Download Topics in Ultracold Atomic Gases: Strong Interactions and Quantum

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.
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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. Zhenhua 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 mentors 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
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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
Abstract . . . . . .
Acknowledgments
Vita . . . . . . . .
List of Figures .
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Page
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. viii
Chapters
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Basics in ultracold atomic gases . . . . . . . . . . . . . . . . . .
2.1 Scattering models in ultracold atoms . . . . . . . . . . . . . . . .
2.1.1 General scattering theory, T -matrix . . . . . . . . . . . .
2.1.2 Low energy scattering, s-wave scattering length and phase
2.1.3 Zero range model and Fermi’s pseudo potential . . . . . .
2.2 Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Probes in cold atoms experiments . . . . . . . . . . . . . . . . . .
2.3.1 Direct Imaging . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
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shift . . .
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1
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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 . . . . . . .
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3.4 Two dimensional Fermi gases with spin imbalance . . . . . . . . . . . . . .
82
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3.4.1
3.4.2
3.5
Fermi gases in two dimensions . . . . . . . .
Thermodynamic quantities of two component
mension . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . .
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Fermi gases
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in two
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di. .
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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
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List of Figures
Figure
2.1
2.2
2.3
2.4
Page
A sketch of phase shifts for different scattering lengths. . . . . . . . . . . . .
A sketch of the s-wave scattering length and the effective range for a square
well potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A schematic diagram of the potentials in open and closed channels near a
Feshbach resonances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental images from time-of-flight (TOF) expansion, in Bose and Fermi
gases in optical lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A sketch of pairing gap and chemical potential for a Fermi gas across unitarity
at T = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The scattering vertex for the particle-particle channel and particle-hole channel. For dilute gases, particle-hole channel is usually negligible. . . . . . . .
3.4 A sketch of contour deformation of Matsubara sum in NSR formalism. . . .
3.5 A sketch of the superfluid transition temperature and the chemical potential
at Tc for a Fermi gas in BCS-BEC crossover. . . . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 The no-pole approximation in terms of excluding the bound-state contribution, from modifying the integral region after doing the Matsubara sum. . .
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. . . . . . . . . . . . . . . .
3.9 A comparison of the calculated transition temperature Tc from different approaches: Leggett BCS mean field, GMB mean field, and NSR. . . . . . . .
3.10 An illustration of the difference between wide and narrow Feshbach resonances in a Fermi gas with Fermi energy EF . . . . . . . . . . . . . . . . . .
3.11 Schematic diagram of the relation between wide and narrow resonances: their
occupations in the space ((kF r∗ )−1 , kF abg ). . . . . . . . . . . . . . . . . . .
13
16
19
25
3.1
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3.12 The scattering phase shift δ(k) as a function of incoming wave vettork for
wide (A) and narrow (B) resonances. . . . . . . . . . . . . . . . . . . . . . .
3.13 A sketch of second virial coefficients −b2 (A) and “interaction energy” int (B)
as a function of magnetic field, across a narrow Feshbach resonance. . . . .
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 . . . . . . .
3.15 Experimental data from the Penn State group for fermionic 6 Li gases across
the 543.25G Feshbach resonance at different temperatures. . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.18 The “phase diagram” of a homogeneous upper-branch Bose gas with fixed
density n. The system is divided into three parts. . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . .
3.20 Three typical density profiles for stable and unstable upper-branch Bose gases
in a harmonic trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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.
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.
3.23 Spin susceptibility and compressibility for upper branch two dimensional
Fermi gas with equal population, at a temperature T = 6TF . . . . . . . . .
3.24 A diagram of “stability” of repulsive branches at different temperatures and
polarizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
A sketch of the origin of integer quantum Hall effects. . . . . . . . . . . . .
A sketch of the energy levels of Fock Darwin states for particles in rotating
traps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . .
Schematic figure of experimental setup in NIST. The Raman coupling is
realized by two counter-propagating laser beams. . . . . . . . . . . . . . . .
The direction of the vector Beff from the effect of Raman coupling plus a
linear detuning in magnetic field. . . . . . . . . . . . . . . . . . . . . . . . .
Profiles of the vector potential and the synthetic magnetic field generated
from the NIST setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic diagrams of energy levels in both Abelian and non-Abelian synthetic gauge fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A sketch of the energy spectrum of two branches for spin-orbit coupled gases.
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105
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113
116
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
Phase diagram of the cloud in rotating anisotropic traps: two phases of the
BEC and QH are identified. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Distortion and transition in vortex lattices of a rapidly rotating BEC in an
anisotropic trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Density profiles at cut x = 0 and y = 0 of the Laughlin wavefunctions at
different positions in figure 4.9. . . . . . . . . . . . . . . . . . . . . . . . . .
Distribution of zeros in the relative coordinate of the ground state of two
body problem in rotating anisotropic traps. . . . . . . . . . . . . . . . . . .
Variation in the distance of splitting zeros as a function of inverse interaction
strength, for fixed anisotropy α = 0.96. . . . . . . . . . . . . . . . . . . . . .
Density profiles after TOF expansion of the gas in a rotating isotropic trap.
Density profiles after TOF expansion of anisotropy α = 0.94. . . . . . . . .
Second order correlation in equation 4.64, in a rotating isotropic (left) and
anisotropic (right) trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A sketch of high-resolution in situ imaging technique in neutral atoms. . . .
In situ images of the quantity |ψ(r/2, −r/2)| for an anisotropy α = 0.96. .
x
124
125
127
128
129
132
133
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135
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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
states change gradually as a−1
s increases continuously. It is found that there is a continuous
crossover between the superfluidity in the BCS limit ((kF as )−1 → −∞) and the BEC limit
((kF as )−1 → +∞). In BCS, the Cooper pairs form the constituent of the condensate, while
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 interesting 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.
1
In 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” |ψin i, which is a solution of
Schrödinger’s equation for H0 . The full wave function |ψi with total energy E consists of
an incident wave part and a scattering wave part |ψsc i, which satisfies the equations below:
|ψi = |ψin i + |ψsc i,
(2.1)
(E − H)|ψi = 0,
(2.2)
(E − H0 )|ψin i = 0.
(2.3)
Using the Lippmann-Schwinger equation, we can define the scattering T -matrix3 in this
2
Even though most cold atom experiments are conducted in harmonic traps, the smoothness of the overall
potential enables one to neglect the broken translational symmetry.
3
In 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
|ψi = |ψin i + |ψsc i = (E − H0 )−1 V |ψi
= (E − H0 )−1 TE |ψin i,
(2.4)
where the scattering T -matrix at energy E is defined as
T |ψin i ≡ V |ψi.
(2.5)
By left multiplying the interaction operator V on both sides of equation 2.4, we have
V |ψi = TE |ψin i = V (E − H0 )−1 TE |ψin i,
TE = V G0 (E)TE ,
(2.6)
(2.7)
where G0 (E) ≡ (E − H0 )−1 is the free Green’s function at energy E. Equation 2.7 above
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:
TE = V + V G0 (E)V + · · · =
V
.
1 − G0 (E)V
(2.8)
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 G0 (E)V 1, the Born
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
k0 can be written as hk|TE |k0 i ≡ TE (k, k0 ). The condition that E = ~2 k 2 /2µ = ~2 k 02 /2µ,
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
hl, k|TE |l0 , k 0 i ≡ TEl,l (k, k 0 ),
7
(2.9)
in which |l, ki is the partial wave with angular momentum l and wave number k. In an
0
0
isotropic potential with spherical symmetry, T l,l = δl,l0 T l,l 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 hr|4 , and assume the incident wave as a plane wave with momentum
k. The full wave function in real space then takes the form
ψ(r) = eikr + f (θ, φ)
eikr
,
r
(2.10)
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 .
dΩ
(2.11)
With spherical symmetry, the scattering wave function can also be expanded by partial
waves, into the form
ψ=
∞
X
Al Pl (cos θ)Rkl (r),
(2.12)
l=0
in which Pl ’s are Legendre polynomials, and the radial part Rkl satisfies the equation
2 0
l(l + 1) 2µ
2
Rkl + Rkl + k −
− 2 V (r) Rkl = 0,
r
r2
~
00
(2.13)
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
4
To get the following form with “out-going” wave only, we have to add an infinitesimal imaginary part
in the Green’s function, i.e. substitute (E − H0 )−1 by the form (E − H0 + i0+ )−1 .
8
Neuman functions jl and nl . The general form of the solution at large distance reads
Rkl (r → ∞) ≈
sin(kr − lπ/2 + δl )
,
kr
(2.14)
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 δl ≡ 0.
By comparing the asymptotic form above and the expression of f (θ) in equation 2.10,
we have
f (θ) =
∞
∞
X
1 X
(2l + 1)fl Pl (cos θ)
(2l + 1)(e2iδl − 1)Pl (cos θ), ≡
2ik
(2.15)
l=0
l=0
with the l-wave scattering amplitude
fl ≡ (e2iδl − 1)/2ik.
(2.16)
The scattering amplitude relates the scattering T -matrix as fl = −Tl µ/2π~2 . And the total
cross section of the scattering is given by
σ=
∞
∞
4π X
4π X
2
(2l
+
1)|f
|
=
(2l + 1) sin2 δl .
l
k2
k2
l=0
(2.17)
l=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
jl (kr) ∼ (kr)l ,
nl (kr) ∼ (kr)−l−1 ,
at kr → 0.
(2.18)
If the boundary condition is enforced at r = r0 where r0 is the range of potential, the
relative ratio between nl and jl will be proportional to (kr0 )2l+1 . In low-energy scattering,
namely kr0 1 limit, it is natural to see tan δl ∼ δl ∼ (kr0 )2l+1 , hence the s-wave scattering
(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 :
δ0 = −kas ,
for k → 0.
(2.19)
The s-wave cross section
σ=
4π
sin2 δ0 = 4πa2s
k2
(2.20)
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 k 2 as the following5 :
f −1 (k) = k cot δ(k) − ik = −
1
r∗
+ k 2 + O(k 4 ) − ik
as
2
(2.21)
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
kn(0) =
nπ
.
R
(2.22)
The range of n is given by the density of the system such that the largest n = N satisfies
kF =
Nπ
R .
First we discuss the “weakly interacting” limit with small scattering length, i.e.
kF as 1. In this limit when we put the interaction in, a nonzero phase shift emerges and
5
The 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 = −kas . The corresponding eigen wave vectors have to satisfy
kn R + δ(kn ) = kn (R − as ) = nπ,
(2.23)
which gives
kn =
nπ
.
R − as
(2.24)
The corresponding shift in energy levels are given by
(0)2
(0)
where ψn =
∆En =
~2 kn2
~2 kn
−
2mr
2mr
1
(2πR)1/2 r
sin
nπr
R
≈
~2
2as
2π~2 as (0)
(nπ)2 3 =
|ψn (0)|2 ,
2mr
R
mr
(2.25)
is the wave function of non-interacting systems. The ap-
proximation δ = −kas we use in the previous derivation is based on the assumption that
the scattering length as Nπ
R
is very small, i.e. the system is in the weakly interacting
limit with kF as 1. In this limit, the effective interaction hamiltonian can be written in
a delta-function form:
V (r) = gδ(r),
g=
2π~2 as
,
mr
(2.26)
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 kF as 1. For the strongly interacting regime, i.e. kF as ∼ 1 or larger than unity,
the approximation δ(k) = −kas cannot be used in the region 0 < k < kF . Instead the
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
X
e−βE1n + z 2
n
X
n
11
#
e−βE2n ,
(2.27)
which is approximated to the second power of fugacity. E1n and E2n are the energy levels of
P
single-particle and two-body states. We have to calculate Z2 = z 2 n e−βE2n and subtract
the value of a non-interacting system to get the contribution from 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
∆Z2 = z 2
X
n
(0)
(0)
e−βE2n − e−βE2n
,
(2.28)
(0)2
with E2n = ~2 kn2 /2mr and E2n = ~2 kn /2mr . As is shown that kn + δn = nπ, we have
dn =
dδ
R+
dk
dk.
(2.29)
And the sum over n becomes an integral over k, such that
∆Z2 = z
2
Z
0
∞
dk dδ −β~2 k2 /2mr
≡ z 2 b2 ,
e
π dk
(2.30)
where b2 is defined as the second virial coefficient for the scattering states6 . 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
k cot δ = −
1
.
as
(2.31)
A plot of phase shifts at different negative scattering lengths is shown in figure 2.1. They
cover a large range of as from the weakly interacting regime kF as = −0.1 to extremely
strongly interacting case kF as = −100. The phase shifts for positive scattering lengths are
mirror symmetric to the horizontal axis in the plot, namely they only differ by a minus
6
Here 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
∆HkL
Π
2
Π
4
0
0
0.5
1
kkF
1.5
2
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 kF as = −0.1, −0.5, −1, −10, −100. For
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: kF as → 0 and kF as → ∞. The former is realized by approximating
∂δ/∂k = −as , and for the latter, the approximation is made as ∂δ/∂k 6= 0 only in a small
region of momentum k < k ∗ kF . Simplify the integral in 2.30, we have the following
expressions:
(0)
b2
b∞
2
as
= −√ ,
2λ
sgn(as )
2
= −
(1 − erf(x))ex ,
2
(2.32)
(2.33)
(0)
and b∞
2 are second virial coefficients for weakly and strongly interacting cases
√
√
respectively. In the expressions, x = λ/( 2πas ) with λ = h/ 2πmr kB T as the thermal
where b2
wavelength of the system, and erf(x) is the error function. For the weakly interacting case,
b2 is asymptotically first order in as /λ 1. And the limit that as → ∞ (x → 0) gives
13
b2 = −sgn(as )/2 = ±1/2, in the order of unity. Thus we conclude the strongly interacting
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
= P0 + 2 2z 2 3 b2 ,
λ "
#
√
2 T ∂b2
3T n
b2
3
= 0 +
,
(nλ ) − √ +
2
3 ∂T
2
P
where P and are the pressure and energy density respectively, subindex
(2.34)
(2.35)
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 |b2 |. The large absolute
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 kr0 1. As in the dilute gases,
since the energy scale is given by the Fermi wave vector kF , and it satisfies kF r0 1, by
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.
V (r) = 0, r > r0 ;
V (r) = −V0 ≡ −
~2 κ2
, r < r0 .
2mr
(2.36)
Solving the Schrödinger’s equation for total energy E = ~2 k 2 /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:
as
tan κr0
=
− 1,
r0
κr0
(2.37)
r∗
1
(κr0 )2
=1−
−
.
r0
2κr0 (κr0 − tan κr0 ) 3(κr0 − tan κr0 )2
(2.38)
and the effective range as
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, as jumps from −∞ to +∞, but (as )−1 changes continuously.
For this scattering model, at low energy the inverse scattering length 2.21 contains
two parts: a momentum independent constant as −a−1
s and a k-dependent term. In this
15
-as r0 r* r0
2
0
-2
-4
0
2Π
Π
3Π
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∗ k 2 /2.
For low energy scattering k (r∗ )−1 , such that |r∗ k 2 /2| | − ik|, one can simplify the
inverse scattering length by f −1 = −a−1
s −ik. Now in the simplified model, we have only one
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:
ψ(r) ∼
sin(kr − kas )
1
1
∼ − ,
kr
r as
r → 0.
(2.39)
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
Vpp =
∂
2π~2 as
δ(r) r.
mr
∂r
(2.40)
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 constant ḡ. The order-by-order expansion
of the T -matrix will have an ultraviolet divergence, since
X
p
G0 (E = 0) ∼ −
Z
dpp2
1
→ −∞
p2
(2.41)
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
T (k, k0 ) = ḡ − ḡ
X 1
T (p, k0 ),
p
p
(2.42)
where p = ~2 p2 /2mr . In the low-energy limit, we approximate the T -matrix for any energy
and equal incoming and outgoing momenta as TE=0 (0, 0) = 2π~2 as /mr ≡ T (0; 0, 0). The
equation above is rewritten as
X 1
X 1
1
1
mr
=
−
=
−
.
ḡ
T (0; 0, 0)
p
2π~2 as
p
p
p
(2.43)
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, the ḡ appearing in final expres17
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 100aB where aB ≈ 0.53Å is the Bohr radius. Few can have scattering lengths
as large as 2000aB . As the most dense stable dilute gases have typical density as large
as 1015 cm−3 , the characteristic scale of the system is given by |kF as | < 1. And as the
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
|Ψi = |ΨP i + |ΨQ i,
and the Schrödinger’s equation is

