Dynamics of Quenched Ultracold Quantum Gases
byJohn P. Corson
B.S., Brigham Young University, 2009
M.S., Brigham Young University, 2011

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2016This thesis entitled:
Dynamics of Quenched Ultracold Quantum Gases
written by John P. Corson
has been approved for the Department of Physics

Prof. John Bohn

Asst. Prof. Jose D’Incao

Date

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

iii
Corson, John P. (Ph.D., Physics)
Dynamics of Quenched Ultracold Quantum Gases
Thesis directed by Prof. John Bohn

Recent advances in the tunability of ultracold atomic gases have created opportunities forstudying interesting quantum many-body systems. Fano-Feshbach resonances, in particular, allow
experimenters to freely adjust the scattering of atoms by controlling an external magnetic field.
By rapidly changing this field near a resonance, it is possible to drive systems out of equilibrium
towards novel quantum states where correlations between atoms change dynamically. In this thesis,
we take a wave-function-based approach to theoretically examine the response of several interesting
systems to suddenly-switched, or “quenched”, interactions.
We first calculate the time evolution of a Bose-Einstein condensate that is quenched to theunitarity regime, where the scattering length a diverges. Working within the time-dependent variational formalism, we find that the condensate does not deplete as quickly as the usual Bogoliubov
theory would suggest. We also make a quantitative prediction for the dynamics of short-range
pair correlations, encoded in Tan’s contact. We then consider the dynamics of these correlations
for quenches to small a, and we find that bound states can cause high contrast oscillations of
the contact. These dynamics can be modelled quantitatively at short times by using a properlychosen two-body model. Finally, we characterize the nonlocal correlation waves that are generated
by an interaction quench in arbitrary dimensionality. Our analysis demonstrates that the largemomentum limit of the post-quench momentum distribution can sometimes include contributions
from both the short range and the long range, depending on the quench protocol.

Dedication

To my one true amor.

v

Acknowledgements

I would first like to thank my advisor, John Bohn. I applied to CU-Boulder with a “clear”
plan of exactly what research I wanted to do, and twenty minutes at a whiteboard with John
changed everything. His intuition-based storytelling is almost as fun to witness as it is informative,
and it quickly kindled my interest in cold quantum gases. Over the years since then, I have felt
inspired by John’s creativity as a problem solver, his enthusiasm as a leader, and his patience
as a mentor. I should specifically mention his rare gift for pointing out miscalculations without
triggering embarassment.
It has been a pleasure to associate with other folks from the Bohn group. In no particularorder, I would like to thank Ryan, Michelle, Brandon, Andrew, James, Goulven, Michael, Lucie, andArnaud. We’ve had good times. I would also like to thank Jose for taking time to chat about threebody physics, and for his competition at ping pong. I also acknowledge the multitude of brilliant
experimentalists at JILA who have taught me lessons about real-world physics. Specifically, theKRb mixture group, the 85 Rb group, and the degenerate Fermi gas group.
Thanks are due also to Justin Peatross and Michael Ware, my research advisors at BYU,
for helping me discover the joy and satisfaction of research. They have been valuable mentors andfriends over the years.
Most of all, I would like to thank my wife Amy for her love, support, and patience. She is
the wind in my sails.

Contents

Chapter
1 Introduction
1.1

1Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Short-Range Two-Body Interactions
2.1

2.2

Alkali-Atom Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
7
8

2.1.1

Low-Energy Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2

Fano-Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Single-Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1

Finite-Range Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2

Zero-Range Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3

Tan’s Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Weakly Interacting Bose-Einstein Condensates
3.1

3.2

The Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1

The Importance of Indistinguishability . . . . . . . . . . . . . . . . . . . . . . 29

3.1.2

Experimental Observation in Dilute Alkali Gases . . . . . . . . . . . . . . . . 31

Many-body Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1

3.3

27

Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Weakly Interacting BECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vii
3.3.1

The Bogoliubov Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.2

Solving the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.3

The Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 BEC Quenched to Unitarity

46

4.1

Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2

Many-Body Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3

Post-Quench Dynamics

4.4

Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Bound-State Signatures in Quenched BECs

61

5.1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2

Many-body Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3

Two-Body Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4

5.3.1

Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3.2

Revisiting the Quench to Unitarity . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3.3

Quenching to Finite Scattering Length . . . . . . . . . . . . . . . . . . . . . . 73

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6 Quench-Induced Correlation Waves

80

6.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2

Ballistic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.3

Arbitrary Quenches

6.4

Lattice Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

101

viii
Bibliography

104

Appendix
A Two-Body Solution after a Quench

115

A.1 Solving the Problem in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.2 Generalizing to 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.3 Generalizing to 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Figures

Figure2.1

Schematic adiabatic curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92.2

Schematic Feshbach Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3

Square-well potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4

Square-well scattering states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5Interactions as boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1

Ballistic expansion of a BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2

Born Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1

Resonant interaction potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2

Dynamical momentum distribution at unitarity . . . . . . . . . . . . . . . . . . . . . 53

4.3

Dynamical momentum distribution at different density . . . . . . . . . . . . . . . . . 55

4.4

Dynamics of a single mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5

Depletion dynamics after the quench . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.6

Contact dynamics after the quench . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1

Bound-state oscillations of the contact in quenched many-body model . . . . . . . . 65

5.2

Bound-state oscillations of depletion in quenched many-body model . . . . . . . . . 66

5.3

Momentum distribution of two-body models . . . . . . . . . . . . . . . . . . . . . . . 71

5.4

Contact oscillations: two-body vs many-body . . . . . . . . . . . . . . . . . . . . . . 74

5.5

Contact oscillations after a downward quench . . . . . . . . . . . . . . . . . . . . . . 76

x
6.1

Ballistic expansion of a bound-state wave function . . . . . . . . . . . . . . . . . . . 83

6.2

Wigner distribution for expanded bound state . . . . . . . . . . . . . . . . . . . . . . 84

6.3

Wigner distribution for a weak quench . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.4

Quenching a bound state to repulsive interactions . . . . . . . . . . . . . . . . . . . . 88

6.5

Quenching a bound state on a lattice site . . . . . . . . . . . . . . . . . . . . . . . . 94

6.6

Lattice transport dynamics after a quench . . . . . . . . . . . . . . . . . . . . . . . . 97

A.1 Integration contour for two-body solution . . . . . . . . . . . . . . . . . . . . . . . . 118

Chapter 1

Introduction

In recent years, ultracold atomic gases have become a fertile ground for developing andtesting ideas of quantum many-body physics. Typical experiments use magnetic or optical forces
to trap a dilute vapor of atoms in high vacuum. To a good approximation, samples are thermallyisolated from the environment and thus represent idealized quantum systems. Atoms can be further
manipulated by external magnetic, electric, or optical fields to realize new and exciting states ofmatter that are governed by the laws quantum mechanics.
One of the most significant breakthroughs in this realm was the experimental creation of
Bose-Einstein condensates (BECs) of Alkali atoms. This novel state of matter was first theorizedby Einstein a century ago [56, 57] based on contemporary work by Bose [16], who had considered
the statistics of photons. The statistics that describe photons apply more generally to any indistinguishable particles that do not satisfy the Pauli exlusion principle, termed “bosons”. Einstein’s
insight was that, below a certain temperature, such particles preferentially occupy the exact same
single-particle quantum state. This many-body ensemble, a BEC, was not observed in the controlled environment of cold atomic gases until 1995, when researchers at JILA [1] and MIT [44]successfully cooled atoms to below the transition temperature (∼ 100nK). This achievement cameon the heels of a couple decades of progress in the realm of atomic trapping and laser cooling[134, 29, 34].
The exquisite control necessary to realize Bose-Einstein condensation in the laboratory paved
the way for a wealth of other studies of quantum phenomena. Experimenters began to explore how2interactions between atoms change the character of a system. These interactions give rise to thecomplexity that makes a single many-body problem intrinsically different from many one-bodyproblems. In the absence of interactions, the many-body Schrödinger equation is completely separable, and the eigenstates of the system can be described completely in terms of single-particle
quantum states. This simplification is what makes the thermodynamics and phase transition of anoninteracting BEC a common textbook problem of statistical mechanics [131]. Nontrivial interactions, on the other hand, introduce particle-particle correlations that alter the many-body statein fundamental ways.
The theory of interacting BECs has been in development for several decades. Most of theearly work focused on the ground state in the weakly interacting limit, where mean-field theory
gives an adequate description of the many-body system [10, 7, 8, 106, 84, 74, 75, 136]. Mean-field
theory makes the assumption that most of the atoms remain at rest in the ground state, despite
interactions. For short-range interactions, this assumption is predicated upon the smallness of thediluteness parameter na3 , where n is the number density and a is the s-wave scattering length forthe interacting particles. The scattering length is defined in terms of the phase shift of an interacting
two-body wave function, and it relates to the low-energy limit of the elastic cross section for thecolliding pair via σ ∼ a2 . In mean-field models, its magnitude is a measure of the interaction
strength and its sign indicates repulsive (a > 0) or attractive (a < 0) character. The condition
that na3 ≪ 1 is easily satisfied in most experimental setups, and as a result, the predictions ofmean-field theory have typically agreed very well with BEC experiments [162, 130, 81, 115, 174].
The discovery and characterization of Fano-Feshbach resonances in ultracold gases [158, 86,
165, 42] has given experimenters the ability to explore exotic, strongly interacting regimes thatcannot be described by the usual mean-field models. These resonances occur at specific values ofthe magnetic field and, in some cases, they can be several Gauss wide. Close to resonance, thescattering length a is large in magnitude, and it diverges on resonance (the “unitarity” regime).
Interactions can be tuned by simply fixing an external magnetic field to a desired value close toa resonance [28]. Experimenters can therefore study the effects of interactions in a given system

3by repeating experiments at various interaction strengths. This advantage is rather unique to thefield of ultracold atoms. For example, the parameters of the Standard Model of particle physics(masses, couplings, etc) are fixed by Nature, and we must live with the consequences. Not so, in
ultracold atoms.
Performing experiments at large scattering length does come at a cost, however. Just as alarge scattering length increases the probability for a two-body collision event, it also increases
the probability for three-body inelastic processes. When three atoms come together, two of themcan bond chemically, and the released binding energy is transformed into the kinetic energy of the
new molecule and third atom in a way that conserves momentum and energy. This extra kinetic
energy is almost always greater in magnitude than the depth of the magnetic or optical trap that isconfining the atoms. As a result, all three atoms are lost from the trap. Even in dilute gases, wherena3 ≪ 1, the total number of trapped atoms always decreases if you wait long enough. It has beenshown that the loss rate scales as a4 multiplied by a species-specific constant [63, 127, 59, 167].
This unfortunate scaling of loss has hindered experimental investigations of strongly-interacting
BECs in equilibrium. The splendor of large scattering lengths is, unfortunately, short lived.
The possibility of tuning interactions via Fano-Feshbach resonance has led to a resurgence oftheoretical efforts to understand strongly interacting BECs, despite their short lifetimes in practice.
Several studies [43, 146, 45] used constrained variational methods to calculate the ground stateenergy and chemical potential at strong interactions. Among the results found was that, in the a →∞ limit, these quantities scale universally with units derived from the number density n, as found
for two-component Fermi gases in the analogous regime [102]. Efimovian three-body contributions[54, 55] to the dilute-gas limit were considered in Ref. [17], and Refs. [14, 177] considered themetastability of the system in the strongly interacting limit. The equilibrium states discussed inthese studies are interesting in their own right, but it is unclear whether it’s possible to transition
to them adiabatically given the catastrophic a4 scaling of the three-body loss rate.
The outlook on the strongly-interacting-BEC problem changed when it was discovered that,when a system is rapidly brought out of equilibrium, interesting dynamics may occur before sub-4
stantial three-body loss sets in. The idea is to take a weakly interacting BEC (na3 ≪ 1) and
rapidly change the scattering length to the unitarity regime (a → ∞) by ramping the magnetic
field quickly to a Fano-Feshbach resonance. This general protocol of rapidly changing a parameterof the Hamiltonian to induce nonequilibrium dynamics is often called a “quench”. The experimentreported in Ref. [112] performed a quench to unitarity and observed that, at typical densities, threebody loss occurs over the course of about 1 ms. Interestingly, their measurements of the dynamic
momentum distribution indicated that their system seemed to reach some kind of steady state inabout 100 µs, with momentum-dependent saturation dynamics occuring along the way. This landmark experiment raises interesting questions about how weakly correlated states develop stronger
correlations when the interaction strength is increased, and about the nature of the nonequilibrium
steady state. Are there simple ways to understand the dynamics that occurs?
The nonequilibrium realm of ultracold quantum gases has become a hot topic in recent years
due to the exotic nature of the excited states that can be generated. For a BEC that is quenched
to unitarity, as discussed in the previous paragraph, the state is remarkably long lived. Whenquenching a one-dimensional BEC from strong repulsion to strong attraction, the resultant stateis correlated in such a way that the attractive forces do not cause clustering [3]. This exotic
state, known as the super-Tonks-Girardeau gas, was recently observed in experiment [76]. Anotherexperiment [85] studied weak interaction quenches in two-dimensional BECs, and the authors foundthat the system shows oscillatory density-density correlations, called Sakharov oscillations [140],
that are similar in nature to what has been famously observed in the distribution of matter in the
known universe [58]. Also of interest are the results reported in Ref. [48]. They observed that a
BEC undergoing a Ramsey sequence of interaction quenches can display coherence between atomic
and molecular states. We also mention the “bosenovae” experiment [49], which observed the violent
collapse dynamics of a three-dimensional BEC whose interactions are suddenly quenched from weak
repulsion to weak attraction. These experiments, and others, serve as an inspiration for the work
presented in this thesis.

5

1.1

OutlineThis thesis represents my graduate work done on the topic of interaction quenches in ultracold

atomic systems, and it does not include my studies of dipolar BECs [41, 40] and the Bose polaron[83]. Most of the presented results deal with three-dimensional Bose-Einstein condensates, althoughsome aspects are more generally applicable. Chapter 2 describes the interactions that typically takeplace in ultracold atoms. Due to the low energies involved at ultralow temperatures, the scattering
is predominantly s-wave in character and is well described by the scattering length a. Althoughatom-atom interactions are generally very complicated on the microscopic scale, one can often
approximate these interactions with simpler short-range models that yield the correct low-energy
scattering behavior. This idea leads to a universality in which the important consequences ofinteractions are model-independent. We introduce Tan’s contact [152, 154, 153], which relates
short-range pair correlations to the energetics of generic few- and many-body systems.
Chapter 3 gives a review of background concepts for BECs in the noninteracting and weaklyinteracting regimes. We discuss the thermodynamics of the condensed state, demonstrating how
the indistinguishability of identical bosons leads to macroscopic occupation of the single-particle
ground state even at nonzero temperature. We then introduce Bogoliubov theory, which describes
the low-energy collective excitations of a weakly-interacting BEC in terms of quasiparticles.
Chapter 4 presents our theoretical work on the topic of a BEC that is quenched to unitarity.
Our approach uses a time-dependent variational wave function to approximate the dynamical manybody state of the system. The ansatz explicitly encodes two-body correlations, which we expect
to be dominant in the early stages of evolution. The short-time dynamics that we compute isindependent of the way in which we model short-range, resonant interactions. As main results,
we find that the BEC does not deplete as rapidly as Bogoliubov theory would suggest, and the
dynamics of the momentum distribution scale universally with units derived from the number
density n. This observation is consistent with the experimental results reported in Ref. [112]. We
are also able to compute the time evolution of Tan’s contact after the quench. Much of this work6
was published in Ref. [150]
In Chapter 5, we describe the dynamics of short-range pair correlations for BECs quenched
to finite scattering length. An important result is that bound-state effects may strongly influence
the dynamics when the quench is diabatic. The condensate fraction and Tan’s contact oscillate at afrequency EB /~, where EB is the binding energy of the bound state. We also find that short-range
pair correlations are strongly enhanced by the bound state. Intriguingly, a diabatic decrease in the
scattering length yields a net increase in short-range pair correlations. We are able to explain theseresults with an intuitive two-body model that yields quantitatively accurate results at short times.
This work was published in Ref. [38].
In Chapter 6, we present results on the nonlocal correlation waves that are generated by aninteraction quench. We investigate these waves at the level of a two-body wavefunction, where thesimplicity of the system allows for an analytic representation of the correlation waves in momentum
space. We find that correlations propagate ballistically and, to leading order, the amplitude of the
wave relates directly to the quench parameters and the initial short-range pair correlations. Although intuition about equilibrium usually associates large-momentum asymptotics exclusively withshort-range behavior, our solution to the quench problem demonstrates that the large-momentum
tail of a nonequilibrium momentum distribution may also have a nonlocal, ballistic contribution.
We find that a semiclassical model based on the ballistic wave function gives accurate predictions
for transport over barriers, such as in an optical lattice. These results have recently been acceptedfor publication and can be found on the arXiv [39].
Chapter 7 provides a summary of our work and describes promising avenues for future research.

