higher-order properties of the atoms such as density-density
correlations can easily be measured with standard quantum optical techniques. The useof higher-order correlation functions of the probe field is a starting point for generalizing
this technique to probe many-body physics in optical lattices.

Chapter 6

Cavity QED Measurements of Many-Body Dynamics
Near the Mott-Insulator Regime

In the previous chapter, we presented a method for probing the dynamics of noninteracting atoms trapped in an optical lattice by monitoring the light field out-coupledfrom an optical cavity. For this particular measurement, we predicted signal-to-noise(SNR) ratios as high as 104 . There were a number of drawbacks in this technique.
First, the condition that the cavity fields act simultaneously as both the probe field andthe lattice field limited the measurement to one where the atomic dynamics induce atleast coherences between bands. Since the signal went to zero as the lattice depth wasincreased, this method was restricted to small lattice depths. In addition, we were notable to vary the probe wavelength relative to the lattice wavelength. Finally, the method
worked only for atomic wave-packets with widths of at least a few lattice constants
In this chapter, we relax the condition that the cavity fields act as both latticeand probe fields, allowing us to optimize the signal as a function of system parameterssuch as probe wavelength even in systems where the lattice depth is large. The detection
scheme is applied to the atomic dynamics of atoms near the Mott-insulator regime thatare subjected to a constant external force. This system is modeled by a single-band
Bose-Hubbard model with a linear energy shift. Atoms are allowed to hop around thelattice when the energy offset between adjacent wells is nearly equal to the two-particle
interaction strength. This physics is closely related to that exploited in atomtronic

103

l+1

U » ÑΩBll-1

Figure 6.1: Schematic of the atomic system. In the Mott-insulator regime with unitfilling, there is exactly one particle per site. When the lattice is tilted so that thedifference in energy ~ωB between adjacent wells is equal to the two-particle interaction
strength U , particles are free to hop one site to the left. Assuming the rest of thelattice wells each contain one atom, the state pictured is the one-dipole state, |1; li ∝b̂†l b̂l+1 |ψMI i.

systems to emulate electronic devices such as diodes [103].
We first develop the theory of atomic dynamics in this system, using the methods
described in reference [112] as a starting point. Following this, we couple this system
to the two counter-propagating modes of a ring cavity and examine the behavior of thelattice fields in this special atomic regime.

6.1

Atomic Dynamics
The Hamiltonian for atoms moving in a one-dimensional optical lattice in thepresence of a constant force is given by
Z
Ĥ =



~2 d2
2
†
dz Ψ̂ (z) −+ V0 cos (kL z) + M az + ~g0 Ψ̂ (z)Ψ̂(z) Ψ̂(z),2M dz 2†

(6.1)

where F = M a is the force acting on the particles of mass M . As in Chapter 3,
we expand the atomic field operators in Wannier states. In the tight-binding regime(V0 ≥ 10ER ), the Hamiltonian reduces to the standard Bose-Hubbard model with an
extra linear shift,
ĤA =

X
j

( + M gZ + M gdj)b̂†j b̂j − J

X †UX † †b̂j b̂k +b̂j b̂j b̂j b̂j ,
2

hj,ki

j

(6.2)

104
where
Z
Z=

dz w∗ (z)zw(z).

(6.3)

We have dropped the band index n since in this regime, all of the dynamics occurs
within the first band, n = 1. This system is illustrated in figure 6.1.
This Hamiltonian is the starting point for the physics investigated in this chapter.
We restrict our attention to parameter regimes near the Mott-insulator phase, U  J,with the linear shift approximately equal to the on-site interaction energy,
U ≈ ~ωB = M gd.

(6.4)

In this regime, if the system is initialized in the Mott insulator state, the atomic dynamics considerably simplifies. This simplification can be effected by deriving an effective
low-energy Hamiltonian in a reduced Hilbert space consisting of states near-resonantwith the integer-filled number state. Since particle numbers are conserved, we can drop
the on-site energy terms involving  and M gZ. In addition, we mainly consider a filling
fraction of 1. The general behavior observed should be sustained for larger integer filling
fractions. Finally, we consider finite-sized lattices with N sites.

6.1.1

Dipole Hamiltonian

As an approximation to the Mott-insulator state, we consider the unit-filled state,!
N
Y†|ψMI i =b̂k |0i.
(6.5)
k=1This state has energy
1
EMI = hψMI |ĤA |ψMI i = N (N + 1)~ωB .
2

(6.6)

105
It is directly connected via the hopping part of the Hamiltonian to the states,
1|1; li = √ b̂†l b̂l+1 |ψMI i,21|1; l+ i = √ b̂†l+1 b̂l |ψMI i,
2

(6.7)
(6.8)

l = 1, . . . N − 1,

(6.9)

1
E1;l = h1; l|ĤA |1; li = N (N + 1)~ωB + U − ~ωB ,2(6.10)

1
E1;l+ = h1; l+ |ĤA |1; l+ i = N (N + 1)~ωB + U + ~ωB ,2(6.11)

which respectively have energies

andThe state |1; li has one particle per site except at sites l and l + 1 where there are twoand zero particles, respectively. The state |1; l+ i is similar except that the number ofparticles on sites l and l+1 are zero and two, respectively. The MI state is also connected
to states where the two-particle and zero-particle sites are not adjacent. However, since
the coupling is second order in the hopping J, and the states are separated in energy,
we can neglect them as long as we consider small enough timescales for the dynamics.
We work in the parameter regime whereU ≈ ~ωB ,(6.12)|J|  U + ~ωB .

(6.13)The states |1; l+ i then couple to |ψMI i with coupling constant,gM I,+ ∼J2
 J,
U + ~ωB(6.14)

so we can neglect these states.
This argument easily generalizes when we consider, for instance, the states that
|1; li are connected to. These consist of states where two particles have been moved one

106site to the left,
j = 1, . . . , l − 2
1|2; l, ji = b̂†l b̂l+1 b̂†j b̂j+1 |ψMI i :.
2
l = 3, . . . , N − 1(6.15)The domains chosen for the site indices j and l ensure that if there is a “hole” at sites
j and l, then on each of the sites l + 1 and j + 1, there are two particles. These states
have energy,
1E1;l = h1; l|ĤA |1; li = N (N + 1)~ωB + 2(U − ~ωB ),2(6.16)

and again, since U ≈ ~ωB , these states are near-resonant with the |1; li states and areconnected with coupling proportional to J.
In general, the near-resonant subspace is spanned by the states,|M ; kM , . . . , k1 i = √1MYb̂†ki b̂ki +1 |,2M i=1(6.17a)ki = 2i − 1, . . . , ki+1 − 2,

(6.17b)kM = 2M − 1, . . . , N − 1.

(6.17c)The parameter, M , counts the number of dipoles—that is, the number of zero-particle/twoparticle adjacent-site pairs—in the state |M ; kM , . . . , k1 i, and the two-particle sites are
given by the site indices {kM , . . . , k1 }. The maximum number of dipoles in an N -sitelattice is bN/2c, where b·c is the floor function. The total number of dipole states in
the M -dipole subspace is given by(N − M )!
nM ==M !(N − 2M )!



N −M
.
M

(6.18)

We enumerate the states asM Xkm − mnkM ,...,k1 = 1 +.
m(6.19)

m=1

A state |M + 1; kM +1 , . . . , k1 i is connected via the hopping terms to the states|M ; lM , . . . , l1 i where we have simply removed one of the ki ’s. Since this connection

107
implies a particle hopping onto an already occupied site, we get an extra factor of√2

due to the bosonic nature of the particles. The Hamiltonian restricted to these subspaces
can be written as
bN/2c

X
1
ĤD = N (N + 1)~ωB +Ĥ (M ) +2
M =0Ĥ (M ) =X

bN/2c−1

X(M →M +1)

ĤJ

+ H.c.,(6.20a)M =0

M (U − ~ωB )|M ; kM , . . . , k1 ihM ; kM , . . . , k1 |,(6.20b)kM ,...,k1(M →M +1)

ĤJ√
= −J 2X|M + 1; kM +1 , . . . , k1 i

kM +1 ,...,k1

×

M
+1
X

hM ; kM +1 , . . . , kn+1 , kn−1 , . . . , k1 |,(6.20c)n=1where it is understood that for M = 0, there is no sum over ki ’s, and the single state isexactly |ψMI i.

6.1.2

Symmetric Hamiltonian

We can get an idea of how to further simplify this Hamiltonian by first considering
the coupling between the Mott-insulator state and the one-dipole subspace, given by
√
(0→1)
ĤJ
= −J 2N−1
X|1; l1 ihψMI |.

(6.21)

l=1

We define a new basis for the one-dipole subspace by|1; li = √

N
−2
X
1
e−i2πml/N −1 |1m i.
N − 1 m=0(6.22)(0→1)

Renaming |ψMI i as |0i (we can think of this state as the dipole “vacuum”), ĤJ
becomes
(0→1)

ĤJ

√
= −J 2|10 ih0|.

(6.23)

Consequently, as far as the dipole vacuum is concerned, the only relevant state in the
one-dipole subspace is |10 i. The states |1m6=0 i do not couple to |0i so they can beneglected.

108
The problem is more complicated for the general coupling problem. Taking theabove as a guide, we define a new basis, |Mm i, for the M -dipole subspace by|Mm i = √|M ; kM , . . . , k1 i = √

1
nM

X

ei2πmnkM ,...,k1 /nM |M ; kM , . . . , k1 i,

kM ,...,k1
nX
M −1

1
nM

e−i2πmnkM ,...,k1 /nM |Mm i.

(M →M +1)

(M →M +1)

(6.25)

m=0

In terms of these new states, ĤJ
ĤJ(6.24)√= −J 2X

becomes|M + 1; kM +1 , . . . , k1 i

kM +1 ,...,l1
nX
M −1
1
×
ei2πmnkM +1 ,...,kn+1 ,kn−1 ,...,k1 /nM hMm |
√
nM
n=1
m=0
r
nX
M −1
√
nM +1
= −J 2(M + 1)|M + 1; ψm ihMm |.
nM
M
+1
X

(6.26)

m=0

where we have defined the states,X|M + 1; ψm i =M+1
X

kM +1 ,...,k1 n=1ei2πmnkM +1 ,...,kn+1 ,kn−1 ,...,k1 /nM|M + 1; kM +1 , . . . , k1 i.
√(M + 1) nM +1(6.27)

This new set of states is not identical with the set defined in equation (6.24). We
can see this by noting that each state in equation (6.24) gets multiplied by one phase
only, whereas each state in the expansion, equation (6.27), is multiplied by a sum ofphases. These new states do not comprise an orthogonal set, and in general they are
not normalized. We note however that |M + 1; ψ0 i = |(M + 1)0 i.
If the system is initialized in the state |0i, then for short times, this state will
evolve into |10 i, which will in turn evolve into |20 i, as we can see from taking the special(1→2)

case ĤJ

above. Since |(M + 1)0 i is not in general orthogonal to |M + 1; ψm6=0 i, it

is coupled to |Mm6=0 i through these states. For instance, |20 i is coupled to the states|1m6=0 i through other states within the two-dipole subspace. If the overlap between
these states is small, however, then we can neglect them as long as we evolve only forshort enough times.

109
We can compute these overlaps as(M )Sm0 m = hM ; ψm0 |M ; ψm i
=

M
1 1 X
M 2 nM 0(6.28)

X

ei2πmnkM ,...,kn+1 ,kn−1 ,...,k1 /nM −1

n ,n=1 kM ,...,k1×e−i2πm0 nkM ,...,k 0

n +1 ,kn0 −1 ,...,k1/nM −1.

(6.29)In order to characterize the total overlap of the |M ; ψm6=0 i’s with |M0 i, we compute thequantity,
(M )
S0 =

s

X

(M ) 2S0,m.

(6.30)

m6=0

In figure 6.2 we have plotted some of these overlaps. For large enough numbers of latticesites, those dipole subspaces that are close to the dipole vacuum have the property thatthe overlaps are small. However, for dipole subspaces far from the dipole vacuum (e.g.
M = bN/2c), these overlaps are not small and in fact are never zero. This means that if
we evolve long enough so that these large dipole-number subspaces get populated, thenwe cannot simplify the Hamiltonian. We note that this is a qualitative argument, anda rigorous, quantitative argument for this approximation requires appealing to secondorder perturbation theory or the variational principle.
For short times, the only states relevant to the dynamics are the states |M0 i. The
hopping Hamiltonians become√(M →M +1)
ĤJ
= −J(M + 1) 2

r

nM +1
|(M + 1)0 ihM0 |.
nM(6.31)The effective tunneling parameter is given by
s
r
√
√
n
(M + 1)(N − 2M )(N − 2M − 1)
M +1
J (M →M +1) = J(M + 1) 2
=J 2
. (6.32)nM(N − M )The size of this reduced Hilbert space is bN/2c + 1, which scales as the number ofsites, N . This is a huge reduction over the full Hamiltonian and even over the dipole
Hamiltonian derived in the previous section.

110

HbL

20
18
16
14
12
10
8
6
4

0.8
0.6

S0HML

N HNumber of sitesL

HaL

0.4
0.2

2 3 4 5 6 7 8 9 10
M HNumber of dipolesL

(M )

0.0
4

6

8

10

12

14

16

18

20

N HNumber of sitesL

Figure 6.2: Overlaps S0
in the M -dipole subspace. (a) As the number of sites isincreased in a fixed dipole subspace, the total overlap decreases, indicating that theextra states, |M ; ψm6=0 i become more irrelevant. (b) The overlaps tend towards zero as
long as M . N/4.

111
6.1.3

Validity of the approximations

In this section, we investigate the general dynamics in various parameter regimes.
We have two main goals: (a) to determine when the approximate Hamiltonians generated in the previous sections are useful, and (b) to determine the large-N behavior of
the system.

6.1.3.1

Comparison of full basis and dipole basis dynamics

We first determine how well the dynamics are approximated using only states inthe dipole subspace. To do this, we vary the interaction strength U at a fixed number ofsites N = 6. Taking the initial state to be the unit-filled state, we evolve the system fora time ∆t = 2000~/ER . The lattice depth is V0 = −15ER , corresponding to a hopping
energy of J = 0.0065ER , so that ∆t ≈ 13~/J. We are therefore evolving over timescales
relevant to the atomic dynamics. The interaction strength U takes on values 5J, 10J,
20J, and 70J. We consider the exactly resonant case, ~ωB = U .
In figure 6.3, we plot the total population PD in the dipole subspace as a functionof time using the full Hilbert space of 6 atoms in 6 sites. As the interaction strength
is increased from 5J to 70J, PD saturates at 1, indicating that the bulk of the atomic
dynamics is occurring within the dipole subspaces once the interaction strength is large
enough. This is consistent with the discussion in previous sections.
However, as we can see from figure 6.4, the population in the dipole subspaces
decreases with the number of sites at a fixed U ; in this figure, PD is plotted for N = 2, 3,
4, 5, and 6 at U = 20J. Apparently, there is a lattice-size dependence in the dynamics.
In making the dipole approximation, it is not enough to stipulate U  J. Equation√
6.32 suggests that the proper condition is U  J N , although this may not be enoughif higher dipole subspaces get populated.

