Topics in Ultracold Atomic Gases: Strong Interactions and Quantum Hall Physics

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Weiran Li, B.S. Graduate Program in Physics

The Ohio State University 2013

Dissertation Committee:

Professor Tin-Lun Ho, Advisor

Professor Eric Braaten

Professor Jay Gupta

Professor Nandini Trivedi c Copyright by

Weiran Li

2013 Abstract

This thesis discusses two important topics in ultracold atomic gases: strong interactions in quantum gases, and quantum Hall physics in neutral atoms.

First we give a brief introduction on basic scattering models in atomic physics, and an approach to adjust the interactions between atoms. We also include a list of experimental probes in cold atom physics. After these introductions, in Chapter 3, we report a few interesting problems in strongly interacting quantum gases. We introduce the BCS-BEC crossover model and relevant many-body techniques at the beginning, and discuss the details of several specific systems. We find the Fermi gases across narrow Feshbach resonances are strongly interacting at low temperature even when the magnetic field is several widths away from the resonance. We also discuss an approach to describe the metastable repulsive branch of Bose and Fermi gases across the resonance, and find a stable region of repulsive Bose gas close to unitarity. Some studies in two dimensional Fermi gases with spin imbalance are also included, and they are closely related to a number of recent experiments.

In Chapter 4, we discuss quantum Hall physics in the context of neutral atomic gases.

After illustrating how the Berry phase experienced by neutral atoms is equivalent to the magnetic ﬁeld in electrons, we introduce the newly developed synthetic gauge ﬁeld scheme in which a gauge potential is coupled to the neutral atoms. We give a detail introduction to this Raman coupling scheme developed by NIST group, and derive the theoretical model of the system. Then we make some predictions on the evolution of quantum Hall states when an extra anisotropy is applied from the external trap. Finally, we propose some experiments to verify our predictions.

ii Acknowledgments

I still remember when I was a kid, on my way to kindergarten every day, my father used to ask me lots of questions about natural phenomena. Most of them ended up being rhetorical questions as expected, but I guess all those lectures gradually generated my passion in maths and sciences. Although seldom expressed, I admire him as my ﬁrst science teacher, and many more of him. I am extremely lucky to have my mother taking good care of me, and to have my wife taking over this in the past few years. These two important women in my life have always been encouraging me to pursue what I am interested in. It was extremely hard time for my wife in the last year of my PhD study when I stayed in China, and I cannot imagine her being more supportive.

I am extremely grateful to my advisor Dr. Tin-Lun Ho, for supporting me in the past

ﬁve years. He has been very generous with his time and energy, trying to help me gain more insights in important physics problems. I really appreciate his eﬀort in educating me in my PhD study. I am also very grateful to all my committee members, Dr. Eric Braaten,

Dr. Jay Gupta, Dr. Nandini Trivedi for all the advice these years. I would like to thank

Dr. Braaten especially, for his help in writing and revising this thesis.

In my PhD study, I really enjoyed the discussions with Dr. Shizhong Zhang, Dr. Zhen- hua Yu, and my collaborator Dr. Xiaoling Cui. They have been very generous with their time, always ready to help me with diﬀerent kinds of problems. I consider them as my men- tors and wish them the best in their future career. I would also like to thank the hospitality from IASTU, especially from Dr. Hui Zhai’s group, during my visit between 2012-2013.

Last but not least, I thank all my friends in physics department, especially the condensed matter theorists on the second ﬂoor, to make my life colorful in Columbus. I will always

iii miss the fun we had.

Finally, I acknowledge the ﬁnancial support from National Science Foundation and

DARPA.

iv Vita

April 24th, 1985 ...... Born—Harbin, China

July, 2008 ...... B.S., Tsinghua University, Beijing, China

Publications

Bose Gases Near Unitarity Weiran Li and Tin-Lun Ho Physical Review Letters 108, 195301 (2012)

Alternative Route to Strong Interaction: Narrow Feshbach Resonance Tin-Lun Ho, Xiaoling Cui and Weiran Li Physical Review Letters 108, 250401 (2012)

Fields of Study

Major Field: Physics

Studies in Theories on degenerate quantum gases: Professor Tin-Lun Ho

v Table of Contents

Page Abstract ...... ii Acknowledgments ...... iii Vita...... v List of Figures ...... viii

Chapters

1. Introduction ...... 1

2. Basics in ultracold atomic gases ...... 5 2.1 Scattering models in ultracold atoms ...... 5 2.1.1 General scattering theory, T -matrix ...... 6 2.1.2 Low energy scattering, s-wave scattering length and phase shift . . . 8 2.1.3 Zero range model and Fermi’s pseudo potential ...... 14 2.2 Feshbach resonances ...... 18 2.3 Probes in cold atoms experiments ...... 23 2.3.1 Direct Imaging ...... 24 2.3.2 Spectroscopy ...... 27

3. Strongly interacting quantum gases across Feshbach resonances . . . . . 30 3.1 Introductions to strongly interacting quantum gases and the BCS-BEC crossover 32 3.1.1 Superﬂuidity across BCS-BEC crossover ...... 34 3.1.2 Critical temperatures and ladder approximation in dilute quantum gases ...... 42 3.1.3 The “upper branch” of the quantum gases ...... 51 3.1.4 Summary ...... 54 3.2 Fermi gases across narrow Feshbach resonance ...... 57 3.2.1 Wide resonance and narrow resonance ...... 57 3.2.2 Strong interactions in Fermi gases across narrow resonance . . . . . 62 3.3 Repulsive Bose gases across Feshbach resonance ...... 69 3.3.1 Three body loss in Bose gases close to unitarity, “low recombination” regime ...... 70 3.3.2 Strongly repulsive Bose gases close to unitarity, “shifted resonance” 72 3.3.3 Equation of state and instabilities of Bose gas in a trap ...... 79 3.4 Two dimensional Fermi gases with spin imbalance ...... 82

vi 3.4.1 Fermi gases in two dimensions ...... 83 3.4.2 Thermodynamic quantities of two component Fermi gases in two dimension ...... 87 3.5 Conclusions ...... 90

4. Rotating gases, synthetic gauge fields, and quantum Hall physics in neutral atoms ...... 93 4.1 Rapidly rotating Bose-Einstein condensates and quantum Hall physics . . . 94 4.1.1 Quantum Hall physics, Laughlin wavefunctions ...... 95 4.1.2 Rotating Bose-Einstein condensates, vortex array and quantum Hall regime ...... 98 4.2 Synthetic gauge field scheme ...... 102 4.2.1 Berry phase in adiabatic states, Abelian gauge field ...... 103 4.2.2 NIST scheme of Abelian synthetic gauge field ...... 106 4.2.3 Non-Abelian gauge fields, spin-orbit coupled gases ...... 112 4.2.4 Summary ...... 117 4.3 Quantum Hall physics of atomic gases with anisotropy ...... 118 4.3.1 Single particle wave functions of particles in rotating anisotropic traps 119 4.3.2 Transition from a condensate to a quantum Hall state ...... 121 4.3.3 Quantum Hall wave functions in broken rotational symmetry . . . . 126 4.3.4 Detection of quantum Hall wave functions in cold gases ...... 130 4.4 Conclusions ...... 134

Bibliography ...... 137

Appendices

A. Path integral formalism of BCS-BEC crossover ...... 145

B. Adiabatic states and their gauge potential in spatially varying magnetic ﬁeld ...... 149 B.1 General adiabatic states in spatially varying magnetic ﬁeld ...... 149 B.2 Gauge potentials associated with the adiabatic states ...... 152

vii List of Figures

Figure Page

2.1 A sketch of phase shifts for different scattering lengths...... 13 2.2 A sketch of the s-wave scattering length and the effective range for a square well potential...... 16 2.3 A schematic diagram of the potentials in open and closed channels near a Feshbach resonances...... 19 2.4 Experimental images from time-of-flight (TOF) expansion, in Bose and Fermi gases in optical lattices...... 25

3.1 A sketch of pairing gap and chemical potential for a Fermi gas across unitarity at T =0...... 38 3.2 Class of diagrams included in the ladder approximation. All the legs of the ladder, i.e. the propagators on both ends of the interaction lines, run in the same direction...... 43 3.3 The scattering vertex for the particle-particle channel and particle-hole channel. For dilute gases, particle-hole channel is usually negligible...... 44 3.4 A sketch of contour deformation of Matsubara sum in NSR formalism. . . . 47 3.5 A sketch of the superfluid transition temperature and the chemical potential at Tc for a Fermi gas in BCS-BEC crossover...... 50 3.6 The phase shifts ζ(E) for scattering in the presence of the bound state: the phase shift of the scattering state is modified such that it starts from 0 at the scattering threshold...... 52 3.7 The no-pole approximation in terms of excluding the bound-state contribution, from modifying the integral region after doing the Matsubara sum. . . 54 3.8 A schematic diagram for the Gor’kov-Melik-Barkhudarov (GMB) correction. The scattering matrix Γ is different to the simple ladder approximation, by including one higher order of density fluctuation...... 55 3.9 A comparison of the calculated transition temperature Tc from different approaches: Leggett BCS mean field, GMB mean field, and NSR...... 56 3.10 An illustration of the difference between wide and narrow Feshbach resonances in a Fermi gas with Fermi energy EF ...... 60 3.11 Schematic diagram of the relation between wide and narrow resonances: their −1 occupations in the space ((kF r∗) , kF abg)...... 61

viii 3.12 The scattering phase shift δ(k) as a function of incoming wave vettork for wide (A) and narrow (B) resonances...... 63 3.13 A sketch of second virial coefficients b (A) and “interaction energy” (B) − 2 int as a function of magnetic field, across a narrow Feshbach resonance. . . . . 66 3.14 An example of the s-wave scattering length (upper panel) and the interaction energy (lower panel) of a Fermi gas near a narrow resonance at low temperatures. The system is quantum degenerate at T = 0.5TF ...... 67 3.15 Experimental data from the Penn State group for fermionic 6Li gases across the 543.25G Feshbach resonance at different temperatures...... 68 3.16 A sketch of how bosonic and fermionic media affect the formation of dimers. The occupation and quantum statistics play important roles in the molecule formations...... 75 3.17 Energy density of an upper branch Bose base across the unitarity, and the critical scattering length for dimer formations in a Bose medium. Both are at T = 4TF ...... 77 3.18 The “phase diagram” of a homogeneous upper-branch Bose gas with fixed density n. The system is divided into three parts...... 78 3.19 “Phase diagram” of an upper branch Bose gas in a trap at fixed temperature T and trap frequency ω. A global view of density profile for any gases can be obtained in this (µ/T )-(λ/as) plane...... 80 3.20 Three typical density profiles for stable and unstable upper-branch Bose gases in a harmonic trap...... 81 3.21 Data of the Cambridge experiment in attractive and repulsive polarons: the interaction energy for both branches, and the lifetime for the repulsive branch. 84 3.22 The interaction energy for the attractive branch (red curve) and the repulsive branch (blue curve) at a high temperature T = 6TF , for equal spin populations. 88 3.23 Spin susceptibility and compressibility for upper branch two dimensional Fermi gas with equal population, at a temperature T = 6TF ...... 90 3.24 A diagram of “stability” of repulsive branches at different temperatures and polarizations...... 91

4.1 A sketch of the origin of integer quantum Hall effects...... 97 4.2 A sketch of the energy levels of Fock Darwin states for particles in rotating traps...... 100 4.3 A sketch of the origin of the gauge field in the presence of the spatial dependent magnetic field: the adiabatic states follow the orientation of the external magnetic field, and thus experience the Berry phase...... 105 4.4 Schematic figure of experimental setup in NIST. The Raman coupling is realized by two counter-propagating laser beams...... 107 4.5 The direction of the vector Beff from the effect of Raman coupling plus a linear detuning in magnetic field...... 109 4.6 Profiles of the vector potential and the synthetic magnetic field generated from the NIST setup...... 110 4.7 Schematic diagrams of energy levels in both Abelian and non-Abelian synthetic gauge fields...... 113 4.8 A sketch of the energy spectrum of two branches for spin-orbit coupled gases. 116

ix 4.9 Phase diagram of the cloud in rotating anisotropic traps: two phases of the BEC and QH are identified...... 124 4.10 Distortion and transition in vortex lattices of a rapidly rotating BEC in an anisotropic trap...... 125 4.11 Density profiles at cut x = 0 and y = 0 of the Laughlin wavefunctions at different positions in figure 4.9...... 127 4.12 Distribution of zeros in the relative coordinate of the ground state of two body problem in rotating anisotropic traps...... 128 4.13 Variation in the distance of splitting zeros as a function of inverse interaction strength, for fixed anisotropy α = 0.96...... 129 4.14 Density profiles after TOF expansion of the gas in a rotating isotropic trap. 132 4.15 Density profiles after TOF expansion of anisotropy α = 0.94...... 133 4.16 Second order correlation in equation 4.64, in a rotating isotropic (left) and anisotropic (right) trap...... 134 4.17 A sketch of high-resolution in situ imaging technique in neutral atoms. . . . 135 4.18 In situ images of the quantity ψ(r/2, r/2) for an anisotropy α = 0.96. . 136 | − |

x Chapter 1 Introduction

The concept of Bose-Einstein condensation (BEC) can be traced back to 1924-1925, when

Bose and Einstein ﬁrst predicted that a macroscopic occupation of a single quantum state occurs below a critical temperature in non-interacting Bose systems. After that, superﬂuid helium was the only example of BEC in nature for decades. However, in helium-4, the density of helium atoms is so high that the system is in the strongly interacting regime.

The fraction of the condensate part is thus less then 10% even when the temperature is far below the critical value. Nevertheless, important theoretical problems have been studied in weakly interacting Bose gases in the condensate phase, even in the absence of a single realistic system of which the models are exactly suitable.

After many years of eﬀort in cooling down the dilute gases of neutral alkali atoms, in

1995, scientists from JILA[1] realized the ﬁrst weakly interacting dilute Bose system, in which almost 100% of the particles are in the condensate phase. The ﬁrst gaseous BEC was in a dilute Rubidium gas at a temperature of 170nK, followed by the realizations of

Lithium[2, 3] and Sodium[4] condensates. This breakthrough of cooling down the quantum gases to degenerate and condensation limit brings the area of ultra cold atomic gases, or

“cold atoms” into the front stage of modern physics. The dilute BECs not only can be used to verify so many well developed theories, but also open up new directions of important physical problems[5]. In the past two decades, BECs are realized in a lot more isotopes of alkali, alkali earth[6], as well as rare earth atoms[7, 8]. And also, there are extensive eﬀorts in cooling down the dilute Fermi gases to degenerate limit[9]. These successes have been used

1 to study the superﬂuidity of paired fermions, analogous to the Cooper pair condensation in conventional charged superconductors.

Implementation of Feshbach resonances in scattering between atoms makes it possible to study the Fermi gases in strongly interacting regime. By applying an external magnetic ﬁeld, the inverse s-wave scattering length as can be tuned continuously through the scattering resonance, at which the scattering length diverges. The resonance point is also referred to as the unitarity. To the two sides of the resonance, the scattering length has diﬀerent signs, and a two body bound state appears on the positive scattering length side. The

Feshbach resonance technique gives us opportunities to precisely control the scattering length, or the interaction strengths between atoms. This provides a platform to study some interesting problems, among which is the well addressed problem back in 1970’s, the

BCS-BEC crossover[10]. In the BCS-BEC crossover problem, the properties of superﬂuid

−1 states change gradually as as increases continuously. It is found that there is a continuous crossover between the superﬂuidity in the BCS limit ((k a )−1 ) and the BEC limit F s → −∞ ((k a )−1 + ). In BCS, the Cooper pairs form the constituent of the condensate, while F s → ∞ in the BEC limit, each two fermions form a deep bound bosonic molecules such they undergo

Bose condensation as a weakly interacting Bose gas. There is no sharp transition between these two types of superﬂuids, since the systems in the region connecting these two limits can be described by a single class of wave functions with gradually changing parameters.

The transition temperature of superﬂuidity close to unitarity has a large ratio of Tc/TF

(almost to 20%) where TF is the Fermi temperature of the system. This system has the

largest Tc/TF value in all the known substances in the universe, and has at least an order

of magnitude higher than other high Tc superﬂuids, e.g. quark-gluon plasma, and cuprates. Besides the large attractions in the Fermi gases close to unitarity, it is also very in-

teresting to study other important questions in quantum gases with repulsive interactions.

Among them there is an interesting question as the existence of the Stoner ferromagnetism

in uniform systems. In the context of cold atoms, it is known that deep in the BEC

side where the scattering length is small and positive, the “true ground state” is the Bose

gas composed of diatomic molecules. However, there is a metastable branch (the “upper

2 branch”) free of these low-lying bound states. Such atomic gas is eﬀectively weakly repulsive if the underlying tight binding dimers are not populated. The diluteness of the quantum gas assures the system persists in this well deﬁned thermodynamic states on a time scale that long enough for the system to equilibrate by two-body collisions. In this time scale, three body recombinations from which bound states are generated are negligible. While there are well developed theories for weakly interacting Bose and Fermi gases, including the

Bogoliubov theory and Lee-Huang-Yang corrections, for larger scattering lengths, there is no well accepted theory in describing this upper branch. Even the existence of the upper branch at ﬁnite temperature is under debate[11, 12, 13, 14].

Also with the capability of precisely controling in the interactions, and the external potential (engineering of overall traps and optical lattices), one may better study some important models in condensed matter physics. In conventional solid state materials, the complications brought by impurity etc makes it diﬃcult to observe the behavior of a “pure model”. This concept of quantum simulation is one of the major worldwide eﬀorts in cold atom laboratories. It aims to build platforms as closely as possible to what a quantum model exactly describes. For instance, it is intriguing to study the low temperature properties of two dimensional Hubbard models and quantum spin models, including their transitions between the normal and superconducting or other magnetic phases. These models are important because they are what many theories in strongly correlated systems are based on. Also, with the tunability of interactions between atoms and the internal degrees of freedom of the atoms, there are potentially some exotic models and quantum phases which are absent in the conventional solid state materials.

On the other hand, apart from studying the broken symmetry phases, cold atoms also enable us to study other nontrivial strongly correlated states, for instance the quantum

Hall states. It has been shown that atoms in rotating quantum gases are analogous to charged particles in the presence of a magnetic ﬁeld, for the Coriolis force in the rotating frame provides the equivalence of the Lorentz force[15]. In fast rotations, as many as a hundred vortices have been observed in a Bose gas[16], and they form a triangular lattice.

It is conceivable if the rotation frequency closely approaches the trapping frequency, the

3 single particle spectrum is almost ﬂat. The number of vortices will be in the same order as the number of particles, and the lattice will melt in this situation. The vortices ﬁnally become invisible as the angular momentum quanta are attached to the particles themselves.

This system with quantum ﬂux bounded to particles is then in a topologically non-trivial quantum Hall (QH) regime. This transition to a quantum Hall state cannot be described by Landau’s phase transition theory, since there is no local order parameter in the quantum

Hall regime. The quantum Hall physics has been mostly studied in two-dimensional electron gases in semiconductor heterostructures, and it is usually very hard to grow a clean sample to verify the predictions of any theoretical models. In cold atoms, one has the advantage of building the sample in a more controllable way. Also, with the ability to implement the internal degree of freedom–the hyperﬁne states of alkali atoms–it is possible to study some exotic quantum Hall states in cold atoms that are absolutely absent in electrons.

This thesis is organized as following. In Chapter 2, we give a brief introduction to some theoretical and experiment backgrounds in cold atomic gases. We include the basic scattering model widely used in dilute quantum gases, the Feshbach resonance technique to tune the interactions, as well as useful probes in cold atoms experiments. In Chapter 3, we focus on an important aspect of quantum gases: the strong interactions. We give a review of studies in modeling BCS-BEC crossover in Fermi gases across the Feshbach resonance, and present a derivation of the important ladder approximation in dilute gases. Based on these knowledge, we illustrate the strong attractive interactions in narrow Feshbach resonances, and predict some thermodynamic properties for Bose gases near unitarity in both upper and lower branches. We also address some issues in two dimensional Fermi gases with spin imbalance which is related to some recent experiments. In the last Chapter 4, we

first introduce how quantum Hall physics comes into the rotating quantum gases, and then briefly review a recent developed synthetic gauge field scheme with the goal to couple the artificial gauge potential to neutral atoms. Finally we study some interesting properties of quantum Hall states in the presence of geometric distortion of external potentials.

4 Chapter 2 Basics in ultracold atomic gases

In this chapter we give an introduction to some important background knowledge of cold atomic gases. These include the basic scattering problems between the neutral atoms, and the model we use in many-body hamiltonian. It is also necessary to understand the

Feshbash resonance, by which the interactions between particles can be tuned to any value.

Finally in this chapter, some useful experimental probes are listed, and they are compared to experimental approaches in conventional condensed matter physics.

2.1 Scattering models in ultracold atoms

The ﬁrst step to theoretically study the macroscopic quantum phenomena in quantum gases, is to understand the few-body atomic system. In quantum liquids like liquid helium, the range of interaction is comparable to the interparticle spacing. In contrast, the class of

“quantum gases” we focus on in this thesis have an interparticle spacing much larger than its interaction range. The diluteness of quantum gases ensures that two-body scattering process is the most important element to building up the interaction hamiltonian1. In this section we will focus on the two-body scattering models between neutral atoms.

1In fact in cold atomic gases, especially bosonic gases, there is a family of Eﬁmov states with interesting discrete scaling invariance. They aﬀect the three body recombination process in the atomic gases and sometimes have to be considered more carefully. However we will not go into details of the three body states in this thesis, and they can be found in this review paper [17], and references therein.

5 2.1.1 General scattering theory, T -matrix

Understanding the basic scattering models for two-body collisions is one of the most important, and still active areas in studies on cold atomic gases. A clear and thorough understanding of two-body physics is a crucial ﬁrst step to building up a many-body hamiltonian, for any system.

Here we start from introducing the general scattering theory for a single-channel scattering model, formulating the problem to solve and relate them to physical observables later.

For a two-body problem, we separate out the center-of-mass degree of freedom as usual.

In a translation invariant environment2, we are only interested in motions in the relative

coordinates. Further, when we approximate the scattering into single-channel model, the

internal degree of freedom is also frozen out, hence we only consider the elastic scattering

processes. The simplest starting point we have now, is an eﬀective single-particle problem

in the relative motion frame, and in the presence of the scattering potential.

For a short-range potential the hamiltonian is written as H = H0 + V , where H0 is the non-interacting kinetic energy and V is the interaction potential which vanishes at large distances. The asymptotic form of the wave function at infinitely large distance in the relative coordinate is characterized by an “incident wave” ψ , which is a solution of | ini Schrödinger’sequation for H . The full wave function ψ with total energy E consists of 0 | i an incident wave part and a scattering wave part ψ , which satisfies the equations below: | sci

ψ = ψ + ψ , (2.1) | i | ini | sci (E H) ψ = 0, (2.2) − | i (E H ) ψ = 0. (2.3) − 0 | ini

Using the Lippmann-Schwinger equation, we can deﬁne the scattering T -matrix3 in this

2Even though most cold atom experiments are conducted in harmonic traps, the smoothness of the overall potential enables one to neglect the broken translational symmetry. 3In this thesis we sometimes use the phrase “scattering matrix” to refer to the T -matrix. Actually the terminology of scattering matrix can be referred to as the unitary S-matrix, which relates to T -matrix as S = 1 − 2ikT .

6 model by

ψ = ψ + ψ = (E H )−1V ψ | i | ini | sci − 0 | i = (E H )−1T ψ , (2.4) − 0 E| ini where the scattering T -matrix at energy E is deﬁned as

T ψ V ψ . (2.5) | ini ≡ | i

By left multiplying the interaction operator V on both sides of equation 2.4, we have

V ψ = T ψ = V (E H )−1T ψ , (2.6) | i E| ini − 0 E| ini

TE = VG0(E)TE, (2.7)

where G (E) (E H )−1 is the free Green’s function at energy E. Equation 2.7 above 0 ≡ − 0 gives a relation between the T -matrix and the interacting potential of the system, and

is the starting point of many useful models. As we can see, the scattering T -matrix can

be expressed in a geometric series with all the orders of interaction operator V , which

corresponds to itinerating the scattering process between the two particles:

V T = V + VG (E)V + = . (2.8) E 0 ··· 1 G (E)V − 0 The approximation that keeps only the ﬁrst order of V in the T -matrix is called the

Born approximation. In the weakly interacting limit, namely when G (E)V 1, the Born 0 approximation provides a good estimate of many scattering quantities. The full expression

above is actually in the matrix form, to the extent that it can be sandwiched by any pair

of states, for instance the scattering between plane-wave states with wave vectors k and

0 0 0 2 2 2 02 k can be written as k TE k TE(k, k ). The condition that E = ~ k /2µ = ~ k /2µ, h | | i ≡ where µ here is the reduced mass of the two particles, is often called the on-shell condition.

Another very useful decomposition of the T -matrix is by partial waves, where the matrix

elements are deﬁned by

0 l, k T l0, k0 T l,l (k, k0), (2.9) h | E| i ≡ E

7 in which l, k is the partial wave with angular momentum l and wave number k. In an | i l,l0 l,l0 isotropic potential with spherical symmetry, T = δl,l0 T is diagonal in angular momentum sectors. In this thesis we mostly focus on the s-wave scattering problems, namely when only the l = l0 = 0 sector of the T -matrix is important.

2.1.2 Low energy scattering, s-wave scattering length and phase shift

From the Lippmann-Schwinger equation 2.4, we write it down in the ﬁrst quantized form by left multiply the bracket r 4, and assume the incident wave as a plane wave with momentum h | k. The full wave function in real space then takes the form

eikr ψ(r) = eikr + f(θ, φ) , (2.10) r where θ, φ are the polar and azimuthal angles with respect to the direction of the incident wave vector. The second term in the full wave function corresponds to outgoing spherical waves. For a spherically symmetric potential, the scattering amplitude f is independent of the azimuthal angle φ, and the diﬀerential cross section for scattering is given by

dσ = f(θ) 2. (2.11) dΩ | |

With spherical symmetry, the scattering wave function can also be expanded by partial waves, into the form ∞ ψ = AlPl(cos θ)Rkl(r), (2.12) Xl=0 in which Pl’s are Legendre polynomials, and the radial part Rkl satisﬁes the equation

00 2 l(l + 1) 2µ R + R0 + k2 V (r) R = 0, (2.13) kl r kl − r2 − 2 kl ~ where V (r) is the ﬁrst quantization form of the scattering potential. At large distance where

the potential vanishes, the solution of Rkl is a linear combination of spherical Bessel and 4To get the following form with “out-going” wave only, we have to add an inﬁnitesimal imaginary part −1 + −1 in the Green’s function, i.e. substitute (E − H0) by the form (E − H0 + i0 ) .

8 Neuman functions jl and nl. The general form of the solution at large distance reads

sin(kr lπ/2 + δ ) R (r ) − l , (2.14) kl → ∞ ≈ kr in which δl is the phase shift for the l-th partial wave. The ratio between the Neuman

function nl and Bessel function jl at large distances is given by tan δl. In the non-interacting limit where the potential is extremely weak, the asymptotic form of wave function at large

distance is a pure Bessel function, and the phase shift vanishes δ 0. l ≡ By comparing the asymptotic form above and the expression of f(θ) in equation 2.10,

we have

∞ ∞ 1 f(θ) = (2l + 1)(e2iδl 1)P (cos θ), (2l + 1)f P (cos θ) (2.15) 2ik − l ≡ l l Xl=0 Xl=0 with the l-wave scattering amplitude

f (e2iδl 1)/2ik. (2.16) l ≡ −

2 The scattering amplitude relates the scattering T -matrix as fl = Tlµ/2π~ . And the total − cross section of the scattering is given by

∞ ∞ 4π 4π σ = (2l + 1) f 2 = (2l + 1) sin2 δ . (2.17) k2 | l| k2 l Xl=0 Xl=0 Consider a simple ﬁnite-range scattering model, with the phase shift δl determined by the boundary condition at small distance. At short distances, the asymptotic form of jl and nl are j (kr) (kr)l, n (kr) (kr)−l−1, at kr 0. (2.18) l ∼ l ∼ →

If the boundary condition is enforced at r = r0 where r0 is the range of potential, the

2l+1 relative ratio between nl and jl will be proportional to (kr0) . In low-energy scattering, namely kr 1 limit, it is natural to see tan δ δ (kr )2l+1, hence the s-wave scattering 0 l ∼ l ∼ 0 (l = 0) is dominant at this energy scale for it is the leading term of kr0 in f. Physically, it corresponds to the fact that a low-energy incident wave, whose wavelength is much larger

than the potential itself, will not be able to probe the detailed structure of the potential.

9 All the information we have here is an almost isotropic cross section of outgoing waves.

In the s-wave channel, since the phase shift is linear in k, we can deﬁne the opposite slope as the s-wave scattering length as:

δ = ka , for k 0. (2.19) 0 − s →

The s-wave cross section 4π σ = sin2 δ = 4πa2 (2.20) k2 0 s is the same as if there is a hard sphere with radius as. A more complete expression which relates the s-wave scattering amplitude and the phase shift can be derived from equation

2.16. And in low energy limit, it can also be expanded in powers of k2 as the following5:

1 r∗ f −1(k) = k cot δ(k) ik = + k2 + O(k4) ik (2.21) − −as 2 −

∗ ∗ where r is called the eﬀective range of the scattering potential. The value of as and r are determined by the microscopic properties of the scattering potential.

The scattering length and phase shift are the most important quantities to determine a lot of low-energy physical properties of the system. For example we illustrate here the energy shift of the system as an example. We take a model that the particles are loaded in a hard-wall spherical container with radius R, and the boundary condition is enforced at the surface of the container that ψ(R) 0. In the absence of the interaction potential ≡ between particles, the spectrum takes a set of levels with

nπ k(0) = . (2.22) n R

The range of n is given by the density of the system such that the largest n = N satisﬁes

Nπ kF = R . First we discuss the “weakly interacting” limit with small scattering length, i.e. k a 1. In this limit when we put the interaction in, a nonzero phase shift emerges and F s 5The series expansion is in powers of k2 in most of the cases, since the scattering matrix is analytical in E. However, for some power-law decaying potentials V ∼ r−n, even when n is large enough such that the phase shift and scattering length are well deﬁned for s-wave (and other partial waves for 2l + 3 < n), the scattering amplitude may have logarithmic dependence on k. See [18, 19] for more details.

10 can be approximated by δ = ka . The corresponding eigen wave vectors have to satisfy − s

k R + δ(k ) = k (R a ) = nπ, (2.23) n n n − s which gives nπ k = . (2.24) n R a − s The corresponding shift in energy levels are given by

2 2 2 (0)2 2 2 ~ kn ~ kn ~ 2 2as 2π~ as (0) 2 ∆En = (nπ) 3 = ψn (0) , (2.25) 2mr − 2mr ≈ 2mr R mr | | where ψ(0) = 1 sin nπr is the wave function of non-interacting systems. The ap- n (2πR)1/2r R proximation δ = ka we use in the previous derivation is based on the assumption that − s the scattering length a Nπ is very small, i.e. the system is in the weakly interacting s R limit with k a 1. In this limit, the effective interaction hamiltonian can be written in F s a delta-function form: 2 2π~ as V (r) = gδ(r), g = , (2.26) mr in which δ(r) is the three dimensional Dirac-delta function, g is the effective coupling constant proportional to the s-wave scattering length. This effective hamiltonian is essentially in the perturbative level.

From the discussion in the previous paragraph, we see the physical quantities, such as the energy level shift is determined by the scattering length or the phase shift in the limit k a 1. For the strongly interacting regime, i.e. k a 1 or larger than unity, F s F s ∼ the approximation δ(k) = ka cannot be used in the region 0 < k < k . Instead the − s F phase shift has to be determined by the full expression in 2.21. In this case, one can look at the change of the total energy by virial expansion to the second order of fugacity z = eβµ[20], which gives a ﬂavor of how interacting thermodynamic quantities behaves in the high-temperature and low-fugacity eβµ 1 limit. The grand thermodynamic potential in this expansion to the second order is

Ω = T log 1 + z e−βE1n + z2 e−βE2n , (2.27) − " n n # X X

11 which is approximated to the second power of fugacity. E1n and E2n are the energy levels of

2 −βE2n single-particle and two-body states. We have to calculate Z2 = z n e and subtract the value of a non-interacting system to get the contribution fromP two-body interactions.

As we have the expressions for the eigenstates of the interacting problem in a spherical box (0) (0) kn and the non-interacting ones kn as in 2.23 and 2.22, the diﬀerence between Z2 and Z2 is

2 −βE −βE(0) ∆Z = z e 2n e 2n , (2.28) 2 − n X 2 2 (0) 2 (0)2 with E2n = ~ kn/2mr and E2n = ~ kn /2mr. As is shown that kn + δn = nπ, we have

dδ dn = R + dk. (2.29) dk And the sum over n becomes an integral over k, such that

∞ dk dδ 2 2 ∆Z = z2 e−β~ k /2mr z2b , (2.30) 2 π dk ≡ 2 Z0 6 where b2 is deﬁned as the second virial coeﬃcient for the scattering states . We will show

later the magnitude of b2 relies on the phase shift structure, and the more abrupt the

δ changes, the large b2 is. From the expansion of k cot δ in 2.21, we can calculate the

second virial coeﬃcient b2 as a function of the scattering length, even at unitarity when as diverges[21].

We simplify the discussion here by approximating the expansion of the inverse scattering

amplitude in 2.21 to the ﬁrst order of k, i.e. to use the following expression to determine

the phase shift 1 k cot δ = . (2.31) −as A plot of phase shifts at different negative scattering lengths is shown in figure 2.1. They cover a large range of a from the weakly interacting regime k a = 0.1 to extremely s F s − strongly interacting case k a = 100. The phase shifts for positive scattering lengths are F s − mirror symmetric to the horizontal axis in the plot, namely they only differ by a minus

6Here we neglected all the possible bound states in the two-body channel when we sum over all the eigenstates of the system, since we focus on the scattering state properties at present.