Chapter 2

Short-Range Two-Body Interactions

Given the complexity of a single many-electron atom, it may seem odd that people consider
doing theoretical calculations for systems that contain thousands or millions of interacting atoms.
There are too many degrees of freedom to keep track of, and modern computational resources arehopelessly out of their league. Nevertheless, a considerable simplification comes about when we
work in the ultracold regime.
An analogy can be drawn to our theoretical understanding of the proton. Particle-accelerator
experiments reveal that protons are composite particles made up of quarks dressed by a field ofgluons. Understanding all of the nuts and bolts on a truly microscopic scale is a complicatedproblem, but these details are not significant at sufficiently low energy scales. It is usually a good
approximation to consider the proton to be a structureless point particle that is characterized by amass, a spin, and an electric charge. To a casual spectroscopist, the energy levels of the hydrogen
atom can’t seem to tell the difference.1
A similar approximation can be made in the theoretical description of ultracold atomic gases.
At sufficiently low energies, we can treat individual atoms as point particles that are defined bytheir mass, some quantum numbers that summarize their internal structure (such as a hyperfine
or Zeeman state), and a single parameter that defines how atoms interact with each other. This
interaction parameter, the scattering length a, has a complicated dependence on the internal statesof the atoms and the magnetic field. The precise determination of a, as well as the location of1

High-precision spectroscopy reveals that the internal structure of the proton leads to slight, but measurable,changes in the hydrogenic energy levels. For an example, see Ref. [60].

8
Fano-Feshbach resonances, requires sophisticated calculations that are fine-tuned with experimental
data. Over the years, experimental physicists have become adept at controlling the internal state oftrapped Alkali atoms using coherent manipulation techniques, such as adiabatic rapid passage andRF pulses. Typical experiments usually trap an ultracold sample whose atoms occupy well-defined
spin states and where an external magnetic field fixes the scattering length to a known value. The
model many-body Hamiltonian is usually easy to write down and includes a just as a parameter,
as will be shown in Ch. 3. The challenge, then, is to solve the many-body problem.
We now turn our attention to describing simple models for the interactions between neutral
Alkali atoms, with a special focus on effects in three spatial dimensions. In what follows, we briefly
review the schematics of more realistic atom-atom interactions, emphasizing the important role
played by the scattering length. We then outline several useful short-range-interaction models thatare used as inputs into our many-body calculations. This chapter then concludes with a discussionof Tan’s contact, which has emerged as a useful concept in understanding how interactions define
the short-range correlations and energetics of few- and many-body systems.

2.1

Alkali-Atom Interactions
The low-energy scattering of Alkali atoms is an intricate process that depends in detail on

how the electrons in each atom respond to the presence of the other atom. The natural language fordescribing these interactions is given by the Born-Oppenheimer approximation, where the atomic
cores are assumed to move slowly in the field produced by the valence electrons. This is justified
by the orders-of-magnitude separation between the mass scales of the atomic core and the valence
electrons. The general idea is to treat the separation between the atomic cores, r, as a slowly
changing parameter and then solve for the electronic energies as a function of r. These energies
define a set of adiabatic potential curves that, if the atoms are asymptotically in their electronic
ground state, are approximately spherically symmetric. Figure 2.1 depicts one of these curves
schematically.
The qualitative behavior of atom-atom interactions is intuitive. We would expect that when9

Repulsive Core0

Van der Waals Attraction

0

Figure 2.1: Schematic representation of an adiabatic potential curve, with van der Waals attractionin the long range and repulsion in the short range.

r is roughly the size of an atom or smaller, the electronic clouds from the two atoms should havesignificant overlap. This leads to repulsion at small interatomic separations on a length scale set by
the Bohr radius, a0 . On large length scales, the instantaneous polarization of one atom can create
an electric field that induces opposite polarization in the other atom. This results in an attractive
van der Waals force that decays as r−6 for large particle separations. It is typical to define a vander Waals length rvdW that quantifies the approximate “range” of the interaction. This range istypically less than 5 nm for Alkali atoms [28].
At intermediate distances, the adiabatic potential smoothly interpolates between the repulsive
and attractive asymptotic regimes. The depth of the well depends strongly on whether the twovalence electrons have a symmetric (triplet) or antisymmetric (singlet) spin configuration, which
determines if covalent bonding is likely to occur. When the atoms are more than a few nanometersapart, their individual hyperfine or Zeeman energies dominate the physics and define the scatteringchannels of the problem. Colliding atoms that are initially in well-defined hyperfine/Zeeman states
project nontrivially onto the electron spin basis that governs interactions at smaller r. This leads
to coupling between the different scattering channels of the problem. An interesting result of thiscoupling is that Fano-Feshbach resonances can occur in ultracold Alkali-atom systems. We discuss
the phenomenology of these resonances in Sec. 2.1.2.

10
2.1.1

Low-Energy Behavior

Scattering events take on a particularly simple description in the low-energy limit. One
can make a partial-wave expansion of the scattering amplitude and exploit the fact that highangular momentum requires high kinetic energy and/or a large impact parameter. For short-ranged
interactions and low energies, we therefore expect that higher partial waves should not contribute
appreciably to the scattering. This expectation is verified in the Wigner threshold laws, which state
the following: For an interaction potential that decays as r−ν in the long range, the contribution of
the ℓth partial wave to the total cross section σ vanishes at least as quickly as σℓ ∼ E min{ν−3,2ℓ} inthe E → 0 limit, where E is the kinetic energy [155]. For van der Waals interactions, where ν = 6,
only s-wave (ℓ = 0) scattering occurs at ultralow temperatures.
The results of s-wave scattering are easy to write down in the low-energy limit. In the absenceof interactions, isotropic waves in 3D take the form
ψ(r) ∝
where k =sin(kr)r(2.1)

p2µE/~ is the wave number and µ is the reduced mass for the noninteracting atoms.

When scattering occurs in the presence of interactions, this wave experiences an energy-dependent
phase shift δ0 (k) for r ≫ rvdW :ψ(r) ∼

sin(kr + δ0 (k))
.
r

(2.2)

It can be shown that the low-energy behavior of δ0 (k) satisfies the effective range expansion [155]
1 1
k cot (δ0 (k)) = − + reff k 2 + O(k 4 ),
a 2

(2.3)

where a is the s-wave scattering length and reff is the so-called “effective range” of the interaction.
It is often (but not always!) the case that the effective range reff is similar in magnitude to theshort range rvdW . Such cases are universal in the sense that, for low energies satisfying kreff ≪ 1,
the s-wave phase shift is determined only by the energy and the scattering length a. Despite thecomplicated physics occuring in the short range, scattered wave functions are remarkably easy to11characterize in the low-energy limit once the scattering length is known. As an added bonus, thetotal cross section has a simple energy dependence in this regime:2
σ≈

4πa2
.
1 + k 2 a2

(2.4)

The magnitude of the scattering length therefore describes the apparent size of interacting atoms
from a low-energy collisional standpoint.
For attractive short-range interactions, the scattering length can be either positive or negative
depending on the behavior of the scattering phase shift. It is easy to show from Eq. (2.3) that
δ0 (k) ≈ −ka for k|a| ≪ 1. Positive scattering length thus corresponds to waves being phaseshifted to larger particle separations, as occurs for purely repulsive potentials. Despite the seeming
repulsion for scattering states, however, attractive short-range interactions always admit a shallow
bound state when a > 0, and the binding energy behaves as
EB = −

~2
2µa2

(2.5)

in the limit of large a, where µ is the reduced mass of the interacting atoms. This counterintuitive
fact will be demonstrated for a simple case in Sec. 2.2.1. We will see in Chapter 3 that, within
typical mean-field theories, small positive (negative) scattering length corresponds to repulsive(attractive) interactions because the interaction energy is proportional to a.

2.1.2

Fano-Feshbach Resonances

The multichannel nature of atomic collisions leads to the useful phenomenon of Fano-Feshbachresonances. These resonances are typically manifested in the fact that, for atoms in a given scattering channel j, the scattering length exhibits a resonance profileµaj (B) = abg,j 1 −∆jB − B0,j¶(2.6)

as a function of the magnetic field B. The parameter B0,j describes the location of a given resonance(of which there might be many), ∆j defines the width of the resonance, and abg,j is the background2

For identical bosons, symmetrization requires that we multiply this formula by a factor of 2

12Bound State

00
Figure 2.2: Schematic representation of a Fano-Feshbach resonance. The cyan curve is the adiabatic
potential for the scattering channel j, the red curve is the potential for some higher-energy channelk, and the horizontal dashed line represents the low energy of the scattering atoms.

scattering length at magnetic fields far away from any resonance. We have made explicit thedependence of the resonance parameters on the scattering channel j.
The basic phenomenology of a Fano-Feshbach resonance can be understood from a twochannel model. We will consider the coupled channels to represent different Zeeman states whose
magnetic moments are different by an amount δµ, and we assume that the atoms scatter in channelj, which we define to have the lower energy in the r → ∞ limit. The adiabatic curves are shown
schematically in Fig. 2.2. The channel k is termed “closed” because energy conservation prevents
atoms from exiting in this channel. Channel j is analogously termed “open”. A Fano-Feshbachresonance occurs when atoms colliding in the open channel have approximately the same energy as
a bound state in the closed channel, as shown. Due to the magnetic-moment differential δµ between
the two channels, their energy difference, and hence the detuning of the bound state relative tothe scattering energy, is sensitive to the magnetic field B in accordance with the Zeeman effect.
The resonance position B0,j is the magnetic field at which the bound-state detuning approximately
vanishes, and its precise location depends on the coupling between the channels. The width of theresonance, ∆j , relates to the strength of the channel coupling, where strong coupling usually results

13
in a broad resonance. The background scattering length, abg,j , is determined by the scattering that
occurs in the potential V (j) (r), ignoring channel coupling [28].
The idea of exploiting magnetic Fano-Feshbach resonances to tune interactions in ultracoldgases was first suggested in 1993 [158], and it was demonstrated experimentally in 1998 [86]. The
experimental protocol was to measure the loss of an ultracold sample of trapped 23 Na atoms asa function of magnetic field. It is predicted that the loss rate should scale as a4 multiplied bya constant [63, 127]. Hence, loss should be strongly enhanced in the vicinity of a Fano-Feshbachresonance. By scanning the magnetic field along a range of about 1000 G, experimenters wereable to identify two loss resonances whose locations agreed with theoretical predictions. They werethen able to infer the scattering length near these resonances by measuring the interaction energyof a Bose-Einstein condensate, which is predicted to scale linearly with the scattering length (seeSec. 3.3.3). Near the loss resonances, the magnetic-field profile of the scattering length matched
the predicted profile of Eq. (2.6).
Although most experiments utilize magnetic Fano-Feshbach resonances to tune interactions,
we also mention the possibility of using optical Feshbach resonances. The underlying physics isnearly identical to that already described, where scattering states from one channel are coupled
to the bound state of another channel. The distinguishing feature in optical Feshbach resonancesis that the coupling is provided by an external laser field that couples scattering states to an
excited quasibound state via an optical transition. Rather than tune the magnetic field B, one cantune both the intensity I and frequency ω of the off-resonant laser beam to alter the scattering
length of collisions [62, 11, 12]. Soon after its prediction, this effect was confirmed in experiments[61, 156]. This scheme suffers the disadvantage of having extra loss when the excited state decays
spontaneously, and the laser field may also induce unwanted dipole forces to the system underobservation. However, recent progress has been made to mitigate these drawbacks [30]. Optical
Feshbach resonances offer the possibility of spatially modulating interactions. Additionally, they
allow experimenters to dynamically tune interactions on shorter timescales than are feasible withmagnetic Fano-Feshbach resonances. This improves the prospects for measuring the nonequilibrium

14
results that we present in Chapters 4-6.

2.2

Single-Channel Models
The discussion in Sec. 2.1.1 shows that the phase shifts and cross sections of ultracold Alkali

collisions can be reduced to a simple form that depends only on the scattering length, a, andthe effective range, reff . The description continues to simplify if we narrow our consideration
to the regime where |a| ≫ |reff |. In this limit, the low-energy condition kreff ≪ 1 is sufficient
to keep only the scattering length term on the right-hand side of the effective-range expansion,
Eq. (2.3). This approximation, in which all scattering is determined by a only, is a valid descriptionof ultracold gases near entrance-channel-dominated resonances. The scattering and bound wave
functions associated with these resonances are composed mainly of the open-channel component,with relatively little contribution from the closed-channel bound state despite the coupling [96].
These resonances are often wider than 1 G and are commonly referred to as “broad” resonances[28]. In the remainder of this thesis, we will only consider interactions that are governed by thisclass of Feshbach resonance. The scattering length a therefore uniquely determines all scatteredwave functions according to Eqs. (2.2)-(2.3), where we ignore the effective range and higher-orderterms.
In our quest to describe the physics of interacting many-body systems, we can exploit the diluteness of typical experimental Alkali-atom samples in the ultracold regime. The nearest-neighborspacing is usually a few orders of magnitude larger than the interaction range rvdW , and thissuggests that only a tiny fraction of the wave function exists within the interaction region. We
therefore expect that all of the interesting physics in such interacting systems is determined by
the part of the wave function that is phase shifted. The scattering length governs this phase shiftand, as a result, should uniquely dictate how interactions affect a dilute many-body system in theultracold limit. To simplify the many-body problem, we represent the microscopically-complicatedatom-atom interactions with simplified models that reproduce the desired scattering length. The
only errors introduced by this approximation occur at small particle separations (r . rvdW ) and at

15

0

0

Bound State

0

2

4

0

a)

2

4

b)

Figure 2.3: Attractive square-well potentials. In (a), the well is shallow. In (b), the well is deep
and admits a bound state, as shown.

large momentum (kreff ∼ 1), both of which compose only a negligible fraction of the wave function
for ultracold, dilute, Alkali gases.
This section describes several useful short-range interaction models. The first class of models
uses a finite-range attractive well whose depth and range are chosen to reproduce the correct
scattering length. The second class invokes the zero-range approximation, where interactions are
replaced with a scattering-length-dependent boundary condition on the wave function at vanishingparticle separation.

2.2.1

Finite-Range Models

The simplest interaction potentials for which the Schrödinger equation can be solved analytically are those that are piecewise constant. The solutions are either real or complex exponentials,
depending on how the energy compares to the local potential height. The piecewise wave functions
can then be joined by assuming continuity and smoothness. As our first example, we solve for the
scattering and bound solutions in an attractive square-well potential
V (r) = −V0 Θ(r0 − r),(2.7)

where V0 > 0 is the depth of the well, r0 is its range, and Θ is the Heaviside step function. Thispotential is depicted in Fig. 2.3.