HP P

HQP



(2.44)

HP Q  ΨP 
ΨP 
  = E  ,
ΨQ
ΨQ
HQQ
(2.45)
in which HP P = P HP 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 Schrödinger’s equation in the open channel manifold:
(E − HP P − HP0 P )|ΨP i = 0,
(2.46)
HP0 P = HP Q (E − HQQ )−1 HQP
(2.47)
where
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 hΨQ |H|ΨQ i, the equation above
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
from the open channel itself, and U2 = HP0 P is the closed-channel induced interaction
potential. With this prescription, we may write down a simplified two channel model
H=
X
k
k c†k ck + U1
X
c†↑ c†↓ c↓ c↑ + αd† c↑ c↓ + h.c. + δd† d,
(2.49)
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
TP P (0; 0, 0) =
|α|2
|α|2
=
,
−δ
∆µ(B∞ − B)
(2.50)
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
at the resonance. The inverse scattering length a−1
s 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
T = G−1
0 GU,
(2.51)
where G = (E − H0 − U )−1 is the full Green’s function. And we further express the full
Green’s function by this Dyson equation
G = G1 (1 + U2 G),
(2.52)
in which G1 = (E − H0 − U1 )−1 is the propagator that includes the background interactions
in the open channel itself. Then 2.51 is transformed into
−1
−1
T = G−1
0 G1 (1 + U2 G)U = G0 G1 U1 + G0 G1 U2 (1 + GU ) ≡ T1 + T2 ,
(2.53)
where T1 ≡ G−1
0 G1 U1 is the scattering T -matrix in which the closed channel is absent, and
the additional term from closed channel is T2 :
−1
T2 ≡ G0−1 G1 U2 (1 + GU ) = G−1
0 G1 U2 GG0 .
(2.54)
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) =
2π~2 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
ares =
mr
−1
hk0 |G−1
0 G1 U2 GG0 |ki,
2π~2
k, k0 → 0,
(2.55)
where |ki is the incoming plane wave with momentum k in the open channel. When GG−1
0
acts on the Dirac ket to the right and G−1
0 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
make the assumption to the first order of U2 , by which we replace GG−1
0 to G1 G0 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
between open channel U1 = 0 in 2.49, G1 G−1
0 = 1 and ares is given by
ares =
mr
|α|2
,
2π~2 ∆µ(B∞ − B)
(2.56)
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
mr
|α|2
∆B
as (B) = abg +
,
≡ abg 1 +
2π~2 abg ∆µ(B∞ − B)
B∞ − B
in which ∆B =
mr |α|2
2π~2 ∆µ
(2.57)
is the distance (with sign) between the position where as diverges
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
7
In 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
b
a
c
d
0.04
e
OD
y
2 hk
x
0
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) = µ0 − m(ω ·
r)2 /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)
∂r /µ.
In the region where k(r) kF (kF is the local Fermi
vector given by the density as n ∼ kF3 ), at each point we can assume there is a set of local
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-twodimensional 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
P (µ, T ) =
Z
µ
dµ0 n(µ0 , T ).
(2.59)
−∞
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
p
R
R
density n(x) ≡ dydzn(x, y, z) = dρ2 n(x, ρ) ∼ P (x), where ρ = y 2 + z 2 . P (x) is
actually the pressure as a function of x at the center of the “tube” y = 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 Hubbard 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 |1i and |2i. There is
another unpopulated hyperfine state labeled |3i. In the large-magnetic-field limit, these
three states are the lowest energy states for this fermionic isotope. The energy difference
between atomic states |3i and |2i is denoted by ω0 . By applying an rf field, state |2i is
connected to the originally unoccupied state |3i. As the atoms typically have momentum
much larger than the rf photons, the selection rule implies that the momentum of |2i and
|3i before and after the transition are the same, and energy is changed by the rf photon
energy ω. In the experiments, the incoming photon frequency is swept in the region around
ω0 . For every incoming energy ω, the population of final state |3i after certain holding time
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 |3i is proportional to
the transition rate and to the original spectral function of A(k, ωf − (ω − ω0 ))8 , where ω is
the rf frequency. By sweeping through ω, one collects the spectral function of the original
|2i state. For a non-interacting system, one expects the final signal as a delta function of
δ(ω − ω0 ), namely the original system has a z = 1 quasiparticle weight, and the dispersion
relation is the same with the final |3i. For an interacting system, the spectral function
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 .
8
Here we make the assumption that the final state |3i is almost non-interacting, such that there is no
“spectrum shift” caused by interactions in |3i, namely within the approximation that ωf (k) ≡ k2 /2m. The
different momentum states are thus with the same energy reference.
9
For 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 quantum 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 stronglyrepulsive 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 BCSBEC 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 similarity 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 superfluidity 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 (kF as )−1 changes from −∞ to +∞ where kF is the Fermi
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 “manybody” 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
kF as → 0− and kF as → 0+ respectively.
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 kF as → 0+ , especially in the Bose condensates[49].
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 = −(kF as )−1 . For most of the Feshbach resonances, by increasing the magnitude
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)
H=
XZ
σ
dr
Z
1
†
†
∇ψσ (r) · ∇ψσ (r) − µψσ (r)ψσ (r) +g drψ↑† (r)ψ↓† (r)ψ↓ (r)ψ↑ (r), (3.1)
2m
where ψσ is the field operator for different spin states with σ =↑, ↓ in two-component Fermi
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
|Ψg i =
Y
(uk + vk c†k↑ c†−k↓ )|0i,
(3.2)
k
where |0i is the vacuum. All the physical quantities of the ground state can be determined
by this trial wave function. The mean-field solution of hamiltonian 3.1 determines the
parameters uk and vk as the following:
1
ξk
1+
,
(3.3)
2
Ek
ξk
1
2
1−
,
(3.4)
vk =
2
Ek
q
where ξk = k 2 /2m − µ and Ek = ξk2 + ∆2 is the dispersion relation for single-particle
u2k =
excitations. The BCS pairing gap ∆ is the order parameter of the system, and is defined
as the mean-field quantity of pairing term
∆ = hΨg |
g X
ck↑ c−k↓ |Ψg i,
V
(3.5)
k
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 X
g X ∆
uk vk = −
.
V
V
2Ek
k
(3.6)
k
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 X 1
.
=−
g
V
2Ek
(3.7)
k
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
2
Here 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 X 1
1
m
1 X 1
−
=
=−
,
g
4πas V
2k
V
2Ek
k
k
1 X 1
m
1
=
,
−
2πas
V
k Ek
(3.8)
(3.9)
k
where k = k 2 /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 X †
(ck↑ ck↑ + c†k↓ ck↓ )|Ψg i
V
k
X
k3
1
k − µ
1−
= F2 .
V
Ek
3π
n = hΨg |
=
(3.10)
k
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 −(kF as )−1 from BCS to BEC limit, even in
the unitary region. The two equations are well defined at unitarity as → ∞, and the BCS
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 X
fk c†k↑ c†−k↓ .
d†0 = √
V k
36
(3.11)
In the grand canonical ensemble, a bosonic condensate can be described as a (unnormalized)
coherent state:
|BECi = exp(αd†0 )|0i,
where α =
√
(3.12)
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:
|BECi =
=
Y
√
exp( nfk c†k↑ c†−k↓ )|0i
k
Y
1+
k
√
nfk c†k↑ c†−k↓ |0i.
(3.13)
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
√
nfk =
vk
.
uk
(3.14)
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
ρ2 = hc†↑ (r1 )c†↓ (r2 )c↓ (r20 )c↑ (r10 )i.
(3.15)
ρ2 is Hermitian and can be diagonalized with real eigenvalues:
ρ2 (r1 , r2 ; r10 , r20 ) =
X
ni χi (r1 , r2 )χ∗i (r10 , r20 ).
(3.16)
i
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(r1 , r2 ) = χ(r1 − r2 ), and the Fourier transform of χ(r) is defined as F (q =
0, k), where q is the center of mass momentum. From the BCS wave function 3.2, F (q =
0, k) = uk vk is the effective BCS pair wave function. As we can see from 3.3, at large
momentum region in the limit where ξk ∆, there is an asymptotic form of uk ∼ 1.
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 − 2ma
2 in the BEC side. The order parameter ∆ is
s
exponentially small in the weakly attractive region. Adapted from reference [54].
and Fk = vk uk 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 superfluidity 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: (kF as ) → 0−
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
∆ = 8EF exp −
2kF |as |
.
(3.17)
We can see the pairing gap scales as the Fermi energy, and is exponentially small in the BCS
limit where
π
2kF |as |
→ ∞. In fact, this expression is analogous to that of the conventional
phonon-induced superconductivity in electron gases, in which
∆ ∼ ~ωD exp −
1
N (0)V
,
(3.18)
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: (kF as ) → 0+
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
with binding energy Eb =
1
.
ma2s
The size of these bound molecules is in the order of
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
µ ∼ −Eb /2 = −~2 /(2ma2s ). This is a reasonable approximation since the chemical potential
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 = −Eb /2 to
39
the number equation 3.10 we have
∆=
4EF
∼ EF (kF as )−1/2 .
3πkF as
(3.19)
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
= √ (kF as )3/2 1
|µ|
3π
(3.20)