112

1.0
0.8

PD

0.6
0.4
0.2
0.0
0

500

1000

1500

2000

t Hunits of ER ÑL

Figure 6.3: Total population PD of the dipole subspaces for a six-site lattice computed
using the full-basis Hamiltonian. As the interaction strength is increased from U = 5J
(blue) through U = 10J (purple) and U = 20J (red) to U = 70J (black), this population
saturates at 1, indicating that the dynamics occur mainly within the dipole subspaces.

1.00
0.95

PD

0.90
0.85
0.80
0.75
0.70

0

500

1000

1500

2000

t Hunits of ER ÑL

Figure 6.4: Total population PD of the dipole subspaces for an interaction strength of
U = 20J computed using the full-basis Hamiltonian. As the number of sites is increased
from N = 2 (purple) through N = 3 (blue), N = 4 (red), and N = 5 (black) to N = 6(green), this population decreases, indicating that there is the dynamics depend in some
non-trivial way on the system size.

113
1.0
0.8

PS

0.6
0.4
0.2
0.0
0

200

400

600

800

1000

t Hunits of ER ÑL

Figure 6.5: Total population PS in the symmetric states as a function of time as computed with the dipole subspace Hamiltonian. The lattice size is varied, taking on valuesof N = 4, 5, 6, 8, 10, 12, 15, and 18 sites. As the lattice size is increased, the symmetric
Hamiltonian approximation gets worse. The timescale over which non-symmetric states
get populated is given by ~/J, indicated as a vertical dashed line.

6.1.3.2

Comparison of dipole basis and symmetric basis dynamics

The situation is not as promising for the symmetric Hamiltonian. In figure 6.5,
we plot the total symmetric-state population at U = ~ωB for lattice sizes ranging fromN = 4 to N = 18 using the dipole subspace Hamiltonian. The approximation that the
dynamics occur within the symmetric subspace is good only for times t  ~/J; thistimescale is indicated as a vertical dashed line in the figure. For lattice sizes N ≥ 18 andtimes greater than t > ~/J, the atomic dynamics occur entirely outside the symmetricsubspace. For this reason, the symmetric approximation made in the previous section
is not a good one. However, there is evidence that such an approximation works ina dilute gas, i.e. one where the number of atoms is much smaller than the number oflattice sites [110].

6.1.3.3

Two-state dynamics for |U − ~ωB |  J

There is one special case where the symmetric Hamiltonian is a good approximation for the atomic system. When |U − ~ωB |  J, the coupling of the unit-filled state|ψMI i to dipole subspaces beyond M = 1 scales as J 2 /|U − ~ωB |  J. The system then

114
1.0
0.8

PS

0.6
0.4
0.2
0.0
0

50

100

150

200

t Hunits of ER ÑL

Figure 6.6: Population in the symmetric states as a function of time for |U − ~ωB |  Jcomputed using the dipole subspace Hamiltonian. The system undergoes off-resonant
Rabi-type oscillations between the unit-filled state (black) and the symmetric one-dipole
state (blue). Other symmetric dipole states get negligibly populated. Parameters here
are N = 10, U = 100J, J ≈ 0.0065ER , and ~ω = U − 20J.

reduces to a two state system with Hamiltonian,Ĥ2 = (U − ~ωB )|1ih1| − Jp2(N − 1)(|1ih0| + |0ih1|),(6.33)and the system displays Rabi-like oscillations between the unit-filled state and the symmetric one-dipole state. These dynamics are particularly simple, and we therefore consider them below in the discussion of the optical detection of the atomic motion in thissystem. An example of these dynamics is plotted in figure 6.6. The general dynamics
of this system is left for future work.

6.2

Detection Scheme
In order to optically probe the atomic dynamics just discussed, we place thesystem inside an optical cavity that supports two counter-propagating running-wave
modes. We pump one of the two cavity modes, and photons from this mode scatter off
the atoms into the second mode. A heterodyne detection scheme at the cavity output
allows us to measure the dynamics of the amplitude of the probe field. From thismeasurement, we can infer certain characteristics of the atomic motion.

115
The Hamiltonians for the bare cavity modes and the cavity-atom interaction are
given byĤC = ~ω N̂C + ~η(â†kC + âkC ),(6.34)Z
ĤA/C = ~g0 N̂A N̂C + ~g0 dz Ψ̂† (z) â†kC â−kC e−i2kC z + â†−kC âkC ei2kC z Ψ̂(z), (6.35)
where
Z
N̂A =

dz Ψ̂† (z)Ψ̂(z),N̂C = â†kC âkC + â†−kC â−kC ,(6.36)(6.37)

are the total number of atoms and total number of photons, respectively. Decay of thecavity modes out of the cavity is given by the Liouvillian,L̂ρ̂ = −~κ X  †âk âk ρ̂ + ρ̂â†k âk − 2âk ρ̂â†k .
2(6.38)

k=±kC

If we assume that the number of photons in both modes is small enough, we can assume
that the atomic dynamics is unaffected by the presence of the cavity. In particular, nointerband transitions are induced by the presence of photons in the cavity. In this case,
we can expand ĤA/C in the Wannier functions from the previous section, resulting in
ĤA/C = ~g0 N̂A N̂C + ~g0

X †b̂j 0 b̂j â†−kC âkC fj 0 ,j (2kC ) + H.c.,(6.39)

j 0 ,jwhereZ
fj 0 ,j (k) =dz w∗ (z − zj 0 )eikz w(z − zj ).

(6.40)

2kC = 2nkL + q0 ,(6.41a)0 ≤ q0 ≤ 2kL ,(6.41b)n ∈ Z,(6.41c)Taking116
the single-particle matrix element becomes
fj 0 ,j (2kC ) = e−iq0 j 0 d 1

XNiq(j−j 0 )d

e

Z


∗
dz ψq+q(z)ei2nkL z eiq0 z ψq (z) ,0(6.42)

qsince eiq0 z has a well-defined quasi-momentum as a sum over Bloch states with the same
quasi-momentum. We define
Iq(n) (q0 ) =Z


∗i2nkL ziq0 zdz ψq+q
(z)e
e
ψ
(z)
.
q
0

(6.43)

If we restrict our attention only to the dipole subspaces, then the only terms that survive
in ĤA/C are those involving b̂†j b̂j and b̂†j b̂j±1 . In this case, the interaction Hamiltonianis!
X †1 X (n)Iq (q0 )b̂j b̂j e−iq0 jd + H.c.
N q
j
!
X
X
1
e−iq0 dI (n) (q0 )e−iqdb̂†j+1 b̂j e−iq0 jd + H.c.
N q q
j
!
X †
1 X (n)eiq0 dIq (q0 )eiqdb̂j−1 b̂j eiq0 jd + H.c.
(6.44)N q

ĤA/C = ~g0 N̂A N̂C + â†−kC âkC ~g0+ â†−kC âkC ~g0+ â†−kC âkC ~g0jThis shows us that the plane-wave index n is irrelevant to the extent that thedependence on n is absorbed into the coupling constants. It may make a difference inthe relative strength of the two terms, but it is clear that the q0 is the more important(0)

quantity. For this reason, we set n = 0 in what directly follows and take Iq (q0 ) = Iq (q0 ).
In figure 6.7, we plot the absolute value of the single-particle matrix elements, f0,0 (q0 )and f1,0 (q0 ) as a function of quasi-momentum for different lattice depths. As the lattice
depth is increased, |f0,0 | increases and |f1,0 | decreases. In addition, |f0,0 | ≈ 1 across all
quasi-momenta, whereas |f1,0 | = 0 at q0 = π/d. For this reason, we choose q0 = π/dso that the second and third terms in equation (6.44) are zero, and the probe field issensitive only to the site-numbers. In this way, the signal depends explicitly on the
dipole subspace in which the dynamics is occurring.
With the balanced heterodyne detection outlined in the previous chapter, we candetect the cavity field amplitude proportional to hâ−kC i. In order to determine what

117

HaL 1.00

HbL 0.06
0.05
0.04

È f1,0 ÈÈ f0,0 È

0.95

0.90

0.03
0.02
0.01

0.85

0.00
0.

0.5

1.

q0 Hunits of ΠdL

1.5

2.

0.

0.5

1.

1.5

2.

q0 Hunits of ΠdL

Figure 6.7: Hubbard model coupling parameters fi,j (q0 ) for the detection scheme plotted
as a function of quasi-momentum. (a) The diagonal coupling f0,0 increases as a functionof lattice depth V0 and is roughly constant across quasi-momenta. (b) The next-tonearest-neighbor coupling f1,0 decreases with lattice depth and is exactly zero at q0 =
π/d.

118
properties of the atomic motion this will measure, we compute the equations of motion
for hâ±kC i. They areXdhâ−kC i κ= δp + g0 hN̂A i − ihâ−kC i + g0 f0,0 (q0 )hâkC b̂†j b̂j ie−iq0 jd
dt2(6.45)XκdhâkC i = δp + g0 hN̂A i − ihâkC i + η + g0 f0,0 (q0 )hâ−kC b̂†j b̂j ieiq0 jd
dt2(6.46)

i

j

and
i

j

Since the cavity linewidth is large, we can adiabatically slave hâkC i to the atomic motion,
writinghâkC i → α =−ηδp + g0 hN̂A i − iκ/2,

(6.47)We have neglected the other terms because they are small compared to both κ andhN̂A i. We can also adiabatically solve for hâ−kC i, resulting inhâ−kC i =−αg0 f0,0 (q0 )δp + g0 hN̂A i − iκ/2hn̂(q0 )i,(6.48)

wheren̂(q0 ) =X †hb̂j b̂j ie−iq0 jd .

(6.49)j

The last quantity is the one that tells us how the field amplitude reflects the atomic
dynamics. In the next section, we explore the behavior of this expectation value for thevarious types of atomic states that show up in our approximations.

6.2.1

Dipole-subspace detection

We need to determine what sort of information about the atomic dynamics that
detection of the field amplitude allows us access to. As a first step, we compute theexpectation values, hM ; kM , . . . , k1 |n̂(π/d)|M ; kM , . . . , k1 i. All other expectation values
vanish, since |M ; kM , . . . , k1 i are site-number eigenstates. Abbreviating |M ; kM , . . . , k1 i

119
as |M ; {ki }i, the matrix elements are
hM ; {ki }|n̂(π/d)|M ; {ki }i =X

hM ; {ki }|b̂†j b̂j |M ; {ki }ie−ijπ

j

=

M
X X

=2

(δj,kn − δj,kn +1 ) + 1 e−ijπ

n=1

j
M
X

!

(−1)kn +n=1

X(−1)j .

(6.50)

j

The second term is clearly zero. The size of the first term depends on where the dipoles
are, and it can range anywhere from 0 to ±2M . For instance, if {ki } = {2, 4, 6, 8}, thenthe first term is equal to −8, whereas if {ki } = {1, 4, 7, 10}, then it is equal to zero.
This matrix element is sensitive to the distribution of dipoles within a dipole subspace.
In the context of the simplified two-state model of the previous section, we computeN −1

h1|n̂(π/d)|1i =

X
1
−2
2
(−1)j =
,
N −1
N −1

(6.51)

j=1

where we have assumed a lattice with an even number of sites. While this quantity
differs from zero, which is the value of h0|n̂(π/d)|0i, it decreases with the lattice size,
indicating that the signal will not be very strong for large lattices. In figure 6.8, we haveplotted F from equation (6.49) as a function of time for the system described in figure
6.6. The field oscillates from 0, which indicates that the atoms are in the unit-filledstates, to 0.0013, which is where the atoms are in a superposition of unit-filled andsymmetric one-dipole states.

6.2.2

Conclusion

We have investigated a system comprised of strongly-interacting bosons confinedin a tilted optical lattice. In the regime where the interaction strength is equal to theenergy difference between adjacent lattice wells, the system dynamics take place in a
restricted subspace that consists of so-called dipole states.

120ÈXn` Hq0 L\ 20.00150.00100.00050.0000
0

50

100

150

200

t Hunits of ÑER L

Figure 6.8: Time sequence for |hn̂(q0 )i|2 . The probe-field intensity is proportional tothis quantity. The signal oscillates between 0 and 0.0015. The maximum value 0.0015
depends on the population of the symmetric one-dipole state.

121
We have also developed a method for measuring this atomic dynamics by coupling
the atoms to two counter-propagating modes of an optical cavity. Pumping one mode
strongly, we detect the light-field out-coupled through the cavity mirrors from the second
mode. The center-of-mass motion is reflected in the dynamics of the probe field, as wedemonstrate for one special case where the atomic motion consists of particles oscillatingback and forth between adjacent wells.
The detection scheme is relatively insensitive to the dynamics in this specialcase, but there is a strong signal for dipole states far from the unit-filled state. One
generalization of this measurement method is to measure higher-order correlations of
the light-field, since they are coupled to higher-order correlation functions of the atoms
such as hn̂i n̂j i. These density-density correlations are sensitive to differences in particle
number on different sites, and therefore this may be a better differentiator of the dipole
states.
Eventually, this system might be utilized as a probe for more complicated manybody systems—for instance, those experiencing long-range interactions.

Chapter 7

Quantum Non-Demolition Measurementof the Photon Number in a Cavity

Up to this point, we have been interested in the properties of single- and manyparticle atomic systems and how these properties might be experimentally probed withlight. In the final chapter of this thesis, we reverse our focus and examine a cavity
QED system in which atoms may be used to probe the state of the light field. In thiscase, we perform a quantum nondemolition (QND) measurement of the cavity photon
number by measuring the internal state of an atom dispersively coupled to a single
mode of the cavity. Besides providing tests of fundamental principles such as quantum
measurement theory, this setup allows for a simple method of preparing non-classicalstates of light such number states. These states are important in precision metrology
applications where the goal is to beat the standard quantum limit of measurement
uncertainty [55, 83].
We imagine an experiment where a cloud of two-level atoms are subjected to a
π/2-pulse. The ground state atoms are split off and allowed to evolve freely, whereas theexcited state atoms are allowed to interact with a single standing-wave mode of a fardetuned optical resonator. We operate in the Bragg regime where the recoil energy of theatom during absorption and re-emission of a photon is much larger than the interaction
strength. This restricts the dynamics to two center-of-mass states undergoing Rabi-likeoscillations with a photon-number dependent frequency. The excited and ground state

123
atoms are recombined and subjected to a second π/2-pulse. The population inversionof each atom is measured, and the photon number is inferred using a Bayesian method.
A similar scheme was proposed for making QND measurements of the photon
number by measuring the momentum distribution of atoms sent through a cavity [56].
This system operated in the Raman-Nath regime [88] where many Bragg peaks appear
in the atomic momentum distribution. This requires that the atom-field interaction
time must be much smaller than the inverse recoil frequency.
Our scheme is also closely connected to the microwave cavity QND measurements
performed in the group of Raimond and Haroche [50]. In these experiments, Rydberg
atoms are sent through a high finesse cavity. The recoil imparted to the atom during absorption or emission of a microwave photon is negligible in these situations. In addition,
the interaction is such that the population inversion of the atoms is unchanged. The
relative phase between excited and ground state changes with a frequency dependent on
the number of photons in the cavity, and Ramsey interferometry at the cavity output
allows to infer the photon number in the cavity.
Quantum non-demolition measurements have in general been applied to highfidelity readout of ionic qubit states [59], to probing of Rabi oscillations in cesium
clocks [26, 131], and to proposals for atomic state preparation in ultracold gases [87].