16
We confine our analysis to solutions that are spherically symmetric, representing the s partial
wave that dominates at ultralow temperatures. In three dimensions, it is easiest to solve for thequantity u(r) ≡ rψ(r), where ψ is the relative wave function. The radial Schrödinger equation isthen given by
Eu = −~2 d 2 u+Vu
2µ dr2(2.8)

where µ is the reduced mass for the atoms and E is the energy. Hermiticity requires that u(r)|r→0 =0 [118]. We will first consider scattering solutions where E = ~2 k 2 /2µ is the relative kinetic energy.
For r > r0 , the potential vanishes and the solution is given by
uk (r) = A sin (kr + δ) ,(2.9)

where A is a normalizaton constant and δ is the yet-to-be-determined phase shift. For r < r0 , thesolution is¡ ¢uk (r) = B sin k ′ rwhere B is a normalization constant, and the wavenumber is shifted to k ′ =(2.10)p

2µ (E + V0 ) /~2 dueto the excess kinetic energy inside the well. We can solve for the phase shift δ by matching the log
derivative u′ /u at r0 . After a small amount of algebra, one finds that−1

δ = tan

µ

¶
¡ ′ ¢
k
tan k r0 − kr0 .
k′(2.11)The low-energy limit of this phase shift should be δ → −ka according to the effective range
expansion, and this fixes the scattering length to bea = r0 − √´³p~
2µV0 r0 /~ .
tan
2µV0

(2.12)

Two sample scattering solutions are shown in Fig. 2.4 for a well whose range and depth are chosen
to yield a scattering length of a = 10r0 . In the long range, they show the usual oscillatory behavior
with a phase shift. This extra phase is acquired in the short range, as shown in the left-hand plot. As
a consistency check of Eqs. (2.11)-(2.12), we see that both the phase shift and the scattering length√
vanish in the V0 → 0 limit. The scattering length in Eq. (2.12) diverges whenever 2µV0 r0 /~ =

17
0

1

-0.1
0

-0.2
-1
0

5

10

0

500

Figure 2.4: Two scattering states at different energies for a square well whose range and√ depth arechosen to yield a scattering length of a = 10r0 . We constrain the depth such that 2µV0 r0 /~ <3π/2, where at most one bound state is allowed. Deeper wells lead to more short-range nodes of
the wave function. The left-hand plot shows a short-range zoom of the wave functions. The plottedfunctions are normalized to behave as sin (kr + δ) for r > r0

(n + 1/2)π, where n is a non-negative integer. As we will see below, this resonant behavior occurs
whenever the well deepens sufficiently to include another bound state.
The bound-state solutions can be calculated in much the same way as already shown, exceptthat we assume the energy satisfies −V0 < E < 0. The solution outside of the well must decay
exponentially as√uE (r) = Ae− −2µEr/~.

(2.13)The solution inside the well is unchanged. Matching log derivatives at r0 , we find that the energy
must satisfy√
No solution exists for

√

³p
´
p
−E = − E + V0 cot
2µ (E + V0 )r0 /~ .

(2.14)

2µV0 r0 /~ < π/2, as the well is not deep enough to support a bound state√in this regime. It is easy to verify that this equation has a zero-energy solution when 2µV0 r0 /~ =√(n + 1/2)π for non-negative integer n. There is a shallow bound state when 2µV0 /~ is slightly

greater than (n + 1/2)π, and this regime corresponds also to large positive scattering length as
evidenced by Eq. (2.12). For a weakly-bound state satisfying |E| ≪ V0 , one can approximate

18
Eq. (2.16) as√−E ≈ −´³pp
V0 cot2µV0 r0 /~

~
≈√
2µa

,

(2.15)

where we have used the large-a limit of Eq. (2.12). For large positive scattering length (a ≫ r0 ),the shallow bound state is given bye−r/aψB (r) ≈ √
r 2πa

,

EB ≈ −~22µa2(2.16)

for r >> r0 . This asymptotic relation between the scattering length and the shallow bound stateis valid more generally [28], as mentioned in connection with Eq. (2.5).
One can simulate short-range interactions with other single-channel potentials besides thesquare well. The square well offers the simplicity of being transparent and analytically solvable,
and the scattering length relation Eq. (2.12) allows one to conveniently choose the two model
parameters V0 and r0 to achieve any desired scattering length. One can also add a repulsive barrier
to the square well [88]. Analytic solutions are also available for a pure van der Waals interaction
with a hard core [71, 66], whose free parameters are the C6 coefficient and the core radius. The lowenergy scattering in a Morse potential can also be solved analytically [123, 94, 93]. Another simple
interaction model is the attractive Gaussian well, although the relation between the scattering
length and its depth and range must be found numerically [157]. The common thread of thesemodels is that they are essentially short ranged and can be tuned to give the correct low-energy
scattering.

2.2.2

Zero-Range Models

From a theoretical standpoint, we are free to choose the interaction range of our model to be
as small in magnitude as we wish. The parameter r0 in the square well, for example, could take anarbitrarily small value. One would simply have to adjust the depth V0 to yield the desired scattering
length a according to Eq. (2.12). As we allow r0 to approach zero for fixed scattering length,
the scattering wave function (2.9) and bound wave function (2.16), valid for particle separations

19

0

-0.1

-0.2

0

5

10

Figure 2.5: Short-range part of the scattered wave functions for a square-well potential chosen toyield a = 10r0 , with the phase shifts extrapolated back to vanishing particle separation. Comparewith the left-hand plot in Fig. 2.4.

satisfying r > r0 , eventually become valid for all r > 0. This is shown schematically in Fig. 2.5,
which should be compared with the left-hand plot in Fig. 2.4. It is in this limiting sense that we
introduce the idea of a zero-range interaction.
The zero-range approximation stipulates that the relative wave function for a pair of interacting atoms satisfies the noninteracting Schrödinger equation, albeit with a rigged boundary
condition at vanishing particle separation. For three-dimensional systems, this boundary conditionis given by¯∂ [rψ] /∂r ¯¯
1
=− ,
¯
rψ
a
r→0+

(2.17)as discussed in a famous paper by Bethe and Peierls [9]. (The basic idea was also presented
in earlier work by Fermi [64].) The quantity on the left-hand side is often called the “logarithmic
derivative”, as it can be expressed as ∂ [ln(rψ)] /∂r. The Bethe-Peierls boundary condition enforcesa scattering-length-dependent relation between the value of the wave function and its slope in ther → 0+ limit. For small particle separations, the wave function must behave linearly ash¡ ¢irrψ(r) = [rψ]|r→0+ 1 − + O r2 .
a

(2.18)The scattering length evidently has the following meaning in the zero-range approximation: If you

20
fit a line to the short-range behavior of rψ, it intercepts the horizontal axis at r = a. This same
geometric interpretation is valid for interactions with a finite range in the limit that |a| ≫ r0 ,although it is only robust in the low-energy regime (kr0 ≪ 1) and the line must be fit outside
the interaction region. This behavior is readily visible in the square-well eigenstates depicted inFig. 2.5, and one can easily verify its validity for the bound-state solution (2.16).
It is easy to find the eigenstates of the Schrödinger equation within the zero-range approximation. For r > 0, the scattering solution must be a superposition of sin(kr) and cos(kr), and theonly superposition that satisfies the log-derivative boundary condition (2.17) isrψk (r) = A [sin(kr) − ka cos(kr)]

,

Ek =

~2 k 22µ(2.19)

where A is a normalization constant. After a simple trigonometric manipulation, one can showthat this is proportional to sin(kr + δ), where the phase shift must satisfy
k cot(δ) = −1a(2.20)

for all k. On comparing this with the effective range expansion (2.3), we see that the zero-range
approximation sets reff → 0 and defines all phase shifts in terms of the scattering length only. A
bound-state solution only exists for a > 0, and it is given bye−r/arψB (r) = √
r 2πa

,

EB = −~2.
2µa2(2.21)

These solutions are useful in that they are simple to write down, and they give a remarkably
accurate description of ultracold scattering in the limit that |a| ≫ rvdW near a broad Feshbachresonance.
Zero-range interactions are often cast in the language of the Fermi pseudopotential [65, 84].
Within this framework, the relative wavefunction for a pair of interacting particles in three dimensions must satisfy
Eψ = −~2 2
2π~2 a (3) ∂ (rψ)
∇ ψ+
δ (r)
,
2µ
µ
∂r

(2.22)

where the last term represents the operation of the Fermi pseudopotential on the wave function.
Integrating Eq. (2.22) over a small sphere of radius ǫ > 0 about the origin, we find that the left-hand21side vanishes in the ǫ → 0 limit. What remains is
~2
0=−
2µ

Z

¯
2π~2 a ∂ (rψ) ¯¯d r∇ ψ +µ∂r ¯r→03r<ǫ2(2.23)

If we assume that ψ(r) = (rψ)|r→0 ( 1r + O(r0 )) for small r, we can exploit the relation ∇2 1r =−4πδ (3) (r) to evaluate the integral. The result is that¯
2π~22π~2 a ∂(rψ) ¯¯,0=(rψ)|r→0 +µµ∂r ¯r→0(2.24)

which is identical to the Bethe-Peierls boundary condition (2.17). The two formulations are therefore equivalent.
Representing short-range interactions with a delta function is obviously a nontrivial task, asevidenced by the unusual partial differentiation that appears in the Fermi pseudopotential (2.22).
It raises the question of whether one can use a delta function without the derivative. Can we
represent interactions with a potential that is constant in momentum space, as would befit theFourier transform of a delta function? It turns out that this is possible in a limiting sense, but
it requires a careful renormalization procedure in which the strength of the delta function must
depend on its width (which is nonzero for a nascent delta function). We begin by defining theFourier transform of a functionZ

d3 re−ik·r f (r)(2.25)

Z

d3 k ik·r ˜
e f (k).
(2π)3(2.26)f˜(k) =and its inverse transformf (r) =

Rather than use a strict delta function for the interaction, whose Fourier transform equals 1 for allk, we consider the analogous quantity up to a momentum-space cutoff Λ:(3)δΛ (r) ≡Z

d3 k ik·re Θ(Λ − k)(2π)3(2.27)

where Θ is the Heaviside function. The quantity Λ−1 can be understood as the range of theinteraction. We will define the interaction potential to be(3)

Vint (r) = UΛ δΛ (r),(2.28)

22
where UΛ is a scattering-length-dependent coupling constant required to give the correct short-range
physics.
The functional form of UΛ can be demonstrated simply by solving the bound-state problem
in momentum space. The momentum-space Schrödinger equation can be written as [117]Z~2 k 2(3)
EB ψ̃B (k) =ψ̃B (k) + UΛ d3 re−ik·r δΛ (r)ψB (r)2µZ~2 k 2d3 k ′=
ψ̃B (k) + UΛψ̃B (k′ − k)Θ(Λ − k ′ )
2µ
(2π)3
Z
d3 k ′~2 k 2
ψ̃B (k) + UΛψ̃B (k′ )
≈
2µ
(2π)3

(2.29)k′ <Λwhere we have invoked the convolution theorem and assumed that Λ is much larger than both kand a−1 . In momentum space, the bound-state solution shown in Eq. (2.21) becomes√
2 2πa3
ψ̃B (k) =1 + k 2 a2,

EB = −~22µa2(2.30)

If we substitute this solution into Eq. (2.29), all of the k dependence vanishes, and we can solve
directly for UΛ . When the smoke clears, we find thatUΛ =2π~2 a/µ1 − π2 Λa(2.31)

in the Λa ≫ 1 limit. This is the coupling constant required to give the correct short-range physics.
It is valid more generally for any value of a and even for Λa ≪ 1 (where no bound state occurs),
as can be proved with Green function techniques [19].
It is interesting that, for fixed scattering length, UΛ decays as Λ−1 in the Λ → ∞ limit. Thisbehavior may seem surprising at first, but it is analogous to what occurs in the r0 → 0 limit of thesquare well calculation at fixed scattering length. In that case, the “volume” of the well (∼ V0 r03 ,analogous to UΛ ) decays as O(r0 ) as the range is reduced.3 Given our interpretaton of Λ−1 as the(3)range of δΛ (r), we see that the scalings are identical in the zero-range limit. In fact, the two model
potentials are not particularly different, except that one is a Heaviside function in position space
3
This is immediately obvious from Eq. (2.12) in the case of infinite scattering length, where we can fix
2µV0 r0 /~ = π/2. It can be shown that this scaling is also valid for arbitrary nonzero scattering length in the
r0 → 0 limit.

√

23
and the other is a Heaviside function in momentum space. Both models are broad in momentum
space up to an effective cutoff Λ ∼ r0−1 .

2.3

Tan’s Contact
There is considerable power in the idea that, once you know the scattering length for a pairof particles with short-range interactions, you can make universally valid statements about the
scattering and bound wave functions to within a good approximation. We have discussed thisparadigm in the two previous sections. The necessary assumption is that relative momenta be low
enough to satisfy kreff ≪ 1, which can be fulfilled in most cold-atom experiments near a broad
Feshbach resonance. It turns out that, in this regime describable by the zero-range approximation,
one can make even stronger statements about system observables based purely on the short-range
nature of the interactions. These statements, called the Tan relations, were introduced by Shina
Tan about a decade ago [152, 154, 153]. They hold true for all equilibrium states in the ultracold
regime, both few-body and many-body, both in the ground state and at finite temperature, whether
we know how to solve the problem or not.
Central to the Tan relations is the idea that global interaction effects should scale with theprobability that atoms are close enough to interact. As an obvious example, the interaction energy
for a pair of atoms is zero if their relative wave function vanishes inside the interaction region,and it should be large in magnitude if the wave function has a significant component in the shortrange. Within the framework of the zero-range approximation, the relative wave function behavesas ψ(r) ≈ (rψ)|r→0 1r in the short range, as written in Eq. (2.18). The operative quantity (rψ)|r→0
is not determined by the log-derivative boundary condition (2.17), but rather by the specific stateof the system. Its magnitude-squared encodes the probability that the atoms are close enoughto interact or, in other words, are in “contact” with each other. This contact probability should
determine the interaction energy and the extent to which the pair will notice a change in theinteractions. By properties of the Fourier transform, it should also determine the large-k behavior¯2¯¯ . Such a probability is well definedof the momentum distribution via nk ∼ |ψ̃(k)|2 ∼ ¯ (rψ)|r→0 4π
k2

24
for an arbitrary system of two or more interacting particles, and Tan’s crucial development was toestablish quantitatively the universal ways in which it relates to the observables of any equilibrium
system for which the zero-range approximation is valid.
The key parameter that appears in the Tan relations is the “contact” C, and it is proportional
to the short-range pair probability of the system. The short-range r−2 behavior of the interacting
two-body probability |ψ(r)|2 carries over to the many-body pair distribution function g (2) (r), whose
short-range behavior defines the contact via
g

(2)

C
(r) =
+O
(4πn)2 r2µ ¶1
r(2.32)

where n is the many-body number density of the system. The pair distribution function quantifies
the relative likelihood of measuring other particles a distance r from any particle of interest.4 In
the absence of correlations, g (2) (r) = 1. Similar to the two-body case, the short-range behavior of
the pair probability also determines the large-k tail of the momentum distribution according toCnk = 4 + O
k

µ

1
k5

¶

,

where nk is assumed to be normalized to the density as n =(2.33)
Rd3 kn .
(2π)3 kAs written so far, thecontact is a volume-averaged intensive quantity. If we were to normalize the density distribution
to the particle number N instead of the density, then the definition (2.33) would imply that C isan extensive quantity. There are several conventions in the literature. When addressing manybody problems in this thesis, we will normalize momentum distributions to the number density n;
therefore, we consider C to be intensive.
We now briefly state some of the Tan relations as they apply to identical bosons of mass m in
the absence of three-body effects.5 Relevant derivations can be found in Refs. [152, 154, 153, 20, 18].
The first relations can be considered to be Eqs. (2.32)-(2.33), either of which can define the contact.
The contact relates intimately to the total energy density E of a homogenous gas in equilibrium
4This likelihood is compared to an uncorrelated gas, where the probability of measuring a particle in a volume
element d3 r is given simply by n × d3 r.
5Three-body effects will be discussed in Ch. 7

25
according to
E=

Z

¸
·
~2
d 3 k ~2 k 2
C
+
n
−
C,
k
(2π)3 2mk48πma

(2.34)

where both the contact and the momentum distribution depend implicitly on a. The contact also
determines how this energy density evolves as the scattering length is changed adiabatically:
dE
~2
=−
C.
d (1/a)8πm(2.35)

Similarly, diabatic changes in the scattering length lead to a contact-dependent change in the energy
density~2
Ef − Ei = −
C
8πm

µ

1
1
−
af
ai

¶

,

(2.36)

where it is assumed that the change occurs quickly enough that the short-range behavior of theinitial quantum state, characterized by C, does not evolve appreciably while the interactions are
switched. Similar relations have been derived for the large-detuning limit of the RF spectrum,
the short-range limit of the one-body density matrix, the pressure, and the virial theorem. It is
important to note that these results provide constraints on system observables, but they do notdirectly predict what these observables are unless the parameter C is known.
Tan’s relations have been verified experimentally in a variety of systems. Several of these relations were measured in a gas of ultracold two-component fermions across the BCS-BEC crossover,
and agreement was found for both weak and strong interactions [147]. Soon afterwards, Tan’s predictions were seen in a weakly-interacting Bose-Einstein condensate [174]. The contact has recently
gained significant interest in the context of nuclear-physics experiments, which appear to display
asymptotic k −4 behavior in the momentum distribution [169, 168, 77]. It is significant that thecontact is relevant to such disparate systems despite the fact that their densities differ by about afactor of 1020 .