in the BEC limit, consistent with the assumption we made above. Also, the lowest quasiparticle excitation energy Ek appears at k = 0, with value
Ek=0 =
p
µ2 + ∆ 2 ,
(3.21)
instead of Ekmin = ∆ 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
∆ = Û K̂∆,
(3.22)
in which Û and K̂ 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
∆k = −
3
1 X
U (k, k0 )K(k0 )∆k0 .
V 0
k
In Appendix A, we will give a brief derivation of path integral formalism at finite temperatures.
40
(3.23)
For a translational invariant system, U (k, k0 ) = U (k − k0 ) is the Fourier transform of U (r).
At finite temperature, the kernel is
K(k) =
1 − 2f (Ek )
tanh(βEk /2)
=
.
2Ek
2Ek
(3.24)
Here, f () = (1 + exp(β( − µ)))−1 is the Fermi function with β = (kB T )−1 being the
q
inverse temperature, Ek = ξk2 + ∆2k is the quasiparticle excitation energy. As we can see,
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
T = U + U G0 T , where G0 = −(~2 k 2 /2m)−1 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:
∆ = −U K∆
∆ − U G0 ∆ = −U K∆ − U G0 ∆
(1 − U G0 )∆ = −U (K + G0 )∆
∆ = −(1 − U G0 )−1 U (K + G0 )∆ = −T (K + G0 )∆
(3.25)
The last equation above is the regularized gap equation implementing the scattering matrix,
and is more explicitly written as
1 X
tanh(βEk0 /2)
m
0
∆k = −
T (k, k )
− 0 2 ∆ k0 .
V 0
2Ek0
(k )
(3.26)
k
As in the large momentum limit, Ek ∼ k 2 /2, the ultraviolet divergence is cancelled on the
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. kF r0 1, where r0 is the potential range. In this limit, we make the
approximation that Uk is a step function, which is constant until some momentum cutoff
Λ ∼ r0−1 . The dilute limit assures that kF Λ.
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 meanfield 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 ≥ Tc , and calculate the grand thermodynamic potential Ω.
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 particleparticle (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 ∼ r0−1 kF , we evaluate the integration of
these two channels:
(a) ∼
(b) ∼
Z
ZΛ
d4 q U (q)G0 (p1 + q)G0 (p2 − q),
(3.27)
d4 q U (q)G0 (p1 + q)G0 (p2 + q),
(3.28)
Λ
43
p1 + q
q
p2
q
p1 + q
p1
p1
q
p2
p2 + q
(a)
p2
(b)
Figure 3.3: The scattering vertex for the particle-particle channel (a) and for the particlehole 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 4dimensional 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) ∼
(b) ∼
X
(1 − f (p1 + q))(1 − f (p2 − q)) + f (p1 + q)f (p2 − q),
(3.29)
q<Λ
X
(1 − f (p1 + q))f (p2 + q) + f (p1 + q)(1 − f (p2 + q)),
(3.30)
q<Λ
where f (p) is the Fermi function for the momentum p state:
f (p) = 1/(e(p
2 /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 p2 inside the Fermi surface, the first (1 − f )(1 − f ) term in (a) remains close to unity
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)
Ω0 = −
2T X 2
ln 1 + e−(p /2m−µ)/T .
V p
(3.33)
The interacting part is calculated by imagining that the interaction potential is multiplied 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
Ωint =
Z
1
0
dλ
hλÛ i,
λ
(3.34)
where Û 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
Π(q, zm ) =
1 X 1 − f (q/2 + k) − f (q/2 − k)
,
V
zm − ξq/2+k − ξq/2−k
(3.35)
k
where ξp = p2 /2m−µ, f is the Fermi function, and zm are the bosonic Matsubara frequencies
4
Here we borrow the usage of polarization, by its analogy to the iterative bubble diagrams widely used
in electronic gases.
45
zm =
2imπ
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 λÛ is given by
hλÛ i = −
T X λgΠ(q, zm )
.
V q,z 1 − λgΠ(q, zm )
(3.36)
m
By substituting the explicit form of hλÛ i into equation 3.34, we have
Ωint =
T X
ln (1 − gΠ(q, zm )) .
V q,z
(3.37)
m
In the thermodynamic limit, we write down the sum over momentum q as an integral:
Ωint = T
XZ
zm
dq
ln (1 − gΠ(q, zm )) .
(2π)3
(3.38)
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:
Nb (ω) =
1
.
eβω − 1
(3.39)
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:
Ωint =
Z
dq
(2π)3
Z
+∞
−∞
dω
Nb (ω) ln[1 − gΠ(q, ω + i0+ )] − ln[1 − gΠ(q, ω − i0+ )] . (3.40)
2π
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:
Ωint
Z
Z +∞
dq
=
(2π)3 −∞
Z
Z +∞
dq
=
(2π)3 −∞
dω
1
+
Nb (ω)Im ln Π(q, ω + i0 ) −
π
g
dω
Nb (ω)Im ln Γ−1 (q, ω + i0+ ),
π
47
(3.41)
where the argument Γ−1 of the logarithm is:
1 X 1 − f (q/2 + k) − f (q/2 − k) 1
−
V
ω + i0+ − ξq/2+k − ξq/2−k
g
k
Z
dk
1 − f (q/2 + k) − f (q/2 − k)
m
m
=
+ 2 −
3
+
2
2
(2π)
ω + i0 + 2µ − k /m − q /4m k
4πas
Z
dk
1 − f (q/2 + k) − f (q/2 − k) m
m
= P
+ 2 −
(2π)3
ω + 2µ − k 2 /m − q 2 /4m
k
4πas
Z
dk
+i
(1 − f (q/2 + k) − f (q/2 − k))δ(ω + 2µ − k 2 /m − q 2 /4m).
(2π)3
Γ−1 (q, ω + i0+ ) =
(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
Z
Z
1
=P
dx
x − A + i0+
dx
1
− iδ(x − A)
x−A
(3.43)
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 − q 2 /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
+
Γ−1
0 (q, ω + i0 ) = −i
in which k0 =
mk0
m
−
,
4π
4πas
(3.44)
p
mω + 2mµ − q 2 /4 is the wave vector of the relative motion. This is exactly
in the same form 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
Ωint = −
Z
dq
(2π)3
Z
+∞
−∞
dω
Nb (ω)ζ(q, ω),
π
(3.45)
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:
n=−
∂Ω
= n0 (µ, T ) +
∂µ
Z
dq
(2π)3
Z
+∞
−∞
dω
∂ζ(q, ω)
Nb (ω)
,
π
∂µ
(3.46)
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 ≥ Tc region. This scheme was first introduced by
Noziéres and 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 ≥ Tc in the functional integral approach.
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
5
In 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 (kF as )−1 ∼
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 + q 2 /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 ωb (q) and ωb (q) < −2µ + q 2 /4m. The zero of
Γ−1 , or the pole of Γ corresponds to a bound state, with binding energy Eb = ωb (q) − ω(q).
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
solution to the bound-state pole gives the asymptotic binding energy ωb (q) − ω(q) = − am2
s
as (kF as
)−1
→ ∞. The point µ = 0 is hence important in that it separates Cooper
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 transition 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 commonly 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 prescription 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 ferromagnetism 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
ωc (q) to +∞. In the absence of a molecule (bound-state) pole, ωc (q) = ω(q) = q 2 /4m − 2µ,
6
Although 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
zHEL
⇣(E)
¯
⇣(E)
zHEL
p
E
p
2
|Eb |
- p2
E
0
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
Z
Z +∞
dq
dω
∂ζ(q, ω)
Nb (ω)
3
(2π) ωb (q) π
∂µ
Z
Z
Z +∞
dq
∂ωb (q)
dq
dω
∂ ζ̄(q, ω)
= −
N
(ω
(q))
+
Nb (ω)
b b
(2π)3
∂µ
(2π)3 ω(q) π
∂µ
n − n0 =
= 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 ζ̄ = 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 ∂ωb /∂µ → −2, since ωb = q 2 /4m − 2µ − |Eb | and Eb is essentially independent of
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
R dq
to the medium. In this limit, we have a simplified nbd = 2 (2π)
3 Nb (ωb (q)): two atoms
per molecule times the Bose distribution function. For the system closer to unitarity, the
expression is generally more complicated. The term ∂ωb /∂µ deviates significantly from −2,
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| |Eb |, T . For an expansion of the fugacity z to the
lowest (second) order:
n
bd
nsc
where λ =
coefficients
p
√ !3
2
dq −βq2 /4m
2
e
= 2z
bbd
= 2z e
2 ,
(2π)3
λ
√ !3
Z
Z ∞
dq
d
2
∂ζ()
2
= z2
e−βq /4m
e−β
= 2z 2
bsc
2 ,
3
(2π)
π
∂µ
λ
0
2 β|Eb |
Z
(3.49)
(3.50)
2π/mT is the thermal de Broglie wave length, and b2 ’s are the second virial
bbd
2
β|Eb |
=e
,
bsc
2
=
Z
0
∞
d −β ∂ζ()
e
.
2π
∂µ
(3.51)
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
er 10.
(a)
(b)
!ιω nplane
molecule
pole
!b (q)
!(q)
branch cut
Pole of F(z)
Branch−cut
C
of F(z)
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 axis are n
even Matsubara
frequencies.
(c)
P (µ, T ) =
Rµ
−∞ dµ
0 n(µ0 , T ).
ιω
With this set of prescriptions, one is able to study many
properties of the upper branch in both Bose and Fermi gases across unitarity, as will be
C’
discussed in the following sections.
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.
Pole of F(z)
We also introduced an approach for excluding the bound state in the effective repulsive
Fermi gas.
54
Branc
of F(z
˜
=
+
=
+
=
+
˜
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 Γ̃ 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 Γ̃. 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. kF as → 0− . The
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 kF , as in exp(−π/(kF as ))
appears in the Tc expression is substituted to as + CkF a2s , 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.
7
This 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
G0 G[61] as the two legs (propagators) of the ladders instead of the G0 G0 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
as (E, B) = abg 1 +
γ∆B
E − γ(B − B∞ )
,
(3.52)
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 = ~2 k 2 /2m (in the
rest of the section, we set ~ = 1). The “theoretical width”8 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,
and it can be either positive or negative. From two-channel models, we have abg γ∆B > 09 .
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:
as (E, B) ≡ abg +
(mr∗ )−1
,
E − γ(B − B∞ )
1/(mr∗ ) ≡ abg γ∆B.
(3.53)
(3.54)
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
resonance part is completely determined by (r∗ )−1 , 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
parameters: abg and (r∗ )−1 . Note from two-channel models, this (r∗ )−1 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
8