7.1

Ramsey Interferometry in the Bragg Regime
For the purposes of this QND measurement scheme, we need the probability thatan atom is in the excited state given there are n photons in the cavity. The Hamiltonian
describing the interaction of a single two-level atom with one quantized standing-wave
mode of a linear cavity and a classical clock laser is
Ĥ =

~δ z
1 2
σ̂ +
P̂ + ~ωâ† â22m
~Ω  + ikL x̂σ̂ e+ σ̂ − e−ikL x̂ ,+ 2~g â† â cos2 (kC ẑ) +
2

(7.1)

124
where P̂ is the center-of-mass momentum of the atom, δ is the detuning between theclock laser frequency and the atomic transition frequency, and kL is the clock-laser wave
vector. We work in the basis where σ̂ z , P̂ , and â† â are diagonal,|σ, p, ni = |σi|pi|ni,(7.2a)

P̂ |pi = ~p|pi,(7.2b)â† â|ni = n|ni,(7.2c)σ̂ z |σi = (δe,σ − δg,σ )|σi.

(7.2d)The matrix elements of the Hamiltonian sans the clock-pulse are
 2 2


~ p
~gn
(n)
Ĥpp0 = δpp0
+ ~(g + ω)n +δp0 ,p+2kC + δp0 ,p−2kC ,
2m
2

(7.3)

where we have left out reference to the internal degrees of freedom.
According to the experiment envisioned above, the initial state of the system is|ψi i = |g, 0, ni.

(7.4)

A π/2-pulse is applied to this state, resulting in
1|ψ(0)i = √ |ni(|0i|gi + |kL i|ei).
2

(7.5)

The ground state is allowed to evolve freely for a time t, resulting in|ψg (t)i = eiδt/2 e−in(g+ω)t |0i|gi|ni.

(7.6)

On the other hand, the excited state atom interacts with the cavity field. We
separate out the free time-dependence of this state and expand in plane-wave modes as|ψe (t)i = e−iδt/2 e−in(g+ω)t |ni|eiX

Cp (t)|pi,

(7.7)

p

Cp (0) = δp,kL .

(7.8)

The equation of motion for the coefficient Cp is given byiĊp =ng~ 2
p Cp +(Cp+2kC + Cp−2kC ) .
2m
2

(7.9)

125
In particular, the initial state CkL couples directly to CkL ±2kC . If we consider a near2 /2m—the condition forresonant clock-pulse, i.e. kL ≈ kC , and we assume ng  ~kC

being in the Bragg regime—then we need only consider two states, CkL and CkL −2kC .
We then have the effective Hamiltonian,


2
ng/2
 kL ~/2m

Heff = ~ 
,
2
ng/2
(kL − 2kC ) ~/2m

(7.10)

in the basis {|kL i, |kL − 2kC i}.
After evolving the initial state for a time t under this Hamiltonian, the state isgiven by1
|ψ(t)i = √ eiδt/2 e−in(g+ω)t |gi|ni|0i
21+ √ e−iδt/2 e−in(g+ω)t |ni|ei (CkL (t)|kL i + CkL −2kC (t)|kL − 2kC i) ,2

(7.11)

where, given the initial condition Cp (0) = δp,kL ,

 
 
Ω
∆
Ω
CkL (t) = e−iΦt cos
t − i sin
t
,
2
Ω
2
 
Ω
−iΦt −iVCkL −2kC (t) = e
sin
t .
Ω
2

(7.12)(7.13)The parameters in these expressions are~ 2(k + (kL − 2kC )2 ),
4m L
~ 2
∆=
(k − (kL − 2kC )2 ),2m LΦ=(7.14a)(7.14b)V = ng,pΩ = V 2 + ∆2 .

(7.14c)(7.14d)We perform a second π/2-pulse and compute the population of the excited state at timet; it is1 1
P (e|n)(t) = − cos((δ + Φ)t) cos
2 2



Ωt
2



1∆
+
sin((δ + Φ)t) sin
2Ω



Ωt
2


.

(7.15)

126
The conditions that this formula is correct were given above as kL ≈ kC and ng 
2 /2m. We can make these conditions more exact by noting that we want |k i to~kC
Lconnect to |kL − 2kC i but not to |kL + 2kC i. The coupling constants between thesestates are given by
gkL ±2kC ∼

(gn)2
2m
m 1 (gn)2.
=~ (kL ± 2kC )2 − kL2
2~ kC kC ± kL

(7.16)

As long as gn  2~kC (kC + kL )/m, then the coupling to |kL + 2kC i is negligible. This
limits the width of the momentum distribution of the initial ground-state atomic gas
along the axis of the cavity.
For simplicity, we set δ = −Φ, resulting in


s 

2
2
1 1
ωR t
ng
kC − kL 
P (e|n)(t) = − cos 
+ 16
,
2 2
2
ωR
kC

(7.17)

where ωR = ER /~ is the recoil angular frequency of the cavity field. If the initial
ground-state gas of atoms has a momentum distribution ψ(~p) along the axis of thecavity, then we replace kL with p + kL in the expression for P (e|n)(t) and integrate itagainst |ψ(~p)|2 .

7.2

Bayesian Analysis
The Bayes theorem for conditional probabilities,
P (A|B) =P (A)P (B|A),
P (B)(7.18)

states that the conditional probability of A occurring given that B occurred is equal tothe probability that A occurs times the conditional probability that B occurs given A,
up to a normalization. Here, we take A to be the number of cavity photons n and B tobe the internal state of the atom σ, which can be either e or g. The quantity P (σ|n)
is called the forward distribution, P (n|σ) is called the updated distribution, and P (n)is called the prior distribution and represents our current state of knowledge before the127next measurement. As measurements are made, our knowledge about the state of thecavity system is updated, as follows.
We assume that we have some prior knowledge of the distribution P0 (n) of photons
in the cavity before any measurements are made. This distribution might be a coherent
state, reflecting our knowledge of the initial cavity system setup, or it might be a uniform
distribution, reflecting our knowledge that the number of photons in the cavity is lessthan some number M . After a single measurement σ1 of the internal state of the atom,
our knowledge is updated via Bayes theorem as
P (n|σ1 ) = N1 P0 (n)P (σ1 |n),(7.19)

where N1 is a normalization. If we make N measurements, obtaining the measurement
record,REC = {σ1 , σ2 , . . . , σN },(7.20)

then the updated probability distribution for having n photons given this record is givenby
P (n|REC) = NN P0 (n)NY

P (σj |n).

(7.21)

j=1

The final state of the field is a number state since P (n|REC) is expected to limit tosome δn,nf .
This is the standard Bayes prescription for making a series of measurements todetermine a probability distribution, and it coincides with our understanding of thesquare of the wave-function (or the diagonal elements of the density matrix) being aprobability distribution. However, quantum mechanics deals in probability amplitudes
first and foremost, and it turns out that there is a quantum Bayes theorem for theseamplitudes [25, 44]. Here, since we are dealing with a QND measurement—i.e. themeasurement of the internal state of the atom does not alter the photon number—the coherences between different photon number states do not affect the atom-field

128
interaction. We can therefore neglect them, allowing us to work with the probabilities
directly rather than the probability amplitudes.

7.2.1

Measurement protocols

In the expression for the forward distribution P (e|n)(t), the time of interaction t
is to this point unspecified. This is intentional, as it determines how fast the Bayesianinference method converges. To see that this is so, we consider the case where weknow that there are either zero or one photons in the cavity. For simplicity, we assume
kC = kL . If we choose t = 4π/g, then P (e|0)(t) = P (e|1)(t) = 1. The forward
distribution then does not distinguish between the 0 and 1 photon states. In particular,
the updated prior does not change.
Intuitively, what we want is that the prior “maximally changes” when it is updated. One way to achieve this is to have the largest spread in possible values, P (e|n)(t).
The smallest time for which this happens is given by τ = 2π/nmax |g|, where nmax isthe maximum number of photons in the cavity based on our prior knowledge of thesystem. In figure 7.1(a), we have plotted the time-dependence of P (e|n) for n = 0 ton = 10. The probabilities achieve a maximum spread, varying between 0 and 1, atexactly τ = 2π/10|g|, as can be seen more readily in figure 7.1(b).
According to this measurement protocol, each atom interacts with the cavity
modes for a time τ = 2π/nmax |g|. This is particularly simple, and it seems that we havegained by the fact that the interaction time is very small. However, there are otherprotocols that promise to converge faster than this simple one.
We can see this by investigating figure 7.2. In figure 7.2(a), we have extended
the P (e|n) time sequences out to t = 2π/|g|. At this time, we can see that the P (e|n)
is zero for half of the photon numbers and 1 for the rest (see black curve in 7.2(b)).
This means that if a measurement is made after an atom is allowed to interact for thisamount of time, then the probabilities for half of the photon numbers are automatically129

HaL 1.

HbL

PHeÈnL

0.8
0.6
0.4
0.2
0.
0.0

0.2

0.4

0.6

t Hunits of 2Π10ÈgÈL

0.8

1.0

0

2

4

6

8

10

n

Figure 7.1: Forward probability distribution P (e|n) for the first measurement protocol.
(a) The forwards for n = 0 to n = 10 are plotted for interaction times from t = 0 tot = 2π/10|g|. (b) The probabilities at t = 2π/10|g| span the entire range from 0 to 1.

130
HaL 1.

HbL

PHeÈnL

0.8
0.6
0.4
0.2
0.
0.0

0.2

0.4

0.6

0.8

1.0

0

2

4

t Hunits of 2ΠÈgÈL

6

8

10

n

Figure 7.2: Forward probability distribution P (e|n) for the first optimized protocol. (a)The forwards for n = 0 to n = 10 are plotted for interaction times from t = 0 to t =2π/|g|. (b) The probabilities at τ = 2π/|g| (black), τ = 2π/2|g| (blue), and τ = 2π/3|g|(orange) are each partitioned into sets of photon numbers where the probabilities are
zero or one.

set to zero. This contrasts with measurements made in the previous protocol, where inany measurement, only n = 0 and n = nmax can be set to zero. This promises to speed
up the convergence process.
We generalize this by noting that there are many times in figure 7.2(a) for which
the photon numbers are partitioned into sets where P (e|n) = 0, P (e|n) = 1, and
0 6= P (e|n) 6= 1. Some examples—τ = 2π/|g| (black), τ = 2π/2|g| (blue), τ = 2π/3|g|(orange), and τ = 2π/5|g| (green)—are shown in figure 7.2(b).
Based on these considerations, we adopt the following measurement protocol.
Each atom interacts with the cavity mode for a different amount of time. In order,these are• τ = 2π/n|g|, where n < nmax is prime.
• τ = 2π/n2 |g|, where n is prime and n2 < nmax · · ·• · · · τ = 2π/nk |g|; k is the maximum power for which there is an n such that n
is prime and satisfies nk < nmax .
• τ = 2π/nmax until the probability distribution is converged.

131
0.25

Pn

0.20
0.15
0.10
0.05
0.00
0

2

4

6

8

10

n

Figure 7.3: A three-measurement Bayesian sequence starting with a coherent state ofmean 3 (black). In the first measurement, the result is that the atom is in the excitedstate; the initial distribution is multiplied by the forward distribution P (e|n) (blue).
The second measurement yields |gi, so the updated distribution is multiplied by P (g|n)(orange). Finally, the third measurement yields |ei, so the updated distribution ismultiplied by P (e|n) (green).

This measurement scheme can be applied to prepare a number state of the cavity
or to perform tomography on an initial photon number distribution.

7.3

Results
In figure 7.3, four photon-number distributions are plotted. The black curveis an initial coherent state distribution with mean 3. The next three curves are the
distributions after successive measurements. Here, the measurement record is REC ={e, g, e}, and therefore the initial distribution gets multiplied first by the distribution in
7.1(b), then by one minus this distribution, and finally by the distribution again. We
can already see how the distribution is narrowing.
In figure 7.4, we show three examples of Bayesian measurement sequences starting from the coherent state distribution with mean 3. These distributions are clearly
converging to different photon numbers, nine, three, and two, respectively. It is also
clear that the convergence is not very fast. That is, even 200 measurements are notenough to converge two of these distributions.
In figures 7.5 and 7.6, we illustrate similar results as in figures 7.3 and 7.4 except

132

N Hnumber of measurementsL

HaL 0

HbL

HcL

25
50
75
100
125
150
175
200
0 1 2 3 4 5 6 7 8 9 10 11

0 1 2 3 4 5 6 7 8 9 10 11

0 1 2 3 4 5 6 7 8 9 10 11n Hphoton numberLn Hphoton numberLn Hphoton numberL

Figure 7.4: Three 200-step Bayesian measurement sequences starting with a coherent
state of mean n = 3. (a) The distribution seems to be converging to n = 9, but there
is still significant population in other photon number states. (b) The distribution has
converged to n = 3 after about 150 measurements. (c) The distribution is converging
to n = 2.

133
0.8

Pn

0.6
0.4
0.2
0.0
0

2

4

6

8

10

n

Figure 7.5: A three-measurement Bayesian sequence starting with a coherent state ofmean 3 (black). Here, the second measurement protocol is used. After the first measurement, all odd-number photon states have zero probability (blue). After the second
measurement all photon numbers except for 0, 4, and 8 have zero probability (orange).
Finally, after the third measurement, the zero-photon state has a zero probability, andthe distribution seems to be converging to n = 4 (green). The forward distributions
used here are illustrated in figure 7.2.

that the optimized measurement protocol is used. The distributions are converging anorder of magnitude faster.
Finally, we perform tomography on an initial coherent state of mean n = 3. The
optimized measurement protocol is used, with a total of 19 measurements made in eachsequence. Five thousand sequences are run, and the converged results are collected andbinned. The resulting distribution is essentially identical to the original distribution.

7.4

Experimental Considerations
We have discussed some of the conditions that need to be satisfied in order forthis scheme to work. To reiterate, these conditions are |kC − kL |  |kC + kL | andgn  2~kC (kC + kL )/m. Assuming an initial atomic state with momentum ~p and a
2 /m.
resonant π/2-pulse, kC ≈ kL , these conditions become |p|  |p+2kC | and gn  4~kC

The essence of these conditions is that both the atom-field coupling and the width of
the initial atomic momentum distribution along the axis of the cavity need to be muchsmaller than the recoil energy of the cavity field.