2.4

Summary
In this chapter, we have given a basic description of Alkali-atom interactions and empha-

sized the important simplifications that occur in the ultracold regime. Due to the short range of26interactions, low-energy physics for both scattering and binding is determined universally by thescattering length. We can use simple models to represent these interactions, including the zerorange approximation, and this leads to a significant reduction in complexity over the multi-channel
representation of true atom-atom interactions. We introduced Tan’s contact, which helps bridge theconceptual gap to the many-body problem, demonstrating the important role played by short-range
pair probabilities even in a many-body context.

Chapter 3

Weakly Interacting Bose-Einstein Condensates

In this chapter, we review the basic phenomenology and theoretical treatment of weaklyinteracting Bose-Einstein condensates in equilibrium. Two-body interactions, as described in theprevious chapter, complicate the many-body picture somewhat; at the same time, however, they
deeply enrich the physics, and we will see that the scattering length plays an important role indetermining the properties of a BEC. This is a precursor to our studies of strongly interacting
condensates out of equilibrium, which will be presented in the later chapters of this thesis.

3.1

The Phenomenon
Bose-Einstein condensation is the abrupt macroscopic occupation of the single-particle ground-state wave function that occurs when a many-boson sample is cooled below a critical temperature.
Those atoms that occupy the ground state constitute the Bose-Einstein condensate (BEC). This
phase of matter was first predicted for noninteracting systems by Einstein almost a century ago[56, 57], and its existence can be intuited from the Bose-Einstein distribution function for identicalnoninteracting bosons:
N (Ej , µ, T ) =1e(Ej −µ)/kB T − 1

,

(3.1)

where Ej is the energy of a single-particle state, kB is Boltzmann’s constant, and T is the temperature of the system. If we measure energies with respect to the single-particle ground state, thechemical potential µ must satisfy µ < 0 so that the occupation numbers are nonnegative and finite.

28
The chemical potential is fixed by the total particle number N according toN=X

N (Ej , µ, T ).

(3.2)

j

Equation (3.1) suggests that the single-particle ground state can be macroscopically occupied [thatis, N (0, µ, T ) = O(N )] if and only if |µ| is much smaller than the spacing between single-particle
energy levels. When this is the case, all excited single-particle states satisfy Ej ≫ |µ|, and we cantherefore approximate µ ≈ 0 in calculating their thermal populations via (3.1).
We can identify the conditions for Bose-Einstein condensation by investigating when µ ≈ 0leads to a consistent solution of Eqs. (3.1)-(3.2). Inasmuch as the ground state may be macroscopically occupied, we can separate it out explicitly and approximate the remaining sum in Eq. (3.2)
in the µ → 0− limit as an integral over energies:
N = N (0, 0, T ) +≈ N (0, 0, T ) +≈ N (0, 0, T ) +X

N (Ej , 0, T )j>0Z∞dE g(E)N (E, 0, T ),

0
Z∞
0

(3.3)

g(E)
dE E/k T
e B −1

where g(E) is the system-dependent density of states.1

For systems in which the integral in

Eq. (3.3) is greater than N , the ansatz µ ≈ 0 is clearly wrong because it violates number conservation [Eq. (3.2)]; such systems are therefore uncondensed. When that integral is less thanN , however, there is a macroscopic ground-state occupation N (0, 0, T ) > 0 for which Eqs. (3.1)(3.3) are satisfied, and the system is Bose condensed. The onset of condensation in such cases isdetermined by the temperature TC for which
N=

Z∞
0

g(E)
dE E/k T,
e B C −1

(3.4)

and there is a nontrivial BEC fraction for T < TC that is determined by number conservation
1
E
R′
0

The density of states encodes the degeneracy of single-particle states, and it is defined such that its integraldE g(E) gives the number of states below energy E ′ .

29according to Eq. (3.3). This fraction increases continuously from zero as the temperature is lowered
below TC .
There are a few useful examples for which the condensation temperature and condensate
fraction can be calculated analytically. Here, we simply state the results and refer the reader toRef. [131] for the derivations. For a homogenous gas of N bosons in volume V, the condensation
temperature has been evaluated analytically as
·
¸2/3
2π~2
N
TC =
mkB V ζ(3/2)(Homogenous Gas),(3.5)

where m is the mass and ζ(z) is the Riemann zeta function. We can similarly find the condensate
fraction to be
N (0, 0, T )
=1−
N

µ

T
TC

¶3/2

(Homogenous Gas)

(3.6)

for T < TC . If the bosons are in a harmonic trap of mean frequency ω̄ = (ωx ωy ωz )1/3 , then thecondensation temperature evaluates to·¸~ω̄ N 1/3
TC =kB ζ(3)(Trapped Gas),(3.7)

and the condensate fraction is given by
N (0, 0, T )
=1−
N

µ

T
TC

¶3

(Trapped Gas)

(3.8)

when T < TC . For homogenous gases in 2D and 1D, the energy integrals in Eqs. (3.3)-(3.4) diverge
and no condensation occurs for T > 0.

3.1.1

The Importance of Indistinguishability

It may, at first glance, seem odd that thermodynamics would allow most of the particles to occupy the ground-state even when the temperature is larger than the spacing between single-particle
energy levels. Consider the case of N particles in a spherically-symmetric harmonic oscillator of
frequency ω. For sufficiently large N , Eqs. (3.7)-(3.8) show that it is possible to have an almost
pure BEC of ground-state particles (T ≪ TC ) at a temperature where many oscillator states are30
thermally accessible to each particle (kB T ≫ ~ω). Why do condensed bosons avoid these excitedstates despite their accessibility? The resolution of this seeming paradox comes from the different
counting statistics that apply to distinguishable and indistinguishable particles. We will illustrate
this heuristically.
For the case of distinguishable particles, we can apply Boltzmann statistics separately toeach particle. Occupation probabilities are then proportional to the Boltzmann factor e−E/kB T .
In the regime where kB T ≫ ~ω, a large number q of modes are thermally accessible, and theprobability of an atom occupying any of these states (including the ground state) is of order 1/q.
We conclude on these grounds that only a tiny fraction of particles can occupy the ground state at
this temperature. We could alternatively arrive at this conclusion by applying Boltzmann statistics
to the eigenstates of the many-body ensemble. From this perspective, each system state has its
own energy and Boltzmann factor. The many-body ground state contributes a Boltzmann factor ofP
unity to the partition function Z = j e−Ej /kB T . States with energy ∼ N kB T (that is, ∼ kB T per
particle) each contribute a tiny Boltzmann factor of approximately e−N ; however, there should be
about q N of these states, and this more than makes up for the miniscule scale of their Boltzmann
factors. These excited states, taken together, are exponentially probable for large N comparedto the ground state. Again, we conclude that only a tiny fraction of particles should occupy theground state.
When the particles are indistinguishable, we must count excited states differently. For example, suppose we take the many-body ground state and excite a single particle to a specific orbital.
This configuration corresponds to only one unique eigenstate in the indistinguishable case, whereas
there are N such eigenstates if the particles are distinguishable. There is, generally speaking, adramatic suppression of entropy when we are dealing with the low-energy eigenstates of indistinguishable particles. Returning to the states of total energy ∼ N kB T , the Boltzmann factors remain
approximately e−N . We can estimate the total number of such states by counting the number of31
ways we can arrange N identical particles among q single-particle states:¶µ
N +q−1(N + q − 1)!
.
=
Number of States =N !(q − 1)!
N

(3.9)

Exploiting Sterling’s approximation [2] and the fact that both N and q are large numbers, we canwrite this approximately as

Number of States ≈



¡ ¢


 qe N
N

³ ´q



 Nqe

if 1 ≪ N ≪ q.

(3.10)

if 1 ≪ q ≪ N

At high temperatures (N ≪ q), the number of states scales exponentially with particle number as ∼q N , and the combinatoric factor again causes many-body states of energy ∼ N kB T to dominate thepartition function. In contrast, the low-temperature limit (q ≪ N ) shows only polynomial scaling
with particle number ∼ N q , and this gets trumped by the exponential decay of the Boltzmann
factors e−N . At low temperatures, the collection of states with energy ∼ kB T per particle isthus exponentially improbable. The net result is that configurations similar to the many-bodyground state have an increased relative weighting in the partition function and, consequently, thethermodynamics [142].

3.1.2

Experimental Observation in Dilute Alkali Gases

Typical cold-atom experiments operate at low number density (n . 1014 cm−3 ) in order tominimize chemically reactive processes, and this leads to low condensation temperatures on theorder of about 100 nK. With reference to absolute zero, that is about 10 million times colder thandeep space. Reaching such low temperatures is a technical challenge, and it took several decades
to develop the tools necessary to trap and cool atoms just to the sub-mK regime. For a review oflaser cooling and trapping, see Refs. [134, 29, 34].
The big breakthrough of Bose-Einstein condensation came when evaporative cooling wasapplied to laser-cooled Alkali atoms. The general idea is to selectively remove the most energetic
atoms from a sample, thereby lowering the temperature of the remaining atoms. This is the same

32
process that occurs in a bowl of hot soup: the hottest soup molecules escape as steam, and theremaining soup cools as a result. In a liquid, the work function determines which molecules are hot
enough to escape. In a cold-atom system, the depth of the confining potential discriminates hot
particles, and one can cool a system by slowly lowering the trap depth [78]. One can also use an RF
resonance condition, where photons are detuned in such a way that only energetic atoms can makethe Zeeman or hyperfine transition [113]. (This is often called an RF “knife”.) For either method
to work well, it is necessary that the thermalization timescale be short compared to the lifetime ofthe sample [91]. Under proper conditions, Alkali-atom systems can be evaporatively cooled fromhundreds of µK to several nK, well below the threshold for the BEC transition. This technique
led to the observations of Bose-Einstein condensation at JILA [1] and MIT [44] in 1995, and it has
since become the standard method for creating degenerate quantum gases.
The first experimental evidence for reaching the Bose-condensed state in cold-atom systems
was seen in ballistic-expansion measurements. The general protocol is to turn off the trap, wait
several dozen milliseconds, and then take a picture via absorption imaging. If the gas has expanded
freely during the wait time, the fastest atoms travel the furthest, and the image represents a momentum distribution. For a gas of uncondensed atoms, this distribution is fixed by the temperature
T à la Boltzmann according to
2

p
− 2mk

n(p) ∝ eBT,(3.11)

where m is the atom mass. In the zero-temperature limit, however, the momentum distribution
should be determined by the momentum-space wave function of the harmonic-oscillator groundstate, shared mutually by all condensed atoms. For nonzero temperatures below TC , Eq. (3.3)
suggests that the distribution should be a sum of the classical and quantum predictions withthe relative weighting determined by the condensate fraction (see Ref. [110] for a less hand-wavy
analysis). Figure 3.1 shows this transition from a thermal gas to a BEC for a system of 87 Rb atoms
that is evaporatively cooled across TC [37].
Once this transition was observed, it became possible to investigate systematically how inter-

33

Figure 3.1: (Color online) Velocity distribution for a ballistically expanded Bose gas of 87 Rb atoms,
as it is cooled across the BEC transition temperature. From left to right, the temperatures are
400 nK, 200 nK, and 50 nK. Used with permission from Ref. [37].

34
actions alter the condensed state, both in and out of equilibrium. These systems have turned into
a proving ground for condensed matter theories, with Fano-Feshbach resonances opening the door
to study strongly correlated systems. Before we delve into resonant interactions, we first review
the present understanding of weakly interacting BECs. This will set the stage for later chapters.

3.2

Many-body Formalism
In this thesis, we will assume that systems are cold enough that thermal effects are negligible

and the system can be described by a pure quantum state instead of a density matrix [131]. For
a many-body system of identical bosons, this occurs at temperatures well below the condensation
temperature, as suggested by Eqs. (3.6) and (3.8). We will also consider the atoms to be spinpolarized such that we can suppress all indices relating to internal structure. In this regime, one
can completely describe the state of an N -body system by the wave function Ψ(r1 , r2 , . . . , rN , t).
The wave function must satisfy the Schrödinger equation
¸
N ·
N
X
∂Ψ X
~2 2
i
−
Vint (ri − rj )Ψ,=∇ + Vext (rj ) Ψ +
∂t
2m j
j=1

(3.12)

i<jwhere m is the particle mass, Vext is an external potential, and Vint is the interaction potential.
The magnitude squared of this wave function represents the probability density of measuring theparticles at positions {r1 , . . . , rN }. For identical bosons, this wave function must be symmetric
under particle exchange.
It is useful to reformulate the many-body problem in the language of second quantization.
One chief advantage of this formulation is that it allows us to easily represent many-body states in
the abstract “ket” space, as opposed to the position-space representation implied by Eq. (3.12). We
enforce symmetrization by simply stating commutation relations for the operators that compose
the theory.2 This streamlines calculations considerably, as we will see shortly.
2

The formulation for fermions is nearly identical, except that anticommutators are used [117].

35
3.2.1

Second Quantization

To understand second-quantized operators, we must first make sense of the Hilbert space inwhich they operate. Basis elements in this space, called “Fock states”, are defined by occupation
numbers of single-particle states. Consider a complete, orthonormal set of states {φj (r)} for asingle quantum particle.3

The quantum numbers that describe a given Fock state are then theoccupation numbers {nj } (non-negative integers) of these single-particle states. In other words,
the Fock state|{nj }i ≡ |n1 , n2 , . . .i

(3.13)

is defined to be the (symmetrized) state where n1 particles occupy state φ1 (r), n2 particles occupy state φ2 (r), and so forth. These states are defined to be normalized, and they satisfy theorthonormality condition­ ′ ¯®
{nj }¯{nj } = δn1 ,n′1 δn2 ,n′2 . . . .