12 Π 2

L Π k H 4 ∆

0 0 0.5 1 1.5 2

kkF

Figure 2.1: A sketch of phase shifts as a function of scattering momentum, for different scattering lengths from expression 2.31. Different curves represents different scattering lengths, and from the lowest to highest they correspond to k a = 0.1, 0.5, 1, 10, 100. For F s − − − − − small kF as, the phase shift is almost linear, while for large kF as, it quickly saturates to π/2. The phase shift structures for positive scattering lengths only differ by a sign in δ, and thus are mirror symmetric to the horizontal axis.

sign as in the case for negative scattering length. As we can see, in the weakly interacting case, the phase shift is linear in k up to several Fermi momentum, while in the strongly interacting regime, it quickly saturates to π/2 at small momentum.

From the analysis above, the second virial coeﬃcient b2 can be calculated analytically in two extreme cases: k a 0 and k a . The former is realized by approximating F s → F s → ∞ ∂δ/∂k = a , and for the latter, the approximation is made as ∂δ/∂k = 0 only in a small − s 6 region of momentum k < k∗ k . Simplify the integral in 2.30, we have the following F expressions:

a b(0) = s , (2.32) 2 −√2λ sgn(a ) 2 b∞ = s (1 erf(x))ex , (2.33) 2 − 2 −

(0) ∞ where b2 and b2 are second virial coefficients for weakly and strongly interacting cases respectively. In the expressions, x = λ/(√2πas) with λ = h/√2πmrkBT as the thermal wavelength of the system, and erf(x) is the error function. For the weakly interacting case, b is asymptotically first order in a /λ 1. And the limit that a (x 0) gives 2 s s → ∞ → 13 b = sgn(a )/2 = 1/2, in the order of unity. Thus we conclude the strongly interacting 2 − s ± regimes has a much larger b2 value than the weakly interacting regime. Finally, the contribution from interactions to the thermodynamic quantities are related to the virial coefficients as following:

T P = P + 2√2z2 b , (2.34) 0 λ3 2

3T n 3 b2 √2 T ∂b2 = 0 + (nλ ) + , (2.35) 2 "−√2 3 ∂T # where P and are the pressure and energy density respectively, subindex 0 stands for non- interacting values. By substituting the explicit form of b2, one ﬁnds that the magnitude of interacting pressure and energy are monotonic functions of b . The large absolute | 2| value of as close to unitary does give a much stronger interaction than small ones in the weakly interacting limit[21]. In the following chapters, we will ﬁnd these from many-body calculations as well.

All the discussions above are about the s-wave scattering, which is based on the fact the l = 0 channel is dominant in the energy scale at kr 1. As in the dilute gases, 0 since the energy scale is given by the Fermi wave vector k , and it satisﬁes k r 1, by F F 0 these discussions we conclude that the physical quantities in such systems are pretty much determined by the s-wave scattering length and the phase shift structure.

2.1.3 Zero range model and Fermi’s pseudo potential

As we have shown in the previous sections that the s-wave scattering length and phase shift are the most important quantities for scattering problems in the dilute limit, in this section we will show how they are related to the microscopics of the interaction potential.

For a real potential between atoms, the details of interactions are usually very complicated. The overall proﬁle between neutral atoms is a r−6 decaying van der Waals potential, coming from the virtual dipole process of polarization in neutral atoms. At very short range, there is a repulsive part that overwhelms the attraction at extremely short distances. In the Leonard-Jones picture of interactions between neutral atoms or molecules, the repulsive

14 part comes from the Pauli repulsions and takes a r−12 form. This r−6 attraction plus r−12

repulsion forms a simple description of the interacting potential between neutral atoms,

however it is still very hard to have an analytical form of the solution to this potential

for ﬁnite-energy scattering. Fortunately, in dilute gases, we show in the previous section

that the low energy physical properties are determined by the asymptotic form of the wave

functions at large distances. This asymptotic form is governed by the s-wave scattering

length and the phase shift, hence it is possible to approximate the real potential with a

model of simple shaped potential, neglecting the complicated details at short distances. As

long as the scattering length and phase shift are preserved, all the low-energy physics can

be given correctly by the simpliﬁed model.

We start from a very simple model with an isotropic square-well interaction, which

resembles the short range r−6 attractive potential, plus a short-range cut oﬀ. The square

well potential has a depth V0 and range r0, i.e.

~2κ2 V (r) = 0, r > r0; V (r) = V0 , r < r0. (2.36) − ≡ − 2mr

2 2 Solving the Schrödinger’sequation for total energy E = ~ k /2mr state by connecting the wave functions in two regions at r = r0, we find the scattering length rescaled by r0 varies as a function of κ: a tan κr s = 0 1, (2.37) r0 κr0 − and the effective range as

r∗ 1 (κr )2 = 1 0 . (2.38) r − 2κr (κr tan κr ) − 3(κr tan κr )2 0 0 0 − 0 0 − 0

As we plot these in ﬁgure 2.2, we can see there are some divergence in as when κr0 reaches some certain values. These values are referred to as resonances, at which a zero energy

bound state appears. The ﬁrst resonance for square well potential appears at κr0 = π/2. At these resonances, a jumps from to + , but (a )−1 changes continuously. s −∞ ∞ s For this scattering model, at low energy the inverse scattering length 2.21 contains two parts: a momentum independent constant as a−1 and a k-dependent term. In this − s

15 2 0 r * r 0 0 r s a - -2

-4 0 Π 2 Π 3 Π

Κ r0

∗ Figure 2.2: A sketch of the s-wave scattering length the as and the eﬀective range r rescaled by the width of the potential r0 for a square-well potential, as a function of the dimensionless potential depth κr0. The blue curve is the scattering length, and it diverges at the resonances and changes abruptly close to them. The dashed red curve is the eﬀective range: it remains the same order as r0, except for some very narrow region close to where as vanishes.

expression, the leading term of momentum is ik, and the next quadratic term is r∗k2/2. − For low energy scattering k (r∗)−1, such that r∗k2/2 ik , one can simplify the | | | − | inverse scattering length by f −1 = a−1 ik. Now in the simpliﬁed model, we have only one − s − relevant parameter: the scattering length. This model is thus universal, to the extent that

the details of the interaction potential are irrelevant, as long as the scattering amplitude is

correctly given at the energy scale we are interested in. Within this spirit, the original van

der Waals potential can be eventually simpliﬁed to a “zero-range potential”, in which the

potential vanishes for any ﬁnite distance r > 0. This can be understood as the following:

we adjust the potential depth and range together, such that the wave function outside the

original potential remains unchanged. The limit that the range of potential goes to zero

can be expressed by that the wave function is extended to r 0 in this form: → sin(kr ka ) 1 1 ψ(r) − s , r 0. (2.39) ∼ kr ∼ r − as →

The scattering model is thus equivalent to a boundary condition at r 0, which is called →

16 Bethe-Peierls boundary condition. From this boundary condition, Fermi proposed a pseudo potential in the form 2 2π~ as ∂ Vpp = δ(r) r. (2.40) mr ∂r The wave function 2.39 is a solution to a hamiltonian with the interaction term written as

2.40. Although it looks similar to the mean-ﬁeld interacting model in the weakly-interacting regime 2.26, this form of interaction is beyond the mean-ﬁeld level and can be used in both weak and strong interactions, namely the value of as can be any value, even much larger than the interparticle spacing.

In the ﬁnal part of this section we introduce the approach to regularize the zero-range model in momentum space, in the context of the T -matrix formalism. Recall the relation between T -matrix and interaction operator V as 2.7 and 2.8, and for zero-range potential, the Fourier transform of the Dirac-delta function gives a constant interacting potential in momentum space, deﬁned as the bare coupling constantg ¯. The order-by-order expansion

of the T -matrix will have an ultraviolet divergence, since

1 G (E = 0) dpp2 (2.41) 0 ∼ − p2 → −∞ p X Z is linearly divergent in the absence of a high-energy cutoﬀ. This sickness of the bare delta potential should be ﬁxed by the following procedure. The zero-energy scattering matrix is

1 T (k, k0) =g ¯ g¯ T (p, k0), (2.42) − p p X 2 2 where p = ~ p /2mr. In the low-energy limit, we approximate the T -matrix for any energy 2 and equal incoming and outgoing momenta as TE=0(0, 0) = 2π~ as/mr T (0; 0, 0). The ≡ equation above is rewritten as

1 1 1 m 1 = = r . (2.43) g¯ T (0; 0, 0) − 2π 2a − p p ~ s p p X X It gives the relation between the bare coupling constant g and physical observable T -matrix, and this procedure is actually a regularization of the scattering vertex. For a calculation based on the zero-range model on any physical quantities, theg ¯ appearing in ﬁnal expres-

17 sions should be linked to the physical parameter as according to the formula in 2.43, in order to get a convergent result. In the later chapters, we will show how this regularization works to ﬁx the ultraviolet divergence in many-body calculations starting from this zero-range model in the context of the BCS-BEC crossover.

2.2 Feshbach resonances

Most of the scattering channels between alkali atoms have a scattering length around the order of 100a where a 0.53A˚ is the Bohr radius. Few can have scattering lengths B B ≈ as large as 2000aB. As the most dense stable dilute gases have typical density as large as 1015cm−3, the characteristic scale of the system is given by k a < 1. And as the | F s| discussion above, it is quite far from the most interesting strongly interacting regime.

The Feshbach resonance approach is implemented to tune the interaction strength, usually by making use of a bound state of pairs in different hyperfine channel. This bound state is referred to as the “closed channel”, in contrary to the scattering states which is called the “open channel”. As there is a coupling between the closed and open channel, a divergent scattering length in the open channel appears when the bound-state energy level in the closed channel matches the scattering threshold of the open channel. For most of the Feshbach resonances in alkali atoms, one chooses a closed channel which has a different magnetic moment from the open channel denoted by ∆µ. The energy difference between the two channels can thus be tuned by an external magnetic field.

We illustrate how Feshbach resonances work from a two-channel model with one open channel and one closed channel. We set the zero point of energy level as the scattering threshold of the open channel, i.e. the interaction energy vanishes at large distance in the open channel. Due to different magnetic moment, the scattering threshold between the two channels have a difference ∆µB, where B is the external magnetic field. We denote the energy difference between the bound state in the closed channel and the open channel threshold as the detuning δ, as shown in figure 2.3. As we will see in the following derivations, a large scattering length appears in the open channel when the closed-channel

18 closed

open

Figure 2.3: A sketch of the potential for both the open and closed channels near a Feshbach resonance. By adjusting the external magnetic ﬁeld B, the energy diﬀerence ∆µB and the detuning δ can be tuned continuously.

bound state approaches the scattering threshold for the open channel, i.e. at small detuning

δ. The point where the scattering length diverges is the resonance of the scattering.

We separate the Hilbert space of the wave functions into two subspaces: the open and closed channel. The projection into the open channel is denoted by the P operator and into the closed channel by the Q operator. The full wave function can be written in

Ψ = Ψ + Ψ , (2.44) | i | P i | Qi and the Schr¨odinger’sequation is

HPP HPQ ΨP ΨP     = E   , (2.45) HQP HQQ ΨQ ΨQ             in which HPP = PHP is the hamiltonian projected into the open channel on both sides,

19 and similarly for the other three block hamiltonians. By doing some manipulations to the matrix equation, we have an eﬀective Schr¨odinger’sequation in the open channel manifold:

(E H H0 ) Ψ = 0, (2.46) − PP − PP | P i where

H0 = H (E H )−1H (2.47) PP PQ − QQ QP is the additional term from the coupling to the closed channel. If the energy level between the P and Q subspaces is well separated, namely E Ψ H Ψ , the equation above h Q| | Qi gives the second-order perturbation result. Then we divide the projected hamiltonian in

the open channel as two parts:

Heﬀ = H0 + U = H0 + U1 + U2, (2.48)

where H0 is the free kinetic energy for open channel, U1 is the interaction potential coming 0 from the open channel itself, and U2 = HPP is the closed-channel induced interaction potential. With this prescription, we may write down a simpliﬁed two channel model

† † † † † H = kckck + U1 c↑c↓c↓c↑ + αd c↑c↓ + h.c. + δd d, (2.49) Xk X where ck is the annihilation operator in the open channel, d is the operator for the closed channel molecules, and α is the coupling between the open and closed channels.

We make a simplest approximation of this two-channel model, where the background

interaction in the open channel vanishes as U1 = 0. Also if we apply the Born approximation T = U to the coupled-channel eﬀective interactions, we have the eﬀective scattering T -

matrix as α 2 α 2 TPP (0; 0, 0) = | | = | | , (2.50) δ ∆µ(B∞ B) − − where δ is the detuning illustrated in figure 2.3 and B∞ is the magnetic field at which the bound-state energy exactly matches the scattering threshold of the open channel. We can see in this simple approximation, the scattering matrix and the scattering length can be tuned by applying an external magnetic field. The scattering length can be tuned to

20 divergence in principle when the magnetic ﬁeld reaches a certain value, which we refer to as the resonance. The sign of the scattering length changes, and a bound state appears

−1 at the resonance. The inverse scattering length as changes continuously as we adjust the magnetic ﬁeld. This is similar to “digging” the square well potential, as the behavior of the scattering length here resembles what happens close to the resonances in Figure 2.2.

Beyond this approximation, we have to calculate the T -matrix more carefully. The spirit is to separate the “background” scattering matrix, which is deﬁned as the quantity in the absence of the closed channel, to the additional contribution that comes from the coupling between the open and closed channel. From the expression of T -matrix 2.8, ﬁrst we rewrite this expression as

−1 T = G0 GU, (2.51) where G = (E H U)−1 is the full Green’s function. And we further express the full − 0 − Green’s function by this Dyson equation

G = G1(1 + U2G), (2.52) in which G = (E H U )−1 is the propagator that includes the background interactions 1 − 0 − 1 in the open channel itself. Then 2.51 is transformed into

T = G−1G (1 + U G)U = G−1G U + G−1G U (1 + GU) T + T , (2.53) 0 1 2 0 1 1 0 1 2 ≡ 1 2 where T G−1G U is the scattering T -matrix in which the closed channel is absent, and 1 ≡ 0 1 1 the additional term from closed channel is T2:

T G−1G U (1 + GU) = G−1G U GG−1. (2.54) 2 ≡ 0 1 2 0 1 2 0

We call T1 the background contribution and T2 the resonance contribution. From this equation above, we can map out the scattering length by the usual relation T (0; 0, 0) =

2 2π~ as/mr. The background scattering length then is denoted by abg which is given by the

model where the closed channel is neglected. The resonance part given by the T2 matrix

21 sandwiched by two low energy state

mr 0 −1 −1 0 ares = k G0 G1U2GG0 k , k, k 0, (2.55) 2π~2 h | | i → where k is the incoming plane wave with momentum k in the open channel. When GG−1 | i 0 −1 acts on the Dirac ket to the right and G0 G1 on the bra to the left, they transform the pure plane-wave states in the open channel to a full wave function including the scattering waves generated by the U and U1 potentials. In principle, from the two-channel hamiltonian 2.49, one can calculate ares analytically.

As the full derivation of ares can be found in [22] and references therein, here we again −1 −1 make the assumption to the ﬁrst order of U2, by which we replace GG0 to G1G0 in 2.55. Within this approximation, we neglect the mixing of the closed channel in the real eigenstate of wave functions. If we make an even further approximation, that the background coupling

−1 between open channel U1 = 0 in 2.49, G1G0 = 1 and ares is given by

2 mr α ares = 2 | | , (2.56) 2π~ ∆µ(B∞ B) − which takes the same form as shown in 2.50. The full form of as, if we include back the background scattering length is thus

2 mr α ∆B as(B) = abg + 2 | | abg 1 + , (2.57) 2π~ abg ∆µ(B∞ B) ≡ B∞ B − − 2 in which ∆B = mr |α| is the distance (with sign) between the position where a diverges 2π~2 ∆µ s and vanishes, which is commonly referred to as the width of the resonance. The positive deﬁniteness of α 2 enforces | | abg∆B∆µ > 0. (2.58)

The expression in equation 2.57 for the scattering length across a Feshbach resonance

is widely veriﬁed in experiments, see reference [22] for details. However, the position of the

resonance and width is not exactly what is given from the approximation we used above,

mostly from the fact that if the closed-channel mixture is counted the positions of resonances

will be shifted. Nevertheless, the physical picture of implementing an additional channel to

22 induce large s-wave scattering length, and hence large interactions is well illustrated by the simple arguments in this section. The Feshbach resonance is an important tool to tune the interaction parameter and study the strongly interacting gases in cold atom experiments.

2.3 Probes in cold atoms experiments

In cold atom experiments, most of the physical observables are measured by approaches well developed in atomic, molecular, and optical (AMO) physics. Even for those targeting at the physical quantities of a many-body atomic system, the detection of them can be very diﬀerent from those in conventional condensed-matter systems. The most diﬀerent aspects between the probes are from the reason that the constituents of the many-body systems are quite distinct to each other: in solid state materials, most macroscopic quantum phenomena come from the electronic properties; while in cold atoms, they are mostly governed by the behavior of trapped atoms.

The most importance differences between atoms and electrons can be classified into two types: charge and mass. In most of the solid state materials, because the electrons can couple to external electro-magnetic fields, it enables us to measure a lot of transport properties. These measurements include longitudinal and Hall conductivities, as well as magnetization and susceptibility. The Fermi surface can also be detected by de Hass-van

Alphen eﬀect. Also because the fast movements of light electrons, some of the spectroscopies can be measured by some tunneling experiments. Other probes including angular-resolved photoemission spectroscopy, and other scattering (X-ray, neutron) are also well developed in electronic systems.

Things are diﬀerent in cold atomic gases, both in the continuum and in the lattices.

The neutral atoms normally do not couple to external electromagnetic ﬁelds, hence the transport behaviors of atomic gases are hard to measure. There is also lack to I-V probe of density of states in atomic tunneling. However, the advantage of cold-atom system is that the heavy atoms move relatively slowly, and as there are bunches of internal states for a single atom, they couple to photons in a controllable manner. The absorption and

23 reemission of photons are usually frequency dependent, from which we can select the species of atoms one would like to image. On the other hand, there are also probes of spectroscopy in cold atoms, but rather implements the transition between diﬀerent hyperﬁne states to see the shifts of transition in order to collect the informations of elementary excitations and the spectral functions in the many-body system. These spectroscopies are usually done by couplings to radio-frequency photons, thus are referred to as rf-spectroscopy.

In this section we will brieﬂy review what these measurements are capable of.

2.3.1 Direct Imaging

For alkali atoms, because of the possible transitions between different internal states, one can use the laser to beam to measure the absorption of the gas, and hence determine the local density of the quantum gas. For this absorption imaging, since the interaction between a certain energy photon and different hyperfine states are not identical, the imaging process can be “spin sensitive”, namely it is possible to map out the positions of different species of atoms. For making reasonable measurements, the requirement of a dense enough sample assures a strong enough signal to be separated from the background shot noise. Currently, there are several variations of the direct-imaging approach, and they focus on different aspects of physical quantities in atomic gases.

Time-of-Flight (TOF) imaging

The time-of-flight (TOF) approach aims to map out the momentum distribution of the original system. It is realized by suddenly turning off all the external confinement and potentials, and let the atomic gas expand freely for a long enough time7. The optimum expansion time has to be controlled to satisfy the following two conditions. On one hand, only in the long-time limit, the expanded density profile matches exactly the original momentum distribution. On the other hand, as a high enough column density is required for

7In principle, one should turn oﬀ the interaction between atoms as well to preserve the single-particle density matrix in momentum space. However, in most of the experiments, as the background scattering lengths between particles are usually very small if it is not in the Feshbach resonance region, the short-time evolution of the original system from scattering processes is negligible.

24 0.04 a b c d e

y OD

2 hk 0 x

Figure 2.4: Experimental figures from TOF expansion in Bose and Fermi gases in optical lattices. The upper panel shows a transition between the superfluid and Mott insulating phases (a to h), as the condensation peaks at reciprocal momenta vanishes. The lower panel shows the shapes of Fermi surfaces of a fermionic tight binding model, for different filling factors. The figures are from [23] and [24] respectively.

any absorption imaging, the expansion time cannot be too long since the diluteness of the

ﬁnite number of particles increases as the holding time for expansion.

There are two good examples of TOF expansion in the pioneering works on both Bose and Fermi systems. One is to locate the quantum phase transitions in Bose-Hubbard models between the Mott insulator and superﬂuid phases[23], while the other one measures the shape of the Fermi surface of a degenerate Fermi gas in a square lattice[24]. The TOF approach has been used widely in determining the momentum distribution, as well as to distinguish some diﬀerent phases in cold-atom experiments.

Density proﬁles in the trap

This is the most direct way to get the density distribution of a trapped quantum gas. In this scheme, a laser is shone from a certain direction to the atomic gas. It leaves a “shadow” which represents the column density, namely an integrated density of atoms along the

25 direction of the laser. A projected two (or one) dimensional density distribution n(x, y) or n(x) is collected using a CCD camera.

For quantum gases in a “smooth” harmonic trap, we first approximate the trapped system as a collection of locally well-defined bulk system, i.e. the thermodynamic quantities at each point in the trap is governed by a local chemical potential µ(r) = µ m(ω 0 − · 2 r) /2. The trap can be anisotropic so we use ω = (ωx, ωy, ωz) as a vector of trapping frequencies. In thermal equilibrium, the temperature T is uniform in the trap. The local density is thus given by n(r) = n(µ(r),T ), and this approximation is called the “local density approximation” (LDA). The LDA is usually a good approximation if it satisfies the following condition. At each point there is a length scale given by the variation of the local chemical potential k(r) = ∂µ(r) /µ. In the region where k(r) k (k is the local Fermi ∂r F F vector given by the density as n k3 ), at each point we can assume there is a set of local ∼ F thermodynamic quantities, i.e. the LDA holds.

At the LDA level, the density proﬁle of a trapped quantum gas contains a large amount

of information. The most important feature is that the equation of state of the system is

completely given by a single shot of the density proﬁle in the trap, if the trap is quasi-two-

dimensional or one-dimensional. Related to other thermodynamic quantities, for instance

the pressure of the system can be deduced by an integral of the local density, namely by

using the relation µ P (µ, T ) = dµ0n(µ0,T ). (2.59) Z−∞ The variation of the chemical potential µ is well determined, provided one has a precise

control of the trapping frequency ω. The variation in the chemical potential can thus be

transformed to a variation in real space dr.

In the case of a three dimensional trap, the incapability of 3D resolution leaves us only

the integrated column density. However in the special situation of a axisymmetric trap

(we assume in y-z plane such that ωy = ωz), one considers the integrated one dimensional density n(x) dydzn(x, y, z) = dρ2n(x, ρ) P (x), where ρ = y2 + z2. P (x) is ≡ ∼ actually the pressureR as a function ofR x at the center of the “tube” y =p 0, z = 0. With this

26 prescription, one can gather the information of a lot of useful thermodynamic quantities in the trapped systems[25].

High-resolution in-situ imaging

Recently as the development of high-resolution imaging technique, it is possible to detect occupations of atoms at a single-site level in optical lattices. This kind of approach is capable of focusing on the spacial distribution of lattice gases, and detecting some correlations of the system to the finest level in the lattice model by taking pictures. Unlike the previously discussed imaging of bulk gases in traps, this single-site imaging is by scattering fluorescence light, namely the photons are absorbed and reemitted by the atoms before finally being collected by detectors. It is different from the previous gathering of information from the shape of shadows.

Currently this technique has been well illustrated in detecting the phases of Bose Hub- bard models[26, 27]. Reports on manipulation of atoms have also been presented by the same groups. It has been identified that there is a coexistence of four different phases in a single trap of optical lattice. Two superfluid plus two Mott-insulating phases are arranged in different regions with corresponding local chemical potentials[26]. Also by taking large enough number of pictures at identical copies of the system, it is conceivable that some correlation functions can be mapped out. Further, the fluctuations in the samples can be used to determine the entropy.

An even more ambitious proposal in high-resolution imaging is to get the distribution in the sub-lattice level. Actually, this is done previously by implementing an electron beam to ionize the atoms in the tight trap, such that they ﬂy to the detectors in the external electric ﬁeld[28]. This technique will be discussed in the later chapters in this thesis, and is very useful for verifying our theoretical predictions.

2.3.2 Spectroscopy

The spectroscopy measurements, mostly aiming to get the spectral function A(k, ω) of the quantum gases, are implemented by the response to radio-frequency electro-magnetic waves.

27 The reason that the incoming probe is in radiofrequencies is that the hyperﬁne splitting of alkali atoms are typically around megaHertz.

The scheme of rf spectroscopy is briefly described in the following. Take an example of a two-component Fermi gas, with the hyperfine states labeled as 1 and 2 . There is | i | i another unpopulated hyperfine state labeled 3 . In the large-magnetic-field limit, these | i three states are the lowest energy states for this fermionic isotope. The energy difference between atomic states 3 and 2 is denoted by ω . By applying an rf field, state 2 is | i | i 0 | i connected to the originally unoccupied state 3 . As the atoms typically have momentum | i much larger than the rf photons, the selection rule implies that the momentum of 2 and | i 3 before and after the transition are the same, and energy is changed by the rf photon | i energy ω. In the experiments, the incoming photon frequency is swept in the region around

ω . For every incoming energy ω, the population of final state 3 after certain holding time 0 | i is counted. The first rf measurement can only resolve the energy of final states ωf , i.e. the signal is proportional to n(ωf ). Later, angular-resolved rf spectroscopy was realized by the JILA group as well[29], which provides the information on the population with momentum k and energy ωf as n3(k, ωf ). This is a counterpart of the angular-resolved photoemmision spectroscopy (ARPES) in electronic systems.

From Fermi’s golden rule, the number of particles transferred to 3 is proportional to | i the transition rate and to the original spectral function of A(k, ω (ω ω ))8, where ω is f − − 0 the rf frequency. By sweeping through ω, one collects the spectral function of the original

2 state. For a non-interacting system, one expects the ﬁnal signal as a delta function of | i δ(ω ω ), namely the original system has a z = 1 quasiparticle weight, and the dispersion − 0 relation is the same with the ﬁnal 3 . For an interacting system, the spectral function | i is broadened naturally. Also there is a collective shift from ω0, with its sign depends on the attractive or repulsive nature of the interactions. The rf spectroscopy thus serves as a method of measuring the interaction energy of the quantum gases9.

8Here we make the assumption that the final state |3i is almost non-interacting, such that there is no 2 “spectrum shift” caused by interactions in |3i, namely within the approximation that ωf (k) ≡ k /2m. The different momentum states are thus with the same energy reference. 9For weak interactions, as the broadening of the spectral function is typically small, the well-defined peak usually has a well-defined position, hence a shift from the non-interacting position ω = ω0 can be clearly

28 Apart from the interaction energy measurement, rf spectroscopy has also been used in identifying the onset of the superﬂuid gap in the Fermi gases across the BCS-BEC crossover[30, 31]. The absence of some loss peaks corresponds to the pairings of the particles, by which the single particle transition is blocked.

All the arguments in this section have been handwaving, but more detailed and rigorous derivations can be found in reference [32].

observed. However in the strongly interacting regime, the wide broadened distribution in energy makes it very hard to integrate over the frequency domain to get the interaction energy. 29 Chapter 3 Strongly interacting quantum gases across Feshbach resonances

One of the most interesting and important aspects of quantum gases is their strong interactions. It is known for a two-component ultracold Fermi gas, the superﬂuid transition temperature can be as high as almost one ﬁfth of its Fermi temperature, when being brought close to a Feshbach resonance. This ratio of Tc/TF is the largest so far, among all the many body systems that have been studied1. In the past, there have been a lot of studies on quan-

tum gases with large scattering lengths, i.e. the “unitary region”. Most of them are based

on the model of BCS-BEC crossover, in which the superﬂuid states in a two-component

Fermi gas changes gradually from a BCS type of pairing state to a bosonic condensate

consisting of tightly-bound dimers. Within this model, the large Tc in the unitary region has been predicted[33].

The search for more examples of a high Tc/TF ratio is still one of the forefront topics in ultracold atomic gases. It is conceptually important to understand the origins of strong interactions, and it could possibly shed light on the mechanism for high-temperature superconductivity. In quantum gases, the phase shifts from two-body scattering are usually considered to be essential to strong interactions. It is useful to look for more examples of special phase-shift structures or resonances which can lock up large amounts of interaction

1 High Tc superconductors, for instance cuprates, have a typical Tc/TF ratio one or two orders of magnitude smaller than the unitary Fermi gas.

30 energy in the system. Later in this chapter, we will discuss recent studies of a Fermi gas with large interaction energy across the narrow Feshbach resonance[34]. On the other hand, another direction in searching for high Tc superfluids may be opened up, by enhancing the low-energy density of states. Recent theoretical works indicates that a largely increased superfluid transition temperature Tc appears in spin-orbit-coupled Fermi gases across the unitary region for this reason[35, 36]. More generally, any approach that modifies the density of states, could be possibly change the transition temperature. For instance, loading the gases into a mixed geometry is one of the candidates to reach this goal.

The superﬂuid phases of the Fermi gases discussed above appear in the “equilibrium branch”, namely the thermodynamic state in which the two-body bound states are occupied.

There is also a metastable branch with eﬀective repulsive interactions between atoms. This branch (commonly referred to as the “upper branch”) is free of bound-state dimers, and has

ﬁnite lifetime governed by the rate of three-body recombination processes. These repulsive gases are also of great interest, including recent studies in Stoner ferromagnetism[37, 38].

However, there has been a lack of mathematical description to this many-body state in the upper branch. In this chapter, we will discuss an approach developed by Shenoy and

Ho[39] which uses a generalized ladder approximation to exclude the bound-state dimers in the upper branch. Based on this approach, we will discuss our recent studies on strongly- repulsive Bose gases in a temperature regime such that its lifetime is long enough to conduct reasonable measurements. Our recent study in two-dimensional Fermi gases with spin imbalance will be discussed in the last part in this chapter. This study is closely related to a recent Cambridge experiment looking at the attractive and repulsive branches of the polaron problem in two dimensions[40].

Not limited to these topics mentioned above, there have also been studies in interesting properties of unitary Fermi gases and other novel phases in ultracold atoms. For instance, close to unitarity where the scattering length between two atoms diverges, apart from its strong interaction, some universal behaviors in thermodynamic quantities are also of great interest[41]. There have also been studies of unconventional superﬂuidity in ultracold atoms, which have emerged from the special resonance structures and tunable scattering properties

31 between atoms. For instance, related to quantum computations, some protocols for building quantum bits have been proposed in p-wave paired fermions[42]. They are suggested to be realized in p-wave-resonance-induced p-wave superﬂuids of atomic gases[43]. Furthermore, other types of novel states regarding time-reversal symmetry and topological properties are discussed in the context of cold atoms as well. However, these topics are outside our focus in this thesis.

The sections of this chapter are organized as following: we will ﬁrst go through some general introductions to strongly-interacting quantum gases and the concept of the BCS-

BEC crossover; then we discuss the strong interactions in Fermi gases across a narrow

Feshbach resonances, followed by the studies of the repulsive branch of Bose gases and two-dimensional Fermi gases with spin imbalance.

3.1 Introductions to strongly interacting quantum gases and the BCS-BEC crossover

Properties of the strongly interacting regime of Fermi gases where the interaction cannot

be treated at the perturbative level have been an important and interesting topic to study.

A class of problems on Fermi gases with large attractive interaction between particles were

ﬁrst raised by Eagles in the context of superconductivity in metals with a very low electron

density, where the attraction between electrons was no longer small compared with the

Fermi energy[44]. Later studies by Leggett on the problem of the BCS-BEC crossover in

Helium 3 address the similar issue in a diﬀerent context[10]. In this BCS-BEC crossover

scheme, it pushes further the BCS theory to a strongly-attractive regime to see the simi-

larity of a Cooper-pair condensate to a BEC of diatomic molecules, even though He-3 is

very much in the BCS limit. The BCS-BEC crossover model has been studied extensively

since then, and strong interaction energy together with high transition temperature for su-

perﬂuidity have been predicted in certain situations[45]. In the BCS-BEC crossover model,

it is relatively well understood that the superﬂuid states in a two-component Fermi gas

undergo a continuous crossover when the inverse interspecies s-wave scattering length is

32 gradually tuned, i.e. the value (k a )−1 changes from to + where k is the Fermi F s −∞ ∞ F wavenumber of the system. The unitary region, where the absolute value of the scattering length is much larger than the inter particle spacing, connects two well-known limits: the

Bardeen-Cooper-Schrieffer (BCS) regime where weakly interacting fermions form “many- body” Cooper pairs whose average size is much larger than the interparticle spacing, and the Bose-Einstein condensate (BEC) regime that every two fermions form deep bound-state dimers with size comparable to the scattering length as. These two limits correspond to k a 0− and k a 0+ respectively. F s → F s → Since the realization of quantum degeneracy of dilute Fermi gases, there have been a lot of studies in the past decades[33, 46] both theoretically and experimentally on the properties of cold atomic Fermi gases across unitarity. This is a very well demonstrated example of using cold atomic gases to study interesting physical problems. The Feshbach resonance provides a tool to adjust the inverse scattering length continuously, throughout the unitary region where the interaction strength is large. This is actually exactly the situation addressed in the original BCS-BEC crossover problem. The presence of the superfluid phases has been experimentally identified in such systems from direct observations of the superfluid transition of Fermi gases across the Feshbach resonance, especially in the unitary region[47, 48].

Another related interesting topic is the nature of strong repulsions of Fermi (and Bose) gases across unitarity. Unlike the attractive “bound-state branch” discussed above, the repulsive branch of the quantum gas is metastable, and it is realized by preparing the system in a state orthogonal to the true ground state. The nature of repulsive dilute gases is only well understood in the limiting case of k a 0+, especially in the Bose condensates[49]. F s → The concept of the repulsive branch (or “upper branch” which is more commonly used in the literatures) is somehow ambiguous when the system moves close to resonance. The lack of theoretical description also aﬀects the interpretation of experimental observations, for instance in some ambitious eﬀorts in studying Stoner ferromagnetism[37].