134

N Hnumber of measurementsL

HaL

HbL

HcL

2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7 8 9 10 11

0 1 2 3 4 5 6 7 8 9 10 11

0 1 2 3 4 5 6 7 8 9 10 11n Hphoton numberLn Hphoton numberLn Hphoton numberL

Figure 7.6: Three 200-step Bayesian measurement sequences starting with a coherent
state of mean n = 3. Here, the second measurement protocol is used. (a) The distribution converged to n = 8 within 6 measurements. (b) The distribution is converging
to n = 3 after 16 measurements. (c) The distribution converged to n = 2 after 4
measurements.

HaL

HbL
0.2

Pn

0.15
0.1
0.05
0
0

2

4

6

n

8

10

12

0

2

4

6

8

10

12

n

Figure 7.7: Reconstruction of the initial coherent state of mean 3 using the Bayesianmeasurement sequence. Here, the optimized protocol is used, where in each sequence 19
measurements are made. (a) The initial coherent state distribution. (b) The converged
results of 5000 sequences are binned and plotted. This distribution reconstructs theoriginal distribution.

135
1.0

PHeÈnL

0.8
0.6
0.4
0.2
0.0
0

2

4

6

8

10

n

Figure 7.8: Forward distribution for various initial momentum distribution widths. We
assume a Gaussian profile for the atomic momentum distribution and consider different
widths, ∆p = 0 (black), ∆p = 0.005kC (blue), ∆p = 0.02kC (orange), and ∆p = kC
(green).

The width of the initial momentum distribution also affects the convergence of
the Bayesian sequence. We can see this by assuming an initial Gaussian momentum
distribution,
ψ(~p) =

π 1/4

1
2
2
p
e−p /2∆p ,∆p/kC(7.22)

for the atoms and integrating the forward distribution—equation 7.17 with kL → p +kC —against |ψ(~p)|2 . In figure 7.8, the modified forward distribution is plotted for∆p = 0, ∆p = 0.005kC , ∆p = 0.02kC , and ∆p = kC . By the time the width is equal tothe cavity momentum, the forward distribution is essentially flat, and it cannot be used
to distinguish different photon number states in a Bayesian sequence.
Another condition we briefly touched on is that the lifetime 1/γ of the atomic
excited state needs to be larger than the time involved in a single Bayes measurement,i.e γτ  1. Similarly, the lifetime 1/κ of photons in the cavity needs to be larger than
the time over which the entire Bayes sequence occurs, i.e κτ N  1 where N is the total
number of measurements made.
As an example, we consider a gas of 87 Sr atoms with a temperature of T = 2µK.
The two levels of the atom correspond to the narrow inter-combination line. Typicalexperimental parameters are mA = 1.45 × 10−25 kg, λA = 698 nm, γ = 2π mHz,

136
λC = 813 nm, and κ = 2π × 10 kHz [78]. In terms of the recoil angular frequency,
ωR = 2π × 3441 Hz, we have g ≈ −10−4 ωR , kB T ≈ 12 ~ωR , 2π/κ = 10−4 s, 2π/γ = 103 s,
and τ = 2π/11|g| = 0.26s.
Although the temperature is very much larger than the recoil energy—implyingthat the width of the momentum distribution of the atoms is too large for the Bayesian
scheme to work—much of this energy can be given to the transverse motion of the atoms
rather than the motion along the axis of the cavity. The temperature of the gas of atoms
is therefore not a limiting property. However, in this particular setup, the lifetime ofphotons in the cavity is much smaller than the time over which a single measurement
is made, rendering the method unfeasible.

7.5

Conclusion
We have demonstrated a Bayesian protocol that measures the number of photons

in a cavity by coupling the field to a sequence of two-level atoms and performing Ramseyinterferometry. With a proper choice of measurement protocol—that is, a choice of
atom-field interaction times—the state of the cavity field can be collapsed onto a number
state within fifteen measurements with a relatively high fidelity. We have demonstrated
that the scheme can be used for both quantum state preparation and quantum state
tomography.
In cases where the cavity linewidth is large, the lifetime of a photon in the cavity
is shorter than the timescale of the measurements required for the Bayesian inference
scheme. However, in this case, the cavity field relaxes to a steady state between measurements of the atomic inversion. In essence, the system is re-prepared after every
single measurement, indicating that we can perform tomography on this steady state.
A possible route to implementing tomography in the bad-cavity limit is outlined inreference [25], where measurements distributed in time are treated. This will be thesubject of future work.

Chapter 8

Conclusion

In this thesis, we have examined a series of physical systems comprised of gases
of ultracold atoms trapped in optical lattices, and we have proposed methods to probethese systems with light.
In the first part of this thesis, we examined the ground-state behavior of a gas ofultracold atoms trapped in a rotating optical lattice. By analogy with quasi-momentum
in translationally invariant periodic systems, the notion of quasi-angular momentum wasintroduced to label the eigenstates of a Hamiltonian that has a discrete rotational symmetry. This symmetry language was translated into a many-body formalism where thecreation and annihilation operators carried quasi-angular momentum. Systems comprised of strongly-interacting bosons and non-interacting fermions in two-dimensionallattices were investigated. It was shown that quasi-angular momentum is useful in analyzing the ground-state properties of quantum gases of bosons or fermions in rotating
optical lattices. In particular, monitoring the quasi-angular momentum of the groundstate as a function of rotation allowed us to identify transitions between different circulation values.
For the case of strongly-interacting bosons, the quasi-angular momentum m cycles through the values m = modN (nl), where l = 0, 1, 2, . . . and n is the numberof particles. For the case of fermions, m cycles through the values m = modN (nl),where l = 0, 1, 2, . . . , for odd numbers of particles and m = modN (n(l + 1/2)), where138
l = 0, 1, 2, . . . , for even numbers of particles. A system of non-interacting fermions isthereby distinguishable from a system of hard-core bosons. Similar behavior obtains for
systems of fermions in two-dimensional square lattices.
We also presented a possible avenue by which the quasi-angular momentum of astate can be experimentally determined. We identified characteristics in the momentum
distribution distinguishing between different quasi-angular momentum states at low rotation speeds. For single-particle systems in two-dimensional square lattice geometries,
such signatures include the existence of a peak at k = 0 only for the m = 0 state andpeak-spacing differences between m = 2 and m = 1, 3 state.
In the second part of this thesis, we proposed a scheme for non-destructively
probing the dynamics of atoms by coupling them to two counter-propagating modesof a ring cavity. This scheme satisfied three main goals. The probe was weak sothat the atoms can be continuously monitored without affecting their dynamics. The
cavity fields act simultaneously as both the external lattice potential and the weakprobe for the atoms, requiring no external interrogation fields. The signal-to-noise ratio
was large enough for feasible experimental detection. This method was applied to themeasurement of the Bloch oscillations that non-interacting atoms in an optical latticeundergo when subjected to a constant, external force. These oscillations were reflectedin the dynamics of the probe field, and balanced heterodyne detection of the probe fieldat the cavity output combined with integration in time and across the atomic cloudallowed for signal-to-noise ratios as high as 104 .
We then relaxed the condition that the cavity fields act both as the probe field
and the lattice field. We applied a variant of this measurement scheme to probe thedynamics of bosonic atoms confined in deep optical lattices and subjected to a constant,external force. Choosing an energy offset Ω between adjacent lattice wells approximately
equal to the interaction energy U in the Bose-Hubbard model, the Hilbert space could
be reduced to include only so-called “dipole states.” These states are generated from the139
unit-filled state by introducing defects that consist of a pair of adjacent sites in which
one particle has been moved one site to the left. We derived an effective Hamiltonianin this low-energy subspace.
By exploiting approximate symmetries of the Hamiltonian, we further reduced
the number of relevant states. The final numerical complexity of the problem scaled
linearly with the number of lattice sites rather than exponentially. However, we showedthat the dynamics of the atoms were described by this simple symmetric Hamiltonian
only for short times, after which the population of these states decayed to zero. Thissuggested that the system could be described by a master equation.
We investigated how these dynamics would be reflected in the behavior of themodes of an optical ring-cavity coupled to the gas of atoms. We proposed the following
experimental setup. One mode of the cavity is pumped, and the other mode is monitored
as photons are scattered off of the atoms. The time-dependence of the probe field
amplitude is given schematically byα∝X †hb̂i b̂j ie−ikjd ,

(8.1)

hi,ji

where b̂j is the operator annihilating a particle on site j, and d is the lattice spacing.
The quantity on the right hand side is reminiscent of the structure factor of the atomicgas, and we can infer that the atomic motion can be measured and analyzed throughthe behavior of the light field.
The future direction of this work involves completely characterizing the dynamics
of the atoms and the probe field coupled to these atoms. This relatively simple manybody system can then be used as a probe for more complicated many-particle systems.
For instance, by characterizing how the dynamics change when long-range interactions
between atoms are introduced, it may be possible to use this scheme to probe long-range
correlations between atoms.
In the final part of this thesis, we reversed the paradigm of using light to study

140
atomic systems and instead proposed a quantum non-demolition measurement of thecavity photon number by entangling the field with two momentum states of long-lived
atoms. By means of a sequence of π/2-pulses, the internal states of the atom arecoupled to its external states. We measure the internal state of the atom at the endof the measurement sequence. By use of quantum Bayesian inference and a series ofmeasurements, the state of the field collapses to a photon number state. This scheme
can be used not only to prepare a number state of the cavity field, but it also allows us
to perform state tomography to reconstruct an initial prepared state of the field.
We note finally that throughout this thesis, there was a strong hierarchy in theatom-field interaction. The experimental exploration of quasi-angular momentum ingases of atoms trapped in rotating lattices comes by way of time-of-flight measurements.
In this case, the dynamics of the atoms is primary, and the light-field plays no part in
the observables of interest. The use of cavity fields to probe the atomic systems in both
chapters 5 and 6 again implies primacy of the atomic dynamics. We spent a lot of time
determining exactly when the dynamics of the probe fields did not affect the atomic
motion. Finally, in the Bayesian scheme for measuring the cavity photon number, theatomic motion affected the photon dynamics only weakly, whereas the dynamics of theatoms were very different for different photon numbers.
As evidenced by recent work where the atoms and light-field must be treated onan equal footing [7, 122], it will be interesting to move beyond these models where there
is a natural hierarchy in the atom-field interaction. This will be the context of futurework.

Bibliography[1] A. Alberti, G. Ferrari, V. V. Ivanov, and G. M. Tino. Coherent transport ofatomic wave packets in amplitude-modulated optical lattices under the effect ofgravity. e-print arXiv:0903.0724, 2009.
[2] A. Albus, F. Illuminati, and J. Eisert. Mixtures of bosonic and fermionic atoms
in optical lattices. Phys. Rev. A, 68(2):023606, 2003.
[3] B. P. Anderson and M. A. Kasevich. Macroscopic Quantum Interference from
Atomic Tunnel Arrays. Science, 282:1686, 1998.
[4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell.
Observation of Bose-Einstein condensation in a dilute atomic vapor. Science,
269(5221):198, 1995.
[5] M. A. Baranov, K. Osterloh, and M. Lewenstein. Fractional Quantum Hall Statesin Ultracold Rapidly Rotating Dipolar Fermi Gases. Phys. Rev. Lett., 94:070404,
2005.
[6] K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger. The Dicke Quantum Phase Transition in a Superfluid Gas Coupled to an Optical Cavity. e-print
arXiv:0912.3261, 2009.
[7] K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger. The Dicke Quantum Phase Transition in a Superfluid Gas Coupled to an Optical Cavity. e-print
arXiv:0912.3261, 2009.
[8] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon. Bloch Oscillationsof Atoms in an Optical Potential. Phys. Rev. Lett., 76:4508, 1996.
[9] R. Bhat, M. J. Holland, and L. D. Carr. Bose-Einstein condensates in rotating
lattices. Phys. Rev. Lett., 96(6):60405, 2006.
[10] R. Bhat, M. Krämer, J. Cooper, and M. J. Holland. Hall effects in Bose-Einsteincondensates in a rotating optical lattice. Phys. Rev. A, 76:043601, 2007.
[11] R. Bhat, B. M. Peden, B. T. Seaman, M. Krämer, L. D. Carr, and M. J. Holland.
Quantized vortex states of strongly interacting bosons in a rotating optical lattice.
Phys. Rev. A, 74:063606, 2006.

142[12] Rajiv Bhat. Bosons in rotating optical lattices. University of Colorado at Boulder,
2008.
[13] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J.
Kimble. Photon blockade in an optical cavity with one trapped atom. Nature,
436:87, 2005.
[14] S. Blatt, A. D. Ludlow, G. K. Campbell, J. W. Thomsen, T. Zelevinsky, M. M.
Boyd, J. Ye, X. Baillard, M. Fouché, R. Le Targat, A. Brusch, P. Lemonde,
M. Takamoto, F.-L. Hong, H. Katori, and V. V. Flambaum. New Limits on
Coupling of Fundamental Constants to Gravity Using 87 Sr Optical Lattice Clocks.
Phys. Rev. Lett., 100:140801, 2008.
[15] I Bloch. Ultracold quantum gases in optical lattices. Nat. Phys., 1:23, 2005.
[16] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of BoseEinstein condensation in an atomic gas with attractive interactions. Phys. Rev.
Lett., 75(9):1687, 1995.
[17] M. Brune, J. Bernu, C. Guerlin, S. Deléglise, C. Sayrin, S. Gleyzes, S. Kuhr, I. Dotsenko, J.-M. Raimond, and S. Haroche. Process Tomography of Field Damping
and Measurement of Fock State Lifetimes by Quantum Nondemolition Photon
Counting in a Cavity. Phys. Rev. Lett., 101:240402, 2008.
[18] A. Buchleitner and A. R. Kolovski. Interaction-Induced Decoherence of Atomic
Bloch Oscillations. Phys. Rev. Lett., 91:253002, 2003.
[19] P. Buonsante, R. Franco, and V. Penna. An algebraic approach to the study of
weakly excited states for a condensate in a ring geometry. J. Phys. A, 38(39):8393,
2005.
[20] P. Buonsante, V. Penna, and A. Vezzani. Attractive ultracold bosons in a necklace
optical lattice. Phys. Rev. A., 72:043620, 2005.
[21] H. J. Carmichael and K. Kim. A quantum trajectory unraveling of the superradiance master equation. Optics Comm., 179:417, 2000.
[22] Howard J. Carmichael. Statistical Methods in Quantum Optics 2: Non-Classical Fields.
Springer, 2008.
[23] I. Carusotto, L. Pitaevskii, S. Stringari, G. Modugno, and M. Inguscio. Sensitive
Measurement of Forces at the Micron Scale Using Bloch Oscillations of Ultracold
Atoms. Phys. Rev. Lett., 95:93202, 2005.
[24] L. Casetti and V. Penna. Vortex structures in a chain of coupled bosonic wells
and the Mott regime. J. Low Temp. Phys., 126:455–460, 2002.
[25] C. M. Caves. Quantum mechanics of measurements distributed in time. A pathintegral formulation. Phys. Rev. D, 33:1643, 1986.
[26] S. Chaudhury, G. A. Smith, K. Schulz, and P. S. Jessen. Continuous Nondemolition Measurement of the Cs Clock Transition Pseudospin. Phys. Rev. Lett.,
96(4):43001, 2006.