(3.14)

Fock space is defined to be the span of all Fock states of the form (3.13). It is an enormous Hilbert
space in which any number of particles can occupy any configuration. The Fock space itself, as
defined, is then independent of the choice of {φj (r)}. This is in the same sense that the Hilbert
space for a single particle in free space exists independently of the basis in which we choose torepresent vectors, be they momentum eigenstates or be they harmonic-oscillator wave functions.
Of course, representing a Fock state in the form (3.13) presupposes a specific single-particle basis.
Given the discrete nature of occupation numbers in Fock space, it is useful to define raisingand lowering operators. For a given single-particle basis {φj (r)}, we can define a set of operators{âj } and {â†j } whose operation on a Fock state isâi |n1 , n2 , . . . ni , . . .i =√ni |n1 , n2 , . . . ni − 1, . . .i√â†i |n1 , n2 , . . . ni , . . .i = ni + 1 |n1 , n2 , . . . ni + 1, . . .i

(3.15)

in direct analogy with the operators that are defined in the one-body harmonic-oscillator problem.
3

These states could be momentum eigenstates, harmonic oscillator wave functions, or any other complete basis.

36
Bosonic symmetrization requires that these operators satisfy the commutation relation given by[âj , â†k ] ≡ âj â†k − â†k âj = δj,k.

(3.16)[âj , âk ] = [â†j , â†k ] = 0
These operators can be used to construct number operators for each single-particle mode j vian̂j ≡ â†j âj , and the Fock states given by (3.13) serve as the eigenvectors of these operators witheigenvalue nj . The state for which nj = 0 for all j is called the particle “vacuum” |0i, andEq. (3.15) implies that âj |0i = 0 for all j. It is possible to generate any Fock state from thevacuum by repeatedly acting on it with raising operators:|{nj }i = √³ ´n1 ³ ´n2
1
â†1
â†2
. . . |0i.
n1 !n2 ! . . .

(3.17)The operators â†j and âj are often called “creation” and “annihilation” operators, respectively, as
they appear to create and annihilate particles in state j according to Eq. (3.15).
It is useful to define operators that create and annihilate particles at position r. We can
exploit the completeness of the arbitrary basis {φj (r)} to define the “quantum field operators”
ψ̂(r) =

Xâj φj (r)j(3.18)X †ψ̂ (r) =âj φ∗j (r)†jwhich must satisfy the commutation relationsihψ̂(r), ψ̂ † (r′ ) = δ (3) (r − r′ )
.
i
i h
h
†
† ′
′
ψ̂(r), ψ̂(r ) = ψ̂ (r), ψ̂ (r ) = 0(3.19)

One can use these fields to construct the particle number density operatorn̂(r) = ψ̂ † (r)ψ̂(r).

(3.20)

It is significant that, in this formulation, the position r appears as a parameter instead of as an
operator. The familiar commutation relation from single-particle quantum mechanics [r̂i , p̂j ] =i~δi,j , in which position and momentum are operators in the Hilbert space, is often referred to as

37“first quantization”. The term “second quantization” refers to using operator-valued functions as
quantized fields,4 in the sense of Eqs. (3.18)-(3.19).
In the representation of second quantization, the quantum field operators ψ̂ and ψ̂ † constituteuseful building blocks for important theoretical quantities. For one thing, they provide a naturalexpression of the position-space wave function Ψ(r1 , r2 , . . . , rN , t) in the abstract “ket” space:Z1|Ψ(t)i = √d3 r1 d3 r2 . . . d3 rN Ψ(r1 , r2 , . . . , rN , t)ψ̂ † (r1 )ψ̂ † (r2 ) . . . ψ̂ † (rN )|0i,(3.21)
N!
where the commutation of the ψ̂ † operators according to Eq. (3.19) ensures that that the state issymmetric under particle exchange. The Hamiltonian that governs the dynamics of |Ψ(t)i according
to the Schrödinger equation (3.12) can be written as
·
¸
Z
Z
~2 2
1
3
†
d3 rd3 r′ Vint (r − r′ )ψ̂ † (r)ψ̂ † (r′ )ψ̂(r′ )ψ̂(r). (3.22)
Ĥ = d rψ̂ (r) −∇ + Vext (r) ψ̂(r) +2m
2
It is worth pointing out that this representation of the Hamiltonian does not presuppose any specificparticle number N ; it is valid for all particle numbers and can even be used for describing systems
in which N is not fixed, such as the grand canonical ensemble [117].
It is often convenient to perform many-body calculations assuming a homogenous (untrapped)
sample. This makes it easy to take the thermodynamic limit, where the particle number N and
the volume V approach infinity with the number density n fixed. Momentum is also conservedin the absence of a trap, which simplifies ground state calculations by allowing us to work in thezero-momentum subspace. We therefore expand the quantum field operator ψ̂(r) in the basis ofsingle-particle plane-wave states:
1 X
ψ̂(r) = √âk eik·r ,V k(3.23)

where the operator âk annihilates a particle with momentum ~k. The operators {âk } and {â†k }satisfy the discrete commutation relations given by Eq. (3.16), with momentum indices. In thisbasis, the Hamiltonian (3.22) takes the compact form
Ĥ =

X
k

4

εk â†k âk +1 X
Ṽint (q)â†k+q â†k′ −q âk âk′ ,2V′(3.24)k,k ,q

For rigor, we should call them operator-valued distributions due to the singular nature of their commutators.

38

Figure 3.2: Diagrammatic representation of the Born series for interacting BEC particles. The
dashed lines represent condensed atoms with zero momentum, the wiggly lines represent interactions
via the potential Ṽint , and the straight solid lines represent intermediate scattering states with
arbitrary momentum. Only the first two terms of this series can be generated by the Bogoliubov
Hamiltonian (3.25).

2 2kwhere εk = ~2m
is the kinetic energy of a particle, and Ṽint (q) is the Fourier transform of theinteraction potential.

3.3

Weakly Interacting BECs
It is notoriously difficult to solve the quantum many-body problem in the presence of non-trivial interactions. When these interactions are weak, however, progress can be made. We expect
generally that the ground state of a weakly interacting BEC should still feature a sizeable condensate fraction, for example. This idea leads to the widely used Bogoliubov approximation [10],which treats the problem in a mean-field sense. In this section, we will focus on the case in which
there are N atoms in a homogenous volume V , whose Hamiltonian is given by Eq. (3.24).

3.3.1

The Bogoliubov Approximation

The Bogoliubov theory of interacting BECs begins with the assumption that the condensate
fraction is approximately equal to unity, with minimal uncertainty or fluctuation. Said another way,
we assume that the ground state is dominated by configurations where the number of ground state
atoms N0 is nearly equal to the total number of atoms N . This suggests that, in the relevant partof the Hilbert space, the operators with the largest matrix elements are those that are composed of√√â0 and â†0 , as these operators introduce factors of N0 and N0 + 1 according to Eq. (3.15). We

39
write the Bogoliubov Hamiltonian by discarding terms in the interaction that have fewer than two
factors of either â0 or â†0 :
Ĥ ≈

³
´
X
Ṽint (0) † †1 X
εk â†k âk +â0 â0 â0 â0 +Ṽint (k) 2â†0 â†k âk â0 + â†k â†−k â0 â0 + â†0 â†0 âk â−k
2V
2V
k6=0

k

+

Ṽint (0) XV.

â†0 â†k âk â0

k6=0(3.25)The neglect of terms such as â†q â†−q âk â−k has important consequences for the theory. This Hamiltonian can evidently excite a pair of atoms from the condensate (via â†k â†−k â0 â0 ), but these atoms
cannot scatter multiple times before returning to the condensate (via â†0 â†0 âk â−k ). In terms of
Feynman diagrams, Eq. (3.25) only allows condensate atoms to scatter up to second order in the
Born approximation, which are the first two terms shown in Fig. 3.2. We can expect this Hamiltonian, as written, to reproduce the correct physics only for cases in which the higher-order terms of
the Born series are vanishingly small at low energies.
A good candidate for the interaction is the regularized contact potential given by Eqs. (2.28)
and (2.31). This potential has the formṼint (k) = UΛ Θ(Λ − k) =4π~2 a/m
Θ(Λ − k).
1 − π2 Λa(3.26)

m
In the limit that Λ|a| ≪ 1, the first Born approximation [117] f (1) = − 4π~
2 Ṽint (0) approximately

reproduces the exact low-energy scattering amplitude f = −a; it can be shown that the higherorder terms of the Born series are smaller by successive powers of Λa at low energies. In this regime,
the coupling constant UΛ depends only weakly on the cutoff Λ, and it is convenient to simply write
it as U0 = 4π~2 a/m. It is worth noting that using an arbitrary short-range potential in (3.25),
or even an effective potential (3.26) that violates Λ|a| ≪ 1, leads to incorrect results due to theneglect of higher-order scattering in the sense of Fig. 3.2. At the same time, we must choose therange of the interaction Λ−1 to be small compared to the mean interparticle separation ∼ n−1/3 sothat range effects remain negligible. These considerations constrain the cutoff Λ such thatn1/3 ≪ Λ ≪ |a|−1 .

(3.27)

40
We immediately see that these approximations, neglecting both higher-order scattering and range
effects, can be consistent only for n|a|3 ≪ 1.
Besides neglecting certain terms in the interaction, the Bogoliubov approximation entails√
approximating â0 and â†0 as N0 . This is motivated by the fact that all relevant matrix elementsthat contain these operators introduce essentially that same factor. In this approximation, theBogoliubov Hamiltonian can be written asΛĤBog =Λ´
N 0 U0 X ³ †N02 U0 X
εk â†k âk ++2âk âk + â†k â†−k + âk â−k
2V
2V
k6=0

N 0 U0
+
V

k6=0

Λ
X

,

(3.28)â†k âkk6=0where we have neglected high-energy modes k > Λ that are not expected to be populated in a
weakly interacting BEC.5 The first and last terms can be combined to yield


Λ
Λ
2
X
X
U0  2N 0 U0 N 0 U0+â†k âk =N0 + 2N0â†k âk 
2V
V
2V
k6=0
k6=0
2

Λ
X
U0 
â†k âk 
N0 +
≈
2V

(3.29)

k6=0

≈

N 2U

0

2V

according to number conservation. To this order of approximation, we can now set N0 /V and N/Vto the number density n, and the Bogoliubov Hamiltonian becomes
Λ

Λ

k6=0

k6=0

´
nU0 X ³ †N nU0 Xεk â†k âk ++2âk âk + â†k â†−k + âk â−k .
ĤBog =
2
2
5

(3.30)

It can be shown that these modes are approximately decoupled in the Bogoliubov ground-state calculation,
which is reason enough to ignore them.

41
3.3.2

Solving the Problem

The Hamiltonian (3.30) can be diagonalized by a so-called Bogoliubov transformation.6 The
trick is to define operators according to†âk = uk β̂k − vk β̂−k,(3.31)â†k = uk β̂k† − vk β̂−kwhere uk and vk are undetermined parameters. It can be checked that β̂k and β̂k† satisfy the bosoniccommutation relationshiβ̂k , β̂k† ′ = δkk′hi hiβ̂k , β̂k′ = β̂k† , β̂k† ′ = 0

as long as(3.32)

u2k − vk2 = 1(3.33)

for all k > 0. We can substitute (3.31) into (3.30), and we find that
ΛĤBog =¤
N nU0 X £ 2
vk (εk + nU0 ) − uk vk nU0
+
2
k6=0

+

Λ
X

β̂k† β̂k

k6=0

+

¤
¢
£¡ 2
uk + vk2 (εk + nU0 ) − 2uk vk nU0.

(3.34)Λ´ £¡¢¤1 X³ † †β̂k β̂−k + β̂k β̂−ku2k + vk2 nU0 − 2uk vk (εk + nU0 )2
k6=0

We can choose the parameters uk and vk such that the last line of (3.34) vanishes:rεk + nU0 1
uk =
+
2~ωk
2
r
εk + nU0 1
vk = sgn [U0 ]
−
2~ωk
2

(3.35)

where we have defined~ωk =pεk (εk + 2nU0 ).

(3.36)

After some algebra, the Hamiltonian can be written as
Λ

Λ

k6=0

k6=0

X †
N nU0 1 X
ĤBog =β̂k β̂k ~ωk .
[~ωk − εk − nU0 ] ++2
2
6(3.37)

Although Bogoliubov popularized the idea [10], it appeared in the literature several years earlier in the context
of magnetic systems [82].

42This form is diagonal in the sense that its operator component, the last term, has the form of anumber operator. However, it is not counting particles, but rather a different form of excitation
defined by Eqs. (3.31) and (3.35). These excitations, which are the approximate normal modes ofthe system, are often called “quasiparticles” [116] and their energies are given by ~ωk in Eq. (3.36).
The excitation energies reveal important physics about interacting BECs. For a < 0, theseenergies are imaginary valued in the small-k limit, signifying that the assumed Bose-condensed
state is dynamically unstable. For a > 0, they are real valued and behave asymptotically as

q


2 na

 4π~
~k if kξ ≪ 1
m2
~ωk ≈
,
(3.38)
2ε + 4π~ na if kξ ≫ 1

k

m

√
where we have defined the length scale ξ = 1/ 8π~2 na, called the “healing length”. The linear

dispersion for small k indicates that these modes are phonons with a sound speed given by c =q
4π~2 na. In this case, the sound speed c also serves as the superfluid critical velocity: impurities
m2

traveling slower than c (relative to the BEC) are energetically forbidden from exciting the BEC.

This interesting result follows from energy-momentum conservation, as first discussed by Landauin the context of superfluid helium [103, 104]. It is worth pointing out that the critical velocity
vanishes in the limit that a → 0, and the BEC ceases to be a superfluid. Hence, interactions are
crucial to the phenomenon of superfluidity.