In this section, we will provide a brief introduction to the high-temperature superﬂuid phases in the BCS-BEC crossover model, including a derivation of the widely-used ladder

33 approximation treating the many-body problem. After this, we focus on a newly developed mathematical approach to describe the repulsive branch of dilute quantum gases, by excluding the bound states in the system[39].

3.1.1 Superﬂuidity across BCS-BEC crossover

Superfluid phases emerge in two-component Fermi gases with attractive interactions at low temperatures. In cold atomic gases, this attractive strength can be continuously tuned by implementing the Feshbach resonance approach. By adjusting an external magnetic field, the interaction strength is modified, reflected as a continuous change in the interaction parameter η = (k a )−1. For most of the Feshbach resonances, by increasing the magnitude − F s of magnetic field, the system can be tuned from the BEC (η ) to BCS (η + ) → −∞ → ∞ limit. Some of them go in the opposite direction[22].

The model hamiltonian we start from in this section is (we set ~ = 1 throughout this section)

1 † † † † H = dr ψ (r) ψ (r) µψ (r)ψ (r) +g drψ (r)ψ (r)ψ↓(r)ψ↑(r), (3.1) 2m∇ σ · ∇ σ − σ σ ↑ ↓ σ X Z Z where ψ is the ﬁeld operator for diﬀerent spin states with σ = , in two-component Fermi σ ↑ ↓ gases. µ is the chemical potential for both species, assuming the population of up and down spins are equal, and g is the bare coupling constant of a contact (short-range) interaction.

The bare coupling constant g has to be regularized later in the calculations in order to relate it to a real physical quantity, which in this case chosen to be the s-wave scattering length as. With this model hamiltonian, we will show its outcomes of describing the Fermi gases in both the BCS and BEC limits, as well as in the unitary region, by adjusting a single parameter: the coupling constant g.

We will first focus on the zero-temperature calculation of the crossover. The approach we use here is in a non-conserved particle number regime, namely we always do the calculations in the grand canonical ensemble by introducing the chemical potential. The conserved number of particles approach is also well discussed[10]. The BCS ground-state wave function first introduced by Bardeen, Cooper and Schrieffer in the grand canonical ensemble is

34 written in the form2

Ψ = (u + v c† c† ) 0 , (3.2) | gi k k k↑ −k↓ | i Yk where 0 is the vacuum. All the physical quantities of the ground state can be determined | i by this trial wave function. The mean-ﬁeld solution of hamiltonian 3.1 determines the

parameters uk and vk as the following:

1 ξ u2 = 1 + k , (3.3) k 2 E k 1 ξ v2 = 1 k , (3.4) k 2 − E k where ξ = k2/2m µ and E = ξ2 + ∆2 is the dispersion relation for single-particle k − k k excitations. The BCS pairing gap ∆q is the order parameter of the system, and is deﬁned as the mean-ﬁeld quantity of pairing term

g ∆ = Ψ c ↑c− ↓ Ψ , (3.5) h g|V k k | gi Xk in which V is the volume of the system. By using the explicit expression of BCS wave

function 3.2, the gap equation 3.5 is transformed into

g g ∆ ∆ = ukvk = . (3.6) −V −V 2Ek Xk Xk As the gap ∆ appears as a constant factor on both sides of the gap equation, it can be further simpliﬁed to the following:

1 1 1 = . (3.7) g −V 2Ek Xk From the discussions in the two-body scattering model, the high-energy divergence in the zero-range model is regularized by connecting the bare coupling constant to the scattering

2Here we only consider zero center-of-mass momentum pairing, which essentially dominates in equal spin systems. The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase[50, 51] is a BCS type of superfluid with finite center-of-mass pairing in systems with spin imbalance. There are experiments recently in cold atoms trying to verify the existence of this type of superfluid[52].

35 length as. The ﬁnal equation takes the form

1 m 1 1 1 1 = = , (3.8) g 4πas − V 2k −V 2Ek Xk Xk m 1 1 1 = , (3.9) 2πas V k − Ek Xk 2 where k = k /2m is the kinetic energy of a single fermion. The regularization of the bare interaction parameter in many body systems is exactly the same for two-particle scattering in this situation, since all the high-energy divergence is a consequence of the lack of a length scale for the short-range cutoﬀ of the potential. The short-range behavior of two particles coming close to each other is independent of the many-body medium of the system.

Two unknown quantities ∆ and chemical potential µ need to be solved by the gap equation 3.9, together with another “number equation”:

1 † † n = Ψ (c c ↑ + c c ↓) Ψ h g|V k↑ k k↓ k | gi k X 3 1 k µ kF = 1 − = 2 . (3.10) V − Ek 3π Xk The set of equations 3.9 and 3.10 together (we refer to as crossover equations) determine the order parameter ∆ and chemical potential µ, and consequently other physical quantities at zero temperature.

From the gap equation 3.9, we have no constraint in the value of scattering length as, namely this equation is valid for all value of (k a )−1 from BCS to BEC limit, even in − F s the unitary region. The two equations are well deﬁned at unitarity a , and the BCS s → ∞ type wave function 3.2 is continuously extended even to the BEC limit when every two

fermions form a deep molecule. The molecular bosonic gas becomes a condensate at low

temperatures. To show this continuation more clearly, we write down the pair wave function

in momentum space for molecules deep in the BEC side as f(k). The ﬁeld operator of this molecule with zero center-of-mass momentum is

† 1 † † d = fkc c . (3.11) 0 √V k↑ −k↓ Xk

36 In the grand canonical ensemble, a bosonic condensate can be described as a (unnormalized) coherent state:

BEC = exp(αd†) 0 , (3.12) | i 0 | i where α = √N is determined by the number of particles in the condensate. As ck’s are fermionic operators such that c2 = 0, it is straightforward to expand the exponential term to the following form:

BEC = exp(√nf c† c† ) 0 | i k k↑ −k↓ | i Yk = 1 + √nf c† c† 0 . (3.13) k k↑ −k↓ | i Yk By comparing the unnormalized BEC and BCS wave functions 3.2 and 3.13, we ﬁnd

they are essentially in the same form, and the relation between uk, vk and fk is

vk √nfk = . (3.14) uk

On the other hand, in the BCS formalism, ﬁrst we write down the pair correlation function, or the reduced two-body density matrix in the pairing channel, as

† † 0 0 ρ = c (r )c (r )c↓(r )c↑(r ) . (3.15) 2 h ↑ 1 ↓ 2 2 1 i

ρ2 is Hermitian and can be diagonalized with real eigenvalues:

0 0 ∗ 0 0 ρ2(r1, r2; r1, r2) = niχi(r1, r2)χi (r1, r2). (3.16) Xi The condensed phase is characterized by a large eigenvalue of ρ2 in the same order of number of particles N, and the eﬀective pair wave function of the condensate is given

by the corresponding eigenfunction. For a translational invariant system, this condensate

eigenfunction χ(r , r ) = χ(r r ), and the Fourier transform of χ(r) is deﬁned as F (q = 1 2 1 − 2 0, k), where q is the center of mass momentum. From the BCS wave function 3.2, F (q =

0, k) = ukvk is the eﬀective BCS pair wave function. As we can see from 3.3, at large momentum region in the limit where ξ ∆, there is an asymptotic form of u 1. k k ∼ Consequently, at short range, the pair wave functions in the BEC and BCS limits fk = vk/uk

37 Figure 3.1: A sketch of the pairing gap and the chemical potential for a Fermi gas across unitarity at T = 0. The chemical potential stays around the Fermi energy EF deep in BCS 1 side, and asymptotically approaches 2 in the BEC side. The order parameter ∆ is − 2mas exponentially small in the weakly attractive region. Adapted from reference [54].

and Fk = vkuk are essentially equivalent. This resemblance is natural since when it comes to the short-range (high-momentum), only the two-body physics is relevant and it does not matter if the pair is a large many-body dimer (BCS) or a small deep molecule (BEC).

From the discussions above, we see that there is a continuous crossover between the su- perﬂuidity in the BCS and BEC limits. The gap order parameter ∆, the chemical potential

µ, as well as the pair wave function all change continuously across the resonance region. The size of the pairs evolve gradually from large BCS Cooper pairs to small bosonic molecules.

The overall behavior of these quantities at zero temperature is solved analytically in certain limits[53], and is sketched in ﬁgure 3.1. Here we brieﬂy derive the limiting cases on both ends in the following.

38 BCS limit: (k a ) 0− F s → In the weakly attractive limit, the crossover equations 3.9 and 3.10 can be simpliﬁed by assuming the chemical potential is the same as in the non-interacting systems µ = EF , where

EF is the Fermi energy of the system determined by density n. Within this approximation, there is only one gap equation to be solved, and the pairing gap takes the form

π ∆ = 8E exp . (3.17) F −2k a F | s| We can see the pairing gap scales as the Fermi energy, and is exponentially small in the BCS limit where π . In fact, this expression is analogous to that of the conventional 2kF |as| → ∞ phonon-induced superconductivity in electron gases, in which

1 ∆ ~ωD exp , (3.18) ∼ −N(0)V where ωD is the Deybe frequency, N(0) is the density of states at the Fermi surface, and V is the attraction strength between electrons induced by phonon. By comparing these two expressions, it also shows that in weakly interacting limit, the coupling strength between atoms is proportional to the s-wave scattering length as.

BEC limit: (k a ) 0+ F s → From the solution of the two-body scattering problem, it is known that in the BEC limit where kF as is a small positive number, there is a deep bound state formed by two fermions 1 with binding energy Eb = 2 . The size of these bound molecules is in the order of mas as, much less then the interparticle spacing, hence the system is essentially a Bose gas of dimers. For the crossover equations in this limit, we can make the approximation that

2 2 µ Eb/2 = ~ /(2ma ). This is a reasonable approximation since the chemical potential ∼ − − s is the energy change of adding a particle to the system, and in this case it should be half the energy of the bound state. Within this approximation, by substituting µ = E /2 to − b

39 the number equation 3.10 we have

4EF −1/2 ∆ = EF (kF as) . (3.19) 3πkF as ∼

For a system with ﬁxed density, the gap changes as the square root of the inverse scattering length in the deep BEC side. And the ratio between ∆ and µ is

∆ 4 3/2 = (kF as) 1 (3.20) µ √3π | | in the BEC limit, consistent with the assumption we made above. Also, the lowest quasi-

particle excitation energy Ek appears at k = 0, with value

2 2 Ek=0 = µ + ∆ , (3.21) p min instead of Ek = ∆ as the lowest energy excitation at the Fermi surface in the BCS limit. The important low energy ﬂuctuations in BEC limit are missing when we use the BCS mean

ﬁeld equations. We will see in the later sections that this is one of the important reasons

why the critical temperature does not scale as the pairing gap in the BEC limit, and why

the simple mean-ﬁeld approach does not give the correct critical temperature.

In the last part of this section, we discuss a more general and systematic way of writing down the gap equation and regularizing the ultraviolet divergence in the zero-range model. The gap equation is given by either the canonical transformation of the hamiltonian, or ﬁnding the saddle point after a Hubbard-Stratonovich transformation in the path integral formalism3. Generally, the BCS zero center-of-mass momentum gap equation at any temperature can be written in a matrix form

∆ = UˆKˆ ∆, (3.22)

in which Uˆ and Kˆ are the interaction operator (matrix) and the “scattering kernel” of the

system. To be more explicit, the equation above can be written in a form

1 0 0 ∆k = U(k, k )K(k )∆k0 . (3.23) −V 0 Xk 3In Appendix A, we will give a brief derivation of path integral formalism at ﬁnite temperatures.

40 For a translational invariant system, U(k, k0) = U(k k0) is the Fourier transform of U(r). − At ﬁnite temperature, the kernel is

1 2f(E ) tanh(βE /2) K(k) = − k = k . (3.24) 2Ek 2Ek

Here, f() = (1 + exp(β( µ)))−1 is the Fermi function with β = (k T )−1 being the − B 2 2 inverse temperature, Ek = ξk + ∆k is the quasiparticle excitation energy. As we can see, q for the zero-range model, Uk is a constant for all momentum, thus the right hand side of equation 3.23 is ultraviolet divergent. To regularize this divergence in a systematic way, we rewrite this gap equation in terms of the T -matrix at low energy instead of the bare coupling parameter U. The relation between the bare potential and scattering matrix is

2 2 −1 T = U + UG0T , where G0 = (~ k /2m) is the free Green’s function at zero energy. − This is essentially the Lippmann-Schwinger equation for two-body scattering. From the gap equation 3.23, we do the following manipulations:

∆ = UK∆ − ∆ UG ∆ = UK∆ UG ∆ − 0 − − 0 (1 UG )∆ = U(K + G )∆ − 0 − 0 ∆ = (1 UG )−1U(K + G )∆ = T (K + G )∆ (3.25) − − 0 0 − 0

The last equation above is the regularized gap equation implementing the scattering matrix,

and is more explicitly written as

1 0 tanh(βEk0 /2) m ∆k = T (k, k ) ∆k0 . (3.26) 0 0 2 −V 0 2Ek − (k ) Xk As in the large momentum limit, E k2/2, the ultraviolet divergence is cancelled on the k ∼ right-hand side in this version of the gap equation. Again, the reason we can do this is that

the high-energy behavior of scattering corresponds to the short-range correlation between

fermions, and thus is completely governed by two-body physics.

41 3.1.2 Critical temperatures and ladder approximation in dilute quantum gases

In the previous section we discussed the mean-field solution to the BCS-BEC crossover problem at zero temperature. In this formalism, fluctuations around the saddle point are not included. In the calculations of the critical temperature, especially in the BEC region, the absence of fluctuations gives a very high Tc, far beyond the estimate for a weakly interacting Bose gases given by its density. In this section, we will consider a more accurate approach to treating many-body problems in the dilute gases, including the calculations of the transition temperature Tc and other thermodynamic quantities at finite temperatures. We generalize a little bit the interaction hamiltonian from the original contact interaction, where the coupling constant g is independent of the momentum transfer. In principle, this interaction potential is momentum dependent as Uk. However, in the Van der Waals type of interaction in dilute gases, the interparticle spacing is much larger than the range of the interaction, i.e. k r 1, where r is the potential range. In this limit, we make the F 0 0 approximation that Uk is a step function, which is constant until some momentum cutoff Λ r−1. The dilute limit assures that k Λ. ∼ 0 F First we calculate the critical temperature of the system across the resonance. This Tc is determined by the temperature that the pairing gap ∆ vanishes in the crossover equations.

After we use the gap equation to connect Tc and µ, the number equation which relates the density of the gas to other variables further ﬁxes the chemical potential and gives the absolute value of Tc. As we stated before, the simple canonical transformation at the mean-

field saddle point approach gives extremely high Tc value in the BEC side (as shown later in figure 3.5). The most obvious reason is that in the mean-field number equation on the BEC side, the lack of consideration on fluctuations underestimated the number of particles, hence gives a much higher estimate in µ (and Tc). The underestimation is because the low-energy excitations of the BEC superfluid are finite center-of-mass momentum molecules, instead of the breaking of pairs as on the BCS side.

Now we consider the case T T , and calculate the grand thermodynamic potential Ω. ≥ c

42 Figure 3.2: All the closed diagrams included in the ladder approximation. All the legs of the ladder, i.e. the propagators on both ends of the interaction lines, run in the same direction. The grand potential is a sum over all this class of closed diagrams, with any numbers of interaction lines.

In the grand canonical ensemble, Ω = Tre−β(H−µN). In the absence of broken symmetry, we choose all the signiﬁcant closed diagrams into Ω. In the ladder approximation, the class of diagrams included are shown in the ﬁgure 3.2. This is a collection of diagrams with iterations of two-particle scatterings. Each scattering vertex, together with the pair of propagators on both ends, forms the rung of a ladder.

In the ladder approximation, we see that for each scattering vertex, only the particle- particle (hole-hole) channel, i.e. the pair of propagators traveling in the same direction are preserved. The particle-hole channel, namely the scattering vertex with counter-propagating lines are neglected. These two channels are shown in ﬁgure 3.3 (a) and (b) respectively. In the dilute limit we discussed above, where the coupling is approximated by a step function with a hard cutoﬀ in momentum space at Λ r−1 k , we evaluate the integration of ∼ 0 F these two channels:

(a) d4q U(q)G (p + q)G (p q), (3.27) ∼ 0 1 0 2 − ZΛ (b) d4q U(q)G (p + q)G (p + q), (3.28) ∼ 0 1 0 2 ZΛ 43 p1 + q p1 + q p1 p1 q q p2 p2 p q p + q 2 2 (a) (b)

Figure 3.3: The scattering vertex for the particle-particle channel (a) and for the particle- hole channel (b) These two pieces diﬀer by the directions of propagators between the two legs. In the ladder approximation for dilute gases, (a) is retained and (b) is always omitted.

in which G0 is the non-interacting Green’s function, and the integrals are done in 4- dimensional space-time. The sign of q is the only diﬀerence between the two expressions.

After doing the Matsubara sum in the imaginary time domain, the ﬁnite temperature results of the particle-particle (a) and particle-hole (b) channels are

(a) (1 f(p + q))(1 f(p q)) + f(p + q)f(p q), (3.29) ∼ − 1 − 2 − 1 2 − qX<Λ (b) (1 f(p + q))f(p + q) + f(p + q)(1 f(p + q)), (3.30) ∼ − 1 2 1 − 2 qX<Λ where f(p) is the Fermi function for the momentum p state:

2 f(p) = 1/(e(p /2m−µ)/T 1). (3.31) −

In the low-density regime, the (a) terms are much larger than the contribution from

(b) terms at all temperatures for the following reasons. At zero temperature (and similar arguments apply for any low temperatures in the quantum degenerate regime), the Fermi function f is only nonvanishing when its momentum is inside the Fermi sea. For ﬁxed p1 and p inside the Fermi surface, the ﬁrst (1 f)(1 f) term in (a) remains close to unity 2 − − in a large region when kF < q < Λ. As for the particle-hole channel (b), as both terms are proportional to f, they can only contribute when q < kF . As a result, since the phase space

44 in which (a) is nonvanishing is much larger than in which (b) is nonvanishing, we conclude

that the particle-particle (hole-hole) channel is dominant over the particle-hole channel.

Similar argument holds for the high temperature regime where that the Fermi function f

is always much smaller than 1. Since (1 f)(1 f) in (a) is the only term that is on the − − order of unity, the particle-particle channel always dominates. We can ﬁnally be convinced

to drop the particle-hole channel and use the ladder approximation in dilute gases at both

low and high temperatures.

With the prescriptions of the ladder approximation, we start to calculate the grand

potential of the crossover model. To calculate the grand potential per volume, we separate

out the contribution from free fermions and the interacting part:

Ω = Ω0 + Ωint, (3.32)

where Ω0 is the contribution from the non-interacting fermions and Ωint is the interacting part. For two-component fermions, the non-interacting part is (from now on, we use units

in which the Boltzmann constant dimensionless is kB = 1)

2T 2 Ω = ln 1 + e−(p /2m−µ)/T . (3.33) 0 − V p X The interacting part is calculated by imagining that the interaction potential is multi- plied by a factor λ. The thermodynamic potential changes as this artiﬁcial factor λ changes, and we know that Ωint vanishes when λ = 0. From the Hellmann-Feynman theorem, the grand potential is an integral 1 dλ Ω = λUˆ , (3.34) int λ h i Z0 where Uˆ is the interaction operator. In the ladders, each rung of them, as sketched in (a)

of ﬁgure 3.3, gives a “polarization”4 Π deﬁned by

1 1 f(q/2 + k) f(q/2 k) Π(q, zm) = − − − , (3.35) V zm ξq/2+k ξq/2−k Xk − − where ξ = p2/2m µ, f is the Fermi function, and z are the bosonic Matsubara frequencies p − m 4Here we borrow the usage of polarization, by its analogy to the iterative bubble diagrams widely used in electronic gases.

45 2imπ zm = β , with integers m. We consider the simplest contact interaction approximation, where U(q) can be reduced to a constant coupling constant g independent of the transferred momentum q. The total interaction is a sum over all diﬀerent orders of (gΠ)n where n = 1, 2, ..., as they correspond to ladders with diﬀerent numbers of interaction lines. Hence the expectation value of λUˆ is given by

T λgΠ(q, z ) λUˆ = m . (3.36) h i −V 1 λgΠ(q, z ) q,z m Xm − By substituting the explicit form of λUˆ into equation 3.34, we have h i T Ω = ln (1 gΠ(q, z )) . (3.37) int V − m q,z Xm In the thermodynamic limit, we write down the sum over momentum q as an integral:

dq Ω = T ln (1 gΠ(q, z )) . (3.38) int (2π)3 − m z Xm Z The Matsubara sum can be carried out by the standard approach of analytical continuing the imaginary frequencies to the whole complex ω-plane, and multiplying the summand by a Bose function which is singular at all the bosonic (even) Matsubara frequencies:

1 N (ω) = . (3.39) b eβω 1 −

This function has simple poles at all zm, with residue β. The Matsubara sum can thus be expressed as a contour integral along the imaginary axis of the complex ω-plane. The contour can also be deformed, as long as it does not pass through other singularities. The integrand here has a branch cut on the real axis and is completely analytic except on the axis, and the contribution from the ω = R circle vanishes. The Matsubara sum in | | → ∞ equation 3.38 can thus be reduced to a pair of integrals inﬁnitesimally close to the real axis in the upper and lower plane, as is shown in the sketch 3.4:

dq +∞ dω Ω = N (ω) ln[1 gΠ(q, ω + i0+)] ln[1 gΠ(q, ω i0+)] . (3.40) int (2π)3 2π b − − − − Z Z−∞

46 Figure 3.4: A sketch of contour deformation of Matsubara sum in NSR formalism. The dashed line contour for sum of Matsubara frequencies is deformed into the solid contour C, and is decomposed to two integrals inﬁnitely close to the real axis in upper and lower plane. The contour is deformed this way since the branch cut of function ln Γ−1 is on the real axis.

By analyzing the integrand ln(1 gΠ), we ﬁnd the real part of it (logarithm of absolute − value of 1 gΠ) is continuous as ω passes through the real axis. Only the imaginary part − of the logarithm, i.e. the phase angle of 1 gΠ contributes to the contour integral. The − equation can thus be further reduced to the following form:

dq +∞ dω 1 Ω = N (ω)Im ln Π(q, ω + i0+) int (2π)3 π b − g Z Z−∞ dq +∞ dω = N (ω)Im ln Γ−1(q, ω + i0+), (3.41) (2π)3 π b Z Z−∞

47 where the argument Γ−1 of the logarithm is:

−1 + 1 1 f(q/2 + k) f(q/2 k) 1 Γ (q, ω + i0 ) = − + − − V ω + i0 ξq/2+k ξq/2−k − g Xk − − dk 1 f(q/2 + k) f(q/2 k) m m = − − − + (2π)3 ω + i0+ + 2µ k2/m q2/4m k2 − 4πa Z − − s dk 1 f(q/2 + k) f(q/2 k) m m = P − − − + (2π)3 ω + 2µ k2/m q2/4m k2 − 4πa Z − − s dk +i (1 f(q/2 + k) f(q/2 k))δ(ω + 2µ k2/m q2/4m). (2π)3 − − − − − Z (3.42)

The notation P indicates the Cauchy principal-value integration, and is valid in the presence

of an inﬁnitesimal imaginary part in the frequency ω. We used the identity

1 1 dx = P dx iδ(x A) (3.43) x A + i0+ x A − − Z − Z − in the last line of the derivation equation 3.42. For a set of pure values of ω without the infinitesimal imaginary part, the integral above is ill-defined. This set of points on the real axis form the branch cut of the integrand, which will be discussed later. The function Γ is actually the on-shell scattering T -matrix in the presence of the medium, with center of mass momentum q and scattering energy of relative motion ω + 2µ q2/4m. This is better − illustrated in the extremely dilute limit, i.e. we ignore the Fermi occupation by putting f(q/2 + k), f(q/2 k) 0 in the numerator. In this limit, the inverse scattering matrix is − ≈ simplified to

−1 + mk0 m Γ0 (q, ω + i0 ) = i , (3.44) − 4π − 4πas in which k = mω + 2mµ q2/4 is the wave vector of the relative motion. This is exactly 0 − in the same formp of the inverse scattering T -matrix (and scattering amplitude f) from the

Lippmann-Schwinger equation, as is shown in 2.21. The phase angle of Γ−1 is consequently

the opposite of the scattering phase shift. The ﬁnal form of the interacting grand potential

is dq +∞ dω Ω = N (ω)ζ(q, ω), (3.45) int − (2π)3 π b Z Z−∞ where ζ(q, ω) = ArgΓ(q, ω + i0+) is the scattering phase shift. With the expression for the

48 grand potential, we have the number equation:

∂Ω dq +∞ dω ∂ζ(q, ω) n = = n (µ, T ) + N (ω) , (3.46) − ∂µ 0 (2π)3 π b ∂µ Z Z−∞ where n0 is the number of non-interacting particles. The gap equation given by

Γ−1(q = 0, ω = i0+) = 0 (3.47) is the Thouless criterion for a superfluid transition5. This is equivalent to the BCS gap equation 3.9, since the addition of fluctuations in the grand thermodynamic potential should not change the position of the saddle point. This pair of equations 3.46 and 3.47 form the new crossover equations in the T T region. This scheme was first introduced by ≥ c Noziéresand Schmitt-Rink (NSR) to study the transition temperature for Fermi gases across unitarity[55]. In the above we use the diagrammatic approach to derive this formalism. In the appendix, we will show this is equivalent to including Gaussian fluctuations of pairing gap at T T in the functional integral approach. ≥ c A sketch of the transition temperature and chemical potential is shown in figure 3.5.

We can see the transition temperature is almost flat in the deep BEC limit, consistent with that determined by the density of composite bosons in the weakly interacting limit. The low-energy excitations in the deep BEC side are the finite center-of-mass momentum pairs and are included in the NSR diagrams in figure 3.2.

In the last past of this section, we discuss the difference between the BCS superfluids and BEC superfluids in the context of NSR formalism. Similar to the momentum position at the minimum of the excitation spectrum as is discussed in the previous section at zero temperature, the nature of pairing is different between positive and negative chemical potential µ.

First go back to the expression for Ωint, to the step of integrating over ω in 3.40. The branch cut of the function ln[1 gΠ] = ln Γ−1 is on the real axis for the ω. From the explicit − 5In general, Γ−1(q, ω = 0) = 0 for arbitrary q corresponds to a condensation of pairs at center-of-mass momentum q. However, for most of the systems, including the model we discuss in this chapter, when the system approaches Tc from higher temperatures, it can be rigorously proven that q = 0 pairs condense ﬁrst. After this condensation of these q = 0 pairs, the crossover equations will not be extended any further.

49 Figure 3.5: A sketch of the superfluid transition temperature and the chemical potential at Tc for a Fermi gas in the BCS-BEC crossover. Tc has a maximum close to unitarity, and the chemical potential remains positive into the positive scattering length side, until (k a )−1 F s ∼ 0.4. The dashed line in the Tc plot is the result from the mean-field approximation without fluctuations. Figure adapted from reference [56].

expression for the scattering matrix Γ(q, ω) in equation 3.42, we ﬁnd that there are two kinds of singularities for ln Γ−1 as ω approaches the real axis which make up the branch cut: the poles and the zeros of Γ−1. The poles appear when ω > 2µ + q2/4m ω(q), as the − ≡ integral 3.42 diverges for pure real ω. These poles are the “scattering continuum”, and they correspond to the positive-energy scattering states (extended states). On the other hand, the zero of Γ−1 appears at a certain value of ω (q) and ω (q) < 2µ + q2/4m. The zero of b b − Γ−1, or the pole of Γ corresponds to a bound state, with binding energy E = ω (q) ω(q). b b − The bound state does not necessarily exist for any set of 1/as, q, µ, T . The Thouless criterion 3.47 will thus be classiﬁed into two situations. For µ > 0, since

ω(q = 0) < 0, the pole of Γ(q = 0, ω = 0 + i0+) is a pole of positive scattering energy, which we refer to as the many-body Cooper pole of the scattering matrix. In contrast, when

µ < 0, the condensation happens in the true bound-state (negative-energy) channel. In the

50 deep BCS side where we approximate µ = EF , it is straightforward to show that the Cooper pole has a positive energy ω ω(q = 0) = 2µ. This means that the pairing indeed happens − around the Fermi surface. In the case of large negative chemical potential µ /T 1, the | | m solution to the bound-state pole gives the asymptotic binding energy ωb(q) ω(q) = 2 − − as as (k a )−1 . The point µ = 0 is hence important in that it separates Cooper F s → ∞ condensation and molecule condensation at the NSR level6.

3.1.3 The “upper branch” of the quantum gases

In the previous section, we discussed the NSR scheme for calculating the superﬂuid tran-

sition temperature and other thermodynamic quantities for the normal state at T > Tc. The thermodynamic state we focused on was the attractive Fermi gas, in which the total

energy is always lower than the non-interacting Fermi gas at the same temperature and

density. It is also very interesting to study the properties of the repulsive branch, or com-

monly referred to as the “upper branch” of the Fermi gas in the strongly interacting regime.

While the nature of weakly repulsive gases is well understood as a metastable state free of

deep underlying bound-state molecules, there is still a lack of a serious mathematical pre-

scription for describing the upper branch in the more intriguing strongly interacting regime

close to unitarity. There has been increasing interest in the theoretical study of this upper

branch[14, 57, 58] after the reports on both the existence and absence of Stoner ferromag-

netism in dilute quantum gases[37, 38]. In this section, we introduce one of the theoretical

approaches to exclude the molecules in thermodynamic functions and other thermodynamic

quantities developed by Shenoy and Ho[39].

As is shown in 3.41 and 3.46, the formulas for the grand potential and the number

of particles contain integrals over the frequency ω. From the argument in the last part of

section 3.1.2, the integration domain over ω in 3.41 and 3.46 can be reduced to a region from

ω (q) to + . In the absence of a molecule (bound-state) pole, ω (q) = ω(q) = q2/4m 2µ, c ∞ c − 6Although it is a continuous crossover between BCS and BEC superfluids in s-wave pairing, many studies address this µ = 0 point and other similar boundary point in related quantities as quantum critical points in other models. For instance in some p-wave resonance models, the superfluids on both sides have different symmetry[42, 43].

51 ⇣z(EE) ⇣¯(E) z E p E

H L p H L 2

- p E 2 E 0 | b| Figure 3.6: The phase shifts ζ(E) for scattering in the presence of the bound state. The phase shift is π in the region ωb(q) < ω < ω(q), and gradually decreases. This complete phase shift can be separated into a π-plateau from ωb which corresponds to a bound-state contribution, and a negative “scattering phase shift” as shown in the right ﬁgure starting from zero, which corresponds to the repulsive scattering-state contribution.

while ωc(q) = ωb(q) when the bound state appears. We now consider the situation in which the bound state exists, and rewrite the number equation as

dq +∞ dω ∂ζ(q, ω) n n = N (ω) − 0 (2π)3 π b ∂µ Z Zωb(q) dq ∂ω (q) dq +∞ dω ∂ζ¯(q, ω) = N (ω (q)) b + N (ω) − (2π)3 b b ∂µ (2π)3 π b ∂µ Z Z Zω(q) = nbd + nsc. (3.48)

We have separated the bound-state and the scattering-state contributions to the number of particles for the interaction part. Here ζ¯ = ζ π is the “modified phase shift” as − illustrated in figure 3.6. Since when a bound state emerges, the phase shift starts from π at the continuum threshold, we shift it down by π such that it starts from 0 to describe the scattering-state contribution. For the number equation, this modification is not necessary, as we only calculate its derivative with respect to the chemical potential; however, this shift of π is essential when we calculate other thermodynamic quantities, for instance the pressure of the system.

If we take a closer look at equation 3.48, the integrand for the bound-state contribution consists of the Bose distribution Nb at ﬁnite temperature of the bound state pole ωb(q), and is integrated over q to include all of these pairs for any center-of-mass momentum. In

52 the deep BEC side, we can show that the partial diﬀerentiation has an asymptotic form that ∂ω /∂µ 2, since ω = q2/4m 2µ E and E is essentially independent of b → − b − − | b| b other external parameters for deeply bound states. This is originated from the fact that the pairing of deeply bound molecules is governed by two-body physics, and is insensitive

bd dq to the medium. In this limit, we have a simplified n = 2 (2π)3 Nb(ωb(q)): two atoms per molecule times the Bose distribution function. For the systemR closer to unitarity, the expression is generally more complicated. The term ∂ω /∂µ deviates significantly from 2, b − coming from many-body effects taking place in the pairing.

This no-pole scheme can be identiﬁed as exact in the high-temperature regime. The expression for nbd and nsc are simpliﬁed in the high-temperature regime when the chemical potential is large and negative µ E ,T . For an expansion of the fugacity z to the | | | b| lowest (second) order:

3 dq 2 √2 nbd = 2z2eβ|Eb| e−βq /4m = 2z2 bbd, (3.49) (2π)3 λ 2 Z ! ∞ 3 dq 2 d ∂ζ() √2 nsc = z2 e−βq /4m e−β = 2z2 bsc, (3.50) (2π)3 π ∂µ λ 2 Z Z0 ! where λ = 2π/mT is the thermal de Broglie wave length, and b2’s are the second virial coeﬃcients p ∞ d ∂ζ() bbd = eβ|Eb|, bsc = e−β . (3.51) 2 2 2π ∂µ Z0 Since we use a universal zero-range model, there is only one possible bound state in the system. The separation of the bound-state and scattering-state contributions in equation

3.48 becomes the exact virial expansion result to second order in the fugacity.