This is sometimes called the “theoretical width” in order to distinguish it from the so-called “experimental width” extrapolated from atom-loss experiments.
9
This can be understood from the second-order perturbation of the Feshbach resonance as well. Since
α2
, the positive definiteness of α2 enforces this property. A more detailed discussion is in
∆as ∼ 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
another energy scale associated with background scattering length as Ebg = 1/ma2bg . Then
in scheme (A) the rescaled dimensionless width of the system is defined by a ratio:
α≡
abg
γ∆B
= ∗.
Ebg
r
(3.55)
The narrow resonance and wide resonance, which are denoted by NA and WA respectively,
are defined as:
NA : |α| 1,
WA : |α| 1.
(3.56)
On the other hand, in the many-body context, the scheme (B) is determined by the
dimensionless number (r∗ kF )−1 . The corresponding narrow and wide resonance, denoted
by NB and WB , are defined as:
NB : (r∗ kF )−1 1,
WB : (r∗ kF )−1 1.
(3.57)
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
length scale kF−1 associated with the interparticle spacing. The narrowness of the resonance
varies continuously as (kF r∗ )−1 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
energy scattering length around the Fermi surface E = EF = kF2 /m (the reduced mass is
half the atom mass, for equal-mass mixtures), the scattering length times the Fermi vector
59
kF as
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
kF as as a function of the external magnetic field, and define the |kF as | > 1 region as the
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
region. The parameters for the two resonances are: kF abg = 0.1, ∆B n /EF = 1 for the
narrow resonance as the blue curve, and ∆B w /EF = 10 for the wide resonance as the red
curve.
is
kF as E=EF ,B=B∞
= kF abg +
kF (mr∗ )−1
1
= kF abg +
.
2
kF r∗
kF /m
(3.58)
In the situation of an off-resonance background scattering length where kF abg 1, this
value is determined by the Feshbach resonance term (kF r∗ )−1 . 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
Figure 3.11: Wide and narrow resonances in the space ((kF r∗ )−1 , kF abg ): Scheme (A) is
determined by the value of xy, and WA and NA are the regions above and below the curve
xy = 1 respectively. From scheme (B) the crossover is a region of vertical stripe: WB and
NB are the regions where x 1 and x 1. The dotted line are contours of constant
∆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
as: x = (kF r∗ )−1 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:
WA : xy = α 1,
NA : xy 1;
WB : x 1,
NB : x 1.
(3.59)
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 abg → 0 will give ∆B → ∞, and systems in such a limit
61
live on the horizontal axis, indicated by thick red lines. Take an example of two resonance
channels in fermionic 6 Li: for 6 Li at density n↑ = n↓ = 5 × 1014 cm−3 , one of its wide
resonances of channel ab at 834.1G has abg = −1405aB , ∆B = −300G, |abg |/r∗ = 2.8 × 103 ,
(r∗ kF )−1 = 1.25×103 , and kF |abg | = 2.3. Its narrow resonance at 543.25G has abg = 61.6aB ,
∆B = 0.1G, abg /r∗ = 0.002, (r∗ kF )−1 = 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
k cot δ(k) = −
1
.
as (k)
(3.60)
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
a(k) = abg +
(mr∗ )−1
.
k 2 /m − γ(B − B∞ )
(3.61)
An explicit expression of tan δ(k), in the approximation of zero effective range is thus given
by
tanδ(k) = −kabg −
k 2 /m
k/mr∗
,
− γ(B − B∞ )
(3.62)
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 abg < 0, as shown in (A) on the
left, we have δ(k) = −arctan(kas (k)). Near resonance, (c and d), δ(k) approaches a step
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(kabg ) ∼ −kabg , and |kabg | 1. For narrow resonance with
positive background scattering length abg , γ, ∆B > 0, as shown in (B) on the right, when
B > B∞ , δ(k) approaches a step function of height π at the position k 2 /m = γ(B − B∞ ),
with a width 1/r∗ . For B < B∞ , δ(k) quickly reduces to −kabg as B∞ − B exceeds ∆B,
and |kabg | 1. The δ(k) for both e and f are essentially identical.
energy shift for different momentum states in a characteristic energy scale of the system,
√
namely kF for low temperatures and mkB T 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 ∼ −kF abg 1. The
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
interacting. In the case that ∆B < B − B∞ < kF2 /m, as (b) in figure 3.12, even though
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
π/2 in wide resonances) in the region where k 2 /m > B − B∞ ≡ k ∗2 /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
√
B − B∞ < kF2 /m (or B − B∞ < mkB T in thermal gases where T > EF ); the repulsive
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 temperatures, 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 interaction energy density of the system takes the form:
3T n
int (T, n) =
2
nλ3
√
2
64
2T ∂b2
−b2 +
,
3 ∂T
(3.63)
where b2 is the second virial coefficient. b2 consists of two contributions from the bound
P |E b |/T
sc
bd
α
states and the scattering (extended) states, b2 = bbd
is the
2 + b2 , where b2 =
αe
partition function of a set of bound states {Eαb } (labelled by α) and bsc
2 is given by
bsc
2 =
Z
0
∞
dk dδ(k) −k2 /mT
e
.
π dk
(3.64)
bsc
2 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
sc
int = bd
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
√
for both including and
virial coefficient b2 and the “interaction strength” int / 3T2 n nλ
2
excluding the bound-state contribution in figure 3.13. The narrow resonance we consider is
for a gas of 6 Li at 543.25G with parameters given above. In this case, TF = 40µK ∼ 3γ∆B
(in temperature units).
The branch that exists only on the positive scattering length side with positive interaction energy and −b2 corresponds to the upper branch of the Fermi gas. This upper branch
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, b2 remains large as long as B − B∞ < T . Also, right at the
resonance, since the phase shift at the narrow resonance saturates to π instead of π/2 in
the wide resonance case, b2 ≈ 1 for narrow resonance is twice as large as that of a wide
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: −b2 (A) and int (B) as a function of magnetic field: b2 remains sizable even
when B − B∞ ∼ 2G ∼ 20∆B. This is because we are at T = 5TF ∼ 15γ∆B or larger,
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
limit ∆B → 0, b2 → 1. On the molecular side, B < B∞ , scattering
(for the upper branch)
int
is given by the flat line, given by the small value δ ∼ −kabg .
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 6 Li 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
66
s
a (0)/a
bg
4
2
0
-2
-4
E/E
0
0.0
-0.2
-0.4
-0.6
-0.2
0.0
B-B
0.2
8
-0.4
0.4
0.6
0.8
1.0
(G)
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 2EF , as discussed previously.
Related experiments
There are several experimental groups trying to study the properties of stable narrow Feshbach resonances for Fermi gases. Recently, Ken O’Hara’s group has performed rf spectroscopy studies on the narrow resonance of 6 Li 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 rel67
m º¹
[a0 ]
2 ¹h n
¹
¡
m º¹
[a0 ]
2 ¹h n
¹
¡
m º¹
[a0 ]
2 ¹h n
¹
¡
m º¹
[a0 ]
2 ¹h n
¹
¡
Figure 3.15: Experimental data from the Penn State group for fermionic 6 Li gases across
the 543.25G Feshbach resonance. The interaction energy is measured by rf spectroscopies.
The four panels are from Fermi gases with different temperatures. The asymmetry of the
interaction energy is clearly seen from the data points, especially in the center two panels,
however the large attraction at magnetic fields outside the width is only vaguely indicated.
The dashed curves are the shifts predicted by mean field contact potentials, and the solid
curves are the shifts predicted by a mean-field theory that includes the energy dependence
of the scattering length.
68
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 threebody 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 < kF as 1, from the analysis of cross sections for the three-body
scattering processes, we have an estimate of the three-body collision rate γ3 as
γ3 = c(4π~as /m)n(na3s ),
(3.65)
where c is a dimensionless constant[67]. For the two-body collision rate, it is in the form
γ2 = na2s v,
70
(3.66)
where v is the typical velocity of the bosons, which for weakly-interacting Bose gases at low
temperatures can be approximated as v 2 ∼ 2gn/m. The ratio between these two rates are
given in the form
γ3 /γ2 =
√
4πcna3s
=
c
2π(na3s )1/2 ,
k ∗ as
(3.67)
where k ∗ = mv/~ is the typical wave number of the bosons. In the extremely weakly
interacting limit that n1/3 as 1, γ3 is sufficiently low that the system is essentially free of
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,
at low temperatures, γ3 increases rapidly in the strongly repulsive regime, i.e. n1/3 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
in a single-component Fermi gas: n =
3
kF
.
3π 2
At temperatures much higher than the con-
densate transition temperature T ∼ TF , the characteristic energy scale of the particles
p
are mostly governed by the temperature, and we have v ∼ 3kB T /m ∼ h/(mλ), where
√
λ = h/ 2πmkB T is the thermal wavelength. When we estimate the collision rates close
to unitarity, as in γ2 and γ3 is replaced by λ in this temperature regime, and we have
√
γ2 = (kB T /~)(nλ3 ), and γ3 = C(kB T /~)(nλ3 )2 , where C = 9 3/π ∼ 4.96[72]. In a Bose
71
gas with low enough density, γ3 /γ2 ∼ nλ3 becomes small. In this situation, when we tune
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:
−N
−1
R
drγ3 n
dN/dt = R
= hγ3 iave ,
drn
(3.68)
where h..iave means spatial average.
The previously discussed density and temperature regime (which we referred to as the