3.3.3

The Ground StateThe ground state of the system |Ψ0 i is defined by the absence of quasiparticle excitations.
This means that it must satisfy β̂k |Ψ0 i = 0 for all k 6= 0. We can use this feature, along with
(3.31), to calculate the momentum distribution of excited particles. We find thatnk = hΨ0 |â†k âk |Ψ0 i†= hΨ0 |(uk β̂k† − vk β̂−k )(uk β̂k − vk β̂−k)|Ψ0 i.
2= vk=

εk + nU0 1
−
2~ωk
2

(3.39)

43
It can be shown that this distribution has a large-k tail that goes as C/k 4 , as anticipated in Sec. 2.3.
We can extract the contact C by noting that"#1
C = lim k × p−
k→∞
2 εk (εk + 2nU0 ) 2


nU0
1
+
1
εk
= lim k 4 ×  q
− 
k→∞
2
0
2 1 + 2 nU
,
εk
" µ
¶2
¶3 #
µ
nU0
1 nU0
+O
= lim k 4 ×
k→∞
4
εk
εk
εk + nU04(3.40)= 16π 2 n2 a2
where we have used the definition of the quasiparticle energy (3.36).
We can sum Eq. (3.39) over k 6= 0 to find the total fraction of excited particles. It is helpful
to convert the sum to an integral via
Λ

Nex
1
1 X
nk ≈
=
N
N
n
k6=0

where we have made the usual replacement

P

Z

d3 k
nk ,
(2π)3

(3.41)

k<Λ

k6=0 → VR

d3 k, and recalled that n = N/V .
(2π)3

This

convergent integral is approximately independent of the cutoff as long as Λξ ≫ 1, which is consistent
with the restrictions in Eq. (3.27) whenever na3 ≪ 1. We are therefore justified in extending the
integration limits to infinity, and the excited fraction evaluates to
r
Nex
8 na3
≈
.
N
3
π

(3.42)

This verifies (after the fact, of course) that the BEC fraction is approximately unity in the limitthat na3 ≪ 1. In contrast to the thermal depletion considered in Eq. (3.6), this depletion occurs
purely as a result of interactions. It is often referred to as the “quantum depletion”.
The ground state energy can almost be read off as the c-number contribution to the Bogoliubov Hamiltonian (3.37), which isΛ

N nU0 1 X+[~ωk − εk − nU0 ] .
E0 →
2
2

(3.43)

k6=0This quantity, as written, grows linearly with Λ for Λξ ≫ 1. This conundrum is rooted in the factthat we have ignored the weak cutoff dependence of the coupling constant UΛ . If we restore this44
cutoff dependence to (3.43), then the resulting energy is approximately cutoff independent. We candemonstrate this by expanding UΛ to lowest order in Λa ≪ 1:¸·
4π~2 a2
2UΛ =1 + Λa + O (Λa)
m
π


Z
3
d k 1
= U0 1 + 4πa+ O (Λa)2 ,(2π)3 k 2
k<Λ


Λ
X
1
U0
+ O (Λa)2 = U0  1 +V
2εk(3.44)

k6=0

where we have exchanged an integral for a sum. Substituting this cutoff-dependent expression into
Eq. (3.43) yields
E0 = N nU¸Λ ·2U 2 
Xn1
00~ωk − εk − nU0 +[1 + O (Λa)]
+
2
2
2εk 

(3.45)

k6=0

after some algebra. This quantity is cutoff independent for ξ −1 ≪ Λ ≪ a−1 , where it evaluates toÃ!
r
2π~2 na
128 na3
E0 /N ≈
1+.
(3.46)
m15
π
To lowest order, the energy per particle scales linearly with both the scattering length and thedensity, and it relates to the contact (3.40) in the way predicted by the Tan relations (2.33)-(2.35)
if one ignores higher-order corrections.7

The small parameter na3 , familiar from the depletion(3.42), appears in the next-order correction to the ground state energy. The expressions (3.42) and(3.46) were first derived in a seminal paper by Lee, Huang, and Yang [106].
Although our approach used a specific potential model, the results are valid more generally
for weakly-interacting, dilute BECs satisfying na3 ≪ 1. Such can be proved rigorously using
the diagrammatic methods of quantum field theory, as done by Beliaev in Refs. [7, 8]. Using the
Bogoliubov method to treat the general case, however, requires altering the Bogoliubov Hamiltonian(3.25) at the first stage of the calculation. One must retroactively account for the neglected termsof the Born series in Fig. (3.2) by replacing the true potential Ṽint (k) in (3.25) with the T-matrix,
which is defined as the effective potential for which the first Born approximation gives the exact
7

The exact Tan relations are thus satisfied approximately by our inexact theory.

45
scattering behavior [21, 132]. In the language of quantum field theory, this is often referred to as“summing ladder diagrams”. It is worth noting that both the Bogoliubov and Beliaev approaches
are ill suited to describing physics occurring uniquely at momentum scales larger than (or equalto) a−1 . This becomes a problem either when a is comparable to or larger than the length scales
set by many-body physics (such as the particle separation ∼ n−1/3 ), or in nonequilibrium scenarios
where the bound state may play a role in the dynamics. These situations will be discussed in thenext two chapters.

Chapter 4

BEC Quenched to Unitarity

Many of the results presented in this chapter were published in Ref. [150].

4.1

Motivation
As shown in the previous chapter, the usual theoretical treatment of Bose-Einstein con-densation is rooted in a perturbative inclusion of interaction effects. The starting assumptionof Bogoliubov theory is that the condensate fraction remains close to unity despite interactions,and Eq. (3.42) indicates that this requirement is self-consistently fulfilled only in the limit thatna3 ≪ 1. This condition is met in most BEC experiments, and thus Bogoliubov theory (with its
Gross-Pitaevskii extension to confined systems [74, 75, 136]) has been enormously successful inquantitatively describing observations [162, 130, 81, 115, 126, 174]. In stark contrast, the stronglyinteracting regime na3 ≫ 1 has remained theoretically elusive for decades, and the prospects
for achieving it in a BEC experiment were bleak due to the a4 scaling of three-body loss rates[63, 127, 167]. Even with the advent of magnetic Fano-Feshbach resonances in Alkali-atom experiments, whereby the scattering length a could be freely tuned, it seemed unlikely that one could
create an equilibrium strongly-interacting BEC that would survive long enough for interesting
physics to be observed.
It was soon realized that the strongly-interacting regime could be probed in an out-ofequilibrium scenario as long as the experimental timescale was short compared to the (alreadyshort) lifetime of the gas. One simple protocol for such an experiment is to rapidly change the47scattering length from an initial value ai to a final value af by ramping the magnetic field neara Fano-Feshbach resonance, and then observe the response of the system. This sequence is often
called an interaction “quench”. One early experiment [32] considered the response of a weaklyinteracting BEC (na3i ≪ 1) of 85 Rb atoms that was quenched to na3f . 1 over the course of 12.5 µs,
and it was observed that the condensate exhibited nontrivial decay dynamics over the course ofabout 100 µs that could not be explained simply by three-body loss. After more than a decade,these measurements still have not been explained quantitatively by theory [145].
More recently, experimenters boldly quenched a weakly-interacting 85 Rb BEC directly to theunitarity regime, where af → ∞ [112]. In this case, the dynamical momentum distribution was measured by ballistically expanding the gas after various hold times at unitarity. It was observed, rather
unexpectedly, that the populations of large-momentum modes saturated on timescales (. 100 µs)
that were much shorter than the lifetime of the gas (∼ 1 ms). This saturation suggested that thegas had reached a kind of metastable quasi-equilibrated state, despite the resonant interactions.
Also significant was the fact that, when they repeated their measurements on a gas with a different
density, there were indications of universality. That is, the saturation and loss dynamics for systems of different density were approximately identical as long as momenta and time were rescaledby n1/3 and n−2/3 , respectively. These scalings can be derived trivially from dimensional analysis
under the assumption that the particle spacing ∼ n−1/3 is the only physically relevant length scale
for the system. This assumption makes some sense, given that the scattering length and thermal
deBroglie wavelength are infinite and cannot be used for dimensional analysis. On the other hand,
three-body Efimov physics [54, 55] can introduce another length scale to the problem, namely, thesize of a three-body bound state. Discovering the role of this effect in the quenched BEC system,and how it might disrupt universality, is an area of active research. At the same time, it is useful
to first consider the dynamical consequences of two-body physics in the quench to unitarity.

48

4.2

Many-Body ModelIn our quest to understand the dynamics of a BEC quenched to unitarity, we consider asimplified system that is spatially uniform with density n = N/V and whose atoms are initiallynoninteracting (ai = 0). We adopt a time-dependent variational approach to solving the problem.
Our variational wave function |Ψvar (t)i is constructed to include the physics that should be mostimportant at short times after the quench. For example, we expect that the jump to resonancedoes not instantaneously deplete the condensate, as the many-body wave function must evolve
continuously from its initial condensed state. We also expect that the dominant process immediately
following the quench should be that multiple pairs of condensate atoms scatter off each other withequal and opposite momentum, as required by momentum conservation. To incorporate these two
notions, we use a time-dependent generalization of the ansatz used in Refs. [146, 67, 128]:




X
1
|Ψvar (t)i = A(t)exp c0 (t)â†0 +gk (t)â†k â†−k |0i,


2

(4.1)

k6=0

where c0 (t) and {gk (t)} are time-dependent variational parameters and




h
i
X
1
ln 1 − |gk (t)|2A(t) = exp − |c0 (t)|2 /2 +


4

(4.2)

k6=0

is a normalization constant. The first term in the exponential in (4.1) represents the condensate
as a coherent state [68]. The relation â0 |Ψvar (t)i = c0 (t)|Ψvar (t)i essentially replaces the operatorâ0 by c0 for all matrix elements in the restricted Hilbert space, in analogy with the Bogoliubov
prescription described above Eq. (3.28). It therefore follows that the condensate population is
given by N0 (t) = |c0 (t)|2 . The term involving gk , assumed to have reflection symmetry gk =g−k , generates correlated pairs of excited atoms. One can show that the dynamical momentum√|gk (t)|2
distribution is nk (t) = 1−|g.
As
initialconditions
for
these
parameters,
we
set
c
(0)
=
N and
0
2
k (t)|
gk (0) = 0.
The Hamiltonian that governs the post-quench (t > 0) dynamics of the system is given by

49
Eq. (3.24), which readsĤ =X

εk â†k âk +k1 XṼint (q)â†k+q â†k′ −q âk âk′ .
2V
′

(4.3)

k,k ,qThe interaction potential Ṽint (k) must be chosen to yield resonant scattering (a → ±∞). As
discussed in Section 2.2, this can be accomplished by using an attractive short-range potential
whose depth is chosen so that there is a two-body bound state at zero energy. It is expected thatthe dynamics, if computed correctly, should not depend on the particular model as long as thepotential is short-ranged and resonant.1

This idea will serve as a useful check of our solution

method.
We derive the equations of motion for the parameters c0 (t) and {gk (t)} by minimizing the action for a suitably-defined Lagrangian that “knows” about quantum mechanics. Such a Lagrangian
can be written as
L=

¯
¯
E´
E D
i~ ³D
¯
¯
Ψ(t) ¯Ψ̇(t) − Ψ̇(t)¯ Ψ(t) − hΨ(t)| Ĥ |Ψ(t)i ,
2

(4.4)

where the dot denotes a time derivative. It can be checked that, for an unconstrained state |Ψi, theδL= 0 leads to the time-dependent Schrödinger equation
Euler-Lagrange equation of motion δδL− δΨΨ̇[101]. In our case, the quantum state of the system is constrained to have the form (4.1), so it does
not strictly satisfy the Schrödinger equation except in the least-action sense.2 This should not be
a problem at short times where the ansatz, by design, includes the dominant physics of the system.
In deriving the equations of motion, we evidently need to know the expectation value of Ĥ
1
Otherwise, we’d be stuck using the honest-to-goodness 85 Rb potentials, and even then, the result would only be
relevant to a single atomic species.
2
This is similar to how static variational calculations usually lead only to approximate solutions to the timeindependent Schrödinger equation. In the dynamical case, we find the optimal solution by minimizing the action,
instead of hĤi.

50
in the variational state |Ψvar i. With a little bit of effort, one finds that [146]hΨvar |Ĥ|Ψvar i =i |g |2|gk |2Ṽint (0)
1 Xh
k
4
Ṽ
(k)
+
Ṽ
(0)
+
|c
|
+
|c0 |2int
int
0
1 − |gk |2
2VV
1 − |gk |2
k6=0
k6=0
µ
¶µ
¶
∗
∗2
2
gk∗
c0 gk + c0 gk
gq
1 X
1 X
+
Ṽint (k)Ṽint (k − q)+.
2V
1 − |gk |2
2V
1 − |gk |2
1 − |gq |2k6=0
k,q6=0i µ |g |2 ¶ µ |g |2 ¶
1 X h
q
k
Ṽint (k − q) + Ṽint (0)+2V
1 − |gk |2
1 − |gq |2X

εkk,q6=0(4.5)

Similarly, one can calculate that
D

¯
¯
E
E D
1 X gk∗ ġk − ġk∗ gk¯¯.
Ψvar ¯Ψ̇var − Ψ̇var ¯ Ψvar = c∗0 ċ0 − ċ∗0 c0 +2
1 − |gk |2

(4.6)

k6=0

These ingredients allow us to find the Euler-Lagrange equations of motion
d
dt

µ

∂L
∂ α̇

¶

−

∂L
= 0,
∂α

(4.7)

where α is any variational parameter. When the smoke clears, we havei~ċ0 =

∂hĤi
∂c∗0

c∗0 gk + c0 |gk |2
1 X
Ṽint (k)= nṼint (0)c0 +V
1 − |gk |2

(4.8)

k6=0

and
³
´2 ∂hĤii~ġk = 1 − |gk |2
∂gk∗
³
´
i
Ṽint (k) h 2
22= 2 εk + nṼint (0) gk +c0 + c∗2
g
+
2
|c
|
g
0
k .
0 k
V
2 |gq |2 gk + gq + gq∗ gk2
1 X
Ṽint (k − q)+V
1 − |gq |2

(4.9)

q6=0

In writing these equations of motion, we have used the fact that they conserve mean total particle³´
P|gk |2→ n. We solve these coupled ODEs
number3 to make the replacement V1 |c0 |2 + k6=0 1−|g2k|numerically, assuming that gk (t) depends only on the magnitude of the momentum k.

It is interesting to note that our equations of motion (derived from an ansatz for the quantum
state) map directly onto the time-dependent Hartree-Fock-Bogoliubov (HFB) formulation. In that3

Proving this requires messy algebra.

51case, one expands the Heisenberg-picture field operator as´
1 X ik·r ³euk (t)b̂k + vk∗ (t)b̂†−k ,ψ̂(r, t) ≈ Φ0 (t) + √V k6=0(4.10)

where Φ0 is the condensate component and the uk (t) and vk (t) are quasiparticle amplitudes. Aftermaking certain mean-field approximations [72], one can write coupled equations of motion forΦ0 (t) and {uk (t), vk (t)}. It is then straightforward to show that our parameter gk (t) and the HFBquantity vk∗ (t)/u∗k (t) satisfy exactly the same equations of motion. This equivalence between our
variational calculation and the HFB formalism was suggested recently in Ref. [138]. Our variational
treatment thus suffers from the same spurious low-energy gap as found in HFB; however, because
this unphysical gap should manifest itself at longer time scales, this should not hinder our study ofshort-time behavior. This is a motivating reason why HFB was able to adequately simulate [99] thecoherent atom-molecule oscillations observed a decade ago [48], the main results of which can be
reproduced by our single-channel variational model. Our formalism also reduces to time-dependent
Bogoliubov theory [125] once the usual approximations are made: replace the operators â0 and â†0√by the constant N0 , neglect quartic interaction terms of noncondensed particles in Eq. (4.3), anduse a coupling constant UΛ → U0 = 4π~2 af /m as in Section 3.3.

4.3

Post-Quench Dynamics
We have solved the equations of motion (4.8)-(4.9) for densities and interaction ranges thatare typical for Alkali-atom experiments. In particular, our simulations assumed a density of n =5 × 1012 cm−3 , which approximately matches the average density probed in the JILA experiment[112]. One can use the density to define useful momentum and energy units as kF = (6π 2 n)1/3 ≈1/(2800a0 ) and ~ωF =

2
~2 kF
2m

≈ ~/(61 µs), respectively. To test the robustness of our theory, we

have employed three different models of resonant interactions: a square well, a Gaussian well, and

52
0

0

0

100

200

300

0

0

100

200

300

0

50

100

150

Figure 4.1: Three potential models for resonant interactions, as in Eq. (4.11) . From left to right,
we have the square well, Gaussian well, and the contact model. In each case, the depth of the wellis chosen so that there is a two-body bound state with vanishing binding energy, depicted in red.

the contact model given by Eqs. (2.28) and (2.31). Their potentials are respectively given byVint (r) = −³ π ´22

Θ(r0 − r)22Vint (r) = −2.68e−r /r0

~2
mr02

~2
mr02

Ṽint (k) = −2π 2 Θ(Λ − k),(4.11)~2mΛ

where the depths have been chosen so that there is a two-body bound state of vanishing binding
energy (a → ±∞). We fixed the range to be r0 = 100a0 for the square and Gaussian wells, andwe chose the momentum cutoff of the contact model to be Λ = 100πkF ≈ 1/(9a0 ) to get a better
sense for the zero-range limit. These potentials are depicted in Fig. 4.1.
Our numerical solution of Eqs. (4.8)-(4.9) was performed with a 4th order Runge-Kutta
P
scheme in MATLAB [143]. We replaced sums with integrals using the usual relation V1 k6=0 →R d3 k. For the Gaussian-well calculations, we discretized momentum into 4095 evenly-spaced grid(2π)3points up to a maximum of 5π/r0 ; for the square-well calculations, we used 16383 grid points

up to the same maximum; for the contact model, we used 16383 grid points up to the cutoff Λ.
The chosen grid sizes all have the form 2m − 1 for some integer m, as this optimizes the speed of
discrete sine transforms in MATLAB. These sine transforms are helpful for spectrally evaluating
the convolution integral in Eq. (4.9) via the convolution theorem [2]. It is worth pointing out thatour discretizations are sufficient to resolve physics on both length scales appearing in the equations

53

0

10

-5

10

10

10

10 -10
0

10

1

10

2

10 0

10

-5

10 -10

10

10

0

10

1

10

2

10

0

10

1

10

2

10 0

-5

10

-10

10

10

0

10

0

10

1

10

2

-5

-10

Figure 4.2: Dynamical momentum distribution of 85 Rb atoms at several times after quenching fromnoninteracting to unitarity, given a density of n = 5 × 1012 cm−3 . The red circles, blue squares, andsolid green lines correspond respectively to the square well, Gaussian well, and contact interaction
models, as in Eq. (4.11). In the first frame, the dashed line denotes generic k −4 scaling, forcomparison.