The essence of this “no-pole-approximation” is trying to exclude the bound state in the Hilbert space. The spirit of the method discussed above is to change the integration region after the Matsubara sum in the ladder approximation, as illustrated in ﬁgure 3.7. By doing this, the equation of state of the upper branch will be reduced to an approximated scattering state only. Other thermodynamic quantities such as the pressure P (or the grand thermodynamic potential Ω), can be deduced from n(µ, T ) by doing the integration

53 Chapter 10. c P. Coleman 07 !

!ιω plane (a) n (b)

molecule pole

!b(q) !(q)

branch cut

Pole of F(z) Branch−cut C of F(z) C Figure 3.7: The no-pole approximation in terms of excluding the bound-state contribution. Instead of starting from the molecule pole of Γ at ωb(q) (dashed line+solid line), the upper branch thermodynamic potential only includes the integration region from the scattering continuum ω > ω(q) (solid line only). The crosses on the imaginary axisιω aren even Matsubara frequencies. (c)

µ 0 0 P (µ, T ) = −∞ dµ n(µ ,T ). With this set of prescriptions, one is able to study many properties ofR the upper branch in both Bose and Fermi gases across unitarity, as will be discussed in the following sections. C’ C’ 3.1.4 Summary

In this section we discussed some general features of the BCS-BEC crossover and the unitary

Fermi gas, including the zero-temperature order parameter, the transition temperature Tc as well as other thermodynamic quantities at ﬁnite temperatures in the ladder approximation. We also introduced an approach for excluding thePole bound state of in theF(z) eﬀective repulsive Fermi gas. Branch−cut 54 of F(z)

Figure 10.3: (a) Contour integration around the poles in the Fermi function enables us to convert a discrete Matsubara sum T F (iωn) to a continuous integral (b) The integral can be distorted around the poles and branch-cuts of F (z) provided that F (z) dies away faster ! than 1/ z at inﬁnity. | |

= f("p), (10.47)

+ recovering the expected result. In this example, the convergence factor ez0 that results from the small negative time increment in the Green function, plays an important role inside the Contour integral, where it gently forces the function F (z) to die away faster than 1/ z in the negative half-plane. Of course the original contour C integral could have been | | made by arbitrarily replacing f(z) with f(z) constant. However, the requirement that the − function dies away in the positive half plane forces us to set the constant term here to zero. In the second example (10.44)

1 1 F (z) = G(p + q, iν + z)G(p, z) = n iν + z " z " n − p+q − p which has two poles at z = " and z = iν + " . The integral for this case is then given p − n p+q 208 = +

˜ = + ˜

= +

Figure 3.8: A schematic diagram for the GMB correction. The ﬁrst two identities are the Dyson equations for the scattering matrix in the ladder approximation and with the GMB correction. The scattering matrix Γ is substituted by Γ˜ in the GMB formalism, which includes a second order interaction in each rung of the ladder. The correction to the single-scattering vertex renormalizes the interaction strength, and gives diﬀerent prefactors in both the zero temperature gap and the transition temperature from those calculated by the ladder approximation.

The ladder approximation we used is also called the “T -matrix” approximation, since all the diagrams included are simply iterative scattering of the same two legs, according to the dilute argument in section 3.1.2. A correction from including a virtual particle-hole excitation into the fluctuation, i.e. the Gor’kov-Melik-Barkhudarov (GMB) approach[59] is diagrammatically shown in figure 3.8. It relates to the NSR regime by substituting Γ to Γ.˜ The GMB scheme gives a much lower Tc in the BCS limit, since it actually gives a different saddle point if we go to the path integral formalism. A sketch of the transition temperatures from different theoretical approaches are shown in figure 3.9.

The GMB correction is only valid in the weakly attractive regime, i.e. k a 0−. The F s → correction is essentially perturbative, by including a second-order scattering vertex in each rung of the ladders (see ﬁgure 3.8). As a bubble gives a factor of k , a in exp( π/(k a )) F s − F s 2 appears in the Tc expression is substituted to as + CkF as, where C is a constant. This modiﬁcation results in the change in the prefactor of the exponential term in both the gap

55 Figure 3.9: A comparison of the calculated transition temperature Tc from different approaches: Leggett BCS mean field, GMB mean field, and NSR. Leggett BCS and NSR agree quite well in the deep BCS side, while GMB is much lower by a factor in BCS limit. Figure adapted from one of the chapters in [45].

(equation 3.17) and the transition temperature expressions. Closer to the resonance, one has to go beyond this perturbative approach, and include the contributions from other kinds of diagrams. In this chapter, however, we will not discuss these in details, but rather stick to the ladder approximation7

For the no-pole approximation, this formalism is illustrated by calculating the grand thermodynamic potential. The validity of this approximation is veriﬁed in a high temperature regime. It is also interesting to ﬁnd out if this approach can be extended to the calculations of other physical quantities, such as the self energy, the spectral function, etc.

We will leave these discussions to further studies. 7This GMB correction and other diagrammatic corrections were first studied in early works on the superfluidity of imperfect Fermi gases, and in nuclear physics. In the context of ultracold atomic gases, there have been recent works aiming to sort out the most significant diagrams, including using GG[60] or G0G[61] as the two legs (propagators) of the ladders instead of the G0G0 introduced earlier in this section. Here G0,G are the non-interacting and fully interacting Green’s function, respectively.

56 3.2 Fermi gases across narrow Feshbach resonance

Most of the studies in strongly interacting Fermi gases in cold atoms are in the context of “wide resonances” (or “broad resonances”). For a many-body system, a wide resonance has a width ∆B (which was deﬁned in section 2.2, and will be readdressed later) much larger than the characteristic energy scales of the system, namely the Fermi energy EF for Fermi gases in the degenerate limit, or the temperature T for thermal (Fermi) gases. In

wide resonances, the scattering length is a constant throughout the whole Fermi sea. The

absolute scale of ∆B in wide resonances is usually in the order of hundreds to a thousand

Gauss[22]. For “narrow resonances”, the width is typically less than a few Gauss, and

makes it very diﬃcult to stabilize the magnetic ﬁeld within this resonance region. The lack

of study in narrow resonances is from such reason: in order to get a strongly interacting

regime of Fermi gases across narrow Feshbach resonances, it is a daunting task to precisely

tune the external magnetic ﬁeld in the narrow window.

However, it has been pointed out recently that there is an experimentally accessible strongly-interacting region of Fermi gases across narrow resonances[34]. We will exploit the following facts about narrow resonances in this section. In both the high-temperature and low-temperature regimes, the interaction energy of a Fermi gas across a narrow resonance is comparable to that of the unitary gas, even when the system is several widths away from the resonance. Also, unlike the wide resonance where the interaction energy of the scattering states is antisymmetric about the resonance, it is highly asymmetric for narrow resonances: strongly attractive on the negative-scattering-length side, and weakly repulsive on the positive side.

3.2.1 Wide resonance and narrow resonance

To discuss the different features of wide and narrow resonances, it is useful to give proper definitions for both of them. Currently there are two schemes to classify all the Feshbach resonances[62]. These two schemes are in different context of two-body scattering and many-body systems. The general expression for a Feshbach resonance from a two-channel

57 model can be written as

γ∆B as(E,B) = abg 1 + , (3.52) E γ(B B∞) − − where abg is the background scattering length when the external magnetic field is far away from resonance, B0 and B∞ are the positions of magnetic fields at which as vanishes and diverges respectively. E is the incoming energy of the scattering as E = ~2k2/2m (in the 8 rest of the section, we set ~ = 1). The “theoretical width” of the resonance ∆B B0 B∞ ≡ − is the distance in magnetic field between zero and divergent as. γ is the magnetic moment difference between a pair of open-channel atoms and the closed-channel molecular state,

9 and it can be either positive or negative. From two-channel models, we have abgγ∆B > 0 . Then there is a well deﬁned positive quantity r∗, as the characteristic length scale in this resonance structure, such that the Feshbach resonance can be written in an alternative form:

−1 (mr∗) as(E,B) abg + , (3.53) ≡ E γ(B B∞) − − 1/(mr∗) a γ∆B. (3.54) ≡ bg

∗ Note that this expression using the r is more general, for it covers the case when abg = 0, in which the width ∆B has to diverge. This expression separates the background part

and “Feshbach part” (or “resonance part”) induced by the closed-channel molecules. The

∗ −1 resonance part is completely determined by (r ) , a parameter independent of abg and ∆B. Nevertheless, the relation between these three parameters is in equation 3.54. We will

use this expression implementing r∗ from now on as the scattering length formalism across

a Feshbach resonance: any Feshbach resonance structure is determined by two independent

∗ −1 ∗ −1 parameters: abg and (r ) . Note from two-channel models, this (r ) is proportional to the modular square of the coupling between the open and closed channels.

From the expressions above, we now classify the wide and narrow resonances in two

8This is sometimes called the “theoretical width” in order to distinguish it from the so-called “experimental width” extrapolated from atom-loss experiments. 9This can be understood from the second-order perturbation of the Feshbach resonance as well. Since α2 2 ∆as ∼ , the positive deﬁniteness of α enforces this property. A more detailed discussion is in E−γ(B−B∞). section 2.2.

58 schemes: one is from two-body physics (denoted by scheme (A)), and the other one makes use of many-body scales kF (denoted by scheme (B)). In the two-body scheme, we deﬁne

2 another energy scale associated with background scattering length as Ebg = 1/mabg. Then in scheme (A) the rescaled dimensionless width of the system is deﬁned by a ratio:

γ∆B abg α = ∗ . (3.55) ≡ Ebg r

The narrow resonance and wide resonance, which are denoted by and respectively, NA WA are deﬁned as:

: α 1, : α 1. (3.56) NA | | WA | | On the other hand, in the many-body context, the scheme (B) is determined by the

∗ −1 dimensionless number (r kF ) . The corresponding narrow and wide resonance, denoted by and , are deﬁned as: NB WB

:(r∗k )−1 1, :(r∗k )−1 1. (3.57) NB F WB F

As we can see, in both scheme (A) and (B), the “narrowness” is proportional to (r∗)−1.

The diﬀerence between them is the reference length scale of the system. In the two-body scheme, the only relevant length scale is given by the background scattering length abg.

In a simpliﬁed model where the background scattering length vanishes as abg = 0, from the deﬁnition 3.56, the system always goes to the narrow limit. It is a consequence of the absence of the comparison to another absolute length scale. However, if we put this resonance structure in the context of a many-particle system, there is always a characteristic

−1 length scale kF associated with the interparticle spacing. The narrowness of the resonance ∗ −1 varies continuously as (kF r ) changes. The scheme (B) can also be understood as the following: the scattering length for zero incoming energy E = 0 scattering is divergent when B = B∞. If we consider the ﬁnite

2 energy scattering length around the Fermi surface E = EF = kF /m (the reduced mass is half the atom mass, for equal-mass mixtures), the scattering length times the Fermi vector

59 kFas 1

EF B

-1

Figure 3.10: An illustration of the difference between wide and narrow Feshbach resonances in a Fermi gas with Fermi energy EF . We plot the scattering length or interaction strength k a as a function of the external magnetic field, and define the k a > 1 region as the F s | F s| strongly interacting region, denoted by a vertical strip bounded by the blue dashed line and the red dashed line for narrow and wide resonances, respectively. The scale of the Fermi energy here is denoted by the length of the black double arrow. We find for a wide resonance, the whole Fermi sea is in the strongly interacting region. In contrast, in the narrow resonance, only a small part of the Fermi sea is contained in the strongly interacting n region. The parameters for the two resonances are: kF abg = 0.1, ∆B /EF = 1 for the w narrow resonance as the blue curve, and ∆B /EF = 10 for the wide resonance as the red curve.

is −1 kF (mr∗) 1 kF as = kF abg + 2 = kF abg + ∗ . (3.58) E=EF ,B=B∞ k /m kF r F

In the situation of an oﬀ-resonance background scattering length where kF abg 1, this ∗ −1 value is determined by the Feshbach resonance term (kF r ) . For a wide resonance, we can see that the scattering length at the Fermi surface is still large; in contrast, for the narrow resonance, it becomes a small value at the Fermi surface. This is better illustrated in ﬁgure 3.10: we would like to see how much the energy-dependent scattering length changes when the incoming scattering momentum varies inside the Fermi sea. For the wide-resonance limit, the scattering length will almost stay unchanged, and we can assume that the scattering length is a constant. However for the narrow resonance limit, the

60 −1 Figure 3.11: Wide and narrow resonances in the space ((kF r∗) , kF abg): Scheme (A) is determined by the value of xy, and and are the regions above and below the curve WA NA xy = 1 respectively. From scheme (B) the crossover is a region of vertical stripe: and WB are the regions where x 1 and x 1. The dotted line are contours of constant NB ∆B. The thick red lines on the x-axis correspond to the simple two-channel models with vanishing abg. This phase diagram is for abg > 0. The full phase diagram including abg < 0 is mirror symmetric about the x-axis.

scattering length can be very sensitive to the incoming energy, and all the formula have to

use energy-dependent scattering lengths.

To show how scheme (A) and (B) relate to each other, we make a schematic diagram as

shown in ﬁgure 3.11. The horizontal and vertical axis are dimensionless quantities deﬁned

−1 as: x = (kF r∗) and y = kF abg. The criterion of wide and narrow resonance in scheme (A) and (B) correspond to the following regions in the (x, y) plane:

: xy = α 1, : xy 1; WA NA : x 1, : x 1. (3.59) WB NB

The contours of constant ∆B are denoted by dotted straight lines, whose slope y/x is the ratio 2EF /(γ∆B). As we can see, a simpliﬁed two-channel model with vanishing background scattering length where a 0 will give ∆B , and systems in such a limit bg → → ∞

61 live on the horizontal axis, indicated by thick red lines. Take an example of two resonance

6 6 14 −3 channels in fermionic Li: for Li at density n↑ = n↓ = 5 10 cm , one of its wide × 3 resonances of channel ab at 834.1G has a = 1405a , ∆B = 300G, a /r∗ = 2.8 10 , bg − B − | bg| × −1 3 (r∗k ) = 1.25 10 , and k a = 2.3. Its narrow resonance at 543.25G has a = 61.6a , F × F | bg| bg B −1 ∆B = 0.1G, abg/r∗ = 0.002, (r∗kF ) = 0.02, and kF abg = 0.1[22].

3.2.2 Strong interactions in Fermi gases across narrow resonance

In the previous studies of wide resonances, the origin of the strong interactions is understood as due to the special structure of phase shifts in two-body scattering. A general prescription of the scattering problem is to write down the inverse scattering amplitude as f −1(k) = k cot δ(k) ik. In the universal model, the real part of it at small wave vector k 0 is − → approximated by 1 k cot δ(k) = . (3.60) −as(k)

Unlike in wide resonances where the scattering length as is almost independent of incoming scattering energy, in narrow resonances, the scattering length is very sensitive to energy, with (mr∗)−1 a(k) = abg + 2 . (3.61) k /m γ(B B∞) − − An explicit expression of tan δ(k), in the approximation of zero eﬀective range is thus given

by k/mr∗ tanδ(k) = kabg 2 , (3.62) − − k /m γ(B B∞) − − with the phase shift δ(k) defined up to modulo π. When we tune the magnetic field, the evolution of the scattering-state phase shift is determined by Levinson’s theorem, namely it jumps by π when a new bound state emerges, such that the phase shift always starts − from zero at zero-energy scattering. In figure 3.12, we plot out the scattering phase shift

δ(k) for a wide resonance (A) and a narrow resonance (B), as a function of the incoming wave number k, when we tune the magnetic ﬁeld.

The origin of strong interactions in quantum gases is from the special phase shift structures which lock up the energy in the system. The phase shift essentially serves as a collective

62 Figure 3.12: δ(k) vs k for wide (A) and narrow (B) resonances: The labels (a,b,c,d,e,f) + + correspond to (a : B B∞; b : B > B∞; c : B = B∞ + 0 ; d : B = B∞ 0 ; − e : B < B∞; f : B B∞). For wide resonance with a < 0, as shown in (A) on the bg left, we have δ(k) = arctan(ka (k)). Near resonance, (c and d), δ(k) approaches a step − s function of height π/2. Far from resonance, (a and f) δ(k) reduces to a linear function with ± a constant slope δ(k) = arctan(ka ) ka , and ka 1. For narrow resonance with − bg ∼ − bg | bg| positive background scattering length abg, γ, ∆B > 0, as shown in (B) on the right, when 2 B > B∞, δ(k) approaches a step function of height π at the position k /m = γ(B B∞), − with a width 1/r∗. For B < B∞, δ(k) quickly reduces to ka as B∞ B exceeds ∆B, − bg − and ka 1. The δ(k) for both e and f are essentially identical. | bg|

energy shift for diﬀerent momentum states in a characteristic energy scale of the system,

namely kF for low temperatures and √mkBT at high temperatures. We take the example of the quantum degenerate regime where T < EF and we only consider the scattering for states inside the Fermi sea. For wide resonances, when the system is oﬀ resonance, the interaction energy is governed by a small phase shift as large as δ k a 1. The ∼ − F bg system will remain weakly interacting until the magnetic ﬁeld is tuned close to as (c) and

(d) in ﬁgure 3.12(A), where the scattering length is almost divergent and the phase shift shows a near-step function behavior at all momentum. The single-particle energy levels in this case, will be shifted collectively by π/2, and go to a very strongly attractive or ± repulsive case in the scattering channel, respectively.

63 For narrow resonances, since the scattering length is energy dependent, we find some different features. In the very off-resonance limit B B∞ or B B∞, the phase shift for those momentum states inside the Fermi sea is very small, and the system is weakly

2 interacting. In the case that ∆B < B B∞ < k /m, as (b) in ﬁgure 3.12, even though − F the magnetic ﬁeld is several widths away from resonance, the system in narrow resonance can be strongly interacting. The reason comes from there being abrupt change of scattering phase shift inside the Fermi sea, such that the system has a phase shift of π (instead of

2 ∗2 π/2 in wide resonances) in the region where k /m > B B∞ k /m. The large phase − ≡ ∗ shift in the momentum shell k < k < kF contributes a large interaction energy to the system. When the system ﬁnally approaches B B∞, all the momentum states inside the → Fermi sea gain the π shift as the step function grows at k 0, and the system will have a → maximum attraction energy for scattering states.

To summarize the diﬀerence between the wide and narrow resonances, the wide resonance is only strongly interacting when B B∞ < ∆B, and the system has a symmetric − attractive and repulsive interaction on diﬀerent sides of the resonance. For narrow resonances, the system can be strongly attractive even when B B∞ > ∆B, as long as − 2 B B∞ < k /m (or B B∞ < √mk T in thermal gases where T > E ); the repulsive − F − B F energy on the repulsive side when B < B∞ will always be weakly repulsive. To show the conclusions above in a more systematic and rigorous manner, one has

to calculate real physical quantities in both the high-temperature and low-temperature

regimes, as is shown in the following parts.

Virial expansions

The eﬀect of interactions can be illustrated by high-temperature expansions. At high tem-

peratures, the leading order of the interaction energy is the second power in fugacity. It

reveals all the two-body physics in the system. In this high-temperature regime, the inter-

action energy density of the system takes the form:

3 3T n nλ 2T ∂b2 int(T, n) = b2 + , (3.63) 2 √2 − 3 ∂T 64 where b2 is the second virial coeﬃcient. b2 consists of two contributions from the bound

b bd sc bd |Eα|/T states and the scattering (extended) states, b2 = b2 + b2 , where b2 = α e is the partition function of a set of bound states Eb (labelled by α) and bsc isP given by { α} 2 ∞ dk dδ(k) 2 bsc = e−k /mT . (3.64) 2 π dk Z0 sc b2 comes from the interacting partition function of the scattering states. As the total interaction energy is a linear function in b2, the interaction itself can be decomposed into

bd sc int = int + int. In the situation of a wide resonance, the second virial coeﬃcients b2 and the interaction energy density to this power have been studied in reference [21]. The

narrow resonance case, in contrast, is addressed in reference [34]. Here we show the second

3 virial coeﬃcient b and the “interaction strength” / 3T n nλ√ for both including and 2 int 2 2 excluding the bound-state contribution in ﬁgure 3.13. The narrow resonance we consider is for a gas of 6Li at 543.25G with parameters given above. In this case, T = 40µK 3γ∆B F ∼ (in temperature units).

The branch that exists only on the positive scattering length side with positive interaction energy and b corresponds to the upper branch of the Fermi gas. This upper branch − 2 is free from all the underlying bound states. The other branch is the true ground state in which the molecules are occupied by atoms. From figure 3.13 we see the system has a large attractive interaction in the attractive branch, even when the magnetic field is still several widths away from the resonance. The upper branch, in contrast, has a very small repulsive energy on the positive-scattering-length side. The origin of these two features are from the special phase shift structures, as illustrated in figure 3.12. For the full b2 including the bound-state contributions, we find that the phase shift jumps to π for scattering states

in the relevant energy scales (which in this case is the temperature T ) for a large region

in magnetic ﬁeld. Hence, b remains large as long as B B∞ < T . Also, right at the 2 − resonance, since the phase shift at the narrow resonance saturates to π instead of π/2 in

the wide resonance case, b 1 for narrow resonance is twice as large as that of a wide 2 ≈ resonance. For the upper branch, because the scattering phase shift is only signiﬁcant in a

very small region of the width of resonance, and otherwise almost vanishes, it provides the

65 Figure 3.13: b (A) and (B) as a function of magnetic ﬁeld: b remains sizable even − 2 int 2 when B B∞ 2G 20∆B. This is because we are at T = 5T 15γ∆B or larger, − ∼ ∼ F ∼ where thermal sampling extends over an energy range of several T . When the system is in equilibrium, it follows the lower curve (or the “lower branch”). At resonance and in the scattering limit ∆B 0, b 1. On the molecular side, B < B∞, (for the upper branch) → 2 → int is given by the ﬂat line, given by the small value δ ka . ∼ − bg weakly repulsive nature in the positive scattering length side.

Interaction energy at low temperatures

To better illustrate the interaction effect, we apply a generalized NSR approach introduced in section 3.1.3 to quantitatively calculate the interaction energy for both the upper and lower branches of the Fermi gas in narrow resonances. The expressions are generalized in a way that the scattering length is energy dependent. We use the same 6Li across 543.25G narrow resonance as the example in figure 3.14, and plot the energy density as a function of the magnetic field for a degenerate gas at T = 0.5TF , where TF is the Fermi temperature. The behaviors of both branches are similar to those found from the virial expansion shown in figure 3.13. The energy scale, however, is very different. One sees that the interaction energy is as much as 50% of the total energy of an ideal Fermi gas right at

2 bg

(0)/a s

-2 a

-4

0.0 0

-0.2

E/E

-0.4

-0.6

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

B-B (G) 8

Figure 3.14: An example of the s-wave scattering length (upper panel) and the interaction energy (lower panel) of a Fermi gas near a narrow resonance at low temperatures. We have T = 0.5TF , TF = 3γ∆B as in ﬁgure 3.13. Other parameters are the same as those given above. E0 is the energy for a non-interacting Fermi gas at the same temperature. The dashed line indicates the width ∆B of the resonance. The downward turning curve and the ﬂat curve are the energies of the lower and upper branch, respectively. The interaction energy of the lower branch reaches 50% of the free fermion energy at resonance, and remains sizable beyond the width of the resonance.

resonance, and can be as high as 30 40% even beyond the width of the resonance. One − also sees that the interaction energy is only signiﬁcant when the distance from resonance

γ(B B∞) is within 2E , as discussed previously. − F

Related experiments

There are several experimental groups trying to study the properties of stable narrow Fes- hbach resonances for Fermi gases. Recently, Ken O’Hara’s group has performed rf spectroscopy studies on the narrow resonance of 6Li at 543.25G and has found the asymmetry of the interaction energy int on different side of the resonance[63]; their data is shown in figure 3.15. However, the quantitative observation of the attractive interaction energy for the lower branch is ambiguous. The rf-spectroscopy method in their approach gives a rel- 67 uvsaetesit rdce yama-edter hticue h nrydependence energy solid the the includes and that potentials, theory contact mean-field length. field a scattering by mean the indicated. predicted by of vaguely shifts panels, only predicted two the is shifts center are width the the curves the the are in outside of curves fields especially asymmetry dashed magnetic points, The The at data attraction the temperatures. large from different the seen however with clearly gases is Fermi energy from interaction are panels four The iue31:Eprmna aafo h enSaegopfrfermionic for group State Penn the from 543 data the Experimental 3.15: Figure . 5 ehahrsnne h neato nryi esrdb fspectroscopies. rf by measured is energy interaction The resonance. Feshbach 25G

m º¹ m º¹ m º¹ m º¹ ¡ [a0] ¡ [a0] ¡ [a0] ¡ [a0] 2h ¹ n¹ 2h ¹ n¹ 2h ¹ n¹ 2h ¹ n¹ 68 6 igssacross gases Li atively large error in the strongly interacting regime. It is pointed out previously[25] that the interaction energy int can also be determined exactly (free of the modeling by speciﬁc theories) from in situ density measurements. This may be used to calibrate the observations in related experiments.

3.3 Repulsive Bose gases across Feshbach resonance

While there are plenty of studies of unitary Fermi gases, Bose gases close to unitarity have not been enough addressed in the previous literature for various reasons. As is observed in experiments, as the bosons approach the large scattering length regime, the system will not persist in a well-deﬁned thermodynamic state, mostly because of two reasons. One is that as the scattering length is tuned towards the divergence from the negative side, the strong attractive interaction between bosons will cause negative compressibility, and the gas will collapse due to mechanical instability. Another important reason is that the existence of a series of Eﬁmov trimers which can have very little binding energy enhances the chance of a three-body recombination process. In such a process, two of the atoms fall into a deeply bound state, while ejecting the third atom by this inelastic scattering. This three- body recombination causes extremely severe atom loss, and the system can never reach equilibrium.

In this section, we focus on a low fugacity regime in which the Bose gas equilibrates via two-body collisions, while three-body collisions can be substantially suppressed. We will show that the loss rate caused by the three-body recombination process can be reduced to a low enough rate that reasonable measurements can be made in this metastable thermodynamic state. Several thermodynamic quantities, including the energy and the equation of state, are calculated for this Bose gas close to unitarity[64].

69 3.3.1 Three body loss in Bose gases close to unitarity, “low recombination” regime

For Bose systems, the absence of quantum degeneracy pressure makes them vulnerable to mechanical instabilities. At zero temperature, any attraction strength between the bosons will lead to a collapse. This is for the reason that the interaction energy and kinetic energy scale differently as functions of the density. The interaction energy is proportional to the density n, so it always wins the competition against the n2/3 dependence of the kinetic energy at high enough densities. The system seeks an infinitely dense “ground state”, and can not be well described by any conventional approach in the thermodynamic limit. At finite temperatures, thermal fluctuations bring in entropy to counterbalance the contribution to the free energy from the attractive interactions. The entropy makes it possible to stabilize the Bose system in some weakly-interacting regime. Some previous theoretical studies in weakly attractive Bose gases (in harmonic traps) show that the collapse temperatures are close to the transition temperatures for Bose-Einstein condensation[65].

However, close to resonance where the scattering length diverges, the strong interaction overrides the entropy and leads the compressibility to a negative value. An alternative for the bosons is that they pair up and form dimers by strong interactions. The remnant eﬀective interaction between these dimers may be smaller attractive, or even repulsive, and the system may persist in a stable state of dimers[66].

In this section, instead of going to strong attractions, we focus on the “upper branch”, i.e. the repulsive branch of the Bose gases free of bound states. As is discussed earlier in this chapter, the repulsive branch of Bose gases is only metastable. For weak repulsion at low temperatures, where 0 < k a 1, from the analysis of cross sections for the three-body F s scattering processes, we have an estimate of the three-body collision rate γ3 as

3 γ3 = c(4π~as/m)n(nas), (3.65)

where c is a dimensionless constant[67]. For the two-body collision rate, it is in the form

2 γ2 = nasv, (3.66) 70 where v is the typical velocity of the bosons, which for weakly-interacting Bose gases at low temperatures can be approximated as v2 2gn/m. The ratio between these two rates are ∼ given in the form 4πcna3 s √ 3 1/2 γ3/γ2 = ∗ = c 2π(nas) , (3.67) k as ∗ where k = mv/~ is the typical wave number of the bosons. In the extremely weakly interacting limit that n1/3a 1, γ is suﬃciently low that the system is essentially free of s 3 molecules.

In the past few years, there are an increasing number of experiments on strongly repulsive

Bose gases, trying to examine their properties at low temperatures[68, 69, 70, 71]. However,

1/3 at low temperatures, γ3 increases rapidly in the strongly repulsive regime, i.e. n as > 1, as the fourth power of the scattering length as. This sharp increase in three-body collisions leads to severe atom loss as the system approaches resonance, and the system is far from equilibrium. The dependence of the three-body loss on scattering length has also been observed experimentally by several groups[68]. Also, close to unitarity, due to the series of Efimov trimers, the three-body recombination process can be greatly enhanced at some specific scattering lengths at low temperatures. While one can still explore strong interaction effects in this “fast loss” regime of Bose gases by bringing the system quickly in and out of the strongly-interacting region, it is not clear how to define equilibrium properties in such situations.

It is in a diﬀerent situation for Bose gases at higher temperatures and lower densities, i.e. in the low-fugacity regime. We consider a Bose gas with density n, and deﬁne the

“Fermi wavenumber” and “Fermi energy (temperature)” in the same way they are related 3 kF in a single-component Fermi gas: n = 3π2 . At temperatures much higher than the condensate transition temperature T T , the characteristic energy scale of the particles ∼ F are mostly governed by the temperature, and we have v 3k T/m h/(mλ), where ∼ B ∼ λ = h/√2πmkBT is the thermal wavelength. When we estimatep the collision rates close to unitarity, as in γ2 and γ3 is replaced by λ in this temperature regime, and we have

3 3 2 γ2 = (kBT/~)(nλ ), and γ3 = (kBT/~)(nλ ) , where = 9√3/π 4.96[72]. In a Bose C C ∼

71 gas with low enough density, γ /γ nλ3 becomes small. In this situation, when we tune 3 2 ∼ the scattering length between the bosons from the weakly repulsive side, the system can equilibrate by two-body collisions, while the three-body recombination rate is substantially suppressed. The loss rate can be low enough that the upper branch has a long enough lifetime to make reasonable measurements. Also in the presence of the trap, in order to reach a global equilibrium, we need the condition that the spatially averaged rate of particle loss falls below the trapping frequency:

drγ n N −1dN/dt = 3 = γ , (3.68) drn 3 ave − R h i where .. means spatial average. R h iave The previously discussed density and temperature regime (which we referred to as the

“low-recombination” regime) where γ < γ , γ < ω, or 3 2 h 3iave

3 3 −1/2 nλ 1, nλ < ~ω/kBT, (3.69) C p where n2 n3/ n. In this regime, very few molecules are formed even at unitarity during ≡ the time whenR theR Bose gas reaches global equilibrium through two-body collisions. Due to

the low loss rate, the system persists in a well deﬁned metastable thermodynamic state with

eﬀective repulsive interactions. The underlying bound states (or Feshbach molecules) are

not populated even though they are able to accommodate atoms in the system. Recently,

this “low-recombination” regime has been realized by Salomon’s group at ENS[73], and a

lot of interesting related problems are being studied, including the third virial coeﬃcient[74]

and the lifetime of this upper branch[11]. In the following sections, we will focus on the

calculation of thermodynamic quantities of this Bose gas in the low-recombination regime.

3.3.2 Strongly repulsive Bose gases close to unitarity, “shifted resonance”

In the low recombination regime described in the previous part, we consider a Bose gas

that is prepared in the weakly repulsive limit, namely the scattering length is small and

positive 0 < k a 1. From the analysis above, we can adiabatically tune the scattering F s length towards resonance at a suitable rate. This rate lies between γ2 and γ3, i.e. it is fast

72 enough that three-body collisions do not generate severe losses, and also slow enough that the system will reach global equilibrium by two-body collisions. In this process, the Bose gas remains in a metastable upper branch, which is free of molecules. This upper branch is (c) eﬀectively repulsive, up to some critical scattering length as < 0 on the negative-scattering length side. In contrast, the “true equilibrium” state of a Bose gas contains both atoms and dimers. This equilibrium branch can be accessed by tuning the scattering length from the negative side toward resonance, at a high enough temperature that the Bose gas itself does not collapse.

To study the homogenous upper-branch Bose gas, we implement the “no-pole approximation” discussed in the ﬁrst section of this chapter. In this approximation for the bosons, the expression for the grand potential is similar to that for the fermions, except that in 3.42 all the Fermi functions f(q) ,are replaced by Bose distribution functions n (k) = 1/(eξk/T 1) evaluated at the single-particle energy ξ = µ where = k2/2m. B − k k − k Explicitly, the scattering matrix in the bosonic medium is

dk 1 + n (q/2 + k) + n (q/2 k) m m T −1(q, ω + i0+) = B B − + (3.70) B (2π)3 ω + i0+ + 2µ k2/m q2/4m k2 − 4πa Z − − s

The phase angle ζ is deﬁned as the argument of this function TB accordingly. The resulting form of the grand potential, and the equation of state consists of two parts:

sc bd n(µ, T ) = n0(µ, T ) + ∆n (T, µ) + ∆n (T, µ), (3.71)

sc where n0(T, µ) = k nB(ξk) is the density of the ideal Bose gas and ∆n (T, µ) and ∆nbd(T, µ) are theP interaction contributions of the scattering states and the bound states

respectively:

1 ∞ dω ∂ζ(q, ω) ∆nsc(µ, T ) = n (ω) , (3.72) −Ω π B ∂µ q ω(q) X Z

1 ∂ω (q) ∆nbd(µ, T ) = n (ω (q)) b , (3.73) −Ω B b ∂µ q X where ω(q) q2/4m 2µ. The integration in the scattering state part is from the threshold ≡ − 73 of the “scattering continuum” ω(q), which arises from the branch cut of the T -matrix. The

other part in the number density is from the bound states, denoted by the pole of the

T -matrix ωb(q), which is the solution of the equation

m 1 γ(k; q) 1 + + = 0, (3.74) −4πa Ω k2 k2 s ω ω(q) m m ! Xk − − where γ(k; q) 1 + n (q/2 + k) + n (q/2 k). ≡ B B − For a pure two-body problem, the T -matrix is obtained by replacing γ = 1 + n + n by 1.