“low-recombination” regime) where γ3 < γ2 , hγ3 iave < ω, or
nλ3 1,
where n2 ≡
R
nλ3 < C −1/2
p
~ω/kB T ,
(3.69)
R
n3 / n. In this regime, very few molecules are formed even at unitarity during
the time when the 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 < kF as 1. From the analysis above, we can adiabatically tune the scattering
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
nB (k) = 1/(eξk /T − 1) evaluated at the single-particle energy ξk = k − µ where k = k 2 /2m.
Explicitly, the scattering matrix in the bosonic medium is
TB−1 (q, ω
+
+ i0 ) =
Z
dk
(2π)3
1 + nB (q/2 + k) + nB (q/2 − k) m
+ 2
ω + i0+ + 2µ − k 2 /m − q 2 /4m
k
−
m
4πas
(3.70)
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:
n(µ, T ) = n0 (µ, T ) + ∆nsc (T, µ) + ∆nbd (T, µ),
where n0 (T, µ) =
P
k nB (ξk )
(3.71)
is the density of the ideal Bose gas and ∆nsc (T, µ) and
∆nbd (T, µ) are the interaction contributions of the scattering states and the bound states
respectively:
1X
∆n (µ, T ) = −
Ω q
sc
∆nbd (µ, T ) = −
Z
∞
ω(q)
dω
∂ζ(q, ω)
nB (ω)
,
π
∂µ
1X
∂ωb (q)
n (ωb (q))
,
Ω q B
∂µ
(3.72)
(3.73)
where ω(q) ≡ q 2 /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
1X
−
+
4πas Ω
k
γ(k; q)
ω − ω(q) −
k2
m
+
1
k2
m
!
= 0,
(3.74)
where γ(k; q) ≡ 1 + nB (q/2 + k) + nB (q/2 − k).
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
as
changes sign from negative
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 γF = 1 − n − n, which corresponds to “Pauli blocking”.
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
=
dimers
linear
combinations
of states
Bose
enhancement:
Fermions:
Pauli blocking
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
aeff (q, ω)
=
=
1
1
+∆
as
a
1
4π X n(ξq/2+k ) + n(ξq/2−k )
∓
,
as
Ω
m(ω − ω(q)) − k 2
(3.75)
k
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
−1
mass momentum q occurs when aeff (q, ω = ω(q)) ≥ 0, or when −a−1
s ≤ −ac (q), where
1
4π X nB (ξq/2+k ) + nB (ξq/2−k )
=−
< 0.
ac (q)
Ω
k2
(3.76)
k
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 centerof-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)
goes back to the original two-body resonance (ac )−1 = 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/ac (q) at different q’s
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
E � E0
1.6
T � 4TF
1.4
1.2
1
0.8
�1
�0.5
0
�1 � kFas
0.5
1
q � kF
15
T � 4TF
10
5
0
0
0.1
�1 � kFac�q �
0.2
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 temperatures 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
Equilibrium
Branch
Upper Branch
4
11
9
Κ=0
3
T  TC
T  TF
5
7
Unstable
2
5
-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
kF as
= −0.2. The repulsion energy remains
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) = nupper (µ − V (r), T ), where µ is the chemical potential at the
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/as , −µ/T ) on this diagram.
Figure 3.19 shows how the Bose gas behaves as it is swept from the positive-scatteringlength 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 = ra − rb , where ra and rb are given by µa = µ − V (ra )
and µb = µ − V (rb ), we can see that the difference that µb − µa becomes very small close to
unitarity. This difference actually controls the width ∆r, and in the limit that ∆r becomes
less than the interparticle spacing, i.e. ∆r < n(r̄)−1/3 , which occurs at a critical ratio λ/a∗s
for given µ/T , ra and rb can be viewed as a single point r̄. The unstable region disappears
in this situation. The critical ratio λ/a∗s can be estimated by setting ∆r = n(ra )−1/3 . (a∗s
is a function of T and µ).
Thus, for an upper-branch Bose gas characterized by the point (−λ/as , −µ/T ) on this
79
5
4
HiLEquilibrium
Branch
HiiLUpper Branch
-Μ  T
3
Μ1
æ
HaL
Μ2
2
æ
Μ3
æ
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 µtrajectory 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/as )∗ = 0.14, 0.18 and 0.21 for the trajectories
−(µ/T )1 = 3, −(µ/T )2 = 2.5 and −(µ/T )3 = 2 respectively. The corresponding particle
numbers are (N1 , N2 , N3 ) = (2.1, 3.4, 5.2) × 104 . All these systems satisfy the condition
to be in the low-recombination regime, equation 3.69 or equivalently equation 3.77, as
hγ3 iave /ω = 0.14, 0.35, and 0.96 respectively, and N1 , N2 , N3 < N ∗ , where N ∗ = 6.5 × 104
for this temperature and trap frequency.
80
�a�
n
Equilibrium
Branch
Upper Branch
r
�b�
n
�c�
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/as < −1/a∗s (T, µ), such that the unstable region–if
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/as exceeds −λ/a∗s (T, µ), such as (b) and (c) in Fig, 3,
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
R
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 on (−µ/T, −λ/as ). We find that A < 1. This
is expected, as the critical number for Bose-Einstein condensation in a harmonic trap with
frequency ω is Nbec = (0.95)−1 (T /ω)3 [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 = 33/4 eµ/T /λ3 , which then implies
N < N ∗ = α(T /ω)2.5 ,
α = 33/4 C −1/2 = 1.024.
(3.77)
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.
single parameter: the bound-state energy in vacuum is Eb ≡ ~2 /ma22d where a2d refers to
the scattering length in two dimensions (2D)[81]. The scattering amplitude for the two-body
problem in 2D is
f
−1
(k) = ln
Eb
E
+ iπ = −2 ln(ka2d ) + iπ.
(3.78)
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
T
−1
+
(ω + i0 , q) =
X
k
γ(q, k)
1
+
+
2
ω + i0 − ω(q) − k
Eb + k 2
,
(3.79)
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) = q 2 /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:
−
X
1
1
=
.
g2d
Eb + k 2
(3.80)
k
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]
p
a2d ∼ l0 exp (− π/2l0 /as ),
(3.81)
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:
nlower (µ↑ , µ↓ , T ) = n0↑ + n0↓ + ∆nsc + ∆nbd ,
(3.82)
nupper (µ↑ , µ↓ , T ) = n0↑ + n0↓ + ∆nsc ,
(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
1X
∆n (µ, T ) = −
Ω q
sc
∆nbd (µ, T ) = −
Z
∞
ω(q)
dω
∂ζ(q, ω)
nB (ω)
,
π
∂µ
1X
∂ωb (q)
nB (ωb (q))
,
Ω q
∂µ
(3.84)
(3.85)
where ζ(q, ω) is the phase angle of the T -matrix in a medium, and ωb (q) is the pole of
the T-matrix which represents the bound state. For q 2 /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 X
∆M (µ, T ) = −
2Ω q
sc
∆M bd (µ, T ) = −
Z
∞
ω(q)
dω
∂ζ(q, ω)
nB (ω)
,
π
∂h
1 X
∂ωb (q)
nB (ωb (q))
.
2Ω q
∂h
(3.86)
(3.87)
The total magnetization is given by the interacting part above plus the non-interacting
86
part, i.e.
Mlower (µ↑ , µ↓ , T ) = M0 + ∆M sc + ∆M bd ,
(3.88)
Mupper (µ↑ , µ↓ , T ) = M0 + ∆M sc ,
(3.89)
where M0 = n0↑ − n0↓ is the magnetization from the non-interacting part. The “polarization” 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
EintêE0
0.25
0
-0.25
-0.5
-1
-0.5
0
lnHkFa2 d L
0.5
1
Figure 3.22: The interaction energy for the attractive branch (red curve) and the repulsive
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 ηc = ln kF ac2d ≈ −0.4 at this
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 twodimensional 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
reaches zero at a critical value of ac2d . This ac2d 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
region a2d < ac2d , 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↑ 6= µ↓ , there is an imbalance
of population in different spin species. By adjusting the two chemical potentials, we can
have gases with different “polarization” defined as m =
n↑ −n↓
n↑ +n↓ .
Similarly, we calculate
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
imbalance will push the stable region further towards the stronger repulsion side (larger ac2d
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
cê c0
1.5
1.25
kêk0
1.
1.
-1.1
-1.
-0.9
-0.8
-0.7
lnHkFa2 d L
-0.6
-0.5
-0.4
-1.1
-1.
-0.9
-0.8
-0.7
lnHkFa2 d L
-0.6
-0.5
-0.4
0.5
0
Figure 3.23: Spin susceptibility (upper panel) and compressibility (lower panel) of the repulsive 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 kF ac2d = −0.39,
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
T  TF
5
æ æ
æ æ
æ
æ
æ
Stable
4
Unstable
æ æ
ææ
3
æ
æ
æ
m=0.8
m=0æ æ
m=0.5
-1.5
-1.
-0.5
lnHkFa2 d L
0
0.5
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 brokensymmetry 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 striking 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 successfully 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 lowestLandau-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 twodimensional 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
RH =
h
.
νe2
(4.1)
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
H=
2m∗
eA
p−
c
2
+ gµB · S,
(4.2)
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 = Bẑ, we choose the “symmetric gauge” in this chapter, namely Ax = −eBy/c
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
Landau levels, separated by the energy spacing ~ωc =
eB~
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
q
~c
classical radius of the cyclotron motion is l = eB
, and this characteristic length scale is
defined as the magnetic length. For each Landau level, it has a degeneracy per unit area as
1
.
2πl2
Thus, the effective “filling factor” of the sample is
ρhc
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
ψm,n = √
1
2πm!n!l2
e|z|
2 /4l2
√
2
2
( 2l)m+n ∂zn ∂z̄m e−|z| /2l ,
(4.3)
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 dielectric constant ∼ 10. Consequently, the cyclotron energy gap for a 10 Tesla magnetic field
is around 10−2 eV (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
p
are rescaled by units of the magnetic length ~c/eB):
Ψ1/m ∼
Y
j<k
1
(zj − zk )m e− 4
P
i
|zi |2
,
(4.4)
where zi = xi +iyi is a complex number that denotes the position of the electron labeled by i
in the x-y plane. This family of wave functions are in the lowest Landau level (LLL) regime,