54
of motion: the interaction range and the particle spacing.
The dynamical momentum distributions are plotted in Fig. 4.2 at several instants in timeafter the quench to unitarity. We can immediately observe that the three interaction models in
Eq. (4.11) lead to the same dynamics over the range of momenta satisfying k ≪ r0−1 . At larger k,range effects predictably seep into the dynamics, as expected from the effective range expansion(2.3); nevertheless, these modes remain relatively unpopulated and thus contribute very little to theoverall momentum distribution. The agreement between models shows that our solution method
is robust against the arbitrariness of how we treat the short-range, resonant interactions. All
calculations indicate that the system rapidly (t & mr02 /~) develops a k −4 tail in the large-k limit thatfirst appears at large momenta and progates into smaller momenta.4 We also observe subleading
oscillatory structure. For the times shown, the low-momentum modes grow in approximate unison.
We reran our square-well simulations at the reduced density n = 2 × 1012 cm−3 to see
if our results displayed universality, as suggested in the experimental study [112]. Figure 4.3
compares these results against those of the previous density. The left-hand plot shows the dynamical
distributions at the same instant in time and as a function of momentum in SI units; the results
are quantitatively different. The right-hand plot shows the distributions after rescaling momenta
and time by kF and ωF respectively, which are different for the two densities. (Note that we are
now plotting the two sets of numerical data at different instants in time.) With this rescaling, thedistributions agree quantitatively. We therefore infer that the dynamics of our variational model
are universal. There is a sense in which this universality is mathematically equivalent to the model
independence shown already in Fig. 4.2. If one writes the equations of motion in these densityderived units, then the only intrinsic parameter of the problem is kF r0 . Thus, the two calculations
discussed in this paragraph would appear to be identical except that they use resonant square wellsof “different” range, due to their different values of kF r0 . As evident in Fig. 4.2, the resulting
dynamics should not depend on these details as long as this range remains small.
4

In analogy with Tan’s results (see Section 2.3), our invoking the “large-k limit” will always entail the stipulation
kr0 ≪ 1.

55

10 0

10 0

10 -5

10 -5

10 -10

10 -10
10 1

10 2

10 0

10 1

10 2

Figure 4.3: Dynamical momentum distribution of 85 Rb atoms after quenching from noninteracting
to unitarity, for two different densities. The blue lines correspond to the density n = 5 × 1012 cm−3 ,and the red lines correspond to the density n = 2 × 1012 cm−3 . Both distributions are calculatedwith the same attractive square well potential of range r0 = 100a0 . On the left, we show these
distributions as a function of momentum in fixed SI units at the same instant in time. On theright, we plot the computed distributions with time and momentum rescaled by the density unitsωF−1 and kF , respectively.

56
×10

-6

10

20

×10

1

2

0.5

1

0

-7

20

40

0
0

0.05

0.1

0

0.05

0.1

Figure 4.4: Population dynamics of a single mode immediately after the quench. On the left, weshow the dynamics for the mode k = 10kF . On the right, we show the dynamics for the modek = 15kF over the same time interval. The bottom horizontal axis shows the time scaled by ωF ,and the top shows the time scaled by 2εk /~.

We now look more carefully at the oscillatory structure of the momentum distribution. Theseoscillations can be viewed more transparently in the population dynamics of a single mode. Figure 4.4 shows the time-dependent population of modes k = 10kF (left) and k = 15kF (right)
shortly after the quench. Comparing the two populations, we find that the mode with larger koscillates with a larger frequency and with a reduced amplitude relative to its (linearly-growing)
offset. These oscillations occur with approximate frequency 2εk /~ in our simulations, as can beseen if one rescales time by this quantity (upper horizontal axis). Analyzing our numerics, we havefound that the amplitude of these oscillations decays as k −5 in the large-k limit at all times after
the quench. Similar oscillations have been observed in time-dependent Bogoliubov calculations forquenched systems [125, 85, 175], and Chapter 6 will demonstrate that they are the signature ofballistic correlation waves. We expect on physical grounds that quasiparticle collisions (which aremostly absent from our ansatz) will lead to damping, although it is difficult to model this quantitatively without introducing a certain amount of arbitrariness to the theory [137, 138, 92, 93, 94].
The general lack of oscillations in the JILA experiment [112] suggests that damping may play animportant role in the late-time dynamics of the unitary Bose gas. In contrast, damping seemed lesssignificant for the weakly-quenched BEC system described in Ref. [85].

57

1

0.5

0
0

5

10

Figure 4.5: Depletion dynamics after the quench to unitarity, calculated using the square wellmodel described in Eq. (4.11).

We calculate the total fraction of excited particles to get a sense for the speed at which
resonant interactions deplete the BEC. This quantity is given bynex (t)n0 (t)1 X
nk (t) = 1 −
=
,
n
nV
n

(4.12)

k6=0

and we plot it in Fig. 4.5. Our numerics suggest that, immediately after the quench, the depletion
fraction grows as ∼ 0.6 (ωF t)3/2 . This should be contrasted with the predictions of time-dependent√
Bogoliubov theory, which suggest a scaling of t after an interaction quench [125, 175, 89]. In
our case, the depletion remains small shortly after the quench, and the heuristics motivating ouransatz (4.1) are self-consistently satisfied. This is no longer the case at late times, however. Ourmodel predicts that all of the atoms are eventually expelled from the condensate. Hence, themain assumptions of our theory are only satisfied at relatively short times. It is worth mentioning,
though, that the Hartree-Fock-Bogoliubov calculations of Ref. [99] gave a good description of atommolecule coherence measurements [48] even when the depletion (4.12) was greater than 50%. This
fact inspires hope that our theory might still capture relevant short-range physics, such as thecontact dynamics, at intermediate times after the quench.
The dynamical contact can be computed by taking the large-k limit of the momentum distribution, similar to Eq. (3.40). As a practical matter, this limit is least ambiguous for the contact
model with large Λ due to the relative lack of range effects in that case. The results are shown in

58
15

10

5

0
0

1

2
w t

3

4

Figure 4.6: Contact dynamics after the quench to unitarity.

Fig. 4.6. As implied by the large-k dynamics in Fig. 4.4, the initial contact growth is linear:C(t) ≈ 26.9 n4/3 ωF t,(4.13)

where the leading factor comes from a numerical fit. This linear growth is already qualitatively
different from what one finds in time-dependent Bogoliubov theory, where the contact is mistakenly
predicted to change discontinuously after a quench [125].5

We can extract the finite saturation

timescale τC by fitting our numerical data to an exponential function ∆C(1 − e−t/τC ), exactly asin the experiment [112]. We find that the contact (and therefore all large-k modes) saturates in atime given by τC ≈ 0.4ωF−1 , which is consistent with the measured equilibration time of the largest
momentum modes reported in Figure 5 of Ref. [112]. For k > 2kF , the observed timescales arescattered within the range τ ∈ [0.1, 0.5]ωF−1 .
At long times, we see that the contact saturates to a value of approximately 12n4/3 . To
give some context, predictions for C in the unitarity ground state can be found in Refs. [45, 149,
161, 36, 139], and they range from 9 − 32n4/3 . The measured momentum distribution in the
JILA experiment did not have enough signal-to-noise at large k to distinguish the contact. We
should note, though, that our prediction is consistent with the measured late-time data shown in
the inset of Figure 3 in Ref. [112]. Our model predicts that their plotted quantity, denoted as
κ4 n(κ),6 should asymptote to 12/(6π 2 )1/3 ≈ 3. This is consistent with the scatter in their data
5
The failure of time-dependent Bogoliubov theory to describe contact dynamics will be discussed in more detailin the next chapter.
R d3 κ
6
They define κ ≡ k/kF and normalize n(κ) such that (2π)
3 n(κ) = 1.

59
at large momentum, which ranges from about 2 to 5 for k > 2.5kF ; unfortunately, there is notenough dynamic range to identify a clear asymptote. The authors of Ref. [144] separately analyzed
the JILA data while making nontrivial assumptions about three-body universality violations,7and they suggested that the late-time contact should be approximately 22n4/3 . Perhaps a futureversion of the experiment will measure the contact directly. There are empirical and theoreticalindications that the large-momentum modes saturate long before three-body loss kicks in, and one
could perform RF spectroscopy [174] after this saturation occurs. Given the large separation insaturation and loss timescales, resolving the late-time contact should be possible even with relatively
long RF pulses.

4.4

Summary and Discussion
Inspired by recent experimental results [112], we have used a time-dependent variational

approach to explore the dynamics of a BEC that is quenched from noninteracting to unitarity.
Our many-body model is based on the idea that the BEC depletes continuously after the quench,with pairs of condensate atoms scattering off each other with equal and opposite momentum. The
calculation encodes two-body correlations while ignoring three-body effects, which are difficult toinclude in a many-body model. We have demonstrated that our approach is robust in the sense thatthe computed dynamics do not depend on the details of how we model the short-range, resonantinteractions. We found that the dynamics scale universally with the appropriate density-dependent
units, a result that seemed present in the published measurements but is still under experimental
investigation. It is expected that any deviations from universality should be due to Efimov physics,
which would be interesting to see in a nonequilibrium scenario.
We focused our attention on the evolution of the momentum distribution after the quench.
We observed the rapid growth of a k −4 tail whose saturation time and value are consistent withthe large-momentum measurements of Ref. [112]; however, further experimental work is necessary
7

These assumptions include the idea that the momentum distribution violates universality, while the two-bodyand three-body contacts do not. It is also assumed that the momentum range k ∈ [1.5, 3]kF is within the asymptotic
regime where populations are governed by short-range correlations via the contact.

60
to identify the contact of the unitary Bose gas. In addition to the k −4 tail, we also saw subleading
oscillations with frequency ∼ k 2 and amplitude ∼ k −5 for large k. The total fraction of excited
particles was found to scale as t3/2 shortly after the quench, indicating that the BEC survives longer
than Bogoliubov theory would suggest [175, 89].
About a year after we published this work [150], another theoretical development came alongin the form of the Hyperbolic Bloch equations [92, 93, 94]. This new formulation is inspired by ideas
from semiconductor physics [109], and it introduces phenomenological damping and dephasing tomean-field-like equations that are similar to the Hartree-Fock Bogoliubov framework. Like ourvariational model, this new formulation of the problem manifestly includes two-body correlations
while ignoring Efimov physics. A major result of that study is that there exists an optimal choiceof phenomenological parameters for which the theoretical momentum distribution agrees quite wellwith the experimental measurement, even at late times and small momentum. We have checked
that, given the proposed free parameters, the model in Ref. [93] yields essentially the same shorttime predictions for the contact and depletion growth as our variational model.8

In the next

chapter, where we focus only on short-time quench dynamics at arbitrary scattering length, we
employ our variational model.

8
The late-time predictions of the newer model depend quantitatively on the choice of phenomenological damping
and dephasing parameters.

Chapter 5

Bound-State Signatures in Quenched BECs

Results presented in this chapter were published in Ref. [38].

5.1

Preliminaries
Rapid interaction quenches represent a possible pathway to creating exotic quantum states.

This much is clear from the unitary BEC experiment [112] discussed in the previous chapter,
which was able to explore the strongly-interacting regime that had been so elusive to both theory
and experiment. The surprisingly-long lifetime of the gas in that setup has been attributed tothe projective nature of the quench: The initial condition projects mainly onto long-lived states,
thereby limiting inelastic loss [150]. Similar ideas play out in a 1D gas of hard-core bosons (a socalled “Tonks-Girardeau gas”) that is suddenly quenched to strong attraction. The resulting state,
known as the “super-Tonks-Girardeau gas” [3], has a long lifetime despite the strong attractive
interactions. This attraction would normally lead to rapid decay into a clustered state; however,
the initial state is anti-clustered and does not project (much) onto eigenstates that are clustered,so minimal decay occurs. Experiments with quasi-1D BECs, in which radial motion is frozen out
by tight confinement, have confirmed this prediction [76].
Interaction-quenched BECs have proven to be fascinating even outside of the the stronglyinteracting regime. As mentioned in Section 3.3, homogeneous condensates are dynamically unstable for a < 0. In a trap, however, the nonzero kinetic energy can stabilize the system against
collapse for small, negative a. In one experiment [49], a weakly-interacting BEC (ai > 0) was62
suddenly quenched across the stability threshhold to a point where the gas was expected to beunstable. Violent collapse dynamics ensued, including an implosion followed by an explosion.1
The sequence was artfully termed “bosenovae”. In another study [48], a multi-quench Ramsey sequence was used to put condensed atoms into a superposition of scattering and bound states. Suchexotic states were found to occur in the dynamics even when the BEC was in the weakly-interacting
regime, where na3 ≪ 1.
These exciting experimental developments have inspired a number of theoretical studies on
the topic of nonequilibrium BECs near Fano-Feshbach resonances. Much of the early work a decadeago used two-channel resonance models embedded in a mean-field-like theory [159, 80, 99, 111, 97,
96, 120, 50, 51, 52, 70, 98, 145]. These models had sufficient flexibility to account for species-specificeffective-range corrections to scattering and binding, which can be important either by narrow
resonances or at large detunings away from a broad resonance. A common theme throughout theseinvestigations was that bound-state physics can be important to the nonequilibrium dynamics
of BECs, a fact that was underscored by the atom-molecule-coherence experiment described inRef. [48]. This idea has been underappreciated in recent years, with many theories of quench
phenomena being based on the time-dependent Bogoliubov (or Popov) formulation [25, 125, 175,
176, 89, 138], which lacks a bound state. These theories can be accurate only in the regime where
quenches are adiabatic with respect to the bound state. Otherwise, pairs of atoms can project
nontrivially onto the bound state of the new Hamiltonian.
In this chapter, we examine the effect of the bound state on the short-time, short-range
correlations of a BEC that is quenched suddenly from one scattering length to another. Our
focus is the dynamical response of Tan’s contact to the quench, and we find that it exhibits highcontrast oscillations occuring at a frequency ωB = −EB /~, where EB < 0 is the binding energyof the Feshbach-molecular bound state. We first approach this phenomenon using the many-bodyvariational model described in the previous chapter. The contact oscillations are shown to be
1

The explosion was on an atomic scale, having only a few thousand atoms reaching “hot” temperatures of a few
hundred nK.