143[27] W. Chen, D. Meiser, and P. Meystre. Cavity QED determination of atomic number
statistics in optical lattices. Phys. Rev. A, 75:023812, 2007.
[28] P. Cladé, E. de Mirandes, M. Cadoret, S. Guellati-Khèlifa, C. Schwob, F. Nez,
L. Julien, and F. Biraben. Determination of the Fine Structure Constant Basedon Bloch Oscillations of Ultracold Atoms in a Vertical Optical Lattice. Phys. Rev.
Lett., 96:033001, 2006.
[29] J. P. Clemens, L. Horvath, B. C. Sanders, and H. J. Carmichael. Collective
spontaneous emission from a line of atoms. Phys. Rev. A, 68:023809, 2003.
[30] C.
Cohen-Tannoudji,J.
Dupont-Roc,
Atom-Photon Interactions. Wiley, 1992.

and

G.

Grynberg.

[31] J. Dalibard, Y. Castin, and K. Molmer. Wave-Function Approach to Dissipative
Processes in Quantum Optics. Phys Rev. Lett., 68(5):580, 1992.
[32] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M.
Kurn, and W. Ketterle. Bose-Einstein condensation in a gas of sodium atoms.
Phys. Rev. Lett., 75(22):3969, 1995.
[33] B. DeMarco and D. S. Jin. Onset of fermi degeneracy in a trapped atomic gas.
Science, 285(5434):1703, 1999.
[34] W. S. Dias, E. M. Nascimento, M. L. Lyra, and F. A. de Moura. Frequency
doubling of Bloch oscillations for interacting electrons in a static electric field.
Phys. Rev. B, 76:155124, 2007.
[35] R. Dum, P. Zoller, and H. Ritsch. Monte Carlo simulation of the atomic masterequation for spontaneous emission. Phys Rev. A, 45(7):4879, 1992.
[36] G. Ferrari, N. Poli, F. Sorrentino, and G. M. Tino. Long-Lived Bloch Oscillations with Bosonic Sr Atoms and Application to Gravity Measurement at theMicrometer Scale. Phys. Rev. Lett., 97:060402, 2006.
[37] A. L. Fetter and J. D. Walecka.
Dover, New York, 2003.

Quantum Theory of Many-Particle Systems.

[38] U. R. Fischer, P. O. Fedichev, and A. Recati. Vortex liquids and vortex quantum
Hall states in trapped rotating Bose gases. J. Phys. B: At. Mol. Opt. Phys.,
37:S301, 2004.
[39] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher. Boson localization and the superfluid-insulator transition. Phys. Rev. B, 40:546, 1989.
[40] T. M. Fortier, N. Ashby, J. C. Bergquist, M. J. Delaney, S. A. Diddams, T. P.
Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts, K. Kim, F. Levi, L. Lorini,
W. H. Oskay, T. E. Parker, J. Shirley, and J. E. Stalnaker. Precision Atomic
Spectroscopy for Improved Limits on Variation of the Fine Structure Constantand Local Position Invariance. Phys. Rev. Lett., 98:070801, 2007.

144[41] M. Foss-Feig, M. Hermele, and A. M. Rey. Probing the Kondo Lattice Model with
Alkaline Earth Atoms. e-print arXiv:0912.4762, 2009.
[42] J. K. Freericks. Quenching Bloch oscillations in a strongly correlated material:
Nonequilibrium dynamical mean-field theory. Phys. Rev. B, 77:075109, 2008.
[43] C. W. Gardiner, A. S. Parkins, and P. Zoller. Wave-function quantum stochastic differential equations and quantum-jump simulation methods. Phys Rev. A,
46(7):4363, 1992.
[44] C. W. Gardiner and P. Zoller. Quantum Noise. Springer-Verlag, 2004.
[45] M. Girardeau. Relationship between systems of impenetrable bosons and fermions
in one dimension. J. of Math. Phys., 1:516, 2004.
[46] S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Deléglise, B. H. Ulrich, M. Brune,
J.-M. Raimond, and S. Haroche. Quantum jumps of light recording the birth anddeath of a photon in a cavity. Nature, 446:297, 2007.
[47] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch. Quantum
phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms.
Nature, 415:39, 2002.
[48] M. Greiner, C. A. Regal, and D. S. Jin. Emergence of a molecular Bose-Einsteincondensate from a Fermi gas. Nature, 426(6966):537, 2003.
[49] A. Griessner, D. Jaksch, and P. Zoller. Cavity-assisted nondestructive laser coolingof atomic qubits. J. Phys. B, 37:1419, 2003.
[50] C. Guerlin, J. Bernu, S. Deléglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune,
J.-M. Raimond, and S. Haroche. Progressive field-state collapse and quantum
non-demolition photon counting. Nature, 448:889, 2007.
[51] S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn. Cavity Nonlinear
Optics at Low Photon Numbers from Collective Atomic Motion. Phys. Rev. Lett.,
99:213601, 2007.
[52] M. Hafezi, A. S. Sorensen, E. Demler, and M. D. Lukin. Fractional quantum Hall
effect in optical lattices. Phys. Rev. A, 76:023613, 2007.
[53] M. Holland, S. Marksteiner, P. Marte, and P. Zoller. Measurement Induced Localization from Spontaneous Decay. Phys Rev. Lett., 76(20):3683, 1996.
[54] M. J. Holland. Unraveling Quantum Dissipation in the Frequency Domain. Phys
Rev. Lett., 81(23):5117, 1998.
[55] M. J. Holland and K. Burnett. Interferometric detection of optical phase shifts atthe Heisenberg limit. Phys. Rev. Lett., 71:1355, 1993.
[56] M. J. Holland, D. F. Walls, and P. Zoller. Quantum Nondemolition Measurementsof Photon Number by Atomic-Beam Deflection. Phys. Rev. Lett., 67(13):1716,
1991.

145
[57] C. J. Hood, M. S. Chapman, T. W. Lynn, and H. J. Kimble. Real-Time Cavity
QED with Single Atoms. Phys. Rev. Lett., 80(19):4157, 1998.
[58] C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble. The
atom-cavity microscope: Single atoms bound in orbit by single photons. Science,
287:1447, 2000.
[59] D. B. Hume, T. Rosenband, and D. J. Wineland. High-Fidelity Adaptive Qubit
Detection through Repetitive Quantum Nondemolition Measurements. Phys. Rev.
Lett, 99:120502, 2007.
[60] F. Illuminati and A. Albus. High-Temperature Atomic Superfluidity in latticeBose-Fermi Mixtures. Phys. Rev. Lett., 93(9):090406, 2004.
[61] M.
Inguscio,W.
Ketterle,
andC.
Salomon,editors.
Ultra-cold Fermi Gases (Proceedings of the International School of Physics ”Enrico Fermi”).
IOP Press, 2007.
[62] V. V. Ivanov, A. Alberti, M. Schioppo, G. Ferrari, M. Artoni, M. L. Chiofalo,
and G. Tino. Coherent Delocalization of Atomic Wave Packets in Driven LatticePotentials. Phys. Rev. Lett., 100:043602, 2008.
[63] J. D. Jackson. Classical Electrodynamics. Wiley, 1999.
[64] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller. Cold BosonicAtoms in Optical Lattices. Phys. Rev. Lett., 81:3108, 1998.
[65] D. Jaksch and P. Zoller. Creation of effective magnetic fields in optical lattices:
the Hofstadter butterfly for cold neutral atoms. New J. Phys., 5:56, 2003.
[66] P. Jordan and E. Wigner. Über das Paulische äquivalenzverbot. Z. Phys., 47:631,
1928.
[67] C. Kittel. Introduction to Solid State Physics. Wiley, 1996.
[68] M. Köhl, H. Moritz, T. Stöferle, K. Günter, and T. Esslinger. Fermionic Atoms ina Three Dimensional Optical Lattice: Observing Fermi Surfaces, Dynamics, andInteractions. Phys. Rev. Lett., 94:80403, 2005.
[69] L. D. Landau and E. M. Lifshitz. Mechanics. Butterworth-Heinemann, Ltd,
Oxford, 1976.
[70] J. Larson, B. Damski, G. Morigi, and M. Lewenstein. Mott-Insulator States of
Ultracold Atoms in Optical Resonators. Phys. Rev. Lett., 100:050401, 2008.
[71] R. H. Lehmberg. Radiation From an N -Atom System. I. General Formalism.
Phys. Rev. A, 2:883, 1969.
[72] D. R. Leibrandt, J. Labaziewicz, V. Vuletic, and I. L. Chuang. Cavity Sideband
Cooling of a Single Trapped Ion. Phys. Rev. Lett., 103:103001, 2009.
[73] I. D. Leroux, M. H. Schleier-Smith, and V. Vuletic. Implementation of Cavity
Squeezing of a Collective Atomic Spin. Phys. Rev. Lett., 104:073602, 2010.

146[74] T. O. Levante, M. Baldus, B. H. Meier, and R. R. Ernst. Formalized quantum mechanical Floquet theory and its application to sample spinning in nuclear magnetic
resonance. Molec. Phys., 86:1195, 1995.
[75] E. H. Lieb and W. Liniger. Exact analysis of an interacting Bose gas. I. The
general solution and the ground state. Phys. Rev., 130(4):1605, 1963.
[76] R. Loudon. The Quantum Theory of Light. Oxford Science Publications, 2003.
[77] W. H. Louisell. Quantum Statistical Properties of Radiation. Wiley, 1990.
[78] A. D. Ludlow, M. M. Boyd, T. Zelevinsky, S. M. Foreman, S. Blatt, M. Notcutt,
T. Ido, and J. Ye. Systematic Study of the 87 Sr Clock Transition in an Optical
Lattice. Phys. Rev. Lett., 96:033003, 2006.
[79] A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G.
de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier,J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke,
K. M. Beck, and C. W. Oates. Sr Lattice Clock at 1x10e16 Fractional Uncertainty
by Remote Optical Evaluation with a Ca Clock. Science, 319:1805, 2008.
[80] M. P. Marder. Condensed Matter Physics. Wiley, New York, 2000.
[81] L. Mathey, I. Danshita, and C. W. Clark. Creating a supersolid in one-dimensional
Bose mixtures. Phys. Rev. A, 79:011602(R), 2009.
[82] P. Mehta and N. Andrei. Nonequilibrium Transport in Quantum Impurity Models:
The Bethe Ansatz for Open Systems. Phys. Rev. Lett., 96:216802, 2006.
[83] D. Meiser and M. J. Holland. Robustness of Heisenberg-limited interferometry
with balanced Fock states. New J. Phys., 11:033002, 2009.
[84] D. Meiser, J. Ye, D. Carlson, and M. J. Holland. Prospects for a mHz-Linewidth
Laser. Phys. Rev. Lett., 102:163601, 2009.
[85] D. Meiser, Jun Ye, and M. J. Holland. Spin squeezing in optical lattice clocks via
lattice-based QND measurements. New J. Phys., 10:073014, 2008.
[86] I. B. Mekhov, C. Maschler, and H. Ritsch. Cavity-Enhanced Light Scattering in Optical Lattices to Probe Atomic Quantum Statistics. Phys. Rev. Lett.,
98:100402, 2007.
[87] I. B. Mekhov and H. Ritsch. Quantum Nondemolition Measurements and State
Preparation in Quantum Gases by Light Detection. Phys. Rev. Lett., 102:020403,
2009.
[88] P. Meystre and M. Sargent III. Elements of Quantum Optics. Springer, 2007.
[89] R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, Boozer A. D., and H. J.
Kimble. Trapped atoms in cavity QED: coupling quantized light and matter. J.
Phys. B: At. Mol. Opt. Phys., 38:S551, 2005.

147[90] J. N. Milstein, S. Kokkelmans, and M. J. Holland. Resonance theory of thecrossover from Bardeen-Cooper-Schrieffer superfluidity to Bose-Einstein condensation in a dilute Fermi gas. Phys. Rev. A, 66:43604, 2002.
[91] O. Morsch, J. H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo. Bloch
Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical
Lattices. Phys. Rev. Lett., 87:140402, 2001.
[92] K.K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel,
S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye. A high phase-space-density
gas of polar molecules. Science, 322:231, 2008.
[93] A. E. B. Nielsen and K. Mølmer. Atomic spin squeezing in an optical cavity. Phys.
Rev. A, 77:063811, 2008.
[94] T. Oka and H. Aoki. Ground-State Decay Rate for the Zener Breakdown in Bandand Mott Insulators. Phys. Rev. Lett., 95:137601, 2005.
[95] W. H. Oskay, S. A. Diddams, E. A. Donley, T. M. Fortier, T. P. Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts, M. J. Delaney, K. Kim, F. Levi, T. E. Parker,
and J. C. Bergquist. Single-Atom Optical Clock with High Accuracy. Phys. Rev.
Lett., 97:020801, 2006.
[96] C. Ospelkaus, S. Ospelkaus, L. Humbert, P. Ernst, K. Sengstock, and K. Bongs.
Ultracold Heteronuclear Molecules in a 3D Optical Lattice. Phys. Rev. Lett.,
97:120402, 2006.
[97] S. Ospelkaus, C. Ospelkaus, O. Wille, M. Succo, P. Ernst, K. Sengstock, andK. Bongs. Localization of Bosonic Atoms by Fermionic Impurities in a ThreeDimensional Optical Lattice. Phys. Rev. Lett., 96:180403, 2006.
[98] K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein. Cold Atoms in
Non-Abelian Gauge Potentials: From the Hofstadter “Moth” to Lattice Gauge
Theory. Phys. Rev. Lett., 95:010403, 2005.
[99] R. N. Palmer and D. Jaksch. High-Field Fractional Quantum Hall Effect in Optical
Lattices. Phys. Rev. Lett., 96:180407, 2006.
[100] B. Paredes, P. Fedichev, J. I. Cirac, and P. Zoller. 12 -anyons in Small Atomic
Bose-Einstein Condensates. Phys. Rev. Lett., 87:010402, 2001.
[101] B. M. Peden, R. Bhat, M. Krämer, and M. J. Holland. Quasi-angular momentumof Bose and Fermi gases in rotating optical lattices. J. Phys. B: At. Mol. Opt.
Phys., 40:3725, 2007.
[102] B. M. Peden, D. Meiser, M. L. Chiofalo, and M. J. Holland. Non-destructive
cavity QED probe of Bloch oscillations in a gas of ultracold atoms. Phys. Rev. A,
80:043803, 2009.
[103] R. A. Pepino, J. Cooper, D. Z. Anderson, and M. J. Holland. Atomtronic Circuitsof Diodes and Transistors. Phys. Rev. Lett., 103:140405, 2009.