since the prefactor in front of the gaussian is analytic in all z’s. Also, we can see that the
weight of the wave function is suppressed when two particles are close to each other by m
powers of their separation, due to the strong Coulomb repulsion. Furthermore, if one of the
electrons circles around another by one single cycle, it gains a phase of mπ. It is effectively
described by m units of quantum fluxes bound to each particle. The wave function is also
an incompressible state, with κ = ∂n/∂µ = 0. This is also reflected by having a vanishing
density fluctuation in the bulk. Finally, we have to notice that the Laughlin wave functions
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 |ψ̃i = eiΩ·Lt/~ |ψi, where L is
the angular momentum operator. The corresponding Schrödinger equation in the rotating
frame gives
i~∂t |ψ̃i = eiΩ·Lt/~ i~∂t |ψi + (i~∂t eiΩ·Lt/~ )|ψi
= eiΩ·Lt/~ He−iΩ·Lt/~ |ψ̃i − Ω · L|ψ̃i ≡ H̃|ψ̃i.
98
(4.5)
From the equation above, one finds the new hamiltonian of the system in the rotating frame
as
H̃ = eiΩ·Lt/~ He−iΩ·Lt/~ − Ω · L.
(4.6)
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:
H̃ =
m(ω 0 · r)2
(p − mΩ × r)2 m(ω · r)2
p2
−Ω·L+
≡
+
,
2m
2
2m
2
(4.7)
in which ω 0 , 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 = ω 2 + Ω2 for i = x, y, and ω
case where Ω is in the z-direction, we have ω0i
0z = ωz . Hence
i
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 ωz Ω, ωx , ωy , the singleparticle 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:
H̃ = H⊥ + Hz =
(p − mu)2
+ (ω0 − Ω)Lz + Hz
2m
(4.8)
where H⊥ and Hz are separable hamiltonians acting on the x-y plane and in the z direction,
respectively. u = (−ω0 y, ω0 x, 0) is the effective velocity field experienced by neutral atoms.
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 (ω0 − Ω)Lz
lifts the degeneracy. The new eigenstates—the Fock Darwin states—and the eigenenergies
99
Em,n
n=1
!0
m=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
ψm,n =
√
2 2
2
2
1
e|z| /2l lm+n ∂zn ∂z̄m e−|z| /l
πm!n!l2
× Θ0 (z),
Em,n = (n + 1/2)~ω0 + (m + 1/2)~(ω0 − Ω),
in which l =
(4.9)
(4.10)
p
~/mω0 is the new defined “magnetic length”, Θ0 is the lowest harmonic
oscillating state 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 intraLandau-level spacing is ω0 − Ω, as is shown in figure 4.2. Within the LLL, the distribution
√
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 noninteracting 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] =
Z
∗
4
dr ψ (r)H̃ψ(r) + g2d |ψ(r)| .
In the mean-field level of the interaction, g2d =
√
(4.11)
8π~2 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] =
Z
∗
4
dr ψ (r)(H̃ − µ)ψ(r) + g2d |ψ(r)| ,
(4.12)
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 Nv N , a large
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
Nv ∼ N , the chemical potential drops below the second Landau level[87]. In this situation,
the rotating BEC is driven into the LLL regime, and the wave function in the x-y plane
can be written as
ψv (z) ∼
Y
i
(z − zi )e−|z|
2 /2l2
,
(4.13)
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 (ω0 − Ω) < ω0 . For a system with 103 particles, it is required to keep
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:
H=
X
~2 X
∇ψα† · ∇ψα + V (r)ψα† ψα +
gµB ψα† B · Fαβ ψβ + Hint ,
2m α
(4.14)
α,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 Hint is the interaction hamiltonian. We decompose the vector
P
(m)
field into a basis of local eigenstates of the Zeeman term as ψα = Fm=−F φ(m) ηα , where
φ is the new field operator and the vectors η are the local eigenvectors of the Zeeman term:
(m)
B · Fαβ ηβ
= mηα(m) .
(4.15)
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 X
∇ψα† · ∇ψα =
2m α
~2 X
∇(φ† ηα† ) · ∇(φηα )
2m α
#
"
X
X
X
~2
ηα† ∇ηα |2 )φ† φ
|(−i∇ − i
ηα† ∇ηα )φ|2 + (
∇ηα† · ∇ηα − |
2m
α
α
α
=
1
|(p − A)φ|2 + Wφ† φ,
2m
=
(4.16)
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~
X
α
W =
~2
2m
ηα† ∇ηα ,
X
α
∇ηα† · ∇ηα − |
(4.17)
X
α
ηα† ∇ηα |2
!
.
(4.18)
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
Hm,n =
X
(∇ηαm† )ηαn · φ†m ∇φn + (∇ηαm† · ∇ηαn )φ†m φn + h.c..
(4.19)
α
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 φ†m φn is proportional to ∇ηm
· ∇ηn . If
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 ∼ q 2 is small compared to the
absolute energy separation ∆ between the lowest-energy adiabatic state to the other ones in
104
B
m=F
…
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
1
1
ηm=1 = ( (1 + cos θ)e−iφ , √ sin θ, (1 − cos θ)eiφ )T
2
2
2
(4.20)
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,
~2
|∇ cos θ|2
1 − cos2 θ
2
W =
+
|∇φ| .
2m 2(1 − cos2 θ)
2
105
(4.21)
(4.22)
A more general expression for spin eigenstates in the spin-F manifold and the consequent
gauge potential for |F, mi will be included in the appendix. In the next section, we will
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: mF = 0, ±1 of the 5S1/2 , F = 1 electronic ground state. The momentum difference
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 ∼ 2kR while the spin is
increased by one unit of angular momentum. The Rabi frequency of this Raman coupling
process is ΩR . A spatially non-uniform magnetic field is applied in the form B = −(B0 +
Gz)ẑ + Gxx̂, 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:
H = T + HR + HB ,
(4.23)
where T is the kinetic energy, HR is the Raman coupling process from the laser beams, and
HB is the Zeeman term. As the particle increases its spin by interacting with two photons,
106
m=
1
m=0
m=1
x
z
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 spatiallydependent Zeeman term.
the Raman term takes a simple form:
HR =
ΩR + −i(2kR ·r+ωt)
(F e
+ h.c.)
2
= ΩR (Fx cos(2kR · r + ωt) + Fy sin(2kR · r + ωt)).
(4.24)
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 = gµB B0 , 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 |χi = e−iωtFz |ψi, and the new time-independent form is
H̃R = ΩR (Fx cos(2kR · r) + Fy sin(2kR · r)),
(4.25)
as the spatially varying field becomes static. In general, the Zeeman term contains a linear
part and a quadratic part as:
HZ = −gµB (B0 + Gz)Fz + GxFx + γFz2 .
(4.26)
In the small quadratic Zeeman splitting limit, we neglect the Fz2 term, and the Zeeman
term in the rotating frame is
H̃Z = −gµB GzFz + GxFx .
(4.27)
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:
H̃ = T + gµB Beff · F,
(4.28)
in which the effective magnetic field is
Beff = G(λ cos φ + x, λ sin φ, −z),
where λ =
ΩR
gµB G
(4.29)
and φ = 2kR · r. We briefly sketch the direction of the effective magnetic
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
z
y
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 Beff vector will almost be the constant vector −ẑ for z λ or ẑ
for z −λ.
magnetic field 4.21 and 4.22, the explicit expression of the gauge field is
2~z|kR |
A= √
ŷ.
λ2 + z 2
(4.30)
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:
B∗ = ∇ × A =
2kR λ2
x̂,
(λ2 + z 2 )3/2
(4.31)
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 λR λ, in which the region of a near uniform gauge field is much larger
than the wavelength of the Raman beam, the remaining potential W takes the form
√
m 2~kR 2 2
1 ~2 2kR 2 2
W=C−
(
) z =C− (
) z ,
2 2m λ
2
mλ
(4.32)
where C is a constant potential. This remaining term is equivalent to an anti-trapping
√
R
potential with frequency 2Ω, where Ω = ~k
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 ydirection 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
H=
p2y
(px − 2mΩy)2
1
1
+
+ m(α2 ω02 − 2Ω2 )y 2 + mω02 x2 ,
2m
2m 2
2
(4.33)
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:
H =
=
(p − mΩ × r)2 1
1
+ m(α2 ω02 − 2Ω2 )y 2 + mω02 x2
2m
2
2
p2
1
1
− ΩLz + m(α2 ω02 − Ω2 )y 2 + m(ω02 + Ω2 )x2 .
2m
2
2
(4.34)
Note that in this hamiltonian, the trapping frequency in the y-direction is reduced from
α2 ω02 to α2 ω02 −2Ω2 by the synthetic field. Thus, there is a limit of the “rotating frequency”,
2Ω2 < α2 ω02 . 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
Hint =
Z
drdr0 V (r − r0 )ψ † (r)ψ † (r0 )ψ(r0 )ψ(r).
(4.35)
In this quasi-two-dimensional cloud, the contact interaction is written as V (r−r0 ) = g2d δ(r−
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:
g2d =
Z
dz|φ0 (z)|2 g3d =
√
8π~2 as /(maz ),
where φ0 (z) is a normalized gaussian with width az in the z-direction. g̃2d =
111
(4.36)
√
8πas /az
is the dimensionless two dimensional coupling constant. The largest g̃2d 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:
H = T + Hsyn ,
where T =
p2
2m
(4.37)
is the kinetic energy. Hsyn covers all the additional “synthetic” terms to
generate the gauge fields, and they are usually spatial dependent as Hsyn (r). We will refer
it to the “synthetic hamiltonian” from now on. The variation of synthetic hamiltonian
over space has a characteristic momentum q, and gives an energy scale q = q 2 /(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
q = q 2 /(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 ∆ q . Depending on the number of
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 Hsyn . In the NIST scheme, as we discussed in the Abelian case
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 gµB B0 . With the quadratic Zeeman term and a tunable photon energy
included, the full expression of the synthetic hamiltonian is
Hsyn = (ΩR cos qyFx + ΩR sin qyFy − (gµB Gz − ω + Ω0 )Fz + νFz2 )
= e−iqxFz (ΩR Fx − (gµB Gz − ω + Ω0 )Fz + νFz2 )eiqxFz ,
(4.38)
where q = 2kR is the momentum transfer and ω is the energy transfer in the Raman
process. Ω0 = gµB B0 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 |m̃i =
e−iqxFz |mi = e−iqxm |mi, where |mi are the eigenstates of Fz with eigenvalue m.
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