The emergence of the bound state in this case occurs when 1 changes sign from negative as to positive. As in the bosonic medium, the additional terms +n + n in γ give an eﬀective

“bosonic enhancement”, as opposed to the situation of pairings in fermionic media. For

Fermi gases, it has the form of γ = 1 n n, which corresponds to “Pauli blocking”. F − − The enhancement and the blocking eﬀects are illustrated in ﬁgure 3.16. As a consequence

of these eﬀects, the threshold for dimer formations shifts to the negative-scattering-length

side in a bosonic medium, and to the positive side in a fermionic medium. Suppose we

carry on the deﬁnition in two-body scatterings that the resonance is the point where the

bound state appears. In the presence of the media, there is a “shifted resonance” from the

point of divergence in the two-body scattering length. The direction of the shift depends

on the quantum statistics: negative for Bose gases, and positive for Fermi gases. Further,

as is discussed in the earlier sections, the metastable upper branch can thus be extended to

the negative-scattering-length side of the resonance, as opposed to in a fermionic medium

where the upper branch terminates before the inverse scattering length hits zero.

Another systematic way to analyze the shifted resonance is to study the expression for

the scattering T -matrix. By comparing the expression for the two-body scattering T -matrix

in vacuum and its counterpart in the medium, an eﬀective scattering length in a many-body

system can be deﬁned as the following:

74 k linear = combinations dimers of states

Fermions: Bose Pauli blocking enhancement: k k kF

Figure 3.16: A sketch of how bosonic and fermionic media aﬀect the formation of dimers. The dimer formation requires the contribution of a series of single particle momentum states. For fermionic media, the existence of the Fermi sea actually blocks the single particle states, which are required for the construction of a bound state, consequently makes it more diﬃcult to form a dimer. For bosonic media, in contrast, at the same interaction strength, the populated single particle states “enhances” the possible occupancy at this energy level, instead of blocking it by Pauli principle.

1 1 1 = + ∆ aeﬀ (q, ω) as a 1 4π n(ξq/2+k) + n(ξq/2−k) = 2 , (3.75) as ∓ Ω m(ω ω(q)) k Xk − − where the minus sign is for the bosonic case and the plus sign is for the fermionic case.

The distribution functions n’s are the Bose or Fermi functions accordingly. As we can see, the modification of the inverse scattering length at the continuum threshold ω = ω(q) is positive definite for bosonic media, and negative definite for fermionic system. This is the mathematical origin of the shifted resonance. This change in effective scattering length is also momentum dependent: in a Bose medium for instance, a bound state with center of

75 mass momentum q occurs when a (q, ω = ω(q)) 0, or when a−1 a−1(q), where eﬀ ≥ − s ≤ − c

1 4π nB(ξq/2+k) + nB(ξq/2−k) = 2 < 0. (3.76) ac(q) − Ω k Xk In other words, if one approaches the resonance from the atomic side, a bound state with total momentum q will emerge at scattering length ac(q) < 0, which is on the atomic side of the original resonance. This “critical scattering length” is not only a function of the center- of-mass momentum q, but it also depends on the temperature (and chemical potential) of the Bose gas. For extremely high temperatures, as the Bose distribution functions in the numerator are suﬃciently suppressed, the critical scattering length for dimer formation

(0) −1 goes back to the original two-body resonance (ac ) = 0. Also, for higher center-of-mass momentum dimers, they are less affected by the background of bosons, since the momentum states for such dimers lie in a high-energy manifold. The values of 1/a (q) at different q’s − c are shown in the bottom panel of figure 3.17, at a temperature T = 4TF . With the prescriptions given above, one can calculate the grand thermodynamic potential, as well as all the thermodynamic quantities from equations 3.72 and 3.73. All the calculations are done in a temperature regime much higher than the condensation and collapse temperature of the Bose gas[75, 65]. For a homogenous Bose gas, the “Fermi temperature” TF defined above is related to the condensation temperature for non-interacting

Bose gases Tc as TF /Tc = 2.3. The “phase diagram” of the upper-branch Bose gas is shown if ﬁgure 3.18. The upper-branch region is where the eﬀectively repulsive gas exists, in which the system can accommodate the underlying bound states. However these states are not populated because of the suppressed three-body recombination rate. The equilibrium branch is the temperature and scattering length regime in which the attraction is not enough to bring dimers into the system. The Bose gas is in its “true ground state” in the equilibrium branch. The boundary between these two branches corresponds to the critical inverse scattering length where the q = 0 bound states emerge, at the corresponding temperatures.

The third “unstable” region is interpreted as the following: mathematically, in this region when we use the upper branch formalism by counting only the number of particles

76 1.6 1.4 T 4T

0 F E

1.2 E 1 0.8 1 0.5 0 0.5 1 1 kFas 15

10 T 4TF F k

q 5

0 0 0.1 0.2 1 kFac q

Figure 3.17: The upper panel is the energy density across the resonance at T = 4EF = 9.2Tc, rescaled by the energy Eo(T ) of a noninteracting system at the same temperature. The jump represents a transition from the upper to the lower branch. Even at this high temperature, the interaction energy and the jump are substantial fractions of the total energy. The lower panel shows ac(q) as a function of q for ﬁxed n. The jump is due to the sudden change in µ as the system switches branches.

from the free part and scattering part as n = n0 + nsc, there is no solution to the number equation for the chemical potential µ. As this region appears more likely at lower temper-

atures and closer to the unitarity region, the physical origin of it is explained as follows.

When the system approaches the more strongly repulsive regime, the compressibility drops

gradually, and ﬁnally becomes zero at the boundary to this “unstable” region. For a naive

picture of hard-sphere atomic gases, this means that the radius of the spheres increases as

the repulsion is strengthened. Finally the Bose gas becomes fully packed as it is not able

77 6 13

5 Equilibrium 11 Upper Branch F Branch C T T 4 9 T T

3 Κ=0 7 Unstable 5 2 -0.4 -0.2 0 0.2 0.4 -1 kFas

Figure 3.18: The “phase diagram” of a homogeneous upper-branch Bose gas with ﬁxed density n: At the blue curve that separates the upper and lower branches, the energy density undergoes a discontinuous jump as shown in Fig. 3.17. The purple dashed curve represents a state with κ = 0. In the “unstable” region, the number equation for the chemical potential does not have a solution.

to accommodate more particles, and it is also not possible to stay in the repulsive branch when the repulsion is further increased. The “leftover bosons”, when we force the system to further increase the density or repulsion, will have to form dimers, and the system will not remain in the upper branch.

The corresponding behavior of the energy density at T = 4TF is shown in the upper panel of figure 3.17. It can be seen clearly that the repulsive upper branch penetrates to the negative scattering length side, up to around 1 = 0.2. The repulsion energy remains kF as − as strong as almost half of the non-interacting kinetic energy of the Bose gas, even at this high temperature. A jump in the total energy appears when the system is tuned across the boundary between the upper and equilibrium branch in figure 3.18. This jump comes from the change in µ as the system switches branches, and is different from the case in Fermi gases, where the energy curve is continuous at high temperatures[39].

78 3.3.3 Equation of state and instabilities of Bose gas in a trap

For the Bose gases in harmonic traps, we would like to study the equation of state n(µ, T )

of the gas. The density proﬁle of the trapped gas, within the local density approximation

(LDA), is given by n(r) = n (µ V (r),T ), where µ is the chemical potential at the upper − center of the trap. A global view of the density proﬁle can be obtained from the “phase

diagram” in the (µ/T )-(λ/as) plane, where λ is the thermal wavelength, as is shown in ﬁgure 3.19. The density proﬁle along a radial direction starting from the trap center corresponds

to a vertical line emerging from µ/T upward. A trapped Bose gas is therefore speciﬁed − by a point ( λ/a , µ/T ) on this diagram. − s − Figure 3.19 shows how the Bose gas behaves as it is swept from the positive-scattering-

length side to the other. Again, there are three regions in this ﬁgure: (i) the equilibrium

branch, (ii) the upper branch, and (iii) the unstable region with compressibility κ < 0.

This unstable region is diﬀerent from the previously discussed instability associated with

the absence of a solution for the chemical potential. The boundary between (i) and (iii)

and that between (ii) and (iii) are denoted as µ(b)(T ) and µ(a)(T ) respectively. µ(a)(T ) is the boundary of zero compressibility, κ = 0. µ(b)(T ) is the boundary where bound pairs with zero momentum begin to form.

As we can see, the unstable region (iii) always intervenes between branches (i) and (ii).

So in principle, any trapped gas with the upper branch at the center and with negative scattering length will suffer from a certain local instability. However, if we analyze the width of the unstable region ∆r = r r , where r and r are given by µ = µ V (r ) a − b a b a − a and µ = µ V (r ), we can see that the difference that µ µ becomes very small close to b − b b − a unitarity. This difference actually controls the width ∆r, and in the limit that ∆r becomes

−1/3 ∗ less than the interparticle spacing, i.e. ∆r < n(¯r) , which occurs at a critical ratio λ/as

for given µ/T , ra and rb can be viewed as a single pointr ¯. The unstable region disappears

∗ −1/3 ∗ in this situation. The critical ratio λ/as can be estimated by setting ∆r = n(ra) .(as is a function of T and µ).

Thus, for an upper-branch Bose gas characterized by the point ( λ/a , µ/T ) on this − s −

79 5

4 HiLEquilibrium HiiLUpper Branch Branch

3 Μ1 æ HaL T

Μ2 æ Μ - 2 Μ æ 3 HbL HcL ΜHbL ΜHaL Κ=0 1 HiiiLUnstable

0 -1.0 -0.5 0.0 0.5 1.0 -Λ as

Figure 3.19: “Phase diagram” in a trap at fixed temperature T and trap frequency ω: µa represents the state of compressibility κ = 0. µ(b) separates the equilibrium branch with the unstable region. The latter has κ < 0. Each point on this diagram denotes a density profile of the Bose gas, with µ being the chemical potential at the trap center. The density profile can be generated by an upward vertical trajectory using LDA, (see text). The three horizontal lines with arrows are trajectories of a Bose gas across resonance into the atomic side at fixed µ. The termination point (denoted by a black dot) on each µ- trajectory indicates the critical scattering length a∗ for that µ. For as < a∗ (as > a∗), the density profile is stable (unstable). Hence, the density profiles (a) with chemical µ1 is stable, whereas the densities (b) and (c), with chemical potential µ3 are unstable. For T = 1µK, and ω = 2π(250Hz), we find ( λ/a )∗ = 0.14, 0.18 and 0.21 for the trajectories − s (µ/T ) = 3, (µ/T ) = 2.5 and (µ/T ) = 2 respectively. The corresponding particle − 1 − 2 − 3 numbers are (N ,N ,N ) = (2.1, 3.4, 5.2) 104. All these systems satisfy the condition 1 2 3 × to be in the low-recombination regime, equation 3.69 or equivalently equation 3.77, as γ /ω = 0.14, 0.35, and 0.96 respectively, and N ,N ,N < N ∗, where N ∗ = 6.5 104 h 3iave 1 2 3 × for this temperature and trap frequency.

80 a

Equilibrium

n Branch Upper Branch

b c n

r r

Figure 3.20: The density profiles for an upper-branch Bose gas in a trap: figure (a), (b), and (c) correspond to the densities (a), (b), and (c) in figure 3.19. Density (a) has an upper-branch core and an outer shell of the equilibrium state. The dashed purple curve is the density of the equilibrium branch at the same T , as, ω, and N. The densities (b) and (c) are unstable as they contain regions where dn/dµ < 0. For density (a), the width of the unstable region becomes less than the interparticle spacing and is therefore non-existent.

diagram, it will only be stable when 1/a < 1/a∗(T, µ), such that the unstable region–if − s − s there is any–with a width ∆r in real space collapses to zero. The vertical line labelled (a) in

Fig.3.19 represents such a density. Its density proﬁle is shown in Fig.3.20a, which consists

of an upper-branch inner core and an equilibrium-branch outer layer, both of which are free

of Feshbach molecules. Compared to the density proﬁle of the lower branch (dashed line

in Fig.3.20a, which is close to the Boltzmann distribution), one sees a discernible kink in

the upper-branch density. When λ/a exceeds λ/a∗(T, µ), such as (b) and (c) in Fig, 3, − s − s the corresponding density proﬁles are unstable, for they will contain a region of negative

compressibility as shown in Fig.3.20b and 3.20c.

From the equation of state for both the upper branch and the equilibrium branch, one

can determine the total number of particles once the chemical potential at the center is

81 specified. We also find that the total particle number N changes little with as for given µ and T . It is straightforward to show that N = drn(r) has the general form N = A(T/ω)3, where A is a dimensionless number depending onR ( µ/T, λ/a ). We find that A < 1. This − − s is expected, as the critical number for Bose-Einstein condensation in a harmonic trap with

−1 3 frequency ω is Nbec = (0.95) (T/ω) [19]. On the other hand, for the trapped gas to be in the low-recombination regime, equation 3.69 imposes constraints on the central density, and hence the total particle number N. To ﬁnd an estimate of this constraint, we approximate the actual density (say that in ﬁgure 3.20a) by the Boltzmann form, which then gives

N eµ/T (T/ω)3[19]. Within the same approximation, we ﬁnd the quantity n in Eq.(3.69) ∼ to be n = 33/4eµ/T /λ3, which then implies

N < N ∗ = α(T/ω)2.5, α = 33/4 −1/2 = 1.024. (3.77) C

3.4 Two dimensional Fermi gases with spin imbalance

In this section, we study the interacting two-component Fermi gases with equal mass in two dimensions, with spin imbalances. These studies are related to recent experiments in

Cambridge studying the attractive and repulsive Fermi polarons in two dimensions[40]. The repulsive polaron problem attracts great attention since it is relevant to the existence of spontaneous ferromagnetism in the presence of large repulsive interactions in two-component

Fermi gases, namely the Stoner ferromagnetism. The question of the existence of a Stoner ferromagnetic phase can be formulated as follows. The minority particles merge into a background of majority particles and interacts repulsively with them. By comparing the self energy of such minority particle to the energy of adding another majority particle (i.e. the chemical potential of the majority) to the system, one determines if the phase separation between species is energetically favorable.

The Cambridge experiment measures several physical quantities of the two-dimensional

Fermi gas with large spin imbalance. The most interesting observation is the interaction energy from rf-spectroscopy measurements. While the attractive branch has a set of high quality data on the attraction energy, the repulsive branch shows a ﬂuctuating repulsive

82 energy that is not even monotonic as the interaction strength increases, as shown in ﬁgure

3.21. Also, the population of the majority species ranges in a region no more than 90% of the total number of particles[76], and it is not certain that the polaron limit works well in this polarization or even below. These motivate us to study the thermodynamic quantities of the dilute gases in two dimensions. From the discussion in the previous section, the equation of state measurement from imaging the density proﬁles in the trap can be used to deduce the interaction energy. It is also possible that one could calibrate the experiments by predicting the behavior of the Fermi gas with arbitrary spin imbalance. Prior to our study, there have been calculations on self energy and spectral functions[77, 78], as well as some pioneered works in trial wave functions for polarons and molecules[79].

In the following, we will show some of our results and part of our conclusions[80].

By using a generalized Nozieres Schmitt-Rink (NSR) method, we examine the properties of both the attractive and repulsive branches in the systems. We study thermodynamic quantities, including the compressibility and spin susceptibility of systems with diﬀerent polarizations, as well as at various temperatures. We ﬁnd from these features that evidence of spontaneous ferromagnetism is absent in the high-temperature regime. We also study the instability of the repulsive branch, and map out the stable region in the interaction strength versus temperature plane.

3.4.1 Fermi gases in two dimensions

Now we consider a two-component Fermi gas conﬁned in a two-dimensional harmonic trap.

Experimentally, this is realized by applying an extremely tight trap in the z-direction with frequency ωz, such that ~ωz EF ,T where EF is the Fermi energy in two dimensions, and T is the temperature. In this limit, the system is eﬀectively two-dimensional, since the motion in the z-direction is frozen to the lowest harmonic oscillator state in the tight trap.

The low-energy subspace is projected to a two-dimensional system.

The zero-range model in two dimensions is diﬀerent from that of a three-dimensional system. It is known that in the vacuum, a two-body bound state exists with arbitrarily small attractive interaction. This scattering problem can be universally expressed by a

83 Figure 3.21: The measurements in the Cambridge experiment: the upper panel shows the interaction energy for the attractive branch; the lower two panels are for the upper branch, with the ﬁrst one being the lifetime of repulsive polarons, and the second one being the interaction energy.

2 2 single parameter: the bound-state energy in vacuum is Eb ~ /ma where a2d refers to ≡ 2d the scattering length in two dimensions (2D)[81]. The scattering amplitude for the two-body

problem in 2D is E f −1(k) = ln b + iπ = 2 ln(ka ) + iπ. (3.78) E − 2d The scattering amplitude has a maximum magnitude at ka2d=1. In the presence of a medium with spin imbalance, we introduce a mean chemical potential µ and an eﬀective

84 magnetic ﬁeld h, such that µ↑ ↓ = µ h/2 are the diﬀerent chemical potentials for the two , ± species of fermions. Using a similar approach as in the previous sections, the T -matrix can be expressed by

−1 + γ(q, k) 1 T (ω + i0 , q) = + 2 + 2 , (3.79) ω + i0 ω(q) k Eb + k Xk − − where γ(q, k) = 1 n↑(q/2 + k) n↓(q/2 k) and n↑, n↓ are the Fermi distribution − − − functions for up and down spins respectively. The expression 1 n↑ n↓ and represents the − − Pauli blocking from the fermionic media. ω(q) = q2/4 2µ is the onset of the scattering − continuum, Eb > 0 is the two-body bound-state energy in vacuum. In two dimensions, the bare coupling constant is regularized in the following way to cancel the divergence:

1 1 = 2 . (3.80) −g2d Eb + k Xk The ultraviolet divergence is logarithmic in two dimensions, instead of being linear as in 3D.

The expression in 3.79 goes back to 3.78 in the extremely dilute limit, i.e. when γ(q, k) 1. → In the experiments for a quasi-two-dimensional system, where the Fermi gas is in a pancake

shape trapped very tight in one of the directions, the eﬀective interaction parameter a2d is related to the s-wave scattering length in three dimension as[82]

a l exp ( π/2l /a ), (3.81) 2d ∼ 0 − 0 s p where as is the scattering length in 3D, and l0 is the harmonic length in the direction of the tight trap. Since the 3D inverse scattering length 1/as is tunable over a large range via

Feshbach resonances, the range of the eﬀective interaction parameter log(kF a2d) is orders of magnitude.

The origins of the attractive and repulsive branches in two-dimensional gases are similar to those in their three-dimensional counterparts. When the system reaches thermal equilibrium, the equation of state consists of the following contributions: a free-fermion part and an interaction part. From the nature of the attractive interaction and the formation of molecules, this “attractive branch” or “equilibrium branch” shows a negative interaction

85 energy and strong stability. The repulsive branch in 2D is also prepared by putting the gas in the scattering states only. This metastable state is free of molecules, and is usually called the “upper branch”. They are distinguished from each other by taking into account diﬀerent contributions to the equation of state, namely:

sc bd nlower(µ↑, µ↓,T ) = n0↑ + n0↓ + ∆n + ∆n , (3.82)

sc nupper(µ↑, µ↓,T ) = n0↑ + n0↓ + ∆n , (3.83)

where n0↑, n0↓ are the densities of the ideal Fermi gas with chemical potential µ↑, µ↓. The interaction part of the particle number can be written as a sum of two parts: ∆nsc is

the scattering part, and ∆nbd is the bound-state part. By running similar routines for

excluding the bound-state molecules in the system, one can address the attractive and

repulsive branches of the 2D fermi gases separately. The explicit expressions for ∆nsc and

∆nbd are

1 ∞ dω ∂ζ(q, ω) ∆nsc(µ, T ) = n (ω) , (3.84) −Ω π B ∂µ q ω(q) X Z 1 ∂ω (q) ∆nbd(µ, T ) = n (ω (q)) b , (3.85) −Ω B b ∂µ q X where ζ(q, ω) is the phase angle of the T -matrix in a medium, and ωb(q) is the pole of

2 the T-matrix which represents the bound state. For q /4 > µ↑ + µ↓ = 2µ, the Pauli blocking eﬀect is not strong enough that molecules with center-of-mass momentum q always exist[83]. As we have spin imbalance in the system, there is another set of equations for the “magnetization”, i.e. for M = n↑ n↓. A corresponding “polarization” m = − (n↑ n↓)/(n↑ n↓) is deﬁned accordingly. The equations for magnetization are − − 1 ∞ dω ∂ζ(q, ω) ∆M sc(µ, T ) = n (ω) , (3.86) −2Ω π B ∂h q ω(q) X Z 1 ∂ω (q) ∆M bd(µ, T ) = n (ω (q)) b . (3.87) −2Ω B b ∂h q X The total magnetization is given by the interacting part above plus the non-interacting

86 part, i.e.

sc bd Mlower(µ↑, µ↓,T ) = M0 + ∆M + ∆M , (3.88)

sc Mupper(µ↑, µ↓,T ) = M0 + ∆M , (3.89)

where M = n ↑ n ↓ is the magnetization from the non-interacting part. The “polariza- 0 0 − 0 tion” m M/n is determined accordingly. ≡ In this section, we use this prescription of equations 3.82, 3.83, 3.88 and 3.89 to determine

the chemical potential and the polarization of the system. Thermodynamic quantities for

both branches at arbitrary temperature and polarization are consequently calculated. The

values of µ↑ + µ↓ are always restricted to negative numbers in our study. This condition is satisﬁed by going to either the high-temperature regime for systems with any spin imbalance,

or very polarized gases at any temperatures (the polaron limit)10.

3.4.2 Thermodynamic quantities of two component Fermi gases in two dimension

In order to explain and calibrate the experiments, we focus on several thermodynamic

quantities of this two-dimensional Fermi gas with spin imbalance. We ﬁrst calculate the

interaction energy for a spin-unpolarized system. From the scattering amplitude expression

3.78, the eﬀective interaction strength is in the logarithm of a2d. In the many-body system,

we choose the interaction strength η = ln(kF a2d). The large value of ln(kF a2d) corresponds

to a weakly-attractive case, since Eb is small. The attraction strength increases as η moves leftward on the η-axis. On the other hand, for the upper branch, the repulsion increases as

η becomes larger, however the system also becomes unstable as it approaches the strongly

repulsive regime. We will address these observations in this section.

Using the approach introduced in the previous section, we calculated the energy density of two-dimensional Fermi gases in the repulsive and attractive branches. For negative chemical potentials µ↑ = µ↓ < 0, bound states appear at all center-of-mass momentum q,

10 It is a mathematical problem that we have to restrict µ↑ + µ↓ < 0, in order to avoid divergences in the number equations for upper-branch calculations.

87 ⌘c 0.25

0 0 E ê int E -0.25

-0.5

-1 -0.5 0 0.5 1 ln kFa2 d

Figure 3.22: The interaction energy for the attractive branch (red curve) and the repulsive H L branch (blue curve) at a high temperature T = 6TF , for a system with equal spin populations. The ending point of the repulsive branch is around η = ln k ac 0.4 at this c F 2d ≈ − temperature and polarization.

and we always have both attractive and repulsive branches. The energy of the attractive branch decreases as a2d decreases, and it is a consequence of the larger attractive interaction from more deeply-bound molecules. As the true ground state of the system, this branch remains stable for all η values. In contrast, the metastable repulsive branch cannot have inﬁnite repulsion energy as η increases. In fact, it becomes unstable at some point as the repulsion strength reaches a certain value. This “critical point” is dependent on the temperature and polarization. To the right of this critical point ηc, the upper branch does not exist in this temperature and polarization regime. Mathematically, it corresponds to the absence of solutions µ and h to the number and polarization equations 3.83 and 3.89.

The energy of both the attractive and repulsive branches is shown in ﬁgure 3.22. The results for the interaction energy of equal-spin-population systems are consistent with previous experimental[40] and theoretical studies[77, 78] on the spectral functions for the two- dimensional polaron problem. In the spectral function calculations for upper-branch polarons, there is an increased broadening as the repulsion increases, and it corresponds to a shorter lifetime of quasiparticles.

We also examined the compressibility of the repulsive branch. In the weakly-repulsive case, a decrease in the compressibility is predicted from perturbation theory. One can

88 think of the interacting gas as becoming more “rigid” as we turn up the repulsion. This decreasing feature extends in the direction of stronger repulsions, until the compressibility

c c reaches zero at a critical value of a2d. This a2d is the same as the critical scattering length for the termination of the upper branch, as discussed in the previous paragraph. The physical interpretation is that when the repulsive interaction is getting larger, the radius of the atoms is getting larger in the hard-sphere picture. The system ﬁnally becomes incompressible as the atoms are closely packed, as is shown in the lower panel of ﬁgure 3.23. The solution for the chemical potential from the number equation 3.83 is absent above this critical value, hence we conclude that the stability of the repulsive branch ends here. Within the stable

c region a2d < a2d, we also calculate the spin susceptibility of the repulsive two-dimensional gases. As in the upper panel of ﬁgure 3.23, we show that the susceptibility rescaled by the susceptibility of the noninteracting gas increases monotonically, corresponding to larger

ﬂuctuations in spins from stronger repulsions. However, the largest value of χ/χ0 is only around 1.5 within the stable region of the repulsive branch, indicating that a ferromagnetic transition is absent in this high temperature regime.

The instability of the upper branch at all temperatures and polarizations is studied in the following. For a system with unequal chemical potentials µ↑ = µ↓, there is an imbalance 6 of population in different spin species. By adjusting the two chemical potentials, we can − have gases with different “polarization” defined as m = n↑ n↓ . Similarly, we calculate n↑+n↓ the interaction energy as a function of the interaction strength for both the attractive and repulsive branches. Since the interaction potential is interspecies only, we expect a suppression in the interaction energy when a spin imbalance is turned on. Also, the spin

c imbalance will push the stable region further towards the stronger repulsion side (larger a2d for finite m). In figure 3.24, we show the stability boundaries for different polarizations at different temperatures. We can clearly see that the system can be more stable at higher temperatures and in the more spin-imbalanced case, because both of these give a suppression in the repulsive energy of the 2D gas.

89 1.5 0 c ê c 1.25

1. -1.1 -1. -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 1. ln kFa2 d

H L 0 k ê

k 0.5

0 -1.1 -1. -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 ln kFa2 d

Figure 3.23: Spin susceptibility (upper panel) and compressibility (lower panel) of the repul- H L sive branch at a high temperature T = 6TF , rescaled by the noninteracting susceptibility χ0 and compressibility κ0 at this temperature. As the repulsive interaction increases, the system becomes more and more rigid with a lower compressibility value. At ln k ac = 0.39, F 2d − it becomes “incompressible” such that the upper branch will collapse if the repulsive interaction is further increased. This is the corresponding critical point for stability in the upper branch. Within the stable region ln kF a2d¡-0.4, χ/χ0 is at most 1.5, indicating the absence of a ferromagnetic transition at this temperature.

3.5 Conclusions

In this chapter we discussed several important questions in strongly-interacting quantum gases. We focused on the origins of strong attractions in Fermi gases, as well as a systematic way to describe the metastable upper branch in both Bose and Fermi systems.

For the attractive Fermi gases, we explored a new possibility in realizing a superﬂuid phase with a large Tc/TF ratio by using narrow Feshbach resonances. We correct the common misbelief that the quantum gases across narrow resonances are weakly-interacting unless being brought very close to the resonance. By calculating the attraction energy of the lower branch at low temperatures, we showed that the systems near narrow resonances

90 6 æ æ æ

æ æ æ

5 æ æ æ

F æ æ æ

T Stable Unstable

T 4 æ æ æ

æ æ æ m=0.8 m=0æ æ 3 m=0.5

-1.5 -1. -0.5 0 0.5 lnHkFa2 dL

Figure 3.24: A diagram of “stability” of the repulsive branch: the three diﬀerent curves correspond to the κ = 0 contour at diﬀerent magnetizations m = 0, 0.5, 0.8. To the left of the curves, the repulsive branch of a two-dimensional gas is stable; the repulsive branch has zero compressibility at the boundaries and collapses to the right. The upper branch has a larger region of stability at higher temperatures or spin imbalances, because of the suppression of repulsive interactions in the system.

can be even more strongly attractive than those near wide resonances, even when they

are several widths away from the resonance. The prediction of an asymmetric interaction

energy in the upper and lower branches has been observed in recent experiments.

For the upper branch of quantum gases, our calculations are based on a recent developed

generalized ladder approximation which excludes the bound-state pole in the scattering T -

matrix. This approach enables one to compute the thermodynamic quantities of the upper

branch in both Bose and Fermi gases. We ﬁnd in Bose gases at high temperatures that the

stable upper branch can penetrate through unitarity from the positive-scattering-length side

in both homogenous and trapped systems. This upper branch will mechanically collapse

when it hits a negative critical value of the inverse scattering length.

In the ladder approximation used in this chapter, the scattering vertex does not include

91 the higher order contribution from the GMB correction illustrated in 3.1.4. Although including these diagrams gives a signiﬁcant change in the transition temperature in the BCS limit, it is unlikely to aﬀect qualitatively the thermodynamic quantities in the upper branch at high temperatures. Also in this chapter, we only calculated thermodynamic quantities from the grand canonical potential. The calculation of spectral functions is left to further studies.

92 Chapter 4 Rotating gases, synthetic gauge fields, and quantum Hall physics in neutral atoms

In the previous chapters, we discussed strongly attractive and repulsive quantum gases and their high-temperature superfluid phases. These superfluids with broken U(1) symmetry are among the most studied areas in cold atoms. In addition to these types of broken- symmetry phases, there is another class of macroscopic quantum phenomena which cannot be classified by any local symmetry breaking, namely a class of nontrivial phases in the absence of a local order parameter in Landau’s phase transition paradigm. The most strik- ing example is the quantum Hall effect studied since the 1980’s, in which it is the global topological properties of the system that distinguish the quantum Hall states from other trivial states[84]. These topologically protected states have been candidates for realizing the basic units for performing quantum computation— the quantum bits (Qubits)[85].

In terms of cold atoms, it was found out shortly after the realization of BEC in dilute gases that by rotating a trapped quantum gas in two dimensions, the single-particle hamiltonian resembles that of an electron conﬁned in 2D in the presence of an external magnetic

field perpendicular to the plane[16, 86, 87, 15]. In such electronic systems, the gauge potential is coupled to the charged particles. Nearly flat bands, i.e. the Landau levels, are formed in the limit that the rotating frequency approaches the trapping frequency. There have been significant efforts to use this setup to study quantum Hall physics in neutral atoms. Not only can the conventional quantum Hall states be studied, but the tunable interactions as

93 well as the hyperﬁne degrees of freedom in cold atoms open up new directions in quantum

Hall physics. Unfortunately, efforts to realize the quantum Hall regime have not yet suc- cessfully reached this regime in cold atomic gases. One of the primary reasons is that it is a daunting task to stabilize the rotation very close to, but not exceeding the trapping frequency such that the particles can still be confined. Recently, there has been a promising proposal of using the Raman coupling scheme to realize a synthetic gauge field that couples to the neutral atoms[88]. In this scheme, the system is free of risks of “over-rotation”.

In this chapter, we focus on properties of quantum Hall states in neutral atoms in the context of the aforementioned NIST scheme for synthetic gauge ﬁelds. After giving a brief review of rotating BECs and quantum Hall physics, we introduce the NIST setup of generating synthetic gauge ﬁelds. We will then report on our work on vortex states in BECs and quantum Hall states with anisotropy. We also propose some experimental methods to clearly identify the existence of quantum Hall states.

4.1 Rapidly rotating Bose-Einstein condensates and quantum Hall physics

In this section, we review the properties of a rotating Bose-Einstein condensate in a trap and how they are related to different quantum Hall states. The integer quantum Hall effects originate from the flat band structure of the single-particle spectrum, and the fractional quantum Hall effects are induced by interparticle interactions. These can be realized in trapped cold atoms as the gauge potential is generated by a rotation. This rotating scheme is described briefly in the following. In a rotating Bose-Einstein condensate, vortices nucleate as one reaches some finite value of the rotation frequency Ω. As Ω increases, more and more vortices appear, and they form a dense (triangular) array of lattices[16]. In the limit that the rotation frequency approaches the radial trapping frequency ω⊥, a simple lowest- Landau-level (LLL) approximation is applicable. The condensate is then destroyed, and the vortex lattice is melted. The system will finally become a strongly-correlated phase analogous to the fractional quantum Hall state of a two-dimensional electron gas in the

94 presence of a strong perpendicular magnetic ﬁeld.

4.1.1 Quantum Hall physics, Laughlin wavefunctions

The classical Hall effect was predicted and observed in the 19th century: in some two- dimensional electronic systems, the transverse resistance is linear as a function of the perpendicular magnetic field that is applied. This transverse resistance is named the Hall resistance. In 1980, it was first observed by von Klitzing[89] that in the presence of a strong perpendicular magnetic field (in the order of Teslas), quantized plateaus in the Hall resistance (RH) appear in two-dimensional electron systems (2DES) at some integer values of ν as h R = . (4.1) H νe2 The emergence of these plateaus is the integer quantum Hall effect (IQHE). In later experiments, plateaus at fractional values of ν (e.g. 1/3, 2/5 etc) have been observed[90], and they are called the fractional quantum Hall effects (FQHE). To understand the origins of these effects, we first write down the single particle hamiltonian of the system

1 eA 2 H = p + gµB S, (4.2) 2m∗ − c · ∗ where m is the effective mass of the electrons in the corresponding materials, µ = e~/2mc is the Bohr magneton, and g = 2.002... is the spin g-factor for electrons. B and A are − the external magnetic field and its gauge potential respectively. This hamiltonian gives the so called “Landau problem”, and there is a gauge degree of freedom in choosing the gauge potential A, as long as it gives the correct physical magnetic field B = A. Different ∇ × choices of gauge give different forms of wave functions, however the energy spectrum and the expectation values of other self-adjoint operators are independent of the gauge choice.