148[104] M. B. Plenio and P. L. Knight. The quantum-jump approach to dissipative dynamics in quantum optics. Rev. Mod. Phys., 70(1):101, 1998.
[105] S.
Reich,C.
Thomsen,J.
Maultzsch,
andM.
Janina.
Carbon nanotubes: Basic concepts and physical properties. Wiley-VCH, 2004.
[106] A. M. Rey, K. Burnett, I. I. Satija, and C. W. Clark. Lifshitz-like transitionand enhancement of correlations in a rotating bosonic ring lattice. Phys. Rev. A,
75:063616, 2007.
[107] Ana Maria Rey. Ultracold Bosonic Atoms in Optical Lattices.
Maryland College Park, 2004.

University of[108] G. Roati, E. de Mirandes, F. Ferlaino, H. Ott, G. Modugno, and M. Inguscio.
Atom Interferometry with Trapped Fermi Gases. Phys. Rev. Lett., 92:230402,
2004.
[109] T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini,
W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams,
W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist.
Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th
decimal place. Science, 319:1808, 2008.
[110] C. Rubbo. Private communication, 2010.
[111] S. Sachdev. Quantum Phase Transitions. Cambridge University Press, New York,
1999.
[112] S. Sachdev, K. Sengupta, and S. M. Girvin. Mott insulators in strong electricfields. Phys. Rev. B, 66:075128, 2002.
[113] H. Sambe. Steady States and Quasienergies of a Quantum-Mechanical System in
an Oscillating Field. Phys. Rev. A, 7:2203, 1973.
[114] I. I. Satija, D. C. Dakin, and C. W. Clark. Metal-Insulator Transition Revisited
for Cold Atoms in Non-Abelian Gauge Potentials. Phys. Rev. Lett., 97:216401,
2006.
[115] M. Schleier-Smith, I. D. Leroux, and V. Vuletic. Squeezing the collective spin ofa dilute atomic ensemble by cavity feedback. Phys. Rev. A, 81:021804(R), 2010.
[116] T. D. Schultz. Note on the one-dimensional gas of impenetrable point-particle
bosons. J. Math. Phys., 4:666, 1966.
[117] V. Schweikhard, I. Coddington, P. Engels, V. P. Mogendorff, and E. A. Cornell.
Rapidly Rotating Bose-Einstein Condensates in and Near the Lowest Landau
Level. Phys. Rev. Lett., 92:040404, 2004.
[118] B. T. Seaman, M. Krämer, D. Z. Anderson, and M. J. Holland. Atomtronics:
Ultracold-atom analogs of electronic devices. Phys. Rev. A, 75:023615, 2007.
[119] J. H. Shirley. Solution of the Schröinger Equation with a Hamiltonian Periodic inTime. Phys. Rev., 138:B979, 1965.

149[120] C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, andE. Arimondo. Resonantly Enhanced Tunneling of Bose-Einstein Condensates in
Periodic Potentials. Phys. Rev. Lett., 98:120403, 2007.
[121] A .E. Siegman. Lasers. University Science Books, 1986.
[122] A. O. Silver, M. Hohenadler, M. J. Bhaseen, and B. D. Simons. Bose-Hubbard
models coupled to cavity light fields. Phys. Rev. A, 81:023617, 2010.
[123] J. Simon, H. Tanji, J. K. Thompson, and V. Vuletic. Interfacing Collective Atomic
Excitations and Single Photons. Phys. Rev. Lett., 98:183601, 2007.
[124] G. A. Smith, S. Chaudhury, A. Silberfarb, I. H. Deutsch, and P. S. Jessen. Continuous weak measurement and nonlinear dynamics in a cold spin ensemble. Phys.
Rev. Lett., 93:163602, 2004.
[125] A. S. Sørensen, E. Demler, and M. D. Lukin. Fractional Quantum Hall States ofAtoms in Optical Lattices. Phys. Rev. Lett., 94:86803, 2005.
[126] M. Ueda and A. J. Leggett. Ground-State Properties of a Rotating Bose-Einstein
Condensate with Attractive Interaction. Phys. Rev. Lett., 83(8):1489, 1999.
[127] D. van Oosten, P. van der Straten, and H. T. C. Stoof. Quantum phases in an
optical lattice. Phys. Rev. A, 63:053601, 2001.
[128] B. P. Venkatesh, M. Trupke, E. A. Hinds, and D. H. J. O’Dell. Atomic Bloch-Zener
oscillations for sensitive force measurements in a cavity. Phys. Rev. A, 80:063834,
2009.
[129] P. Vignolo, R. Fazio, and M. P. Tosi. Quantum vortices in optical lattices. Phys.
Rev. A, 76:023616, 2007.
[130] N. K. Wilkin and J. M. F. Gunn. Condensation of “Composite Bosons” in a
Rotating BEC. Phys. Rev. Lett., 84:6, 2000.
[131] P. J. Windpassinger, D. Oblak, P. G. Petrov, M. Kubasik, M. Saffman, C. L.
Alzar, J. Appel, J. H. Muller, N. Kjærgaard, and E. S. Polzik. Nondestructive
Probing of Rabi Oscillations on the Cesium Clock Transition near the Standard
Quantum Limit. Phys. Rev. Lett., 100(10):103601, 2008.
[132] P. Wolf, P. Lemonde, A. Lambrecht, S. Bize, A. Landragin, and A. Clairon. From
optical lattice clocks to the measurement of forces in the Casimir regime. Phys.
Rev. A, 75:63608, 2007.
[133] A. Yariv. Optical Electronics. CBS College Publishing, 1985.
[134] H. Zoubi and H. Ritsch. Superradiant and dark exciton states in an optical lattice
within a cavity. Europhys. Lett., 87:23001, 2009.
[135] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle.
Vortices and superfluidity in a strongly interacting Fermi gas. Nature, 435:1047,
2005.

Appendix A

Numerical Implementation

A.1

Lattices
The Hamiltonian for an atom trapped in a one-dimensional sinusoidal lattice isH(z) = −~2 d2+ V0 cos2 (kz).
2m dz 2

(A.1)

Scaling energies by the recoil energy, ER = ~2 k 2 /2m, momenta by the lattice momentum, ~k, and positions by the inverse lattice wave vector, 1/k = d/π, this becomes


d2
V0
V0
H(z) = ER − 2 ++cos(2kz) .
dz
2ER 2ER(A.2)

Defining,
E
V0−,
ER 2ER
V0
t=
,
4ER

a=

(A.3)

the time-dependent Schrödinger equation can be written as0=d2
ϕ(z) + (a − 2t cos(2z))ϕ(z).
dz 2

(A.4)

This equation is known as Mathieu’s equation.

A.1.1

Mathieu functions

The solutions, ϕ, can be written according to Bloch theory as
ϕk (φ) = eikφ uk (φ) : uk (φ − π) = uk (φ),

(A.5)

151
for special values of a that make ϕk (φ) periodic with period π. These are known as
Mathieu characteristic values, ak (t) and bk (t), and they have the following important
properties:
ak (t) = bk (t) : k ∈/ Z,(A.6a)

a0 (t) < b1 (t) < a1 (t) < b2 (t) < a2 (t) < · · · : t 6= 0,(A.6b)a−k (t) = ak (t),(A.6c)b−k (t) = bk (t).

(A.6d)

From now on, we will leave off the parameter t as it is fixed by the lattice depth. We
may decompose ϕk (φ) into parts,
ϕk (φ) + ϕk (−φ),
2ϕk (φ) − ϕk (−φ)
Sk (φ) =,2i

Ck (φ) =

(A.7)
(A.8)

which are respectively even and odd functions of φ. These two functions, known as
the Mathieu cosine and sine functions, respectively, are also eigenstates of Mathieu’s
equation with respective eigenvalues, ak and bk . When k is not an integer, they aredegenerate. If k = m is an integer, we note thatlim Ck (φ) = 0,

(A.9)

lim Sk (φ) = 0,

(A.10)k→m−k→m+so that C and S are exactly equal to the Bloch solutions in this case, corresponding to
ak and bk , respectively.
With these facts in hand, the Mathieu cosine and sine functions can be used toconstruct the Bloch solutions in a systematic way. It is clear from equation A.5 that thequasi-momenta, q, are limited to the interval, −1 ≤ q ≤ 1. Equations A.6c and A.6d

152
allow us to consider only k ≥ 0. Excluding integer values for now, write k as
 n + |q|
if n = evenk → kn,q =, n + |q| − 1 if n = odd(A.11)n ∈ Z+ ∪ {0},(A.12)q ∈ (−1, 0) ∪ (0, 1).

(A.13)

Defining
Cq(n) (φ) = Ckn,q (φ),(A.14)

Sq(n) (φ) = Skn,q (φ),(A.15)(n)u(n)
q (φ) = ukn,q (φ) = u|q| (φ),(A.16)

and
(n)
(n)
ϕ(n)
q (φ) = Cq (φ) + iSign(q)Sq (φ)ϕkn,q (φ) + ϕkn,q (−φ)ϕk (φ) − ϕkn,q (−φ)+ i Sign(q) n,q
2
2i
(n)
(n)
(n)
(n)
u (φ) + Sign(q)u|q| (φ)
u (−φ) − Sign(q)u|q| (−φ)
ikn,q φ |q|−ikn,p φ |q|
=e
+e
2
2
=

(n)

= eSign(q)ikn,q φ u|q| (Sign(q)φ),(A.17)

we have that(n)Sign(q)ikn,q (φ−π)ϕ(n)
u|q| (Sign(q)(φ − π))q (φ − π) = e(n)= e−Sign(q)ikn,q π eSign(q)ikn,q φ u|q| (Sign(q)φ)= e−Sign(q)ikn,q π ψq(n) (φ).
The exponential factor out front is−Sign(q)i(n+|q|)π
if n = even  e−iqπe−Sign(q)ikn,q π = , =e e−Sign(q)i(n+|q|−1)π if n = odd(A.18)(A.19)

so that we can write
ϕq(n) (φ − π) = e−iqπ ϕ(n)q (φ).

(A.20)

153
For k ∈ Z, we let
if n = even  n + |q|k → kn,q = , n + |q| − 1 if n = odd(A.21)n ∈ Z+ ∪ {0},(A.22)q ∈ [−1, 1),(A.23)and define(2p+1)(φ) = ϕ0

(2p+1)(φ) = ϕ−1(2p)(φ) = ϕ0

C0

S−1

S0

(2p+1)(φ) = C2p (φ),(A.24)(2p+1)(φ) = S2p+1 (φ),(A.25)(2p)(2p)(φ) = S2p (φ),(A.26)(2p)

C−1 (φ) = ϕ−1 (φ) = C2p−1 (φ),(A.27)(2p)

which is consistent with the definition of kn,q except for ϕ−1 (φ) which we must reserve
as a special case. Implicitly, this assumes that
(2p+1)

S0(2p+1)(φ) = C−1(2p)(φ) = C0

(2p−1)(φ) = S−1(φ) = 0,(A.28)

It is simple to show that these still satisfy
ϕq(n) (φ − π) = e−iqπ ϕ(n)q (φ).

(A.29)

Thus, we can interpret q as the quasi-momentum and n as the band index of this state.
Finally, the eigenvalues are given by the expression

V0
E = ER a +.
2ER(A.30)

154(n)

For Eq , we take
Eq(n) : a → akn,q : q ∈/ Z,(A.31)(2p)(A.32)

E0: a → b2p ,(2p)

E−1 : a → a2p−1 ,(A.33)(2p+1): a → a2p ,(A.34)(2p+1): a → b2p+1 .

(A.35)

E0

E−1
A.1.2

Hamiltonian decomposition in momentum space

Instead of using the Mathieu sine and cosine functions – which are special functions implemented in, for instance, Mathematica – we can expand the Hamiltonian,equation A.1, in momentum eigenstates. Implementing periodic boundary conditions
as an approximation, this generates a finite-dimensional matrix that can be numericallydiagonalized to construct the Bloch eigenstates.
We choose a set of momentum eigenstates,
hz|n, qi = N ei(2kn+q)z ,(A.36)n ∈ Z,(A.37)

0 ≤ q ≤ 2k,(A.38)

where N is a normalization, and we have indexed the states to reflect the periodicity of
the lattice. The matrix elements of the operators in the Hamiltonian are given by,~
hn0 , q 0 | D̂z |n, qi = δnn0 δkk0 ~(2kn + q),i
2(~(2kn + q))2−~
hn0 , q 0 |D̂z2 |n, qi = δnn0 δqq0
,
2m
2m
V0
hn0 , q 0 | cos(2kz)|n, qi =
δqq0 (δn,n0 +1 + δn,n0 −1 ).
4(A.39)(A.40)(A.41)

Thus,
hn0 k 0 |Ĥ|nki =(~(2kn + q))2V0
δkk0 δnn0 + δkk0 (δn,n0 +1 + δn,n0 −1 ).
2m
4(A.42)

155
This matrix is block-diagonal in the quasi-momentum q. However, n can be any integer,so this is still an infinite-dimensional matrix. To get around this, we note that theharmonic trapping term arising from the kinetic energy allows us to truncate the basis
for low-energy states.
Finally, if we scale energies, momenta, and positions as we did in previous sections,and we apply periodic boundary conditions to get an N -site lattice, the matrix elementsof the Hamiltonian become00

hn m |Ĥ|nmi =22v0(N n + m) δmm0 δnn0 + δmm0 (δn,n0 +1 + δn,n0 −1 ),N
4(A.43)

where
Nq= 0, . . . , N − 1,
2 k
1
N =√
,
Nπ
V0
.
v0 =
ER

(A.44)

m=(A.45)(A.46)The eigenstates can be written asψp,q (z) = hz|ψp,q i =Xψn(p,q) hz|n, qi =n∈ZXψn(p,q) N ei(2kn+q)z ,(A.47)n∈Zorψp,m (z) =nXmax

ψn(m,q) √n=nmin(p,m)

where p is a band-index, and ψn1
ei2(N n+m)z/N ,Nπ(A.48)

are the coefficients of the eigenvectors of the matrix

defined by equation A.43.