√
0
 −(gµB Gz − ω + Ω0 ) + ν ΩR / 2

√
√
Hsyn = 
0
ΩR / 2
ΩR / 2


√
0
ΩR / 2 (gµB Gz − ω + Ω0 ) + ν



.


(4.39)
In the following discussions, we take the gµB Gz = 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 + (ν 2 − Ω2R − (ω − Ω0 )2 )λ + νΩ2R = 0.
(4.40)
The separations of these three states are determined by these three energy scales Ω0 −
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 ΩR ν, (ω − Ω0 ), q , the equation for eigenvalues becomes
λ3 − Ω2R λ = 0.
(4.41)
The three eigenvalues are λm = −mΩR , with m = 0, ±1. They have equal spacing ΩR , and
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 = ~qŷ. This is a constant curlfree gauge potential, which only shifts the dispersion minimum in momentum space. The
114
hamiltonian in the laboratory frame projected to this adiabatic state is
Had =
(p − A)2
.
2m
(4.42)
Note that in the previous section the field gradient term gµB Gz 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 ΩR , ω − Ω0 + ν, equation
(4.40) can be approximated as
λ3 − 2νλ2 = 0.
The three eigenvalues are λ1 = λ2 = 0, λ3 =
(4.43)
√
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 U = e−iqy(Fz −1/2) . The effective single-particle
hamiltonian in this basis is equivalent to a spin-1/2 system with spin-orbit coupling:
Hsyn =
σz
ΩR
h
1
(p − qŷ )2 + √ σx + σz ,
2m
2
2
2
(4.44)
in which we include back the small terms ΩR and h = ν −(gµB Gz−ω+Ω0 ) as perturbations,
and the σ’s are Pauli matrices. The gauge field is non-Abelian, since the
Ω
√R σx
2
term in the
“potential” energy does not commute with the gauge potential qŷ σ2z . In general, we classify
the gauge field by looking at the hamiltonian for a multicomponent system in the form
H=
1
(p̂ − Â)2 + Ŵ ,
2m
(4.45)
where the gauge potential  and the “single particle potential” Ŵ are operators. The
gauge field  is non-Abelian, if there is at least one non-vanishing commutator between
the three components of vector  and Ŵ , i.e. when [Âi , Âj ], [Âi , Ŵ ] are not all identically
√
zeros. For the above case, the commutator [Ây , Ŵ ] = i~ 2qΩR σy 6= 0 makes the gauge
field non-Abelian.
115
| #i
-2
| "i
E
| #i
-1
1
| "i
2
kqyêq
Figure 4.8: A sketch of the energy spectrum for spin-orbit coupled gases: 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
√R σx , the spectrum becomes two branches without a level crossing. For small
coupling Ω
2
√
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 Sz eigenstates | ↑i and | ↓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 (kz σz + ky σy ) and Dresselhaus (kz σz − ky σy )
type of spin-orbit coupling. Note here that the expression deviates from the convention in
solid state physics by the transformation σy → −σz and σz → σy . For ΩR = 0, Sz is still a
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 σx
2
couples the up and down spin component with momentum difference q. This coupling
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 ΩR = 0 and ΩR 6= 0. This spin-orbit coupling brings new physics in both bosons and
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 ky = ±q/2,
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-orbitcoupled 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 BoseEinstein 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
Z
1 †
ψ (r)(−i~∇ − mΩ × r)2 ψ(r)
2m
Z
m
(ωx2 − Ω2 )x2 + (ωy2 − Ω2 )y 2 ψ † (r)ψ(r)
+
dr
2
Z
+
drdr0 g2d (r − r0 )ψ † (r)ψ † (r0 )ψ(r0 )ψ(r).
H =
dr
(4.46)
As discussed in the previous section, the interaction strength g2d (r − r0 ) when the scattering
length is off resonance could be written as a simple delta function
0
g2d (r − r ) =
√
8π~2 as
δ(r − r0 ),
maz
(4.47)
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 as g̃ = g2d /( ~m ). When we scale the
energy by the rotation frequency Ω, we could further define dimensionless effective trapping
frequencies as ω̃x2 = (ωx2 − Ω2 )/Ω2 and ω̃y2 = (ωy2 − Ω2 )/Ω2 . Finally, we define an effective
anisotropy of the trap as given by the form α2 = (ω̃y2 − ω̃x2 )/(ω̃y2 + ω̃x2 ), for ωx ≤ ωy . α = 0
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:
n,m =
1
1
~ω− + m +
~ω+ .
n+
2
2
(4.48)
And for the n = 0 “lowest Landau level”, the wave functions are written as
c m/2
1
ψ0m (x, y) = p
Hm
πax ay m! 2
ζ
√
2c
x2
y2
× exp − 2 − 2 + iκxy .
2ax 2ay
(4.49)
where Hm ’s are Hermite polynomials, ζ = x + iβy, and β, ω+ , ω− , c, ax , ay , κ are all determined by Ω, ω̃x and ω̃y . 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
ω±
q
2
2
4
= Ω ω̃⊥ + 2 ∓ (αω̃⊥ ) + 4ω̃⊥ + 4 ,
2
(4.50)
2 = (ω̃ 2 + ω̃ 2 )/2 is the mean squared “remaining rotating frequency”. In the
where ω̃⊥
x
y
isotropic case α = 0, ωx = ωy = ω, the expression goes back to the form ω± = ω ∓ Ω.
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.
increasing expectation values of x2 and y 2 . The long axis is in the x-direction for ωy > ωx
traps (and we will always keep ωy ≥ ωx as the assumption of notations from now on). The
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. ω̃x 1. Also the repulsive interaction cannot push the
particles to the second LL, which from the simplest mean-field estimate is g2d n < ω− . With
these conditions enforced, it is sufficient to write the Gross-Pitaevskii (GP) wave function
of the condensate in the LLL space as:
φ=
X
cm ψ0m
m
x2
y2
= f (ζ) exp − 2 − 2 + iκxy ,
2ax 2ay
(4.51)
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. ω̃x 1 and ω̃x /ω̃y 1, the asymptotic behavior
of the Thomas-Fermi profile is determined by
s
g2d N
∼ (ω̃x )−1/2 ,
A = 2π
πmγω+
χ =
Rx
= β+ ∼ (ω̃x )−1 ,
Ry
121
(4.52)
(4.53)
where A = πRx Ry gives the area of cloud, and χ is the aspect ratio of the condensate.
They behave as different powers of ω̃x , a value which describes how close the system is to
criticality. From this Thomas-Fermi assumption of the density profile, we relate the interaction 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 = 2N h,
2Ry ≤ L.
(4.54)
(4.55)
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 − b0 )/b1 , τ ) exp ηζ 2 − 2 − 2 + iκxy ,
2ax 2ay
(4.56)
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 −b0 )/b1 , τ )
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
EBEC =
Z
dxdy
|(~∇ − mΩ × r)ψ|2
g2d 4
2
+ V (r)|ψ| +
|ψ| .
2m
2
(4.57)
And for the quantum Hall state, we use the trial wave function
Y
x2
y2
2
ψL ∼
(ζi − ζj ) exp − 2 − 2 + iκxy .
2ax 2ay
(4.58)
i<j
This is analogous to a ν = 1/2 bosonic quantum Hall state in the absence of the external
122
traps, by substituting z = x + iy to ζ. This many-body wave function is in the LLL since
it is analytic for all ζ 0 s. 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 y 2 :
1
1
1
hH0 iLLL = ~ω− + ~ω+ (ββ+ + 1/(ββ+ )) + mγω+ (β+ hx2 i + hy 2 i/β+ ).
2
4
2
(4.59)
Consequently, a classical Monte Carlo sampling of a large number of particles in the trial
“Laughlin wavefunction” gives the expectation values hx2 i, hy 2 i, thus determines the energy
of this “Laughlin state”. We calculated a specific case with 100 particles and interaction
strength g̃ = 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-onedimensional 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-axis ω̃y . 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
b
c
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
isotropic case has a density plateau of ρν =
ν
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
2πax ay
in the bulk. We keep track of the density profiles
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
moderate strength of interaction g2d πa1x ay
~Ω, we can set a cutoff of the complete
set of basis within the lowest Landau level only. This is always reasonable in the cases
where g̃ ω̃x 1. We separate the two-body hamiltonian into a center-of-mass part and a
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
B
C
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 + g2d ψm
(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 (ζ 2 ) 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 (ζ 2 ) 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:
2
2
ψ(ζ1 , ζ2 ) ∼ (ζ1 − ζ2 ) + x21 + x22 y12 + y22
2
× exp A(ζ1 − ζ2 ) −
−
+ iκ(x1 y1 + x2 y2 ) ,
2a2x
2a2y
(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/g̃ < 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
p
regions we are interested in. We observe the split distance between the zeros has a 1/g̃
behavior in a large region, as shown in figure 4.13. The zero limit goes to our trial 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:
ψ({ζi }) ∼
Y
i<j