For the system we are interested in where the magnetic ﬁeld is perpendicular to the x-y plane as B = Bzˆ, we choose the “symmetric gauge” in this chapter, namely A = eBy/c x − and Ay = eBx/c. In the strong-ﬁeld limit, the spin degree of freedom is frozen by the large Zeeman

95 splitting, hence we can project the hamiltonian to a spinless space. The solution to the reduced single-component Hamiltonian (neglecting the Zeeman term) is a set of discrete

eB~ Landau levels, separated by the energy spacing ~ωc = mc . This energy is given by the frequency of the electronic cyclotron motion, and it is natural since microscopically the electrons are doing small circular motions in the 2D plane caused by the Lorentz force. The

~c classical radius of the cyclotron motion is l = eB , and this characteristic length scale is deﬁned as the magnetic length. For each Landauq level, it has a degeneracy per unit area as

1 ρhc 2πl2 . Thus, the eﬀective “ﬁlling factor” of the sample is eB , where ρ is the two dimensional density of electrons. The integer ν at which the IQHE appears is exactly the same number

of the ﬁlling factor. The general expression for the eigenstate wave functions takes the form

1 |z|2/4l2 m+n n m −|z|2/2l2 ψm,n = e (√2l) ∂z ∂z¯ e , (4.3) √2πm!n!l2 where the complex number z = x+iy denotes the two dimensional position of the electrons.

The energy eigenvalue is Em,n = (n + 1/2)~ωc. The n = 0 Landau level is called the lowest Landau level (LLL).

In real experiments, these 2DES’s are usually realized by semiconductor heterostructures with high purity. The carrier density is made very dilute and the carriers have long mean free paths. In these semiconducting materials, the eﬀective mass of an electron is typically around one-tenth of the bare electron mass, and they have a relatively high dielec- tric constant 10. Consequently, the cyclotron energy gap for a 10 Tesla magnetic ﬁeld ∼ is around 10−2eV (10K). The experiments are always done at low temperatures(less than

1K).

The simplest picture of IQHE is given by the flat band structure of the single-particle spectrum, as well as the presence of some bulk disorders or the surface effect. The spectrum of Landau levels is broadened by disorders, and the degeneracy is lifted into a continuous band near the unperturbed energy levels, as is shown in figure 4.1. These extended bands are separated by localized states which do not contribute to transport, hence IQHE is presented when the chemical potential lies in between these “mobility gaps”[91, 92]. A more careful

96 Figure 4.1: the origin of integer quantum Hall eﬀects. In the presence of impurities (disorders), the originally degenerate Landau levels are broadened in energy space. Only the spacial extended states contribute to the transport (Hall conductance), and they are separated by some localized states, i.e. the mobility gap. When the chemical potential changes in the region of the mobility gap, the Hall conductance remains unchanged. An IQHE plateau is consequently observed. Figure adapted from [91].

argument is given by the topological properties of the band structure[93]: the Landau levels are topologically non-trivial as they have integer Chern numbers. At the edge (surface) of the samples, they are connected to topologically trivial vacuum, and the corresponding number of level crossings at the chemical potential determines the plateau in Hall resistance

ν in 4.1.

The fractional QHEs are generally much more complicated, and there are still a great amount of open questions in this field. What makes them more sophisticated is that different fractionals may come from different microscopic origins. In general, the emergence of the fractional plateaus are attributed to interparticle Coulomb repulsion. The repulsion between electrons further split the many-body spectrum inside the Landau levels, and each stable many-body “sub-level state” may correspond to FQHEs with different filing factors. We will not go into theoretical details in FQHEs, as they can be found in reference [91] and references therein. Instead, we limit our discussions to a “phenomenological” level by introducing a class of trial wave functions which are proposed to be the ground states of fractional quantum

Hall states.

This class of wave functions was ﬁrst introduced by Laughlin[94] to explain the existence

97 of the ν = 1/3 QH state. We write down the ground state wave function at ﬁlling number

ν = 1/m for electron gases where m is an odd integer (quantities with dimension of length are rescaled by units of the magnetic length ~c/eB): p m − 1 P |z |2 Ψ (z z ) e 4 i i , (4.4) 1/m ∼ j − k j

4.4 has odd integers m for fermionic systems, and even m for bosonic systems.

4.1.2 Rotating Bose-Einstein condensates, vortex array and quantum Hall regime

The rotating cold atomic gases can assume the aforementioned quantum Hall states in suitable situations. For a trapped Bose-Einstein condensate in the presence of a constant rotation with frequency Ω, it is more appropriate to switch to the co-rotating frame with the same frequency instead of the laboratory reference frame. The wave functions in these two reference frames are connected by a unitary transformation ψ˜ = eiΩ·Lt/~ ψ , where L is | i | i the angular momentum operator. The corresponding Schr¨odingerequation in the rotating frame gives

˜ iΩ·Lt/~ iΩ·Lt/~ i~∂t ψ = e i~∂t ψ + (i~∂te ) ψ | i | i | i = eiΩ·Lt/~ e−iΩ·Lt/~ ψ˜ Ω L ψ˜ ˜ ψ˜ . (4.5) H | i − · | i ≡ H| i

98 From the equation above, one ﬁnds the new hamiltonian of the system in the rotating frame as

˜ = eiΩ·Lt/~ e−iΩ·Lt/~ Ω L. (4.6) H H − · The ﬁrst term is the unitary transformation in the time domain, and will usually be static

(time-independent) in the rotating frame, while the second additional term is the counterpart of the Coriolis force in classical mechanics. This second term clearly favors non-zero angular momentum states in the rotating frame. Usually in the cold atom experiments, an overall trap is applied to conﬁne the atoms, and in the presence of the trapping potential, the full single-particle hamiltonian is written as:

p2 m(ω r)2 (p mΩ r)2 m(ω r)2 ˜ = Ω L + 0 · − × + · , (4.7) H 2m − · 2 ≡ 2m 2 in which ω0, ω are vector forms of the original and eﬀective remaining trapping frequencies.

The vector form of the trapping frequencies is understood as ω = (ωx, ωy, ωz). For a speciﬁc

2 2 2 case where Ω is in the z-direction, we have ω0i = ωi + Ω for i = x, y, and ω0z = ωz. Hence in order to keep the gas conﬁned, the rotating frequency cannot exceed the original trapping frequency in any of the x, y directions. In the situation that ω Ω, ω , ω , the single- z x y particle motion in the z-direction is frozen to the lowest harmonic state, and the system

is eﬀectively in two dimensions. First we take an example of an axisymmetric trap with

ω0x = ω0y. The hamiltonian 4.7 in this case, can be rewritten in a way that is similar to the Landau problem 4.2:

(p mu)2 ˜ = ⊥ + = − + (ω Ω)L + (4.8) H H Hz 2m 0 − z Hz where ⊥ and are separable hamiltonians acting on the x-y plane and in the z direction, H Hz respectively. u = ( ω y, ω x, 0) is the effective velocity field experienced by neutral atoms. − 0 0 This velocity field is the same as the gauge potential coupled to charged particles. The

“cyclotron motion” frequency here is 2ω0. The eigenstates of this hamiltonian are the same as those of the Landau problem as shown in 4.3, however the additional term (ω Ω)L 0 − z lifts the degeneracy. The new eigenstates—the Fock Darwin states—and the eigenenergies

99 Em,n n =1

m =0 !0 m =1 ...... µ

n =0 ! ⌦ 0

Figure 4.2: A sketch of the energy levels of Fock Darwin states. Different lines denote the different quantum numbers of n and m, and the intersections are the eigenstates. n is the Landau level label, and the spacing between them is ~ω0; m is the angular momentum label, and their splitting is ~(ω0 Ω). In the rapidly-rotating limit, namely when ω0 Ω ω0, − − each Landau level is almost flat, resembling the quantum Hall regime in electronic systems.

are

1 |z|2/2l2 m+n n m −|z|2/l2 ψm,n = √ e l ∂ ∂ e Θ0(z), (4.9) πm!n!l2 z z¯ ×

Em,n = (n + 1/2)~ω0 + (m + 1/2)~(ω0 Ω), (4.10) − in which l = ~/mω0 is the new deﬁned “magnetic length”, Θ0 is the lowest harmonic oscillating statep in the tightly trapped in z-direction. The n = 0 states are still denoted by the lowest Landau level, and the good quantum number m is the angular momentum of the eigenstates. In this problem, the inter-Laudau-level energy spacing is ω0, and the intra- Landau-level spacing is ω Ω, as is shown in ﬁgure 4.2. Within the LLL, the distribution 0 − of wave functions for large enough m is close to a ring with radius √ml. The larger angular momentum m~ it has, the larger the ring is and since there is a remaining trapping frequency, it has a larger energy.

In the situation of rapid rotations, if the number of states in the LLL which have lower

100 energies than the second Laudau level is comparable to the number of particles, a non- interacting fermionic system reaches the quantum Hall regime. This is illustrated in ﬁgure

4.2. For a repulsive Bose condensate, the LLL regime requires that the chemical potential is lower than the lowest-energy point of the second Landau level, for instance as the position of the horizontal dashed line in ﬁgure 4.2. The chemical potential for such a system with weak repulsions at zero (low) temperature is estimated by the Gross-Pitaevskii energy functional.

The energy of a condensate is written by

E[ψ] = dr ψ∗(r) ˜ψ(r) + g ψ(r) 4 . (4.11) H 2d| | Z 2 In the mean-ﬁeld level of the interaction, g2d = √8π~ as/maz, where as is the s-wave scattering length for repulsive fermions, and az as the harmonic length in the z-direction. The ground state of a condensate is given by minimizing the following quantity:

K[ψ] = dr ψ∗(r)( ˜ µ)ψ(r) + g ψ(r) 4 , (4.12) H − 2d| | Z where the chemical potential is determined by another number equation for a gas with total number of particles N. For a system with a small number of vortices N N, a large v number of Landau levels in the single-particle space are occupied. With faster and faster rotations, Nv increases and as soon as it becomes comparable to the number of particles N N, the chemical potential drops below the second Landau level[87]. In this situation, v ∼ the rotating BEC is driven into the LLL regime, and the wave function in the x-y plane can be written as

2 2 ψ (z) (z z )e−|z| /2l , (4.13) v ∼ − i Yi in which zi = xi + yi are the positions of the vortices in two dimensions. By implementing this form of trial wave function to the minimization of 4.12, one ﬁnds that the triangular lattice of vortices has the lowest energy[95]. The triangular vortex array similar to the

Abrikosov lattice in type-II superconductors in the presence of an external magnetic ﬁeld has been experimentally observed[16]. Finally, when the number of vortices approaches the number of particles, the vortex lattice melts, and all the vortices become invisible. This

101 crossover or transition to the quantum Hall states, in which the quantum vortices are bound to the particles themselves, has not been clearly observed.

In the previous paragraphs, we related the trapped rotating condensate to the quantum Hall states. We note that this can only be realized by tuning the rotating frequency extremely close to the trapping frequency. A back-of-the-envelope estimate from the Fork

Darwin spectrum is given in the following. In order to accommodate N particles in the

LLL, one needs at least N states in the LLL to have lower energy than the second LL. This gives a criterion that N(ω Ω) < ω . For a system with 103 particles, it is required to keep 0 − 0 the rotating frequency stable in a region that 0.999ω0 < Ω < ω0. So far the experimental capability of rotating is only to approach to within about 99% of the trapping frequency[15].

4.2 Synthetic gauge ﬁeld scheme

Facing the difficulties in rotating trapped atoms to make the flat single-particle bands that are crucial for quantum Hall states, there are many alternative proposals to create a synthetic gauge field[96]. The major advantage of these approaches is that they are free of the risk of “over-rotation”: if the control on rotation is not precise enough, the centrifugal potential may exceed the original trapping potential and the particles will no longer be confined. Among these methods the two-photon Raman coupling synthetic gauge field scheme developed by the Spielman group in NIST (which we will refer to as the “NIST scheme” from now on) is a promising approach to generate both Abelian and non-Abelian gauge fields in neutral atoms[88, 97].

In this section, we will ﬁrst introduce a general scheme for generating Abelian gauge

fields in neutral atoms with spin from the Berry phase, and then discuss the NIST scheme, in which both Abelian and non-Abelian gauge fields can be realized. In the Abelian case, neutral atoms will act like particles with electric charge in a magnetic field, and vortices in such condensates have been observed experimentally. For the non-Abelian case, we will show that multi (two)-component atoms will experience an effective spin-orbit coupling.

102 4.2.1 Berry phase in adiabatic states, Abelian gauge ﬁeld

In this part, we discuss a general approach to generating an Abelian gauge field coupled to neutral atoms. The idea is best illustrated by considering the motion of a particle with spin in a spatially varying magnetic field. In the presence of such a magnetic field, the low-energy manifold of the spin system, in some cases, may consist of only one state that is either the locally aligned or anti-aligned spin state to the external magnetic field. In the large-field limit, this single state is well separated energetically from the other states, and will “adiabatically” follow the orientation of the external magnetic field. This “adiabatic state” then experiences a Berry phase, which is equivalent to a gauge potential coupled to the neutral atoms. We will derive the general form of the Berry phase in these “adiabatic states”, and give an example of the explicit form of the gauge field for a spin-1 particle.

Consider a system of total spin-F particles in a spatially varying magnetic ﬁeld. The hamiltonian takes the form:

2 ~ † † † = ψ ψ + V (r)ψ ψ + gµ ψ B F ψ + , (4.14) H 2m ∇ α · ∇ α α α B α · αβ β Hint α X Xα,β where ψ’s are the vector ﬁeld operators with 2F +1 components, and the sum of α, β is from

F to F . F is the matrix representation of the spin operator, B = B(r) is the spatially − varying magnetic field, and is the interaction hamiltonian. We decompose the vector Hint F (m) (m) field into a basis of local eigenstates of the Zeeman term as ψα = m=−F φ ηα , where φ is the new field operator and the vectors η are the local eigenvectorsP of the Zeeman term:

B F η(m) = mη(m). (4.15) · αβ β α In the situation with a slow varying magnetic ﬁeld that is such that the spin degree of

freedom is frozen by a large Zeeman splitting, the low-energy manifold of the system consists

of one single local eigenstate of the spatially varying Zeeman term. Although the state

is spatial dependent, it remains an eigenstate of the local Zeeman term by adiabatically

following the variation of the external ﬁeld. Within this “adiabatic approximation”, the

103 kinetic part projected to this only state m = F , for instance, in the system is

2 2 ~ † ~ † † ψ ψ = (φ η ) (φη ) 2m ∇ α · ∇ α 2m ∇ α · ∇ α α α X X 2 ~ † † † † = ( i i η η )φ 2 + ( η η η η 2)φ φ 2m | − ∇ − α∇ α | ∇ α · ∇ α − | α∇ α| " α α α # 1 X X X = (p A)φ 2 + φ†φ, (4.16) 2m| − | W in which we already dropped the index m = F . As we see, the original kinetic term is modified, due to Berry’s geometric phase term acquired from the self-adjustment of the adiabatic state in its spin space. Figure 4.3 is a sketch of how the “adiabatic state” tries to follow the orientation of the external field and thus experience the gauge field, or “force” in the classical picture. There is an effective gauge field term A from this Berry phase and also an additional potential : W

† A = i~ η ηα, (4.17) α∇ α X 2 ~ † † = η η η η 2 . (4.18) W 2m ∇ α · ∇ α − | α∇ α| α α ! X X Besides the diagonal matrix elements in the same spin states, the decomposition of the multi-component ﬁeld operators also generates a cross term, which corresponds to the coupling between diﬀerent local eigenstates. The explicit form of the cross term between m and n adiabatic states is

= ( ηm†)ηn φ† φ + ( ηm† ηn)φ† φ + h.c.. (4.19) Hm,n ∇ α α · m∇ n ∇ α · ∇ α m n α X This off-diagonal hamiltonian above will generate a matrix element between the different adiabatic states, and trigger a transition between them. By evaluating these matrix elements, one can estimate the transition rate from one adiabatic state to another. For † † example, the pre-factor of the direct transition term φmφ is proportional to ηm η . If n ∇ · ∇ n the characteristic wave vector of the varying magnetic field is q, the adiabatic approximation is only valid when the energy scale of this transition R q2 is small compared to the ∼ absolute energy separation ∆ between the lowest-energy adiabatic state to the other ones in

104 m=F B …

m=-F

Figure 4.3: A sketch of the origin of the gauge field in the presence of the spatial dependent magnetic field (bold blue arrows): There are 2F + 1 locally defined eigenstates in the spin space (denoted by all the thin red arrows). As the magnetic field smoothly varies, in the case that only the completely aligned maximum spin state (thin green arrows) are populated in the system, the adjustment of this adiabatic state in spin space acquires a phase over the course of a cycle, which gives the gauge field in the kinetic term. This is similar to the fictitious (inertial) force in a non-inertial reference frame in classical mechanics.

the high-energy manifold. However, in the extreme case that the magnetic ﬁeld is uniform and η vanishes, although no transition happens, the eﬀective gauge potential A vanishes ∇ as well. Fortunately the ratio ∆/R can be tuned to be large by increasing the magnitude

of B, while keeping its variance in orientation in this scheme.

Now take a speciﬁc example of system with spin-1 bosonic atoms, in a spatially-dependent

magnetic ﬁeld B(r) = B(sin θ cos φ, sin θ sin φ, cos θ), where the Euler angles θ = θ(r) and

φ = φ(r) are spatially dependent. We assume that the local maximum spin m = 1 eigenstate

1 −iφ 1 1 iφ T ηm=1 = ( (1 + cos θ)e , sin θ, (1 cos θ)e ) (4.20) 2 √2 2 − is the only adiabatic state prepared in the system. From the previous derivation of the form of the gauge ﬁeld and the remaining potential equation (4.17) and (4.18), we have the explicit expressions of the gauge ﬁeld and the remaining potential as

A = ~ cos θ φ, (4.21) ∇ ~2 cos θ 2 1 cos2 θ = |∇ | + − φ 2 . (4.22) W 2m 2(1 cos2 θ) 2 |∇ | − 105 A more general expression for spin eigenstates in the spin-F manifold and the consequent gauge potential for F, m will be included in the appendix. In the next section, we will | i make use of the expression above for the F = 1, m = 1 adiabatic state to derive the eﬀective gauge potential in the experimental setup in the NIST scheme.

4.2.2 NIST scheme of Abelian synthetic gauge ﬁeld

In the two-photon Raman coupling scheme, a bosonic isotope of Rubidium 87 is used. There are two counter-propagating laser beams illuminating the cloud and coupling the three spin states: m = 0, 1 of the 5S ,F = 1 electronic ground state. The momentum diﬀerence F ± 1/2 of the two laser beams provides a momentum kick while ﬂipping the spin of the atoms,

which is the origin of the gauge ﬁelds. We will show the experimental setup of this Raman

coupling scheme, and derive the eﬀective hamiltonian for the system in a special case in

quasi-two-dimensional geometry.

The setup of the experiment is shown as ﬁgure 4.4. The two counter-propagating laser

beams along the y-direction with similar wavelength kR and kR+δk have linear polarizations perpendicular to each other such that they do not interfere. The laser beams couple different spin states in the F = 1 manifold. It transfers a momentum q 2k while the spin is ∼ R increased by one unit of angular momentum. The Rabi frequency of this Raman coupling process is Ω . A spatially non-uniform magnetic field is applied in the form B = (B + R − 0 Gz)ˆz + Gxˆx, with a field gradient G along both z and x directions. This form of the static magnetic field satisfies the equations B = B = 0. ∇ · ∇ × Taking this configuration of setup, the single-particle Hamiltonian of this F = 1 spin

system is decomposed into three parts:

= + + , (4.23) H T HR HB

where T is the kinetic energy, is the Raman coupling process from the laser beams, and HR is the Zeeman term. As the particle increases its spin by interacting with two photons, HB

106 m = 1 m =0 m =1

kR kR k y

Figure 4.4: In the NIST scheme, the two counter-propagating laser do not interfere when their polarization are perpendicular to each other. The three states in the F = 1 spin space are coupled together by the two-photon Raman process, and the transferred momentum is the difference between the red and blue laser beams. Also an external magnetic field (not shown) is applied with a field gradient along the z and x directions, providing a spatially- dependent Zeeman term.

the Raman term takes a simple form:

Ω = R (F +e−i(2kR·r+ωt) + h.c.) HR 2 = Ω (F cos(2k r + ωt) + F sin(2k r + ωt)). (4.24) R x R · y R ·

This process shifts the momentum of the particle by 2kR (2kR + δk to be precise), which is the momentum diﬀerence between the two laser beams, and the energy is shifted by

ω = gµBB0, if the Zeeman term and photon energy are ﬁnely tuned to this value. Together with the conjugate term, which is the process that the particle decreases its spin, the Raman term looks like a magnetic ﬁeld in the x-y plane, with both spatial and time dependence.

We absorb the time-dependence of the Raman hamiltonian, by transforming into the

107 rotating frame as χ = e−iωtFz ψ , and the new time-independent form is | i | i

˜ = Ω (F cos(2k r) + F sin(2k r)), (4.25) HR R x R · y R · as the spatially varying ﬁeld becomes static. In general, the Zeeman term contains a linear part and a quadratic part as:

= gµ (B + Gz)F + GxF + γF 2. (4.26) HZ − B 0 z x z

2 In the small quadratic Zeeman splitting limit, we neglect the Fz term, and the Zeeman term in the rotating frame is

˜ = gµ GzF + GxF . (4.27) HZ − B z x

By adding together the explicit forms of the two separate terms of (4.25) and (4.27), the

full single-particle hamiltonian can be written in a form:

˜ = + gµ B F, (4.28) H T B eﬀ ·

in which the eﬀective magnetic ﬁeld is

B = G(λ cos φ + x, λ sin φ, z), (4.29) eﬀ −

ΩR where λ = and φ = 2kR r. We briefly sketch the direction of the effective magnetic gµB G · field Beff in figure 4.5, in the x = 0 plane. The aligned spin states m = 1 are shown in the figure, which follow the orientation of the varying magnetic field in the adiabatic approximation.

For the quantum Hall physics, we are looking for an analogy of a two dimensional electron gas with magnetic ﬁeld perpendicular to the plane. For this goal, we assume at this point that the neutral atoms are conﬁned to a quasi two dimensional trap, with the single-particle state in the x-direction projected to the lowest harmonic oscillator state by the very tight harmonic potential.

For this quasi-2D atomic gas lying in x = 0 plane, the spatially varying ﬁeld is G(λ cos φ+ x, λ sin φ, z). From the general form of the adiabatic Berry phase for the non-uniform − 108 x

Figure 4.5: The direction of the vector Beff . On the axis of z = 0, the magnetic field (spin) direction is rotating in the x-y plane, with a wavelength q = 2kR. The vector will leave the x-y plane as the particle deviates from the z = 0 axis. The vector has a finite component along the z-direction, and this component grows linearly as a function of the coordinate z. When z > λ, the z component of the vector becomes most important, and Bz starts to dominate such that the B vector will almost be the constant vector ˆz for z λ or ˆz eff − for z λ. − magnetic field 4.21 and 4.22, the explicit expression of the gauge field is

2~z kR A = | | ˆy. (4.30) √λ2 + z2

This vector potential is along the y-direction, and varies as a function of position in the z-direction. The curl of the vector potential is then the synthetic magnetic ﬁeld:

2k λ2 B∗ = A = R ˆx, (4.31) ∇ × (λ2 + z2)3/2 which is indeed perpendicular to the 2D atomic gas. The synthetic field decays algebraically away from the z = 0 center line, with a characteristic width L = λ. The profiles of the velocity field defined as u = A/m and its curl are shown in figure 4.6. They agree with the experimental field profile observed by the Spielman group[88].

In the limit λ λ, in which the region of a near uniform gauge ﬁeld is much larger R than the wavelength of the Raman beam, the remaining potential takes the form W 2 1 ~ 2kR m √2~kR = C ( )2z2 = C ( )2z2, (4.32) W − 2 2m λ − 2 mλ

where C is a constant potential. This remaining term is equivalent to an anti-trapping

√ ~kR potential with frequency 2Ω, where Ω = mλ , and has to be taking into account in the experimental consideration.

109 Figure 4.6: Profile of the vector potential (a) and the synthetic magnetic field (b): the quantity we plot is u = A/m and u = B∗/m. The vector potential is along the y- ∇ × direction while the “magnetic field” is along the x-direction perpendicular to the 2D gas. The blue line is the magnitude of the physical fields derived from equations (4.30) and (4.31), and the red line describes a linear velocity field in the left panel and corresponds to a large uniform magnetic field on the right. The vertical dashed lines give a cutoff of the region where we could approximate the synthetic magnetic field as a constant. The size of the strip in the z-direction is L = λ.

From now on, we transform the orientation of the gas to the more conventional notation from (y, z, x) to (x, y, z), such that the quasi-two-dimensional gas is confined in the x-y plane, with the vector potential in the x-direction and the synthetic magnetic field in the z-direction. The region where an almost constant synthetic magnetic field appears is a strip of λ < y < λ. − We consider an atomic gas originally confined by any shape of harmonic potentials with aspect ratio α, and frequencies ω0 and αω0. After turning on the Raman beams and holding them still, the gases reach equilibrium. When fully loaded to the adiabatic state, the effective full single-particle Hamiltonian for the maximum spin adiabatic state is

(p 2mΩy)2 p2 1 1 = x − + y + m(α2ω2 2Ω2)y2 + mω2x2, (4.33) H 2m 2m 2 0 − 2 0

where mΩ = ~kR/λ. This form is identical to the trapped electron gas in the Landau gauge in the presence of a vector potential. If we apply a gauge transformation to this

hamiltonian into the symmetric gauge, the Hamiltonian turns into a form identical to the

110 atoms in rotating traps:

(p mΩ r)2 1 1 = − × + m(α2ω2 2Ω2)y2 + mω2x2 H 2m 2 0 − 2 0 p2 1 1 = ΩL + m(α2ω2 Ω2)y2 + m(ω2 + Ω2)x2. (4.34) 2m − z 2 0 − 2 0

Note that in this hamiltonian, the trapping frequency in the y-direction is reduced from

α2ω2 to α2ω2 2Ω2 by the synthetic field. Thus, there is a limit of the “rotating frequency”, 0 0 − 2 2 2 2Ω < α ω0. This condition is usually satisfied by other physical considerations in the NIST scheme for the following reason. As we have shown previously in figure 4.6, the “magnetic

ﬁeld” has a maximum in a strip. In the cases we are interested in, in which the majority of the cloud has to be loaded in this narrow strip, the gas must be conﬁned in a relatively narrow region in the y-direction. The system must be far from criticality in the y-direction

consequently. The original anisotropy of the trap and the presence of the Raman beam

breaks the rotational symmetry of the system in general, hence the Fock Darwin states

are not the eigenstates. In the next section, we will discuss the single-particle spectrum in

the anisotropic geometry, which corresponds to “Landau levels” in an anisotropic external

potential, as well as the properties of both the condensate and quantum Hall states in very

elongated gases.

The interacting part of the hamiltonian is taken in the form

= drdr0V (r r0)ψ†(r)ψ†(r0)ψ(r0)ψ(r). (4.35) Hint − Z In this quasi-two-dimensional cloud, the contact interaction is written as V (r r0) = g δ(r − 2d − r0). This bare delta function interaction is only a valid approximation for weak interactions,

and it is on the mean-ﬁeld level. When the harmonic oscillator length az is at least several

times the three dimensional scattering length as, the coupling constant g2d is calculated by an integral of the lowest harmonic oscillator state over the z-direction:

2 2 g2d = dz φ0(z) g3d = √8π~ as/(maz), (4.36) | | Z

where φ0(z) is a normalized gaussian with width az in the z-direction.g ˜2d = √8πas/az

111 is the dimensionless two dimensional coupling constant. The largestg ˜2d value would be around 0.5 for the harmonic conﬁnement, taking the ratio az = 10as. The validity of the bare delta function potential in the trap can be tested, by comparing to the solution of an s-wave scattering problem with Fermi’s pseudo-potential model in a quasi-two-dimensional geometry. The solution for the exact energy eigenvalues from the pseudo-potential model can be found in reference [98, 99]. Such results can be used to calibrate those from the bare delta potential in the 2D case, in order to calibrate the calculations.

4.2.3 Non-Abelian gauge ﬁelds, spin-orbit coupled gases

The previously-discussed Abelian synthetic gauge field has one single adiabatic state relevant to the system. In this part we will discuss a more general scheme for generating synthetic gauge fields, and show in the NIST scheme a family of hamiltonians will cover the creation of both Abelian and non-Abelian gauge fields. Also, we will show that an effective spin-orbit coupling can be realized in some non-Abelian cases.

Consider a class of hamiltonian:

= + , (4.37) H T Hsyn

2 where = p is the kinetic energy. covers all the additional “synthetic” terms to T 2m Hsyn generate the gauge ﬁelds, and they are usually spatial dependent as (r). We will refer Hsyn it to the “synthetic hamiltonian” from now on. The variation of synthetic hamiltonian

2 over space has a characteristic momentum q, and gives an energy scale q = q /(2m). The low-energy manifold is deﬁned as the Hilbert space of the states, which is isolated by an

energy scale much larger than q, to all the higher energy states, as is shown in figure 4.7. In this scheme, depending on the number of states n considered in the low-energy manifold, the gauge potential will be either Abelian for n = 1, or possibly non-Abelian for n 2, as sketched in figure 4.7. In this section we will discuss the non-Abelian case of the ≥ NIST scheme, and will show an effective spin-orbit coupling from the non-Abelian gauge potential[97].

112 n =1 ✏q n 2

Figure 4.7: A sketch of the energy spectrum of the synthetic hamiltonian . Hsyn Hsyn varies in space as a characteristic wavelength q, which provides a characteristic energy scale 2 q = q /(2m). The low-energy manifold is well-defined in the situation that there is a subspace of states in which their energy difference with the ground state is much less than q (the dashed line is the threshold). In some cases, this low-energy manifold is isolated from the other high-energy states by an energy gap ∆ . Depending on the number of q states n in the low energy manifold, there will be an abelian gauge field in this system if n = 1, or a non-abelian gauge field for n 2. The adiabatic approximation is only good ≥ when all the low-energy states are included in the projected hamiltonian.

The local eigenstates, and the structure of the energy spectrum is determined by the details of the synthetic term . In the NIST scheme, as we discussed in the Abelian case Hsyn in the previous section, the linear Zeeman term is large and the quadratic Zeeman term is negligible. Also the energy transfer of the Raman process is tuned to be very close to the linear splitting gµBB0. With the quadratic Zeeman term and a tunable photon energy included, the full expression of the synthetic hamiltonian is

= (Ω cos qyF + Ω sin qyF (gµ Gz ω + Ω )F + νF 2) Hsyn R x R y − B − 0 z z = e−iqxFz (Ω F (gµ Gz ω + Ω )F + νF 2)eiqxFz , (4.38) R x − B − 0 z z where q = 2kR is the momentum transfer and ω is the energy transfer in the Raman process. Ω0 = gµBB0 is the energy of the linear Zeeman term. The second line of the equation shows that under a spatially-dependent rotation in the spin space, the synthetic

hamiltonian becomes spatially uniform. The basis will undergo a transformation m˜ = | i e−iqxFz m = e−iqxm m , where m are the eigenstates of F with eigenvalue m. | i | i | i z One diagonalizes this synthetic hamiltonian to ﬁnd the eigenvalues (energy spectrum)

and the eigenstates of the system. It can be further determined by looking at the number of

states in the low energy manifold if the gauge potential is Abelian or non-Abelian. For any 113 given q, there is a class of hamiltonian with Ω0, ω, ΩR, ν as tunable parameters. By adjusting the set of four parameters, one can access the regions where the adiabatic approximation

is valid. The corresponding number of low-energy states included determines if the gauge

ﬁeld is Abelian or non-Abelian.

Take an example of the NIST scheme with Rubidium 87 bosons in the F = 1 manifold.

In the matrix representation, this rotated synthetic term is

(gµ Gz ω + Ω ) + ν Ω /√2 0 − B − 0 R   syn = Ω /√2 0 Ω /√2 . (4.39) H R R      0 ΩR/√2 (gµBGz ω + Ω0) + ν   −    In the following discussions, we take the gµBGz = 0 limit, which corresponds to the situation that the energy scale from the external magnetic ﬁeld gradient in the region we

are interested in is much less than other energy scales Ω0, ω, ΩR, ν. The energy of the three states in this simpliﬁed hamiltonian is the solution to the following equation:

λ3 2νλ2 + (ν2 Ω2 (ω Ω )2)λ + νΩ2 = 0. (4.40) − − R − − 0 R

The separations of these three states are determined by these three energy scales Ω 0 − ω, ν, ΩR. We will discuss how the spectrum looks like in some special cases in the following.