A.1.2.1

Bose-Hubbard parameters

We can compute Bose-Hubbard (BH) parameters as a function of lattice depth
using the numerical solutions just derived. Since the definition of the Wannier functions
used in the BH derivation depends on the relative phases between Bloch functions, we(p,m)

need to be careful about how the coefficients, ψn, are defined. Luckily, the matrix

156
defined by equation A.43 is real symmetric, which means the eigenvectors can be chosen
to purely real. We then need only worry about an overall minus sign.
With that caveat in hand, we suppose that the coefficients are such that thefunctions1(p)wj (z) = √

N
−1
X

N m=0

ei2πmj/N ψp,m (z)(A.49)

are the Wannier functions in band p. Then, the hopping parameter is
(p)

(p)

−J (p) = hwj |Ĥ|wj+1 i
=

=

1
N
1
N

N
−1
X

(A.50)
0

ei2πm(j+1)/N e−i2πm j/N hψp,m0 |Ĥ|ψp,m im,m0 =0N−1X0ei2πm(j+1)/N e−i2πm j/N Ep,m hψp,m0 |ψp,m im,m0 =0N −1=

1 X i2πm/N
eEp,m .
N(A.51)

m=0

Similarly, the on-site energy is
N −1

(p) =

1 X
Ep,m .
N(A.52)

m=0

The parameters used in chapter 6 for p = 1 and 2kC = k2(N r + l)/N are given by
fj 0 ,j (k) = hwj0 |ei2(N r+l)ẑ/N |wj i
=

=

1
N
1
N

N
−1
X

(A.53)0 0

ei2π(mj−m j )/N hψm0 |ei2(N r+l)ẑ/N |ψm i

m,m0 =0N−1
Xm,m0 =00 0ei2π(mj−m j )/NX(m0 )∗

ψs0

ψs(m) hs0 , m0 |ei2(N r+l)ẑ/N |s, mi

s,s0

N −1

=

1 X i2π(mj−(m+l)j 0 )/N X (m+l)∗ (m)eψs+r ψs .
N
s(A.54)

m=0

A.1.3

Hard-core boson approximation

Here we make explicit the two-state approximation from chapter 3. We write thestates as|l; {nk }i = | . . . , l + nk−1 , l + nk , l + nk+1 , . . . i,

(A.55)

157
where nj ∈ {0, 1} and stipulate the following conditions:âj |l; {nk : nj = 0}i = 0,âj |l; {nk : nj = 1}i =√(A.56a)l + 1|l; {nk : nj = 0}i,â†j |l; {nk : nj = 1}i = 0,â†j |l; {nk : nj = 0}i =√(A.56b)(A.56c)l + 1|l; {nk : nj = 1}i,

(A.56d)â†j âj |l; {nk : nj = 0}i = l|l; {nk : nj = 0}i,

(A.56e)â†j âj |l; {nk : nj = 0}i = (l + 1)|l; {nk : nj = 0}i.

(A.56f)

These conditions imply that the action of the hopping terms is given byâ†j âj±1 |l; {nk : nj = 0, nj±1 = 1}i = (l + 1)|l; {nk : nj = 1, nj±1 = 0}i,

(A.57)

with all other terms such as â†j âj±1 |l; {nk : nj = 0, nj±1 = 0}i equal to zero.
We define new operators b̂j that are related to the old ones via
âj
,
l+1

(A.58)b̂†j b̂j = â†j âj − l.

(A.59)b̂j = √The relations above imply that the site-number and hopping operators are given byâ†j âj = b̂†j b̂j + l,

(A.60a)â†j âj±1 = (l + 1)b̂†j b̂j±1 ,(A.60b)and that these operators satisfy the relations,ihb̂j , b̂†j = 1,

(A.61a)ihb̂j , b̂j = 0,

(A.61b)i(A.61c)+−hb̂i6=j , b̂†j

−

= 0.

The particles apparently have boson-like properties except for the fact that no morethan one particle can occupy a single site, which is enforced by equation A.61a. Finally,

158
we use equation (A.61a) to re-write the interaction term as
N

N

j=1

j=1

 U XUX †  †âj âj âj âj − 1 =b̂†j b̂j + l b̂†j b̂j + l − 1
22NU X †
=b̂j 1 − b̂†j b̂j b̂j
2
j=1

N

N

j=1

j=1

 UX
U X †+
lb̂j b̂j + (l − 1)b̂†j b̂j +l(l − 1)
2
2
= Ul

N
X
j=1

A.1.4

1
b̂†j b̂j + U N l(l − 1).
2(A.62)

Jordan-Wigner fermions

To do this we need the following properties. The operator b̂†k b̂k is either zero orone, so††††eiπb̂k b̂k = e−iπb̂k b̂k = eiπb̂k b̂k .

(A.63)In addition, since the site-number operators commute, we can combine these exponentials as
eiπ

P

†
k<m b̂k b̂k

eiπ

P

†
l<n b̂l b̂l

= eiπ

P

P
†
†
k<m b̂k b̂k +iπl<n b̂l b̂l.

(A.64)

Finally, sincehib̂k6=j , b̂j = 0,b̂2j = 0,b̂j b̂†j b̂j = 1 − b̂†j b̂j b̂j = b̂j ,(A.65)(A.66)

we have that††eiπb̂j b̂j b̂j6=k = b̂j6=k eiπb̂j b̂j ,†eiπb̂j b̂j b̂j =iπ b̂†j b̂j

b̂j e

=

(A.67a)

∞
∞

n
X
X
(iπ)n  † n(iπ)nb̂j b̂j b̂j =δn,0 b̂†j b̂j b̂j = b̂j ,
n!
n!

n=0
∞
X

n=0

(A.67b)

n=0

(iπ)n
n!



b̂j b̂†j b̂j

n

=

∞
X
(iπ)n
n=0

n!

b̂j = eiπ b̂j = −b̂j .

(A.67c)159
These three conditions imply the relations,h†b̂j6=k , eiπb̂j b̂jh†b̂j , eiπb̂j b̂j

i
−

i
+

= 0,

(A.68)= 0.

(A.69)

With these relations in hand, we can show that the ĉj operators are fermionic.
First,hĉj , ĉ†j



i
+

†
k<j b̂k b̂kiπ= b̂j ePh= b̂j , b̂†j

i

−iπ

,e

P

†

k<j b̂k b̂k b̂†
j


+

+

eiπ

P

†
k<j b̂k b̂k

e−iπ

P

†
k<j b̂k b̂khi= b̂j , b̂†j = 1.

(A.70)+

Second,PP†iπ k<j b̂†k b̂k −iπ i<l b̂i b̂i †[ĉj<l , ĉl ]+ = b̂j e
,e
b̂l

+



P
P
P
†
†
†
†
−iπ b̂j b̂j= b̂j , eb̂†l e−iπ n<j b̂n b̂n e−iπ j<i<l b̂i b̂i eiπ k<j b̂k b̂k+= 0.

(A.71)ih
The same computation goes through for ĉj<l , ĉ†l , since owing to
+

eiπ

P

†
k<j b̂k b̂k= e−iπ†k<j b̂k b̂kP,(A.72)

the only difference is the substitution b̂l → b̂†l . Finally,hiiPN −1 †
PN −1 † h†[ĉN , ĉ1 ]+ = b̂N eiπ k=1 b̂k b̂k , b̂†1 = b̂N eiπ k=1 b̂k b̂k eiπb̂1 b̂1 , b̂†1
+

= 0.

+

(A.73)

Summarizing, we have derived the anti-commutation relations,hĉj , ĉ†ji

= δij ,

(A.74)[ĉj , ĉl ]+ = 0,

(A.75)+

160
which implies that the ĉ’s are fermionic operators.
Now, this transformation is only really useful if the resulting transformed Hamiltonian is quadratic in these operators, since in this case it is known that the excitations
are then fermionic. We need to verify that the exponential factors cancel out. First, wenote that the site-number operators are given byĉ†j ĉj = b̂†j e−iπP†k<j b̂k b̂k

eiπ

P

†
k<j b̂k b̂kb̂j = b̂†j b̂j .

(A.76)

This allows us to easily invert the JW transformation as†
k<j ĉk ĉkb̂j = ĉj eiπP†k<j ĉk ĉkP.

(A.77)

Equation A.67 also holds for ĉj .
The hopping terms becomeb̂†j≤N b̂j+1 = ĉ†j e−iπ

P

eiπ

†
k<j+1 ĉk ĉk†ĉj+1 = ĉ†j eiπĉj ĉj ĉj+1= ĉ†j ĉj+1 ,(A.78)

andb̂†N b̂1 = ĉ†N e−iπ

PN −1 †
k=1 ĉk ĉk

†

†

†

†

†

ĉ1 = ĉ†N eiπĉN ĉN e−iπĉN ĉN e−iπĉ1 ĉ1 ĉ1 e−iπ††

= ĉ†N eiπĉN ĉN e−iπĉ1 ĉ1 ĉ1 eiπĉ1 ĉ1 e−iπĉ1 ĉ1 e−iπ

PN −1 †
k=2 ĉk ĉkPN −1 ††
k=2 ĉk ĉk −iπĉN ĉNe= ĉ†N (−ĉ1 )e−iπN̂= ĉ†N ĉ1 (−1)N̂ +1 ,(A.79)

where
N̂ =

N
N
X
X
b̂†k b̂k =ĉ†k ĉk
k=1(A.80)

k=1

is the total number of particles minus N l. The Hamiltonian can then be written as
N

N

j=1

j=1

X †
X
1
ĉj ĉj +(ω)ĉ†j ĉjĤ = U N l(l − 1) + U l
2− t(l + 1)N−1X

e−iπωN/4 ĉ†j ĉj+1 + H.c.

j=1

− t(l + 1)e−iπωN/4 ĉ†N ĉ1 (−1)N̂ +1 + H.c.

(A.81)

161

A.2

Tilted Lattice Problem

A.2.1

Dipole subspace Hamiltonian

We first prove by induction that the number of states in the M -dipole subspace
for a N -site lattice is given by the expression,


N −M
nM =
.
M

(A.82)For M = 0, this formula yields 1: there is exactly one MI state. For M = 1, thisformula yields N − 1, which is clearly the exact number of one-dipole states. We cancount the number of M = 2 states by taking all of the M = 1 states and adding an
extra dipole. To avoid double-counting, we only add dipoles to the left of the existing
dipole in the one-dipole subspace. There are N − 3 two-dipole states that we can makefrom |1; N − 1i since by having a dipole between sites N − 1 and N , we have effectively
removed 2 sites from the lattice. There are N − 4 two-dipole states that we can makefrom |1; N − 2i, by the same argument. We can do the same thing until we reachthe state |1; 3i which accommodates exactly one dipole to the left of site 3. Thus, thenumber of states is in the two-dipole subspace is
n2 =

N
−1
X

N
−3
X

j=3

n=1

(N − j) =1n = (N − 3)(N − 2) =
2



N −2
.
2

(A.83)

In general, we consider the subspace spanned by the vectors,|M + 1; kM +1 , kM , . . . , k1 i = √

1

M
+1
Y

2M i=1ki = 2i − 1, . . . , ki+1 − 2b̂†ki b̂ki +1 |ψM I i : kM = 2M − 1, . . . , kM +1 − 2 ,kM +1 = 2M + 1, . . . , N − 1(A.84)

for some 2 ≤ M ≤ bN/2c. We choose a specific kM +1 ; this effectively reduces the lattice
for the remaining dipoles to be of size kM +1 − 1, since we have a dipole between sites
kM +1 and kM +1 + 1. By inductive assumption, there are
kM +1 − 1 − M
nkM +1 =M(A.85)

162
ways of fitting M dipoles on a lattice of with kM +1 −1 sites. Letting kM +1 vary between
2M + 1 and N − 1, we can count the total number of states as
nM +1 =

N
−1
X

N
−1
X

nkM +1 =kM +1 =2M +1

kM +1 =2M +1 N −2M
X −2 k + M 
kM +1 − 1 − M
=
.
M
M
k=0

(A.86)
With repeated uses of the identity,

 
 

x+y
x+y−1
x+y−1
=
+
.
x
x−1
x

(A.87)

 N −2M
X −2 k + M 
N − (M + 1)
=
,
M +1
M

(A.88)

it is easy to show thatk=0

and the inductive proof is finished.
This proof also gives us a way to enumerate the states in each dipole subspace.
We label the state |M ; kM , . . . , k1 i as

nkM ,...,k1 = 1 +M X
km − mm=1

m:ki = 2i − 1, . . . , ki+1 − 2.

(A.89)

kM = 2M − 1, . . . , N − 1

This enumerates the states in order from 1 to

N −M
M



. Finally, then, we can write

down an effective Hamiltonian that includes only the dipole subspaces. The state|M + 1; kM +1 , . . . , k1 i is connected to the states |M ; lM , . . . , l1 i where we have simply
removed one of the ki ’s. In terms of the indices, the state labeled by
nkM +1 ,...,k1 = 1 +

M
+1 
X
m=1


km − mm(A.90)

in the (M + 1)-dipole subspace is connected to the states labeled by
nkM +1 ,...,kn+1 ,kn−1 ,...,k1 = 1 +

n−1
X
m=1



M 
X
km − mkm+1 − m
+1+
m
m
m=n

(A.91)

in the M -dipole subspace. Since this connection implies a particle hopping onto an√
already occupied site, we get an extra factor of 2 due to the bosonic nature of theparticles.

163
A.2.2

Symmetric Hamiltonian

For the purposes of analyzing the dynamics and the detection scheme, we needto know how to numerically implement the quadratic operators, b̂†j b̂j and b̂†j+1 b̂j . We
first note thatbN/2cb̂†j b̂j →X|M ihM |b̂†j b̂j |M ihM |,(A.92)|M − 1ihM − 1|b̂†j+1 b̂j |M ihM |.