2
2
X
X
xi
y
(ζi − ζj )2 + 2 exp A
(ζi − ζj )2 −
− i2 + iκ(xi yi ) . (4.62)
2
2ax 2ay
i<j
i
A test of overlap between this trial wave function and the exact diagonalization result for
few-particle systems is left to further study.
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:
ρ(r, t) ∼ hψk (
mr
mr †
) ψk (
)i.
~t
~t
130
(4.63)
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
g2 (0) =
Z
drhψ † (r)ψ † (r)ψ(r)ψ(r)i
(4.64)
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
cut
Larger
Interaction
Strength
Quantum Hall state:
y
x
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
x
No
zeros:
not QH
y
Larger
Interaction
Strength
x
Far apart zeros
y
x
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 ) =
D
E
n(r) − n(r) n(r0 ) − n(r0 )
(4.65)
= hn(r)n(r0 )i − hn(r)ihn(r0 )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
hn(r)n(r0 )i = hψ † (r)ψ(r)ψ † (r0 )ψ(r0 )i
= hψ † (r)ψ † (r0 )ψ(r0 )ψ(r)i + δ(r − r0 )hψ † (r)ψ(r)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
q
√
hψ † (r/2)ψ † (−r/2)ψ(−r/2)ψ(r/2)i = 2|ψ(r/2, −r/2)|,
(4.67)
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 Coulomblike 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
y
x
“Weak QH”
y
x
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
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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
Z=
Z
Z
Dψ̄Dψ exp −
0
β
dτ
Z
dr ψ̄σ
∇2
∂
−
− µ ψσ + g ψ̄↑ ψ̄↓ ψ↓ ψ↑ ,
∂τ
2m
(A.1)
where ψ̄ 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:
Z
Z
Z
¯
Z
Z
∆∆
¯
¯
exp − dτ dr|g|ψ̄↑ ψ̄↓ ψ↓ ψ↑ = D∆D∆ exp − dτ dr
− ψ̄↑ ψ̄↓ − ∆ψ↓ ψ↑ ,
|g|
(A.2)
¯ ∆ are bosonic fields such that ∆
¯ is simply the complex conjugate of ∆. The
where ∆,
expression above transforms the four-Fermi operator term into a quadratic form, and the
145
partition function is rewritten into
Z =
Z
Z
¯
D∆D∆
Dψ̄Dψ



Z
Z
∇2
∂
¯

 ∆∆
 ∂τ − 2m − µ
exp − dτ dr 
+ ψ̄↑ ψ↓ 
|g|
¯
−∆
The inverse of the interacting Green’s function is

∇2
∂
 ∂τ − 2m − µ
−1
G ≡
¯
−∆
−∆
∂
∂τ
+
∇2
2m
+µ
∂
∂τ
  
−∆
 ψ↑ 
  
∇2
+ 2m
+µ
ψ̄↓
(A.3)



(A.4)
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
Z=
Z
Z
Z
¯ τ )∆(r, τ )
∆(r,
−1
¯
+ log(detG )
D∆D∆
exp − dτ dr
|g|
(A.5)
We can switch to the momentum and Matsubara frequency domain, in which the field
operator is transformed into
ψ(r, τ ) = √
1 X
ψk,zν eik·r−zν τ .
βV z ,k
(A.6)
ν
2
∇
The ∂/∂τ goes to zν and − 2m
− µ goes to ξk =
k2
2m
− µ. The action from A.3 in momentum
space is


X
X
¯
¯
∆ 
βV ∆∆
zν + ξk
¯ ∆] = − βV ∆∆ + tr
S[∆,
log 
+
log(zν2 − Ek2 ), (A.7)
=−
|g|
|g|
¯
∆
z ν − ξk
zν ,k
zν ,k
in which Ek =
q
ξk2 + ∆2k , and we used an identity log detG−1 = tr logG−1 . The saddle
point is given by
∂S
= 0,
∂∆
146
(A.8)
and it gives the BCS gap equation
1 X
1
1
=− .
βV
|g|
zν2 − Ek2
(A.9)
zν ,k
Summing over the Matsubara frequencies, the equation takes the form
1 X tanh(βEk /2)
1
=
|g|
V
2Ek
(A.10)
k
is exactly same as 3.23. By taking the saddle point of the gap, the BCS mean-field partition
function is
ZBCS = e−S[∆0 ] ,
(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
G−1 = G−1
0 + M,
where


 0 δ∆
M =
.
¯ 0
δ∆
(A.12)
(A.13)
The appearance of tr log G−1 can be expanded in orders of δ∆ (or M ). To the second order,
it is
1
−1
2
3
Tr log G−1 = Tr log G−1
0 +Tr log (1+G0 M ) = Tr log G0 +Tr(G0 M )− Tr(G0 M ) +O(M ).
2
(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
¯
− Tr(G0 M )2 = −G↑↑ G↓↓ δ ∆δ∆
2
147
(A.15)
So the full partition function in the gaussian fluctuation regime is decomposed to
Z = ZBCS
Z
¯
¯
Dδ ∆Dδ∆
exp −G↑↑ G↓↓ δ ∆δ∆
(A.16)
Take the explicit expressions of G in the absence of the pairing order parameter ∆ is
G↑↑ = (zν + ξk )−1 and G↓↓ = (zν − ξk )−1 . We find this unit of G↑↑ G↓↓ is the same as each
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
For a spin-F system, usually we use a set of basis with eigenstates of operators F 2 and Fz
as |F, mi such that
F 2 |F, mi = F (F + 1)|F, mi,
(B.1)
Fz |F, mi = m|F, mi,
(B.2)
where m = −F, −F + 1, ..., F . These states are the eigenstates of a linear Zeeman term
HZ = gµB B · F, with a magnetic field in the z-direction B = Bẑ. In the following, we will
derive the expression for the eigenstates in the |F, mi basis of a Zeeman term in arbitrary
149
direction B = B(sin θ cos φ, sin θ sin φ, cos θ). In other words, we will find the eigenvectors
η m to the matrix representation of B · F, such that
X
β
(B · F)αβ ηβm = mηαm .
(B.3)
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 − = Fx − iFy = b† a,
Fz = a† a − b† b,
(B.4)
and the value of the total spin F is related to the total number of a, b bosons by
1
F = ha† a + b† bi.
2
(B.5)
In a subspace of spin-F , the 2F + 1 eigenstates of F 2 and Fz in this representation can be
written as:
(a† )F +m (b† )F −m
|F, mi = p
|0i,
(F + m)!(F − m)!
(B.6)
where |0i is the vacuum. In the matrix representation, the eigenstates of F 2 , Fz are represented by eigenvectors:
|F, mi = (0, 0, ..., 0, 1, 0, ..., 0)T
(B.7)
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
gµB Bz Fz . From now on, we use an abbreviated form of |F, mi as |mi, where we drop F
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 H̃Z = gµB B · F. The
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 |m̃i, in the original basis |F, mi, i.e. to find out the coefficients Cnm in the
150
expansion:
|m̃i =
X
n
Cnm |ni.
(B.8)
These coefficients are related to the eigenvectors η m as
m
m
m T
η m = (C−F
, C−F
+1 , ..., CF ) .
(B.9)
To find these coefficients, we note that the two Zeeman terms are connected by a rotation
in space
H̃Z = RHZ R−1 ,
(B.10)
in which the unitary transformation operator R is
R = eiSz φ eiSy θ eiSz χ ,
(B.11)
where Sx , Sy , Sz are the spin-1/2 operators. They are generators of the SU (2) rotation
group. The last factor eiSz χ 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 frame ㆠ, b̃† will experience
the same unitary transformation. They are thus related to the original Schwinger boson
operators a† , b† as



ã†
b̃†




 = R
a†
b†


 −1 
R = 
u
−v ∗

a†

v 


,
∗
†
u
b
(B.12)
where u = cos 2θ eiφ/2 and v = sin 2θ e−iφ/2 . Thus the new set of eigenstates are
|m̃i =
(ㆠ)F +m (b̃† )F −m
p
|0i
(F + m)!(F − m)!
(ua† + vb† )F +m (−v ∗ a† + u∗ b† )F −m
p
|0i
(F + m)!(F − m)!
X
=
Cnm |ni,
=
(B.13)
n
which satisfies
H̃Z |m̃i = m|m̃i.
151
(B.14)
The explicit form of the coefficients C are:



p
FX
+m
(F + n)!(F − n)!
 F + m  F − m 
Cnm = p



(F + m)!(F − m)! k=0
k
F +n−k


×uk (u∗ )−m−n+k v F +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 |m̃i is
A = i~hm̃|∇|m̃i = i~η m† ∇η = i~
X
n
Cnm∗ ∇Cnm .
(B.16)
We have
hm̃|∇|m̃i = ∇(hm̃|m̃i) − (∇hm̃|)|m̃i = −(∇hm̃|)|m̃i = −hm̃|∇|m̃i∗ ,
(B.17)
which makes the hm̃|∇|m̃i a pure imaginary number. Consequently, the gauge potential is
pure real. Also, this fact simplifies our calculation since we can focus on the imaginary part
of hm̃|∇|m̃i only.
Now we consider a specific state |m̃ = 0i state (for integer F ). The gauge field associated
with this state is
A0 /(i~) =
X
n
Cn0∗ ∇Cn0 .
(B.18)
The general formula (B.15) in the previous section gives an explicit expression as
Cn0
min(F,n+F )
X
p
= F ! (F + n)!(F − n)!
k=max(0,n)
uk (u∗ )k−n v F −k (−v ∗ )F +n−k
.
k!(k − n)!(F − k)!(F + n − k)!
(B.19)
By comparing term by term in the sum, we find such relation between n and −n as Cn0 =
0∗ and C 0 is real. Pairing up the C 0∗ ∇C 0 and C 0∗ ∇C 0 , the imaginary part of the
(−1)n C−n
n
n
−n
−n
0
152
whole sum in (B.18) is
Im
X
n
Cn0∗ ∇Cn0 = Im
= Im
X
0∗
0
(Cn0∗ ∇Cn0 + C−n
∇C−n
)
n>0
X
(Cn0∗ ∇Cn0 + (−1)2n Cn0 ∇Cn0∗ )
n>0
= Im
X
(ReCn0 − iImCn0 )∇(ReCn0 + iImCn0 )
n>0
+(ReCn0 + iImCn0 )∇(ReCn0 − iImCn0 )
X
2(ReCn0 ∇ReCn0 + ImCn0 ∇ImCn0 ) = 0.
= Im
(B.20)
n>0
Thus the gauge field associated with this |m̃ = 0i state is identically zero. We can then
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
s
(2F )!
CnF =
uF +n v F −n ,
(F + n)!(F − n)!
s
(2F )!
Cn−F =
(−v ∗ )F +n (u∗ )F −n ,
(F + n)!(F − n)!
(B.21)
(B.22)
which both contain only one term. Calculating the gauge potential for these two states, we
have
A−F /(i~) =
X
n
=
X
n
=
X
n
=
=
X
Cn−F ∗ ∇Cn−F
(2F )!
(−v)F +n (u)F −n ∇((−v ∗ )F +n (u∗ )F −n )
(F + n)!(F − n)!
(2F )!
(v)F −n (u)F +n ∇((v ∗ )F −n (u∗ )F +n )
(F − n)!(F + n)!
∗
(2F )!
(v∗)F −n (u∗)F +n ∇((v)F −n (u)F +n )
(F − n)!(F + n)!
n
∗
AF /(i~) = −AF /(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 i and |m = −F i are
“time-reversal partners”. In fact, similar to the derivations above, one can show the gauge
potentials of any pair |mi and | − mi have the same magnitude and different signs. The
mathematics is straightforward, but it is more complicated and will not be included here.
154