Abelian

As in the scheme we discussed in the previous section, when these energy scales are tuned

as Ω ν, (ω Ω ), , the equation for eigenvalues becomes R − 0 q

λ3 Ω2 λ = 0. (4.41) − R

The three eigenvalues are λ = mΩ , with m = 0, 1. They have equal spacing Ω , and m − R ± R the ground state is the Fx eigenstate with eigenvalue mx = 1. Also, since the transformation

e−iqyFz is spatial dependent and does not commute with the kinetic energy in the laboratory

frame, the adiabatic state will acquire a gauge ﬁeld A = ~qˆy. This is a constant curl- free gauge potential, which only shifts the dispersion minimum in momentum space. The

114 hamiltonian in the laboratory frame projected to this adiabatic state is

(p A)2 = − . (4.42) Had 2m

Note that in the previous section the ﬁeld gradient term gµBGz plays a crucial role in generating a nontrivial gauge potential whose curl is nonvanishing.

non-Abelian

For the situation with large quadratic Zeeman term such that ν Ω , ω Ω +ν, equation R − 0 (4.40) can be approximated as

λ3 2νλ2 = 0. (4.43) −

The three eigenvalues are λ1 = λ2 = 0, λ3 = √2ν. There are two nearly degenerate low- energy states with an energy gap to the third one as √2ν. The two low-energy eigenstates are the Fz = 1, 0 subspace. Thus we project the hamiltonian into this two-component space and perform another transformation = e−iqy(Fz−1/2). The eﬀective single-particle U hamiltonian in this basis is equivalent to a spin-1/2 system with spin-orbit coupling:

1 σz 2 ΩR h syn = (p qˆy ) + σx + σz, (4.44) H 2m − 2 √2 2 in which we include back the small terms Ω and h = ν (gµ Gz ω+Ω ) as perturbations, R − B − 0 and the σ’s are Pauli matrices. The gauge ﬁeld is non-Abelian, since the Ω√R σ term in the 2 x

σz “potential” energy does not commute with the gauge potential qˆy 2 . In general, we classify the gauge ﬁeld by looking at the hamiltonian for a multicomponent system in the form

1 = (ˆp Aˆ )2 + W,ˆ (4.45) H 2m − where the gauge potential Aˆ and the “single particle potential” Wˆ are operators. The gauge field Aˆ is non-Abelian, if there is at least one non-vanishing commutator between the three components of vector Aˆ and Wˆ , i.e. when [Aî, Aˆj], [Aî, Wˆ ] are not all identically ˆ ˆ zeros. For the above case, the commutator [Ay, W ] = i~√2qΩRσy = 0 makes the gauge 6 field non-Abelian.

115 E |#i |"i |#i |"i

kqy q -2 -1 1 2

Figure 4.8: A sketch of the energy spectrum for spin-orbit coupled gases:ê the dispersion relation with a cut in the y-direction for kx = kz = 0. The dashed lines are the two branches with zero ΩR: the purple for down spin and the blue for up spin. Since the only component of the gauge potential is in the y-direction, the lowest-energy points are shifted in diﬀerent directions on the y-axis. With ﬁnite mixing between two states from Raman coupling Ω√R σ , the spectrum becomes two branches without a level crossing. For small 2 x momentum, a gap opens up between the two eigenstates and it is ∆ = 2√2ΩR at ky = 0. For large momentum, the two branches gradually evolve into dispersions for non-coupling regime, as the eigenstates asymptotically go to the S eigenstates and . z | ↑i | ↓i

In equation (4.44), the original kinetic energy term is modiﬁed in a way which can be viewed as an equal contribution of Rashba (k σ + k σ ) and Dresselhaus (k σ k σ ) z z y y z z − y y type of spin-orbit coupling. Note here that the expression deviates from the convention in

solid state physics by the transformation σ σ and σ σ . For Ω = 0, S is still a y → − z z → y R z good quantum number such that the eigenstates to this hamiltonian are two branches with

up and down spins. The lowest kinetic energy points in these two branches are shifted by

q /2 in two opposite directions in momentum space, as is shown in figure (4.8). The finite | | Ω√R σ couples the up and down spin component with momentum difference q. This coupling 2 x avoids the level crossing between the two branches, and the spin part of the wave function varies as a function of the momentum.

A sketch of the spin-orbit coupled spectrum is shown in ﬁgure 4.8, in the limit h = 0, for both Ω = 0 and Ω = 0. This spin-orbit coupling brings new physics in both bosons and R R 6

116 fermions, since the dispersion relation and low energy degeneracy are dramatically modiﬁed.

In the bosonic case, the minimum energy manifold contains two lines with k = q/2, y ± instead of a single point of k = 0 in the absence of spin-orbit coupling. There are serious consequences including the existence of Bose-Einstein condensates in three dimensions[100].

There are also new types of condensates in this regime[101, 102], fragmentation of them for instance. It is also very interesting in the fermonic case, that by engineering the dispersion relation and tuning the chemical potential such that an eﬀective single-component Fermi gas with relatively strong attractions[35]. There are a number of studies of the possible emergence of Majorana fermions in the type of p-wave superﬂuids based on this spin-orbit- coupled Fermi gas.

4.2.4 Summary

In this section, we started from introducing a general approach to generating synthetic gauge

fields in neutral atoms by implementing a spatially varying magnetic field to particles with spins. In this spirit, we discussed a scheme using a two-photon Raman process to couple different spin states developed by the Spielman group in NIST. We showed that it is possible to generate both Abelian and non-Abelian gauge fields, with the experimental capability of tuning the parameters in the setup. In some non-Abelian regimes, the system is analogous to spin-1/2 bosons or fermions in the presence of spin-orbit coupling.

Despite the heating problem and other experimental issues in this approach, there are still many open theoretical questions triggered by the success of this NIST scheme for creating a synthetic gauge ﬁeld. In the Abelian case, it is important to study the eﬃciency of driving the condensate to a quantum Hall regime. Also, as the shape of the cloud is elongated in order to make the maximum use of the synthetic gauge potential, there are motivations to study the properties of both condensates and quantum Hall wave functions in an anisotropic geometry. In fact, the possible new quantum Hall states in distorted geometry is a conceptually important topic itself. We will discuss these two questions in detail in the next section.

In the non-Abelian case, there are intensive studies on new types of condensates in

117 spin-orbit coupled Bose gases, and searching for Majorana fermions in p-wave superﬂuids emerged from spin-orbit coupled Fermi gases. These topics in non-Abelian case will not be further discussed in this thesis.

4.3 Quantum Hall physics of atomic gases with anisotropy

In this section, we discuss the quantum Hall physics of atomic gases in the presence of anisotropic traps. Part of the motivation for these studies is the fact that in the synthetic gauge field scheme we discussed in the previous section, the synthetic magnetic field is limited to a narrow region. Consequently it requires an elongated cloud to maximize the vorticity of the system. An anisotropic trap is naturally applied to realize this goal. Also, recent progress in preparing degenerate dipolar molecules makes it possible to study the properties of different quantum phases in the presence of (long-range) anisotropic interactions between particles.

This section will be organized as following: ﬁrst we summarize the previous works on solving single particle problems in rotating anisotropic two-dimensional traps. Then we report our studies in vortex states in BEC, as well as a possible transition from the Bose-

Einstein condensate to a ν = 1/2 bosonic quantum Hall state. We also study the properties of both phases in the quasi-one-dimensional geometry, based on a setup of the NIST scheme of synthetic gauge ﬁeld. In the last part, we show the features of a class of new quantum

Hall states within the anisotropic geometry, and experimental methods for detecting and verifying their properties.

118 4.3.1 Single particle wave functions of particles in rotating anisotropic traps

In the previous section, we concluded that the hamiltonian of a quasi two dimensional rotating system in the co-rotating frame can be written as

1 † 2 = dr ψ (r)( i~ mΩ r) ψ(r) H 2m − ∇ − × Z m + dr (ω2 Ω2)x2 + (ω2 Ω2)y2 ψ†(r)ψ(r) 2 x − y − Z + drdr0g (r r0)ψ†(r)ψ†(r0)ψ(r0)ψ(r). (4.46) 2d − Z As discussed in the previous section, the interaction strength g (r r0) when the scattering 2d − length is oﬀ resonance could be written as a simple delta function

2 0 √8π~ as 0 g2d(r r ) = δ(r r ), (4.47) − maz −

where az is the harmonic length of the tight trap in the direction perpendicular to the 2D

~2 plane. A dimensionless interaction strength is defined asg ˜ = g2d/( m ). When we scale the energy by the rotation frequency Ω, we could further define dimensionless effective trapping frequencies asω ˜2 = (ω2 Ω2)/Ω2 andω ˜2 = (ω2 Ω2)/Ω2. Finally, we define an effective x x − y y − anisotropy of the trap as given by the form α2 = (˜ω2 ω˜2)/(˜ω2 +ω ˜2), for ω ω . α = 0 y − x y x x ≤ y corresponds to a completely isotropic trap.

In the presence of this anisotropy of the external trap, the rotational symmetry is broken

and the angular momentum is no longer a good quantum number. Thus the Darwin-Fock

spectrum is no longer valid and the lowest Landau level does not consist of a set of wave

functions characterized by a homogenous function in z apart from the gaussian factor. A

general class of eigen wave functions for any anisotropy α has been worked out in reference

[103]. Similar to the Fock-Darwin states, the eigenstates in the anisotropic traps can also be

labelled by n and m, where n is the counterpart of the Landau level index and m corresponds

to the angular momentum index in axisymmetric systems. While some of the details in the

derivation can be found in the reference, here we write down the spectrum and the general

119 form of wave functions in the “lowest Landau level” of this anisotropic trap:

1 1 n,m = n + ~ω− + m + ~ω+. (4.48) 2 2 And for the n = 0 “lowest Landau level”, the wave functions are written as

1 c m/2 ζ x2 y2 ψ0m(x, y) = Hm exp 2 2 + iκxy . (4.49) πaxaym! 2 √2c × −2ax − 2ay p where Hm’s are Hermite polynomials, ζ = x + iβy, and β, ω+, ω−, c, ax, ay, κ are all determined by Ω,ω ˜x andω ˜y. The explicit expressions of these parameters can be found in reference [103]. Apart from the gaussian factor, the wave functions are entire functions of ζ.

The inhomogeneity in the Hermite polynomial for the “lowest Landau level” is the consequence of non-conserved angular momentum. And the fact that the variable is ζ = x + iβy instead of z = x+iy reﬂects the diﬀerent scaling of harmonic lengths in two directions, from unequal trapping frequency along the x and y axes. The characteristic energies in such a hamiltonian ω+ and ω− are given by

2 2 2 4 2 ω = Ω ω˜ + 2 (αω˜⊥) + 4˜ω + 4 , (4.50) ± ⊥ ∓ ⊥ q 2 2 2 whereω ˜⊥ = (˜ωx +ω ˜y)/2 is the mean squared “remaining rotating frequency”. In the isotropic case α = 0, ω = ω = ω, the expression goes back to the form ω± = ω Ω. x y ∓ Despite the fact that the single-particle eigenstates are not the eigenstates of the angular momentum operator Lz, similar to the wave functions in the real LLL, this set of wave functions have an increasing expectation value of Lz as m increases. Also the density distributions of these wave functions are a set of elliptical rings with increasing radii, i.e.

2 2 increasing expectation values of x and y . The long axis is in the x-direction for ωy > ωx traps (and we will always keep ω ω as the assumption of notations from now on). The y ≥ x density profile of a fully-filled “lowest Landau level” in the anisotropic trap resembles the one for the symmetric system: a plateau is seen in the bulk with zero density fluctuation

(zero compressibility). This is one of the features of the quantum Hall regime. In the following discussions, we will borrow the terminology of LLL (and second Landau level ...) to label the single-particle states in the rotating anisotropic traps.

120 4.3.2 Transition from a condensate to a quantum Hall state

We consider a Bose-Einstein condensate in the rotating anisotropic trap: the LLL projection of the condensate is valid in the situation that the chemical potential is (far) below the energy of the second Landau level. In the cold Bose condensates, it requires that the system be rapidly rotating, i.e.ω ˜ 1. Also the repulsive interaction cannot push the x particles to the second LL, which from the simplest mean-ﬁeld estimate is g2dn < ω−. With these conditions enforced, it is suﬃcient to write the Gross-Pitaevskii (GP) wave function of the condensate in the LLL space as:

x2 y2 φ = c ψ = f(ζ) exp + iκxy , (4.51) m 0m −2a2 − 2a2 m x y X where f(ζ) is an entire function of ζ. The zeros of f(ζ) give the positions of the vortices in the condensate, and the number of vortices inside the cloud corresponds to the angular momentum injected. The system can undergo a transition from a condensate with vortices, to a quantum Hall state where the fluxes are bound to the particles. A back-of-the-envelop estimate of the transition point is obtained by matching the angular momentum between the vortices and the quantum flux in QH states. In the condensate phase, each vortex carries a unit of angular momentum ~. The total angular momentum in the BEC cloud is thus Nv~, where Nv is the number of vortices. One the other hand, a ν = 1/2 bosonic quantum Hall state has two units of fluxes per particle, and its total angular momentum is 2N~. As the total vorticity is proportional to the rotating frequency and the area the cloud covers, we use a Thomas-Fermi analysis to estimate the area of the cloud. For the fast rotating limit when the remaining rotating frequency in the x-direction approaches zero while that in the y-direction remains relatively strong, i.e.ω ˜ 1 andω ˜ /ω˜ 1, the asymptotic behavior x x y of the Thomas-Fermi profile is determined by

g2dN −1/2 A = 2π (˜ωx) , (4.52) sπmγω+ ∼

Rx −1 χ = = β+ (˜ωx) , (4.53) Ry ∼

121 where A = πRxRy gives the area of cloud, and χ is the aspect ratio of the condensate.

They behave as diﬀerent powers ofω ˜x, a value which describes how close the system is to criticality. From this Thomas-Fermi assumption of the density proﬁle, we relate the inter-

action parameter g2d and the trapping and rotating frequencies to the possible transition point from a condensate to a quantum Hall state by the following two equations:

mΩA = 2Nh, (4.54)

2R L. (4.55) y ≤

The second equation enforces that the gas is conﬁned to the strip where the gauge ﬁeld

exists, for the speciﬁc NIST scheme discussed previously.

For a more rigorous calculation of the transition point, we would like to compare the

energy between the condensate with vortex lattice, and the quantum Hall wave function in

this rotating anisotropic trap. For the condensate phase, we use a Jacobi-theta function

form of trial wave function for the vortex lattice:

x2 y2 ψ(x, y) Θ((ζ b )/b , τ) exp ηζ2 + iκxy , (4.56) ∼ − 0 1 − 2a2 − 2a2 x y

in which τ = b2/b1, and b1, b2 are the complex numbers describing Bravais lattice vectors in

the 2D plane, b0 is the position of a vortex. Because of the anisotropy, η is another trial parameter to adjust the shape of the condensate. As the Jacobi-theta function Θ((ζ b )/b , τ) − 0 1 is an analytical function of ζ in the whole complex plane, this trial wave function satisﬁes

the condition that the condensate is conﬁned in the LLL. The energy of this condensate is

given by the GP non-linear energy functional as

2 2 2 x y ψL (ζi ζj) exp 2 2 + iκxy . (4.58) ∼ − −2ax − 2ay Yi

122 traps, by substituting z = x + iy to ζ. This many-body wave function is in the LLL since it is analytic for all ζ0s. And for the contact interaction, it has a vanishing interaction energy since the wave function goes to zero as two particles come together. The kinetic

(single-particle) energy, for both the condensate and the quantum Hall state, can be simply calculated by the density proﬁle of the cloud. This simpliﬁcation is enforced by the fact that the single-particle hamiltonian projected to the LLL can be written as a linear combination of x2 and y2:

1 1 1 2 2 0 LLL = ~ω− + ~ω+(ββ+ + 1/(ββ+)) + mγω+(β+ x + y /β+). (4.59) hH i 2 4 2 h i h i

Consequently, a classical Monte Carlo sampling of a large number of particles in the trial

“Laughlin wavefunction” gives the expectation values x2 , y2 , thus determines the energy h i h i of this “Laughlin state”. We calculated a speciﬁc case with 100 particles and interaction

strengthg ˜ = 0.5. A boundary between the two states is shown in ﬁgure 4.9: the GP-type

condensate wavefunction 4.56 has lower energy to the right of the boundary, while the

“Laughlin” wavefunction 4.58 is favorable to the right.

As the quantum Hall phase can be characterized by having a plateau of density proﬁle

and zero ﬂuctuation in density, we are interested in how the quantum Hall state evolves in

this speciﬁc scheme when the trapping potential gets extremely anisotropic. The tight trap

in the y-direction makes the gas become very much elongated. In the case where the trap in

the y-direction is extremely tight, a large ωy provides an eﬀectively separable hamiltonian, as the bosons always occupy the lowest harmonic oscillator state in the tight direction.

Consequently the density reﬂects a gaussian proﬁle in the y-direction. In this quasi-one-

dimensional regime, both the condensate and the QH state deviate from the above-discussed

description. The condensate cannot be approximated by Thomas-Fermi, and the quantum

Hall wave function no longer has the quantization of the bulk density. Both of these two

states crossover to the same one-dimensional ground state of a Bose gas.

We examine this evolution by studying a system with a moderate number of particles

(in the order of a hundred). Staring from the condensate phase, by minimizing the energy

of the condensate within the trial GP wave function (4.56), we track the evolution of the

123 Figure 4.9: Phase diagram of the cloud in rotating anisotropic traps: the red curve is the boundary between the QH and BEC states. It is extrapolated by a connection of points at which the QH and BEC are energetically equal. In this case we calculate four different values of the trapping frequency along the y-axisω ˜y. The dashed black curve is a rough boundary of L = 5l contour: above the dashed line, the width of the cloud is less than 5 magnetic lengths. The shaded area is the crossover between a 2D quantum Hall state and a Laughlin wave function remaining in 1D. The blue dashed line is the approximate transition of the vortex lattice. The letters noted in the figure are the positions where density profiles will be shown in figure 4.10 and 4.11.

vortex lattice as the gas becomes more and more anisotropic. In the less anisotropic case,

vortices form visible two-dimensional triangular lattices. As the anisotropy is turned up, a

single line of vortices appear in the center of the tight direction of the cloud, and a pair of

lines of vortices are located symmetrically on the two sides close to the edge of the cloud,

as shown in (a) and (b) of ﬁgure 4.10. If we further squeeze the cloud in the y-direction,

the elongated condensate will eventually experience a transition of the vortex arrangement.

Instead of a line of vortices staying on the y = 0 axis, the line in the middle of the cloud

124 a

Figure 4.10: Distortion and transition in vortex lattices: (a), (b), and (c) give the positions in frequency space in ﬁgure 4.9. A line of vortices lie in the center of the cloud in (a) and (b). As the anisotropy of the trap increases, the cloud becomes narrower and narrower. A transition occurs when the vortices jump outside of the cloud in (c). In this case, the vortices are almost invisible, however there is evidence of modulation on the edges. (Length in the ﬁgures are scaled by the magnetic length l, and the scale of the x and y directions are not the same.)

splits into two lines symmetric to the y = 0 axis, as is shown in (c) of ﬁgure 4.10. In

the extremely elongated cloud, these vortices are almost invisible, leaving the condensate a

gaussian proﬁle in the narrow direction. This agrees with the physical picture of a separable

hamiltonian, that the system only occupies the lowest harmonic-oscillator state in the tight

direction. The oscillator length (gaussian width) is close to the magnetic length. This one

dimensional feature is also characterized by the cloud no longer changing its width in the

y-direction, as the trap in the x-direction is further relaxed[104].

For the quantum Hall state, there is a similar crossover to the one-dimensional bosonic

ground state. We track the evolution starting from a usual quantum Hall state, which in the

ν isotropic case has a density plateau of ρν = πl2 in the bulk. For a system with anisotropy from the external traps, the ν = 1/2 bosonic quantum Hall state described as equation

125 4.58 has a fixed density of ρ = 1 in the bulk. We keep track of the density profiles 2πaxay of this class of “Laughlin wavefunctions” 4.58, with different anisotropy of the external trap. A crossover between two distinct ground states appears, as shown in figure 4.11.

When the cloud is extended several magnetic lengths in the tight direction, a plateau in the density profile is still recognized, as in (A) of figure 4.11. It gradually crossovers to the quasi-one-dimensional profile of (C) in the same figure, featured by the loss of quantized density profile and almost inverted parabola shape in x-direction. In the tight y-direction, it resembles the gaussian profile of the lowest harmonic oscillator state in the tight trap, which again indicates that the hamiltonian is almost separable in the quasi-one-dimensional limit.

Throughout the whole crossover, the wave function is always described by the “Laughlin wave function” 4.58.

4.3.3 Quantum Hall wave functions in broken rotational symmetry

In the previous sections, we discussed the properties of condensate and quantum Hall states in anisotropic traps. The trial wave function we used for the QH state is the modiﬁed

Laughlin state 4.58. In fact, unlike for the contact interaction in the axisymmetric traps, this Laughlin trial wave function here is not the eigenstate of the full hamiltonian. As the single-particle eigenstates are Hermite polynomials in the pre-gaussian factors, a true ground state should have a inhomogenous prefactor in ζ for the quantum Hall wave functions. To study how the exact solution deviates from our trial wave functions 4.58, we start from a simple two-body calculation of the ground state of the hamiltonian 4.46.

We do an exact diagonalization of two particles with the hamiltonian of 4.46. For 2 1 moderate strength of interaction g2d ~Ω, we can set a cutoff of the complete πaxay set of basis within the lowest Landau level only. This is always reasonable in the cases whereg ˜ω˜ 1. We separate the two-body hamiltonian into a center-of-mass part and a x relative-coordinate part: the spectrum and the eigenstates of the center-of-mass part are trivial; the relative-coordinate part mixes different single-particle states by an effect of the presence of a delta-function-like potential at the origin. Thus a diagonalization is done in

126 A

Figure 4.11: Density profiles at cut x = 0 and y = 0 of the Laughlin wavefunctions at different positions in figure 4.9 (length is scaled by magnetic length l): (A) the density is quantized in both directions, similar to a conventional quantum Hall state in a bulk system; (B) as the cloud becomes narrower, we see no quantized profile in the y-direction, and a roundoff is shown in the x-direction. This is in the crossover region shaded in figure 4.9; (C) the size of the system in the y-direction goes to a magnetic length, thus loses quantization in both directions. This is a demonstration of the one-dimensional limit of a Laughlin wavefunction. The numerical data are from Monte Carlo sampling of the many-body wave functions.

the relative coordinate space only, with the matrix elements

∗ Hmn = m~ω+δmn + g2dψm(0)ψn(0). (4.60)

For a bosonic system, the solution contains even-order of polynomials only. With a cutoﬀ of a certain number of states, the ground state is a polynomial P (ζ2) times the gaussian

127 Figure 4.12: Distribution of zeros in the relative coordinate of the ground state: the upper ﬁgure is four distributions from results with 30, 40, 50, 80 states diagonalization. The only unchanged zeros are the pair close to the origin which stay in the dashed red box. The lower panel zooms in the red box, and clearly shows the positions of the zeros in the small region. The length of the red box in the ﬁgure is twice the harmonic length in the relative coordinates. For the gas elongated in the x-direction, this physical pair of zeros split along the y-axis.

part. The number of zeros in the polynomial P (ζ2) actually depends on the number of

cutoﬀ in the eigenstate space. As the diagonalization result converges, we can sort out all

the “physical zeros”, i.e. whose positions do not change with different cutoffs. In figure

4.12, we show the distributions of zeros in the relative coordinate wave functions of the

ground states for ﬁxed trap and rotation frequencies. From the ﬁgures, we see there are

only two zeros close to the origin, symmetrically arranged on the imaginary axis. All the

other zeros vary their positions in the 2D plane with diﬀerent cutoﬀs, and they are far away

from the origin by several magnetic lengths. The changing positions of the zeros far away

indicates they are not physical for the lowest-energy ground state. Thus, our conjecture on

the ground state wave function has a characteristic form of:

x2 + x2 y2 + y2 ψ(ζ , ζ ) (ζ ζ )2 + 2 exp A(ζ ζ )2 1 2 1 2 + iκ(x y + x y ) , 1 2 ∼ 1 − 2 × 1 − 2 − 2a2 − 2a2 1 1 2 2 x y (4.61) where is the split of the zeros in relative coordinates. A is another trial parameter which deforms the shape of the system. We also compared the overlap between the trial

128 Figure 4.13: Variation in the distance of splitting zeros as a function of inverse interaction strength, for ﬁxed anisotropy α = 0.96: the splitting is almost linear in the region where 0.5 < 1/g˜ < 4.

wavefunction which has minimized energy within this family and the exact diagonalization results: there are more than 99.5% overlaps between our guess and the exact results in the regions we are interested in. We observe the split distance between the zeros has a 1/g˜ behavior in a large region, as shown in figure 4.13. The zero limit goes to our trialp wave functions (4.58). We find instead of the distinction between the ground states with different angular momentum in an isotropic system, there is a gradual crossover in this case. The competition between rotational energy of the trap and the repulsive interaction between bosons in this anisotropic case makes the change gradual. This is a consequence of broken discrete symmetry in angular momentum, which brings in a continuous flow between the rotational energy and repulsive interaction. The = 0 limit is analogous to the original

Laughlin wave functions which minimize the interaction energy in the isotropic case, while the goes to inﬁnity limit in the spherically symmetric case corresponds to all the particles staying in the lowest angular momentum (m = 0) state. We thus call this pair of two-body solutions as “quantum Hall” state and “condensate” state respectively.

From the two-body study, we have a ﬂavor of the possible form of the quantum Hall wave functions in the presence of an anisotropic external trap. As the short-range part of the

129 Jastrow factor is likely to be dominated by two-body physics, it is subject to further study if the true ground state for a many-body system will preserve the features we discussed. Also, we have not yet related quantitatively the short-range splitting distance , and the trial parameter A in the wave function 4.61. In general, we can propose a trial wave function in the form:

x2 y2 ψ( ζ ) (ζ ζ )2 + 2 exp A (ζ ζ )2 i i + iκ(x y ) . (4.62) { i} ∼ i − j  i − j − 2a2 − 2a2 i i  i

4.3.4 Detection of quantum Hall wave functions in cold gases

The validity of the trial wave functions discussed above for the quantum Hall states in anisotropic traps can be tested experimentally in several ways, for both two-body and many-body systems. These experimental techniques include time-of-ﬂight expansion, photo association, high-resolution in situ imaging, and possible scattering spectroscopy for many body systems.

We first focus on the simplest two-body discussion. A cluster consisting of a large number of identical two-body systems is required in order to give a strong enough signal for any reasonable measurements. This is realized by first preparing an optical lattice with relatively large overall trapping potential, such that there can be a dominating region of two-particle-per-site Mott insulator phase. Also the anisotropy of the effective trapping frequency for each site can be adjusted both by tuning the laser intensity and wavelength.

Within this Mott region, we can either turn on the synthetic gauge ﬁeld or a rotation, to

ﬁnish the preparation procedure.

The density proﬁle image from Time-Of-Flight (TOF) expansion gives the one-body density in momentum space, or the momentum distribution of the original state:

mr † mr ρ(r, t) ψk( ) ψk( ) . (4.63) ∼ h ~t ~t i

130 We would like to compare the final density profiles between the quantum Hall wave functions in the isotropic traps and that of the wave functions in the anisotropic traps. This is an indirect evidence of the modified wave function, however is indicative to reveal the splitting zeros. For the isotropic case, the overall density profile of the quantum Hall state has a dip at the center, while the condensate state is simply a gaussian, as is shown in figure 4.14.

Although the appearance of zeros is absent in the expansion proﬁles, the relative strength of the dip shows how strong the particles repel each other.

The anisotropic case is similar, as shown in ﬁgure 4.15. There is a dip in the density proﬁles at the origin, sandwiched by two maxima along the x-axis. However, the rest of

the features are very diﬀerent from those of the isotropic case. Firstly, a gradual change in

the dip depth shows a continuous evolution in zero splitting: the density proﬁle can be an

anisotropic gaussian, corresponding to a condensate for the original state with the pair of

zeros at inﬁnity. As we pull the pair of zeros closer to the origin, a dip in the density at

the origin appears. This dip becomes more and more pronounced as the splitting between

the pair gets smaller. Also, for some anisotropy and splitting, the dip at the origin is not

the local minimum as it is for the isotropic case: it becomes a saddle point while the proﬁle

in the y-direction remains a gaussian. These diﬀerent features all indicate the splitting

of the pair of zeros in the relative coordinate wave function, as predicted in the previous

discussions.

In cold atomic gases, photoassociation is a process in which two colliding atoms are

optically excited to an electronically excited bound state. This rate can be measured by the

atom loss in the experiments, and it gives the probability of two particles coming together

in the system[105]. In a ﬁrst approximation, photoassociation rates are proportional to

the zero-distance pair correlation function. Speciﬁcally in an experiment of the rotating

clusters[106], it is deﬁned as

g (0) = dr ψ†(r)ψ†(r)ψ(r)ψ(r) (4.64) 2 h i Z For the conventional m = 2 Laughlin wave functions in the isotropic traps, this quantity

should be zero from the vanishing weight in the Jastrow factor when two particles are on

131 Non quantum Hall state: y

x Larger cut Interaction Strength

Quantum Hall state: y

Figure 4.14: Density profiles after TOF expansion of the gas in a rotating isotropic trap: the upper figure represents the pattern for “two-body condensate” (infinite limit), the bottom one is the two-body ”Laughlin state” equation ( = 0 limit). The right one-dimensional plots are the density cuts through the x-axis. As one increases the strength of the repulsion, a transition (level crossing) between a “two-body condensate” and a “Laughlin state” occurs. The QH state is characterized by the emergence of a pair of doubly degenerate zeros at the origin, as illustrated in the cartoon on the left.

top of each other. A ﬁnite value indicates the pair of splitting zeros. We consider the diﬀerence in g2(0) between the rotating isotropic and anisotropic traps, as we ramp up the rotating frequency close to criticality in the experiments. If the system is populated mostly

in its ground state, there will be a sharp drop in 4.64, as shown in the left panel of 4.16.

It reveals a level crossing between a zero-angular-momentum state and a state with two

units of angular momentum. In contrast, for particles in the anisotropic trap, the ﬁnite

splitting weight at the origin gives nonvanishing g2 values. g2 thus changes monotonically as a function of the rotational frequency, as is shown in the right panel of ﬁgure 4.16.

Also the recent progress in high-resolution imaging in cold atoms provides a promising

method to detect the spatial structure of the wave functions. It is able to serve as a more

direct way of observing the zeros in the pair wave functions. Among these techniques, an

in situ imaging technique is capable of mapping out the Wannier wave function of a tight

132 y

No Larger zeros: Interaction not QH y Strength

Far apart zeros

Close zeros: QH

Figure 4.15: Density profiles after TOF expansion of anisotropy α = 0.94: the three profiles describes the splitting distances = 2.6l, 0.43l, 1.0l from top to bottom, where l is the magnetic length. This modification corresponds to an increasing repulsion. As the splitting distance decreases gradually, the depth in the dip at the center of the expansion changes continuously, as shown in the cuts through the x-axis in the right figures. The dip at the origin becomes more pronounced.

harmonic trap in a single site in optical lattices, by using electron beams to ionize the neutral atoms[28]. The experimental setup is shown in ﬁgure 4.17. We consider a detector making use of this technique by measuring the density ﬂuctuation correlator.

Experimentally, by taking a large number of pictures of the atomic cloud, we collect the information of the density-ﬂuctuation correlation function:

g(r, r0) = n(r) n(r) n(r0) n(r0) (4.65) − − D E = n(r)n(r0) n(r) n(r0) . h i − h ih i

The expectation value of the density n is given by an average of numerous measurements,

133 Figure 4.16: Second order correlation in equation 4.64, in a rotating isotropic (left) and anisotropic (right) trap. In the isotropic case, as the rotation reaches a critical frequency, a sharp transition is clearly shown, by a sharp drop in g2. As a contrast, the value of g2 changes gradually from a constant to zero, as the rotation frequency increases in an anisotropic trap. This is a consequence of the variation of with diﬀerent interaction and anisotropy.

and by separating this piece out we have the data for

n(r)n(r0) = ψ†(r)ψ(r)ψ†(r0)ψ(r0) h i h i = ψ†(r)ψ†(r0)ψ(r0)ψ(r) + δ(r r0) ψ†(r)ψ(r) h i − h i = 2 φ(r, r0) 2 + δ(r r0)n(r). (4.66) | | −

The delta function part is a subtraction of the density from double counting the ﬁrst

particle at the same place (the counting of atom number in the same pixel). In the real

experiments, we can then extrapolate the quantity

ψ†(r/2)ψ†( r/2)ψ( r/2)ψ(r/2) = √2 ψ(r/2, r/2) , (4.67) h − − i | − | q to locate the zeros in the relative coordinate in the wavefunctions directly. In figure 4.18, we see the splitting of zeros for three different interactions for fixed anisotropy. The distance between the pair of zeros could be clearly seen from the figures.

4.4 Conclusions

In this chapter we discussed how to realize the quantum Hall states in ultracold quantum gases by either rotations or the synthetic gauge ﬁeld scheme. The synthetic gauge ﬁeld

134 Figure 4.17: High-resolution in situ imaging of neutral atoms: high-energy electron beams hit the atoms and ionize them, such that the atoms will move in the presence of the applied electric ﬁeld. The high-resolution camera in the detector collects the information on the original position of the ionized atoms, and maps out the initial wave function. On the right side, we can see a Gaussian proﬁle, extrapolated from a histogram with grid size as the single pixel of the camera. It shows the lowest band Wannier function in an optical lattice, as expected.

scheme has certain advantages over the rotating scheme, as it does not need a precise control of the rotating frequency to reach the quantum Hall regime. However, currently the NIST scheme still suﬀers from strong heating problem caused by couplings to two laser beams. We studied the beautiful properties of quantum Hall wave functions in the presence of an anisotropic external potential. We also provided a conjecture of the Jastrow factor in the wave functions in such systems, and proposed experimental approaches to detect the existence of QH states as well as their new features we presented.

In the above discussions, we only focused on the possible quantum Hall states in the anisotropic external potential. The feature of splitting zeros in the pairwise functions can be carried on to the anisotropic interactions between particles. There are previous studies in the context of two-dimensional electronic gases in magnetic ﬁelds with anisotropic Coulomb- like interactions[107], and in the context of rotating dipolar molecules in tightly conﬁned quasi-two-dimensional traps[108]. In these studies, similar structures of splitting zeros in

135 y

x No QH

“Weak QH”

Figure 4.18: In situ images of the quantity ψ(r/2, r/2) for an anisotropy α = 0.96. The | − | three images are with splitting distances = 2.6l, = 1.0l and = 0.43l, from the top to bottom. These are direct observations on the splitting zeros in the pair wave functions.

the Jastrow factors at short range are discussed.