(A.93)

M =0bN/2cb̂†j+1 b̂j →X
M =1

Investigating b̂†j+1 b̂j first, we can see thathM − 1|b̂†j+1 b̂j |M i = √1
nM nM −1XX

kM ,...,k1 lM −1 ,...,l1

× hM − 1; lM −1 , . . . , l1 |b̂†j+1 b̂j |M ; kM , . . . , k1 i.
The innermost matrix element is non-zero and equal to√(A.94)

2 exactly when kn = j forsome n and {lM −1 , . . . , l1 } = {kM , . . . , k1 }\{kn }. Equivalently, we count the number of
M -dipole states that have a dipole between sites j and j + 1. By removing this dipole,
we generate all of the (M − 1)-dipole states that are coupled to the M subspace viab̂†j+1 b̂j . The matrix element is then equal toshM − 1|b̂†j+1 b̂j |M i =2× (# of M -dipole states with dipole at j and j + 1).
nM nM −1
(A.95)

To compute this coupling, we fix j, N , and M , and count the total number ofways we can put M − 1 dipoles in the remaining sites. There are j − 1 sites to the leftof j. There are N − j − 1 sites to the right of site j + 1. If we let m be the numberof dipoles to the left of site j in a particular configuration, then there are M − 1 − mdipoles to the right of site j + 1. The total number of configurations is equal to the
product of the number of ways we can fit m dipoles in j − 1 sites times the number of164
ways we can fit M − m − 1 dipoles in N − j − 1 sites. This is
;j)
n(M
=
m




j − 1 − m N − j − 1 − (M − m − 1).
m
M −m−1(A.96)

Summing over m will give us the quantity we are seeking. The maximum numberof dipoles that we can fit in N − j − 1 sites is b(N − j − 1)/2c. Thus, the minimum
number of dipoles we need to put to the left of site j is max(0, M − 1 − b(N − j − 1)/2c).
This ensures that we do not attempt to cram too many dipoles to the right of site j + 1.
Similarly, the maximum number of dipoles that we can fit in j − 1 sites is b(j − 1)/2c.
This cannot be bigger than M − 1; the largest m can be is then min(M − 1, b(j − 1)/2c).
With all of this in hand, the total number of M -dipole states with a dipole betweensites j and j + 1 ismin(M −1,b(j−1)/2c)
(M )
nj =



X
m=max(0,M −1−b(N −j−1)/2c)
j − 1 − m N − j − 1 − (M − m − 1)
. (A.97)
M −m−1mThus, in the restricted basis, we can write
s
bN/2c
X
2
(M )
b̂†j+1 b̂j →n |M − 1ihM |.
nM −1 nM j

(A.98)

M =1

For the computation of b̂†j b̂j , we fix again fix j, N , and M . The total number of(M )states with two particles at site j is exactly nj. The total number of states with zero(M )particles at site j is the same as the total number of states, nj−1 , with two particles at(M )site j − 1. Here, nN(M )and n0are understood to be zero. The total number of states(M )with one particle at site j is nM − nj

(M )− nj−1 . The average number of particles on site

j within the M -dipole subspace is then
(M )
(M )
(M )
(M )
(1)(nM − nj − nj−1 ) + (2)(nj ) + (0)(nj−1 )†hM |b̂j b̂j |M i =
nM
(M )
(M )
nM + nj − nj−1

=

nM

.

(A.99)

Appendix B

Atoms Interacting with Quantized Fields

The description of atoms interacting with quantized light-fields relies heavily on
the material in references [30, 63, 71, 77, 88].
For simplicity, we here consider the interaction of a hydrogen atom with an external electromagnetic field. The generalization to other one-electron atoms is straightforward since the energy levels can be treated via quantum-defect theory, and the sizesof the atoms are all much smaller than the wavelength of optical light. The Hamiltonian
for a hydrogen atom interacting with an electromagnetic field is given by the minimal
substitution, taking the form,
H (re , rp ; pe , pp ) =11(pp − eA (rp ))2 +(pe + eA (re ))2 + V (|re − rp |) + HEM ,
2mp
2me
(B.1)

where the first two terms are the Hamiltonians for the nucleus and the electron, respectively, V is the Coulomb interaction between electron and nucleus, and HEM is the
Hamiltonian of the free electromagnetic field. We work in the Coulomb gauge, where∇x · A (x) ψ (x) = A (x) · ∇x ψ (x) − [∇x · A (x)] ψ (x) = A (x) · ∇x ψ (x) ,(B.2)

and we move to center-of-mass coordinates,R=

mp rp + me re,mp + mer = re − rp .

(B.3)
(B.4)

166
In this case, the Hamiltonian becomes
~2 2
~2 2
∇R −
∇ + V (r) + HEM
2M
2µ r


 
mp
~ e
me
A R−
−
r ·
∇R − ∇ r
i mp
µ
µ


 
mp
me
~ e
A R+
+
r ·
∇R + ∇r
i me
µ
µ




mp
e2 2
me
e2 2
+
A R−
A R+
r +
r .
2mp
µ
2me
µ

H (R, r) = −

(B.5)

where
mp me
,
mp + me(B.6)

M = mp + me ,

(B.7)

µ=

are respectively the reduced mass and the total mass of the atom.
Wavelengths of optical fields are typically on the order of hundreds of nanometers,roughly one hundred times the size of the atom. The electric fields do not appreciably
vary in the region over which the wave-function of the atom is non-zero. We can thenmake the dipole approximation, which at this point amounts to making the replacement,




mp
me
r ,A R +r → A (R) .
(B.8)
A R−
µ
µ
The Hamiltonian becomes
H(R, r) = HCOM (R) + ĤA + Hint (r) + H2 + HEM
=−

~2 2
~2 2
~e
e2 2
∇R −
∇r + V (r) +A (R) · ∇r +A (R) + HEM .
2M
2m
iµ
2µ

(B.9)

For many atoms interacting with the electromagnetic field, we define field operatorsΦ̂ (R, r) – which may be bosonic or fermionic – and write the Hamiltonian as
Z
Ĥ =

d3 R

Z

d3 rΦ̂† (R, r) H (R, r) Φ̂ (R, r) .

The field operator can be expanded in single-particle eigenfunctions asΦ̂ (R, r) =Xm,nĉmn ψn (R) ϕm (r) ,(B.10)

167
where {ψn (R)} and {ϕm (r)} are respectively diagonalize HCOM and Hint .
DefiningΨ̂m (R) =X

ĉmn ψnR (R) ,(B.11)nthe various pieces of the Hamiltonian can be written as
Z
Z
X
~2 X
Em d3 RΨ̂†m (R) Ψ̂m (R) , (B.12a)d3 RΨ̂†m (R) ∇2R Ψ̂m (R) +
2M m
m
 Z

XZ
e
†
3
∗
3
Ĥint =d RΨ̂m0 (R) A (R) Ψ̂m (R) ·ϕm0 (r) pr ϕm (r) d r ,(B.12b)
µ
m,m0
Z
e2 X
H2 =
d3 RΨ̂†m (R) A2 (R) Ψ̂m (R) ,(B.12c)2µ m
ĤA = −

where Em is the eneregy eigenvalue corresponding to the internal-state wave function,
hr|mi = ϕm (r).

B.1

Hamiltonian of the free electromagnetic field
We expand the vector potential, A, in normal modes assX~A (x, t) =ak,λ êk,λ uk (x) e−iωk t + a∗k,λ êk,λ u∗k (x) eiωk t ,2ωk 0(B.13a)k,λωk = c |k| = ck,(B.13b)k̂ × êk,1 = êk,2 , k̂ × êk,2 = −êk,1 , êk,1 × êk,2 = k̂,(B.13c)whereωk = c |k| = ck,(B.14)λ indexes two polarization vectors perpendicular to k, and uk (x) is a complete, orthonormal set of normal modes of the electromagnetic field, satisfying proper boundary
conditions and the Helmholtz equation,∇2 + k 2 A (x, t) = 0.

(B.15)

168
We promote the expansion coefficients to operators satisfying bosonic commutation
relations,hiâk0 ,λ0 , â†k,λ = δk,k0 δλλ0 ,(B.16)

(B.17)

âk0 ,λ0 , âk,λ = 0,in which case the free Hamiltonian,
Z

1
EEM =
2


1
2
|B| ,
d x 0 |E| +µ0(B.18)


1†~ωk âk,λ âk,λ +.
2(B.19)

3

2

can be written as
ĤEM =Xk,λThe equation of motion for the annihilation operator isi~idâk,λ h= âk,λ , ĤEM = ~ωk âk,λ ,dt(B.20)

âk,λ (t) = âk,λ (0) e−iωk t .

(B.21)

which implies that

The electric field operator is
r
dA X
~ωk
E (x, t) = −=iâk,λ êk,λ uk (x) e−iωk t + h.c.
dt
20
k,λ

Equation B.21 tells us that this operator is written in the Heisenberg picture. The
Schrödinger picture operator is then given byrX~ωkâk,λ êk,λ uk (x) + h.c.
E (x) =i
20(B.22)k,λWe may make the identification,(+)(+)
Ek,λ = Êk,λ êk,λ = ir~ωkâk,λ êk,λ ,20(B.23)(+)

where Ek,λ is the complex amplitude associated with the positive frequency part ofmode (k, λ). The amplitudes of the electric field an vector potential can be related via
s
~
1 (+)
(+)
Ak,λ =
âk,λ êk,λ =E .
(B.24)
20 ωkiωk k,λ169

B.2

Optical dipole potential
We can simplify equation (B.12b) considerably in the optical regime by way of therotating wave approximation. In order to make our way towards implementation of thisapproximation, we first justify making the (approximate) identification, Ĥint = −d · E,
where d is the dipole moment of the atoms. The matrix element µe hm0 |pr | mi can be
re-written by way of the equation,i~dr= [r, Hint (r)] = pr ,dtµ

(B.25)

em0 |pr | m = −ieωmm0 m0 |r| m .
µ(B.26)i~as

The interaction Hamiltonian can then be written as
XZ

Ĥint = −d3 RΨ̂†m0 (R)m,m0X ωmm0(+)
e
m0 r · êk,λ |mi Êk,λ uk (R) Ψ̂m (R) + H.c.
ωkk,λ(B.27)Up to the factor of

ωmm0ωk , this is exactly −d · E.

We now suppose that we are dealing with field modes that are nearly resonantwith the atomic transition; we will make the term “nearly resonant” more precise below.
In addition, the matrix element hm0 | r · êk,λ |mi is zero if m0 = m since the eigenstatesof HA (r) can be chosen to be angular momentum eigenstates, in which case r does
not connect states of the same total angular momentum. Focusing on only two stateswith ωm0 > ωm for the moment, we move into an interaction picture where the free
time-dependence of the internal state and the electromagnetic field are made explicit.
In this case,Z
Ĥint = −d3 RΨ̂†m0 (R) e

ωmm0
(+)
m0 r · êk,λ |mi Êk,λ uk (R) Ψ̂m (R) e−i(ωk −(ωm0 −ωm ))tωk+ (m ↔ m0 ) + H.c.

(B.28)

170
If we enforce the condition,(m0 ,m)|Ωk,λ|  |ωk + (ωm0 − ωm )|,r~ (m0 ,m)
ωm − ωm0 ~ωkm0 r · êk,λ |mi ,
Ω
= −ie
2 k,λ
ωk
20

(B.29)(B.30)

which is what we mean by nearly resonant and is not very difficult to satisfy for opticalfrequencies, then we can neglect terms involving e±i(ωk +(ωm0 −ωm ))t , since they average
to zero due to rapid oscillations. The interaction Hamiltonian can be then written as
ZXX
X ~ (m0 ,m)(m0 m)
iδktĤint =
Ωk,λ âk,λ e
d3 RΨ̂†m0 (R) uk (R) Ψ̂m (R) + H.c.
20m m(ωm <ωm0 ) k,λ(B.31)
where(m0 ,m)δk= ωm0 − ωm − ωk .

(B.32)

It is important to keep in mind that although equation (B.31) seems to have asum over all internal states of the atom and all modes of the field, this can obviously
not be case. Not only does the condition in equation (B.29) break down for different
transitions but the same field mode, but the dipole approximation can also break down
if the field frequencies get too high. In this thesis, we consider coherent dynamics only
when we have two level atoms and at most two modes of the electromagnetic field.
Coupling of atoms to a continuum of free-space modes will be treated through a masterequation describing spontaneous emission.

B.3

Adiabatic elimination of high-lying excited states
There is one final simplification we will make for the atom-field interaction. In

the case where the field is far enough detuned from the atomic transition while stillsatisfying equation (B.29), high-lying excited state can be adiabatically slaved to thelower-lying states. This gives rise to an effective position-dependent Stark shift. Not
only does this decouple the internal states of the atom, but it also gives rise to an
external sinusoidal potential for the atoms.

171
In the following, we consider only two atomic states, a ground state and an
excited state, and we consider at most only two electromagnetic field modes with thesame frequency. We have in mind one of two situations. First, the atoms interact witha single standing-wave mode that is pumped by a laser, and the rest of the modes aretreated via a master equation. Second, the atoms interact with two counter-propagating
modes of an optical cavity – where the mode are isolated from each other in frequency.
In either case, we can make the temporary replacement,(e,g)Ωk,λ → Ω;

(eg)

δk

= δ.

(B.33)

Finally, we assume that the fields are “far-enough” detuned from the atomic transition.
We will make more precise what this means below.
The equation of motion for the high-lying, excited-state field operator isi~dΨ̂e X ~Ω iδt=e uk âk,λ Ψ̂g .
dt
2(B.34)k,λWe formally integrate this equation for a time τ  1/δ, resulting inZ τ +t0Ψ̂e (τ + t0 ) =

dt0

t0

≈
=

k,λ

X −iΩ
k,λ

2

X −Ω
k0 ,λ0

X −iΩ

2δ

2

0

eiδt uk âk,λ Ψ̂g t0
Z t+t0uk âk,λ Ψ̂g (τ + t0 )

0

dt0 eiδtt0
uk0 âk0 ,λ0 Ψ̂g (τ + t0 ) eiδ(τ +t0 ) − eiδt0 .

(B.35)

The approximation holds when, schematically,|δ|  |Ω| ,(B.36)

since in this case the time-evolution of Ψ̂g (t) is slow compared to eiδt .
Taking t = τ + t0 and plugging this back into the interaction Hamiltonian, we

172
have

†


X~ Z
X −Ω
Ĥint =
Ω d3 R 
uk0 âk0 ,λ0 Ψ̂g eiδt − eiδt0  uk âk,λ eiδt Ψ̂g + H.c.
22δk,λk0 ,λ0Z
X −~ΩΩ∗ †=âk0 ,λ0 âk,λ d3 RΨ̂†g (R) u∗k0 (R) uk (R) Ψ̂g (R) + H.c.
4δk,λ,k0 ,λ0
 
Ω
iδt
+ Ωe O+ H.c.
(B.37)
δWe drop the last terms by appealing to an iteration of this previous argument, which
implies that  2 !
Ω
Ω
Ωe O
= ΩO
.
δ
δ
iδt

(B.38)

By imposing the condition |δ|  |Ω|, we can drop this term. This is in fact the precise
characterization of “far-enough” detuned. Now, this process can be repeated for any ofthe excited states that the state |gi is connected to. We then sum over all of these state
to get an effective coupling,(g)
gk,k0 ,λ,λ0 =(σg)
X Ωk(σg)∗0 ,λ0 Ωk,λ
(σg)

σ

.

(B.39)

2δk

In addition, this same process can be performed for the excited state in a two-levelatom. In this case, we haveĤint =

X

X

~gk0 ,λ0 ,k,λ â†k0 ,λ0 âk,λ(m)

Zd3 RΨ̂†m (R) u∗k0 (R) uk (R) Ψ̂m (R) . (B.40)

m∈{e,g} k,λ,k0 ,λ0Finally, we consider our two special cases. For the two-level atom interacting withthe single-mode field, this becomesĤint = â†k,λ âk,λ

X

(m)
~gk,λ

Z

d3 RΨ̂†m (R) |uk (R)|2 Ψ̂m (R) .

(B.41)m∈{e,g}

For the case of atoms interacting with two counter-propagating, running-wave modes ofan optical cavity, we have, assuming the same polarization,
Z
X X
(m) 1
d3 RΨ̂†m (R)Ĥint =~gV0m∈{e,g} k,k
× â†k âk + â†k âk + e−i2k·R â†k â−k + ei2k·R â†−k âk Ψ̂m (R) .

(B.42)