On the other hand, the internal degrees of freedom, i.e. the multiple choices of hyperﬁne states in neutral atoms, makes it possible to study multi-component quantum Hall physics in cold atoms. From the previous studies in quantum Hall bilayers (chapter 6 of [92]), we know that new types of quantum Hall physics emerge from the extra degrees of freedom in electron gases. In alkali atoms, as there are high spin isotopes, e.g. 7/2 in Na and 9/2 in K

Fermi gases, one may dig into much richer macroscopic quantum phenomena in quantum

Hall regime. We leave these directions to further studies.

136 Bibliography

[1] Mike H Anderson, Jason R Ensher, Michael R Matthews, Carl E Wieman, and Eric A Cornell. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science, 269(5221):198–201, 1995.

[2] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of Bose- Einstein condensation in an atomic gas with attractive interactions. Phys. Rev. Lett., 75:1687–1690, Aug 1995.

[3] CC Bradley, CA Sackett, and RG Hulet. Bose-Einstein condensation of lithium: Observation of limited condensate number. Physical Review Letters, 78(6):985–989, 1997.

[4] KB Davis, M-O Mewes, MR van Andrews, NJ Van Druten, DS Durfee, DM Kurn, and W Ketterle. Bose-Einstein condensation in a gas of sodium atoms. Physical Review Letters, 75(22):3969–3973, 1995.

[5] Anthony J. Leggett. Bose-Einstein condensation in the alkali gases: Some fundamental concepts. Rev. Mod. Phys., 73:307–356, Apr 2001.

[6] Simon Stellmer, Meng Khoon Tey, Bo Huang, Rudolf Grimm, and Florian Schreck. Bose-Einstein condensation of Strontium. Phys. Rev. Lett., 103:200401, Nov 2009.

[7] Kenichi Kasamatsu and Makoto Tsubota. Static and dynamic properties of multicomponent Bose–Einstein condensates of Ytterbium atoms. Journal of Low Temperature Physics, 150(3-4):599–604, 2008.

[8] Mingwu Lu, Nathaniel Q Burdick, Seo Ho Youn, and Benjamin L Lev. Strongly dipolar Bose-Einstein condensate of Dysprosium. Physical review Letters, 107(19):190401, 2011.

[9] Brian DeMarco and DS Jin. Onset of Fermi degeneracy in a trapped atomic gas. Science, 285(5434):1703–1706, 1999.

[10] A. J Leggett. Diatomic molecules and Cooper pairs. In Modern trends in the theory of condensed matter, pages 13–27. Springer, 1980.

137 [11] BS Rem, AT Grier, I Ferrier-Barbut, U Eismann, T Langen, N Navon, L Khaykovich, F Werner, DS Petrov, F Chevy, et al. Lifetime of the Bose gas with resonant interactions. arXiv preprint arXiv:1212.5274, 2012.

[12] Richard J Fletcher, Alexander L Gaunt, Nir Navon, Robert P Smith, and Zoran Hadzibabic. Stability of a unitary Bose gas. arXiv preprint arXiv:1307.3193, 2013.

[13] Shizhong Zhang and Tin-Lun Ho. Atom loss maximum in ultra-cold Fermi gases. New Journal of Physics, 13(5):055003, 2011.

[14] David Pekker, Mehrtash Babadi, Rajdeep Sensarma, Nikolaj Zinner, Lode Pollet, Martin W. Zwierlein, and Eugene Demler. Competition between pairing and ferromagnetic instabilities in ultracold Fermi gases near Feshbach resonances. Phys. Rev. Lett., 106:050402, Feb 2011.

[15] Alexander L Fetter. Rotating trapped Bose-Einstein condensates. Reviews of Modern Physics, 81(2):647, 2009.

[16] JR Abo-Shaeer, C Raman, JM Vogels, and Wolfgang Ketterle. Observation of vortex lattices in Bose-Einstein condensates. Science, 292(5516):476–479, 2001.

[17] Eric Braaten and H.-W. Hammer. Universality in few-body systems with large scattering length. Physics Reports, 428(56):259 – 390, 2006.

[18] LD Landau and EM Lifshitz Quantum Mechanics. Non-relativistic theory. Landau, LD, Lifshitz, EM: Quantum Mechanics, Course of Theoretical Physics, 1, 1958.

[19] Christopher J Pethick and Henrik Smith. Bose-Einstein condensation in dilute gases. Cambridge university press, 2002.

[20] LD Landau and EM Lifshitz. Statistical physics, vol. 1. Oxford: Pergamon, 24:31–57, 1980.

[21] Tin-Lun Ho and Erich J. Mueller. High temperature expansion applied to fermions near Feshbach resonance. Phys. Rev. Lett., 92:160404, Apr 2004.

[22] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga. Feshbach resonances in ultracold gases. Rev. Mod. Phys., 82:1225–1286, Apr 2010.

[23] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W H¨ansch, and Immanuel Bloch. Quantum phase transition from a superﬂuid to a Mott insulator in a gas of ultracold atoms. Nature, 415(6867):39–44, 2002.

[24] Michael Köhl,Henning Moritz, Thilo Stöferle,Kenneth Günter, and Tilman Esslinger. Fermionic atoms in a three dimensional optical lattice: Observing Fermi surfaces, dynamics, and interactions. Phys. Rev. Lett., 94:080403, Mar 2005.

[25] Tin-Lun Ho and Qi Zhou. Obtaining the phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases. Nature Physics, 6(2):131–134, 2009.

138 [26] Jacob F Sherson, Christof Weitenberg, Manuel Endres, Marc Cheneau, Immanuel Bloch, and Stefan Kuhr. Single-atom-resolved ﬂuorescence imaging of an atomic Mott insulator. Nature, 467(7311):68–72, 2010.

[27] Waseem S Bakr, Jonathon I Gillen, Amy Peng, Simon F¨olling,and Markus Greiner. A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature, 462(7269):74–77, 2009.

[28] Tatjana Gericke, Peter W¨urtz,Daniel Reitz, Tim Langen, and Herwig Ott. High- resolution scanning electron microscopy of an ultracold quantum gas. Nature Physics, 4(12):949–953, 2008.

[29] JT Stewart, JP Gaebler, and DS Jin. Using photoemission spectroscopy to probe a strongly interacting Fermi gas. Nature, 454(7205):744–747, 2008.

[30] C Chin, M Bartenstein, A Altmeyer, S Riedl, S Jochim, J Hecker Denschlag, and R Grimm. Observation of the pairing gap in a strongly interacting Fermi gas. Science, 305(5687):1128–1130, 2004.

[31] Cheng Chin and Paul S Julienne. Radio-frequency transitions on weakly bound ultracold molecules. Physical Review A, 71(1):012713, 2005.

[32] Qijin Chen, Jelena Stajic, Shina Tan, and Kathryn Levin. BCS–BEC crossover: From high temperature superconductors to ultracold superﬂuids. Physics Reports, 412(1):1–88, 2005.

[33] Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari. Theory of ultracold atomic Fermi gases. Rev. Mod. Phys., 80:1215–1274, Oct 2008.

[34] Tin-Lun Ho, Xiaoling Cui, and Weiran Li. Alternative route to strong interaction: Narrow Feshbach resonance. Phys. Rev. Lett., 108:250401, Jun 2012.

[35] Jayantha P Vyasanakere, Shizhong Zhang, and Vijay B Shenoy. BCS-BEC crossover induced by a synthetic non-abelian gauge ﬁeld. Physical Review B, 84(1):014512, 2011.

[36] Hui Zhai. Spin-orbit coupled quantum gases. International Journal of Modern Physics B, 26(01), 2012.

[37] Gyu-Boong Jo, Ye-Ryoung Lee, Jae-Hoon Choi, Caleb A. Christensen, Tony H. Kim, Joseph H. Thywissen, David E. Pritchard, and Wolfgang Ketterle. Itinerant ferromagnetism in a Fermi gas of ultracold atoms. Science, 325(5947):1521–1524, 2009.

[38] Christian Sanner, Edward J Su, Wujie Huang, Aviv Keshet, Jonathon Gillen, and Wolfgang Ketterle. Correlations and pair formation in a repulsively interacting Fermi gas. Physical Review Letters, 108(24):240404, 2012.

[39] Vijay B. Shenoy and Tin-Lun Ho. Nature and properties of a repulsive Fermi gas in the upper branch of the energy spectrum. Phys. Rev. Lett., 107:210401, Nov 2011.

139 [40] Marco Koschorreck, Daniel Pertot, Enrico Vogt, Bernd Fr¨ohlich, Michael Feld, and Michael K¨ohl. Attractive and repulsive Fermi polarons in two dimensions. Nature, 485(7400):619–622, 2012.

[41] Tin-Lun Ho. Universal thermodynamics of degenerate quantum gases in the unitarity limit. Physical review letters, 92(9):090402, 2004.

[42] N. Read and Dmitry Green. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall eﬀect. Phys. Rev. B, 61:10267–10297, Apr 2000.

[43] V. Gurarie, L. Radzihovsky, and A. V. Andreev. Quantum phase transitions across a p-wave Feshbach resonance. Phys. Rev. Lett., 94:230403, Jun 2005.

[44] DM Eagles. Possible pairing without superconductivity at low carrier concentrations in bulk and thin-ﬁlm superconducting semiconductors. Physical Review, 186(2):456, 1969.

[45] Wilhelm Zwerger. The BCS-BEC crossover and the unitary Fermi gas, volume 836. Springer, 2011.

[46] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger. Many-body physics with ultracold gases. Rev. Mod. Phys., 80:885–964, Jul 2008.

[47] Martin W Zwierlein, Christian H Schunck, Andr´eSchirotzek, and Wolfgang Ketterle. Direct observation of the superﬂuid phase transition in ultracold Fermi gases. nature, 442(7098):54–58, 2006.

[48] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and W. Ketterle. Condensation of pairs of fermionic atoms near a Feshbach resonance. Phys. Rev. Lett., 92:120403, Mar 2004.

[49] TD Lee, Kerson Huang, and CN Yang. Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Physical Review, 106:1135– 1145, 1957.

[50] Peter Fulde and Richard A. Ferrell. Superconductivity in a strong spin-exchange ﬁeld. Phys. Rev., 135:A550–A563, Aug 1964.

[51] AI Larkin and IUN Ovchinnikov. Inhomogeneous state of superconductors(production of superconducting state in ferromagnet with Fermi surfaces, examining Green function). Soviet Physics-JETP, 20:762–769, 1965.

[52] Yean-an Liao, Ann S. Rittner, Tobias Paprotta, Wenhui Li, Guthrie B. Partridge, Randall G. Hulet, Stefan K. Baur, and Erich J. Mueller. Spin-imbalance in a one- dimensional Fermi gas. Nature, 467(7315):567–569, September 2010.

[53] M. Marini, F. Pistolesi, and G.C. Strinati. Evolution from BCS superconductivity to Bose condensation: analytic results for the crossover in three dimensions. The European Physical Journal B, 1:151–159, November 1998.

140 [54] Shizhong Zhang. Strongly interacting Fermi gases, radio frequency spectroscopy and universality. 2009. [55] Ph Nozieres and S Schmitt-Rink. Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity. Journal of Low Temperature Physics, 59(3-4):195–211, 1985. [56] C. A. R. Sáde Melo, Mohit Randeria, and Jan R. Engelbrecht. Crossover from BCS to Bose superconductivity: Transition temperature and time-dependent Ginzburg- Landau theory. Phys. Rev. Lett., 71:3202–3205, Nov 1993. [57] Soon-Yong Chang, Mohit Randeria, and Nandini Trivedi. Ferromagnetism in the upper branch of the Feshbach resonance and the hard-sphere Fermi gas. Proceedings of the National Academy of Sciences, 108(1):51–54, 2011. [58] S. Pilati, G. Bertaina, S. Giorgini, and M. Troyer. Itinerant ferromagnetism of a repulsive atomic Fermi gas: A quantum Monte Carlo study. Phys. Rev. Lett., 105:030405, Jul 2010. [59] LP Gorkov and TK Melik-Barkhudarov. Contribution to the theory of superfluidity in an imperfect Fermi gas. Sov. Phys. JETP, 13(5):1018, 1961. [60] R Haussmann. Properties of a Fermi liquid at the superfluid transition in the crossover region between BCS superconductivity and Bose-Einstein condensation. Physical Re- view B, 49(18):12975, 1994. [61] Qijin Chen, Ioan Kosztin, BoldizsárJankó,and K. Levin. Pairing fluctuation theory of superconducting properties in underdoped to overdoped cuprates. Phys. Rev. Lett., 81:4708–4711, Nov 1998. [62] V Gurarie and L Radzihovsky. Resonantly paired fermionic superfluids. Annals of Physics, 322(1):2–119, 2007. [63] E. L. Hazlett, Y. Zhang, R. W. Stites, and K. M. O’Hara. Realization of a resonant Fermi gas with a large effective range. Phys. Rev. Lett., 108:045304, Jan 2012. [64] Weiran Li and Tin-Lun Ho. Bose gases near unitarity. Phys. Rev. Lett., 108:195301, May 2012. [65] Erich J. Mueller and Gordon Baym. Finite-temperature collapse of a Bose gas with attractive interactions. Phys. Rev. A, 62:053605, Oct 2000. [66] Dmitry S Petrov. The few-atom problem. Many-Body Physics with Ultracold Gases: Lecture Notes of the Les Houches Summer School: Volume 94, July 2010, 94:109, 2012. [67] P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov. Three-body recombination of ultracold atoms to a weakly bound s level. Phys. Rev. Lett., 77:2921–2924, Sep 1996. [68] Scott E Pollack, Daniel Dries, and Randall G Hulet. Universality in three-and four- body bound states of ultracold atoms. Science, 326(5960):1683–1685, 2009. 141 [69] S. E. Pollack, D. Dries, M. Junker, Y. P. Chen, T. A. Corcovilos, and R. G. Hulet. Extreme tunability of interactions in a 7Li Bose-Einstein condensate. Phys. Rev. Lett., 102:090402, Mar 2009.

[70] R. J. Wild, P. Makotyn, J. M. Pino, E. A. Cornell, and D. S. Jin. Measurements of Tan’s contact in an atomic Bose-Einstein condensate. Phys. Rev. Lett., 108:145305, Apr 2012.

[71] Nir Navon, Swann Piatecki, Kenneth Günter, Benno Rem, Trong Canh Nguyen, FrédéricChevy, Werner Krauth, and Christophe Salomon. Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas. Phys. Rev. Lett., 107:135301, Sep 2011.

[72] J. P. D’Incao, H. Suno, and B. D. Esry. Limits on universality in ultracold three-boson recombination. Phys. Rev. Lett., 93:123201, Sep 2004.

[73] Nir Navon. private communications.

[74] Yvan Castin and F´elixWerner. Third virial coeﬃcient of the unitary Bose gas. arXiv preprint arXiv:1212.5512, 2012.

[75] Arnaud Koetsier, P. Massignan, R. A. Duine, and H. T. C. Stoof. Strongly interacting Bose gas: Nozi`eresand Schmitt-Rink theory and beyond. Phys. Rev. A, 79:063609, Jun 2009.

[76] Jesper Levinsen. private communications.

[77] Richard Schmidt, Tilman Enss, Ville Pietil¨a,and Eugene Demler. Fermi polarons in two dimensions. Physical Review A, 85(2):021602, 2012.

[78] Vudtiwat Ngampruetikorn, Jesper Levinsen, and Meera M Parish. Repulsive polarons in two-dimensional Fermi gases. EPL (Europhysics Letters), 98(3):30005, 2012.

[79] F. Chevy. Universal phase diagram of a strongly interacting Fermi gas with unbalanced spin populations. Phys. Rev. A, 74:063628, Dec 2006.

[80] Weiran Li and Tin-Lun Ho. Thermodynamics of dilute Fermi gases in two dimensions. unpublished.

[81] Mohit Randeria, Ji-Min Duan, and Lih-Yir Shieh. Bound states, Cooper pairing, and Bose condensation in two dimensions. Phys. Rev. Lett., 62:981–984, Feb 1989.

[82] D. S. Petrov and G. V. Shlyapnikov. Interatomic collisions in a tightly conﬁned Bose gas. Phys. Rev. A, 64:012706, Jun 2001.

[83] S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein. Pairing in two dimensions. Phys. Rev. Lett., 63:445–448, Jul 1989.

[84] Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen. Symmetry- protected topological orders in interacting bosonic systems. Science, 338(6114):1604– 1606, 2012.

142 [85] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma. Non-abelian anyons and topological quantum computation. Rev. Mod. Phys., 80:1083–1159, Sep 2008.

[86] KW Madison, F Chevy, W Wohlleben, and J Dalibard. Vortex formation in a stirred Bose-Einstein condensate. Physical Review Letters, 84(5):806–809, 2000.

[87] Tin-Lun Ho. Bose-Einstein condensates with large number of vortices. Physical review letters, 87(6):060403, 2001.

[88] Y-J Lin, Rob L Compton, K Jimenez-Garcia, James V Porto, and Ian B Spielman. Synthetic magnetic ﬁelds for ultracold neutral atoms. Nature, 462(7273):628–632, 2009.

[89] Klaus von Klitzing. The quantized Hall eﬀect. Physics: 1981-1990, 2:316, 1993.

[90] Daniel C Tsui, Horst L Stormer, and Arthur C Gossard. Two-dimensional magneto- transport in the extreme quantum limit. Physical Review Letters, 48(22):1559–1562, 1982.

[91] Jainendra K Jain. Composite fermions. Cambridge University Press, 2007.

[92] D Yoshioka. The Quantum Hall Eﬀect. Springer.

[93] M Zahid Hasan and Charles L Kane. Colloquium: topological insulators. Reviews of Modern Physics, 82(4):3045, 2010.

[94] Robert B Laughlin. Anomalous quantum hall eﬀect: an incompressible quantum ﬂuid with fractionally charged excitations. Physical Review Letters, 50(18):1395–1398, 1983.

[95] Makoto Tsubota, Kenichi Kasamatsu, and Masahito Ueda. Vortex lattice formation in a rotating Bose-Einstein condensate. Phys. Rev. A, 65:023603, Jan 2002.

[96] Jean Dalibard, Fabrice Gerbier, Gediminas Juzeli¯unas,and Patrik Ohberg.¨ Collo- quium : Artiﬁcial gauge potentials for neutral atoms. Rev. Mod. Phys., 83:1523–1543, Nov 2011.

[97] Y-J Lin, K Jimenez-Garcia, and IB Spielman. Spin-orbit-coupled Bose-Einstein condensates. Nature, 471(7336):83–86, 2011.

[98] Thomas Busch, Berthold-Georg Englert, Kazimierz Rza˙zewski,and Martin Wilkens. Two cold atoms in a harmonic trap. Foundations of Physics, 28(4):549–559, 1998.

[99] Zbigniew Idziaszek and Tommaso Calarco. Analytical solutions for the dynamics of two trapped interacting ultracold atoms. Physical Review A, 74(2):022712, 2006.

[100] Qi Zhou and Xiaoling Cui. Fate of a Bose-Einstein condensate in the presence of spin-orbit coupling. Phys. Rev. Lett., 110:140407, Apr 2013.

[101] Tin-Lun Ho and Shizhong Zhang. Bose-Einstein condensates with spin-orbit interaction. Physical Review Letters, 107(15):150403, 2011. 143 [102] Chunji Wang, Chao Gao, Chao-Ming Jian, and Hui Zhai. Spin-orbit coupled spinor Bose-Einstein condensates. Physical review letters, 105(16):160403, 2010.

[103] Alexander L Fetter. Lowest-Landau-level description of a Bose-Einstein condensate in a rapidly rotating anisotropic trap. Physical Review A, 75(1):013620, 2007.

[104] Amandine Aftalion, Xavier Blanc, and Nicolas Lerner. Fast rotating condensates in an asymmetric harmonic trap. Phys. Rev. A, 79:011603, Jan 2009.

[105] William C Stwalley and He Wang. Photoassociation of ultracold atoms: A new spectroscopic technique. Journal of molecular spectroscopy, 195(2):194–228, 1999.

[106] Nathan Gemelke, Edina Sarajlic, and Steven Chu. Rotating few-body atomic systems in the fractional quantum Hall regime. arXiv preprint arXiv:1007.2677, 2010.

[107] Hao Wang, Rajesh Narayanan, Xin Wan, and Fuchun Zhang. Fractional quantum Hall states in two-dimensional electron systems with anisotropic interactions. Phys. Rev. B, 86:035122, Jul 2012.

[108] R.-Z. Qiu, Su-Peng Kou, Z.-X. Hu, Xin Wan, and S. Yi. Quantum hall eﬀects in fast- rotating Fermi gases with anisotropic dipolar interaction. Phys. Rev. A, 83:063633, Jun 2011.

144 Appendix A Path integral formalism of BCS-BEC crossover

In this appendix, we use path integrals to derive the crossover equations for two-component

Fermi gases across unitarity. The hamiltonian we start from is 3.1, and the corresponding action and partition function can be derived. For equal spin system at ﬁnite temperatures, the thermodynamic partition function is

β ∂ 2 Z = ψ¯ ψ exp dτ dr ψ¯ ∇ µ ψ + gψ¯↑ψ¯↓ψ↓ψ↑ , (A.1) D D − σ ∂τ − 2m − σ Z Z0 Z where ψ¯ and ψ are Grassmann numbers in this coherent-state path integral formalism, the

imaginary time integral is from 0 to inverse temperature β. g < 0 is the coupling constant from the contact interaction. By Hubbard-Stratonovich transformation, an auxiliary ﬁeld is introduced such that the four-operator term in the action can be decomposed and integrated out by the following identity:

∆∆¯ exp dτ dr g ψ¯↑ψ¯↓ψ↓ψ↑ = ∆¯ ∆ exp dτ dr ψ¯↑ψ¯↓ ∆¯ ψ↓ψ↑ , − | | D D − g − − Z Z Z Z Z | | (A.2)

where ∆¯ , ∆ are bosonic ﬁelds such that ∆¯ is simply the complex conjugate of ∆. The

expression above transforms the four-Fermi operator term into a quadratic form, and the

145 partition function is rewritten into

Z = ∆¯ ∆ ψ¯ ψ D D D D Z Z ∂ ∇2 ∆∆¯ µ ∆ ψ↑ exp dτ dr + ¯ ∂τ − 2m − −   ψ↑ ψ↓  2    − g ¯ ∂ ∇ ¯ Z Z | | ∆ ∂τ + 2m + µ ψ↓    −         (A.3)

The inverse of the interacting Green’s function is

2 ∂ ∇ µ ∆ G−1 ∂τ − 2m − − (A.4)  2  ≡ ¯ ∂ ∇ ∆ ∂τ + 2m + µ  −    The auxiliary ﬁeld ∆ brings in the anomalous (oﬀ-diagonal) terms in Green’s function. We

integrate out the fermion operators ﬁrst. The partition function is then

∆(¯ r, τ)∆(r, τ) Z = ∆¯ ∆ exp dτ dr + log(detG−1) (A.5) D D − g Z Z Z | | We can switch to the momentum and Matsubara frequency domain, in which the ﬁeld operator is transformed into

1 ψ(r, τ) = ψ eik·r−zν τ . (A.6) √βV k,zν zXν ,k 2 2 The ∂/∂τ goes to z and ∇ µ goes to ξ = k µ. The action from A.3 in momentum ν − 2m − k 2m − space is

¯ z + ξ ∆ ¯ ¯ βV ∆∆ ν k βV ∆∆ 2 2 S[∆, ∆] = + tr log   = + log(zν Ek), (A.7) − g ¯ − g − | | zν ,k ∆ zν ξk | | zν ,k X  −  X   2 2 −1 −1 in which Ek = ξk + ∆k, and we used an identity log detG = tr logG . The saddle point is given byq ∂S = 0, (A.8) ∂∆

146 and it gives the BCS gap equation

1 1 1 2 2 = . (A.9) βV zν Ek − g zXν ,k − | | Summing over the Matsubara frequencies, the equation takes the form

1 1 tanh(βE /2) = k (A.10) g V 2Ek | | Xk is exactly same as 3.23. By taking the saddle point of the gap, the BCS mean-ﬁeld partition function is

−S[∆0] ZBCS = e , (A.11) where ∆0 is the saddle point solution to A.8. If we go beyond the mean-field theory, Gaussian fluctuations of the pairing order parameter ∆ can be considered near the saddle point ∆0 as ∆ = ∆0 + δ∆. From expressions A.3 and A.4, the Green’s function is modified as

−1 −1 G = G0 + M, (A.12) where 0 δ∆ M =   . (A.13) δ∆¯ 0     The appearance of tr log G−1 can be expanded in orders of δ∆ (or M). To the second order, it is

1 Tr log G−1 = Tr log G−1 +Tr log (1+G M) = Tr log G−1 +Tr(G M) Tr(G M)2 +O(M 3). 0 0 0 0 − 2 0 (A.14)

Since G0 is given by the saddle point of the system, the ﬁrst order in M vanishes in Tr log G−1.

Now we restrict our discussion to the normal phase where ∆ = 0. In this case, the second order term above is

1 2 Tr(G M) = G↑↑G↓↓δ∆¯ δ∆ (A.15) −2 0 −

147 So the full partition function in the gaussian ﬂuctuation regime is decomposed to

Z = Z δ∆¯ δ∆ exp G↑↑G↓↓δ∆¯ δ∆ (A.16) BCS D D − Z Take the explicit expressions of G in the absence of the pairing order parameter ∆ is

−1 −1 G↑↑ = (z + ξ ) and G↓↓ = (z ξ ) . We ﬁnd this unit of G↑↑G↓↓ is the same as each ν k ν − k rung of the ladders in the ladder approximation. After the integration of the ﬂuctuation

ﬁeld, the expression is the same as 3.38. This concludes the equivalence between the ladder

approximation and the gaussian ﬂuctuation in the normal phase.

148 Appendix B Adiabatic states and their gauge potential in spatially varying magnetic field

In this appendix, we derive a general form of the adiabatic states in the total spin-F system in a varying magnetic ﬁeld, and the gauge potential associated with them. In the

first section, we express the local eigenstates of the Zeeman term with spatially varying magnetic field in the basis of eigenstates in the stationary reference frame. In the second part, we derive the gauge field associated with different states, and show some specific cases with the maximum spin states m = F , and the m = 0 state as examples. ±

B.1 General adiabatic states in spatially varying magnetic ﬁeld

2 For a spin-F system, usually we use a set of basis with eigenstates of operators F and Fz as F, m such that | i

F 2 F, m = F (F + 1) F, m , (B.1) | i | i F F, m = m F, m , (B.2) z| i | i where m = F, F + 1, ..., F . These states are the eigenstates of a linear Zeeman term − − H = gµ B F, with a magnetic ﬁeld in the z-direction B = Bˆz. In the following, we will Z B · derive the expression for the eigenstates in the F, m basis of a Zeeman term in arbitrary | i

149 direction B = B(sin θ cos φ, sin θ sin φ, cos θ). In other words, we will ﬁnd the eigenvectors

ηm to the matrix representation of B F, such that ·

(B F) ηm = mηm. (B.3) · αβ β α Xβ We introduce the Schwinger representation of spin: the spin operators are decomposed into two species of bosonic operators a and b as follows:

+ † F = Fx + iFy = a b,

F − = F iF = b†a, x − y F = a†a b†b, (B.4) z − and the value of the total spin F is related to the total number of a, b bosons by

1 F = a†a + b†b . (B.5) 2h i

2 In a subspace of spin-F , the 2F + 1 eigenstates of F and Fz in this representation can be written as: (a†)F +m(b†)F −m F, m = 0 , (B.6) | i (F + m)!(F m)!| i − where 0 is the vacuum. In the matrixp representation, the eigenstates of F 2,F are repre- | i z sented by eigenvectors:

F, m = (0, 0, ..., 0, 1, 0, ..., 0)T (B.7) | i with the (F m + 1)th component as 1 and all the other terms vanishing. This set of states − are the eigenstates of a linear Zeeman term when the magnetic ﬁeld in the z-direction gµ B F . From now on, we use an abbreviated form of F, m as m , where we drop F B z z | i | i since all the discussions in the following will be conﬁned to the manifold of total spin equals

F .

Now we will derive a general form of the set of eigenstates, in this subspace with total spin F , of a Zeeman term with magnetic field in any rotated direction H˜ = gµ B F. The Z B · direction of the magnetic field is B = B(sin θ cos φ, sin θ sin φ, cos θ). We will represent this new set of states m˜ , in the original basis F, m , i.e. to find out the coefficients m in the | i | i Cn 150 expansion:

m˜ = m n . (B.8) | i Cn | i n X These coeﬃcients are related to the eigenvectors ηm as

ηm = ( m , m , ..., m)T. (B.9) C−F C−F +1 CF

To ﬁnd these coeﬃcients, we note that the two Zeeman terms are connected by a rotation in space

H˜ = H −1, (B.10) Z R Z R in which the unitary transformation operator is R

= eiSzφeiSyθeiSzχ, (B.11) R where Sx,Sy,Sz are the spin-1/2 operators. They are generators of the SU(2) rotation group. The last factor eiSzχ in equation B.11 corresponds to a gauge degree of freedom in the spin space. It can be gauge-transformed away, and thus will not be discussed in the following. The new Schwinger boson operators in the rotated framea ˜†, ˜b† will experience the same unitary transformation. They are thus related to the original Schwinger boson operators a†, b† as

a˜† a† u v a† −1   =   =     , (B.12) ˜b† R b† R v∗ u∗ b†      −            θ iφ/2 θ −iφ/2 where u = cos 2 e and v = sin 2 e . Thus the new set of eigenstates are

151 The explicit form of the coeﬃcients are: C

F +m F + m F m m (F + n)!(F n)! n = − − C (F + m)!(F m)!     p − k=0 k F + n k X    −  p uk(u∗)−m−n+kvF+m−k( v∗)F +n−k. (B.15) × −

n The binomial coeﬃcient   is equal to 0 for n < m or m < 0. This equation gives the m   general expression of the local eigenstates in the spatially varying magnetic ﬁeldx in the basis of the stationary lab frame.

B.2 Gauge potentials associated with the adiabatic states

The gauge ﬁeld associated with the local eigenstate m˜ is | i

m† m∗ m A = i~ m˜ m˜ = i~η η = i~ . (B.16) h |∇| i ∇ Cn ∇Cn n X We have

m˜ m˜ = ( m˜ m˜ ) ( m˜ ) m˜ = ( m˜ ) m˜ = m˜ m˜ ∗, (B.17) h |∇| i ∇ h | i − ∇h | | i − ∇h | | i −h |∇| i which makes the m˜ m˜ a pure imaginary number. Consequently, the gauge potential is h |∇| i pure real. Also, this fact simplifies our calculation since we can focus on the imaginary part of m˜ m˜ only. h |∇| i Now we consider a specific state m˜ = 0 state (for integer F ). The gauge field associated | i with this state is

0 0∗ 0 A /(i~) = . (B.18) Cn ∇Cn n X The general formula (B.15) in the previous section gives an explicit expression as

min(F,n+F ) uk(u∗)k−nvF −k( v∗)F +n−k 0 = F ! (F + n)!(F n)! − . (B.19) Cn − k!(k n)!(F k)!(F + n k)! p k=max(0X ,n) − − − By comparing term by term in the sum, we ﬁnd such relation between n and n as 0 = − Cn ( 1)n 0∗ and 0 is real. Pairing up the 0∗ 0 and 0∗ 0 , the imaginary part of the − C−n C0 Cn ∇Cn C−n∇C−n

152 whole sum in (B.18) is

Im 0∗ 0 = Im ( 0∗ 0 + 0∗ 0 ) Cn ∇Cn Cn ∇Cn C−n∇C−n n n>0 X X = Im ( 0∗ 0 + ( 1)2n 0 0∗) Cn ∇Cn − Cn∇Cn n>0 X = Im (Re 0 iIm 0) (Re 0 + iIm 0) Cn − Cn ∇ Cn Cn n>0 X +(Re 0 + iIm 0) (Re 0 iIm 0) Cn Cn ∇ Cn − Cn = Im 2(Re 0 Re 0 + Im 0 Im 0) = 0. (B.20) Cn∇ Cn Cn∇ Cn n>0 X Thus the gauge ﬁeld associated with this m˜ = 0 state is identically zero. We can then | i conclude that this “minimum spin” adiabatic state will not generate any gauge potential in the spatially-varying magnetic ﬁeld.

There are also other special states. The “maximum spin” states m = F and m = F − are worth discussing. For these two states, their eigenstates have the coeﬃcients

(2F )! F = uF +nvF −n, (B.21) Cn (F + n)!(F n)! s − (2F )! −F = ( v∗)F +n(u∗)F −n, (B.22) Cn (F + n)!(F n)! − s − which both contain only one term. Calculating the gauge potential for these two states, we

have

−F −F ∗ −F A /(i~) = Cn ∇Cn n X (2F )! = ( v)F +n(u)F −n (( v∗)F +n(u∗)F −n) (F + n)!(F n)! − ∇ − n X − (2F )! = (v)F −n(u)F +n ((v∗)F −n(u∗)F +n) (F n)!(F + n)! ∇ n X − (2F )! ∗ = (v )F −n(u )F +n ((v)F −n(u)F +n) (F n)!(F + n)! ∗ ∗ ∇ n − XF ∗ F = A /(i~) = A /(i~). (B.23) − In the derivation, we have used the fact that A is real. From the equation above, we proved that the two maximum spin states will experience gauge potentials with the same magnitude

153 but with diﬀerent signs. This is expected, since the states m = F and m = F are | i | − i “time-reversal partners”. In fact, similar to the derivations above, one can show the gauge potentials of any pair m and m have the same magnitude and diﬀerent signs. The | i | − i mathematics is straightforward, but it is more complicated and will not be included here.

154