63significant for realistic quench times near the 85 Rb Fano-Feshbach resonance at 155 G. We thenpresent exactly solvable two-body models that paint a simplified picture of the contact dynamics.
We introduce an intuitive calibration scheme that unambiguously links these models to many-body
physics; our prescription unifies the few-body predictions across a broad class of exactly-solvablemodels, while yielding analytic formulas that agree with the less-transparent, many-body numerics.
Strikingly, we find that the dynamical correlations are larger than Bogoliubov theory would suggest,
and a diabatic reduction of the scattering length is shown to increase pair correlations. Thisfeature is surprising, considering that a smaller scattering length is often associated with “weaker”
interactions. Our two-body models illustrate that these exciting results are a direct consequence ofbound-state physics.

5.2

Many-body Phenomenon
We consider a homogeneous BEC of density n = 1012 cm−3 that is quenched from noninter-acting (ai = 0) to af = 700a0 . These values are reasonable for a hypothetical 85 Rb experiment.
Within the zero-range approximation, this final scattering length corresponds to a binding energy of2~~ωB = ma
2 ≈ ~/(2 µs). The dynamics are computed using the time-dependent variational formuf

lation presented in Section 4.2, with an interaction potential chosen to yield the desired scattering
lengths. We showed in the previous chapter that the dynamics for modes kr0 ≪ 1 are independentof the choice of potential model;2 hence, without loss of generality, we use the contact potentialof Eqs. (2.28) and (2.31) as this gives us better access to the physics at large k. We account for
the finite ramp rate of experimental quenches by letting the scattering length of our model betime dependent. We model these ramps using the scattering length profile of 85 Rb atoms near the
resonance at B0 = 155.04 G:µ
a(t) = abg 1 −2∆
B(t) − B0

¶

,

(5.1)

At finite scattering length, we find that this independence requires a ≫ r0 or Λa ≫ 1, where range effects do
not change the binding energy appreciably.

64
where the resonance width is ∆ = 10.7 G and the background scattering length is abg = −443a0[33]. We assume linear ramps in the magnetic field. This results in a nonlinear evolution of a(t) andUΛ (t) during the ramp. The equations of motion (4.8)-(4.9) are solved numerically with the same
method as described in Section 4.3, except that we use a momentum grid of 8191 evenly-spaced
points up to a cutoff of Λ = 200πkF ≈ 1/(8a0 ). At all instants in time, we find that there is awell-defined k −4 tail, which we use to extract the dynamical contact.
Figure 5.1 shows the time evolution of Tan’s contact for several different quench speeds. We
see that the contact oscillates at approximately the frequency of the bound state, ωB , and thecontrast of these oscillations is strikingly large even when we account for the finite ramp rate ofḂexp = 1.6 G/µs reported in the JILA experiment [112]. The nature of the interference leading
to these oscillations will become apparent in the careful two-body calculation of Section 5.3. We
stress that these dynamics are quite different from those predicted by time-dependent Bogoliubov
theory.
For the case of an instantaneous quench, it is difficult to define a contact in Bogoliubov
theory because the momentum distribution does not have a well-defined k −4 tail [125]. However,such a tail exists as long as the ramp time tR is nonzero, and it occurs at momenta such that~k 2 tR /m ≫ 1. Large-momentum quasiparticles adiabatically follow the scattering length in thiscase, and the contact thus saturates quickly to a new equilibrium value [see Eq. (3.40)]
C0 = 16π 2 n2 a2f(5.2)

over the arbitrarily-small time scale of the quench, regardless of the initial scattering length. These
trivial dynamics are plotted as the horizontal red line in Fig. 5.1, and they are in stark contraast to
the strong oscillatory behavior predicted by our variational theory. The peak-to-trough oscillation
amplitude remains as large as C0 itself when the experimental ramp rate is decreased by a factor of
10. Further reducing the ramp rate eventually results in a quench that is adiabatic with respect tothe bound state, in which case the variational and time-dependent Bogoliubov theories agree andgive a nonoscillating contact.

65

10

C(t)/C 0
C(t)/C 0
5

0

0

10

ωBBtt

20

Figure 5.1: Contact dynamics following a quench from noninteracting to 700a0 , for several ramp
speeds near the 85 Rb Fano-Feshbach resonance at 155.04 G. We assume a density of 1012 cm−3 .
As a reference, the red (horizontal) line represents the prediction from Bogoliubov theory, C0 =16π 2 n2 a2f , which specifies the units of the plot. The blue (oscillating, solid) line is the many-bodyvariational prediction for a quench that is completely diabatic, the cyan (dashed) line is for theexperimental ramp speed of Ḃexp = 1.6 G/µs [112], the green (dot-dashed) line is for a ramp
speed of Ḃexp /10, and the black (dotted) line is for a ramp speed of Ḃexp /50. In each case, thetime t = 0 defines the end of the magnetic field ramp.

66

0.01

(t)/n
nnex
n
ex(t)/
0.005

0
0

25

ω t 50

B tB

75

Figure 5.2: Excitation fraction following a diabatic quench from noninteracting to 700a0 , for a
BEC of density n = 1012 cm−3 . The blue line is the prediction of our many-body-variational
formulation. The magenta line is the prediction from Bogoliubov theory.

One can sense the limitations of the Bogoliubov description of diabatic quench experimentsby considering momentum cutoffs heuristically. Recall from Section 3.3 that using the coupling
constant UΛ → 4π~2 af /m in the theory, as required within the Bogoliubov approximation, requiresthat the momentum cutoff satisfy Λaf ≪ 1. There is no bound state in this limit, and any important−1
physics occuring uniquely on the time scale ωB
and length scale af of the bound state is therefore

absent in time-dependent Bogoliubov theory [25, 125, 175, 176, 89, 138]. When the quench isadiabatic with respect to the bound state [105], the ramp time is at least consistent with the¡¢−1−1shortest timescale describable by the theory, where tR > ~Λ2 /m≫ ωB. That is precisely theregime in which Bogoliubov theory correctly describes the contact dynamics, as shown in Fig. 5.1.
These bound-state oscillations also occur in the depletion of the condensate, defined by
Eq. (4.12). Figure 5.2 shows the time evolution of this quantity after a sudden quench, computed
both within the variational and Bogoliubov theories. As mentioned in the previous chapter, Bogoli√ubov theory predicts that the depletion grows as t at short times [125]. This correctly captures
the initial growth of the depletion except for the large-amplitude oscillations. The nonoscillatory
contribution to this growth comes mainly from momenta that are too small to probe bound-state
effects (kaf ≪ 1), whereas the oscillating component is due to the aforementioned contact dynamics(occurring at kaf & 1). Similar oscillations have been discussed previously in the theory literature

67[159, 80, 50, 52], and they received only a brief mention in the experimental results of Ref. [48]. To
date, there is no published data on the subject, although some preliminary observations of thesecondensate oscillations can be found in Ref. [31].

5.3

Two-Body Models
Two-body models afford another intuitive description of BEC quench dynamics. In somecases, they are exactly solvable [154, 22], and they can paint relatively transparent pictures of
nonequilibrium physics [15, 70, 119, 13, 69] that are sometimes obscured by the mathematics ofmore sophisticated, many-body models. It was recently suggested that such models might evenbe made quantitatively accurate in their descriptions of short-time, large-momentum dynamics inquenched BEC systems [150], although an unambiguous, universal link to the many-body BEC
problem has been absent in the literature. In this section, we establish such a link and derive
analytic predictions for the contact dynamics following a diabatic quench of the scattering length
near a broad Fano-Feshbach resonance.

5.3.1

Calibration

We consider the quantum dynamics of a pair of free-space atoms whose scattering length isquenched from an initial value ai ≥ 0 to a final value af ≥ 0. Assuming zero momentum for thecenter of mass (as is the case for any pair of atoms that scatter out of a BEC), the post-quench
dynamics are governed by the time-dependent Schrödinger equationi~

2π~2 af∂ψ(r, t)~2∂
= − ∇2 ψ(r, t) +δ(r) [rψ(r, t)]
∂t
2µ
µ
∂r

(5.3)

where ψ(r, t) is the wavefunction for the relative coordinate r, µ = m/2 is the reduced mass, andwe model the short-range interactions of the system with the Fermi pseudopotential discussed in
Section 2.2.2.
We can time-evolve an arbitrary spherically-symmetric initial condition by expanding in thebasis of energy-normalized s-wave eigenfunctions of the Hamiltonian shown in Eq. (5.3). These68
eigenfunctions aresin(kr) − kaf cos(kr)(S)ψk (r) = qr 4π 2 ~2 k(k 2 a2f + 1)/m

,

Ek =

~2 k 2m(5.4)

for the scattering states ande−r/afψB (r) = pr 2πaf,

EB = −

~2
ma2f

(5.5)

for the bound state. Given an initial condition ψ0 (r), the solution to Eq. (5.3) is [117]ψ(r, t) =

Z∞
0

(S)
dEk′ e−iEk′ t/~ψk′ (r)Z(S)
d r ψk′ (r′ )ψ0 (r′ ) + e−iEB t/~ψB (r)3 ′Z

d3 r′ ψB (r′ )ψ0 (r) . (5.6)

We then evaluate the momentum distribution by taking the Fourier transform of Eq. (5.6):ψ̃(k, t) =Z

d3 re−ik·r ψ(r, t).

(5.7)

Following our intuition from mean-field theory, we relate this two-body problem to the manybody system by considering the combined effect of a background, dilute BEC on the momentum
distribution of a single particle. Assuming that this time-dependent, single-particle momentum
distribution is normalized in the continuum via
1=

Z

¯2
d3 k ¯¯¯¯ψ̃(k, t)¯ ,
(2π)3

(5.8)

we compute the full momentum distribution by multiplying by the total density n [119, 70]. The
combined effect of the dilute background gas is modeled by an appropriate choice of initial condition
ψ0 (r). Previous calculations of this type have placed the two-body system in a (fictitious) tightharmonic trap, whose frequency is chosen to reproduce either the total density n [13] or the approximate nearest-neighbor separation [150].3 Both of these prescriptions are limited in the sensethat their quantitative predictions for short-distance dynamics depend strongly on the harmonic
nature of the fictitious trap. In this sense, they are intrinsically semi-quantitative.
3

This reference estimated the mean nearest-neighbor spacing using the Wigner-Seitz radius (4πn/3)−1/3 , whichis correct for an uncorrelated gas up to a factor of Γ(4/3) ≈ 0.89. Including this factor in that model would change
the prediction for the contact slope upon quenching to unitarity by a factor of Γ−3 (4/3) ≈ 1.4. See Ref. [26] for an
exact calculation of the nearest-neighbor distribution.

69
The new prescription that we propose is motivated by the fact that a quench of zero-range
interactions signifies a quench of a log-derivative boundary condition at r = 0:
lim

r→0

∂r (rψ(r))
1=− ,(rψ(r))
a(5.9)

as discussed in Section 2.2.2. As a result, the contact dynamics immediately following a quench
occurs entirely in the short range. The most important feature of an initial condition ψ0 (r) is
therefore its behavior as r → 0. Our first requirement is that ψ0 (r) satisfy Eq. (5.9) for the
initial scattering length of the system, ai . (All of the eigenstates in our post-quench expansion
basis, Eqs. (5.4)-(5.5), satisfy this log-derivative condition for the final scattering length, af .)Importantly, this log-derivative condition does not fix the absolute magnitude of ψ0 (r) for smallr; any such scaling cancels in Eq. (5.9). We propose that this absolute scaling of the short-range
wavefunction be fixed by the many-body problem. The quantity |ψ0 (r)|2 represents the probability
density of finding a background particle a distance r from the particle of interest, and this is givenby ng (2) (r) in the many body-problem, where g (2) (r) is the two-body correlation function [131].
For a pure, noninteracting BEC, there are no correlations and g (2) (r) = 1. If the initial scattering
length is nonzero, however, short-range correlations are determined exlusively by the contact via
Eq. (2.32). We thus calibrate the short-range behavior of the two-body wavefunction as follows:
lim |ψ0 (r)|2 = n

r→0Ci|ψ0 (r)| →+O
16π 2 nr2
2

µ ¶
1
r

(ai = 0)(5.10a)(ai > 0)(5.10b)where Ci = 16π 2 n2 a2i is the contact for the initial BEC, as derived in Eq. (3.40). Equation (5.10)
guarantees that the initial probability of finding another particle near the particle of interest (within
our simple model) matches that same probability for the many-body system before the quench. We
hypothesize that these probabilities remain approximately equal in the dynamics that occurs shortly
after the quench.
We now choose a set of initial two-body wavefunctions to test the robustness of our calibration
scheme. We have been able to analytically evaluate the integrals in Eqs. (5.6)-(5.7) for the initial70
conditions


³
´


L0 ai
−r/L0

A
(a
,
L
)
1
−

0 i
0
(L0 −ai )r e




h
i
¢
¡
ψ0 (r) = A1 (ai , L1 ) 1 − ai e−r/L1 1 + r
r
L1



·

³ ´ ¸


A (a , L ) ¡1 − ai ¢ e−r/L2 1 + r + 1 r 2

2 i
2
r
L2
2 L2

(5.11)

where Lj is a free parameter for each initial condition, and Aj (ai , Lj ) is a constant chosen to
give unit normalization. The leading factor in parentheses enforces the log-derivative boundary
condition, and the bracketed polynomial factors have been chosen to add variety to our treatmentof the long-range wavefunction. The calibration given by Eq. (5.10) then completely fixes the shortrange behavior, along with the free parameter Lj . The necessary integrations in Eqs. (5.6)-(5.7)
can be carried out with a combination of contour integration and symbolic mathematical software,such as Mathematica. Appendix A lays out the steps of the solution.

5.3.2

Revisiting the Quench to Unitarity

A useful figure of merit for nonequilibrium physics is the momentum distribution after aquench from noninteracting (ai = 0) to unitarity (af = ∞). We computed this quantity in the previous chapter using a many-body variational formulation. For our two-body models, the formulae
for the exact momentum-space wave functions ψ̃(k, t) are too complicated to usefully write down.
Here, we include only the simplest formula as an example. The first initial condition in Eq. (5.11)
evolves as
√
8 L0

(

"r

#
r
¡
¢
~k 2 t~k 2 t2 2ψ̃(k, t) = ¡ii+ 1 + k L0¢2mm
k 1 + k 2 L20
,
#)
"s
¶
µ
¡
¢
i ~t2 √
~kt
~t
i
erfc
+e mL0 π kL0 − i 1 + k 2 L20
mL0
mL20
¡
¢
1 − k 2 L20 DawsonF(5.12)

where the Dawson function is defined by−z 2

DawsonF(z) ≡ e

Z z

2dy ey .

0This wave function evolves continuously from its initial condition.

(5.13)

71

10

10

0

−5

n
nk(t)
(t )
k

10

10

−10

−15

10

0

1

10 kk/k
/kF

10

2

10

3

F

Figure 5.3: Momentum distributions at fixed time ωF t = 0.01 after a quench from noninteracting
to unitarity. The thick black line is the numerical data from a many-body variational calculation atdensity n = 5 × 1012 cm−3 , using the contact interaction model. The thin lines are the analyticallycomputed two-body results. The cyan, green, and magenta lines respectively correspond to theproperly-calibrated initial conditions in the order listed in Eq. (5.11).

72
The predicted momentum distributions from our two-body models are plotted in Fig. 5.3 at
a fixed time shortly after the quench, and they are compared with the many-body numerics from
Chapter 4. With the free parameter of each initial condition chosen in our prescribed manner, allresults agree favorably at large momentum. Despite the various functional forms for the initialconditions in Eq. (5.11), all of our two-body wavefunctions predict that
C(t) =

128π(6π 2 )2/3

n4/3 ωF t,(5.14)

at short times, which agrees with the many-body variational prediction (4.13) to within less than
two percent. Equation (5.14) also follows from applying our prescription to the Gaussian initialcondition of Ref. [150], for which only the contact growth can be calculated analytically. The
contact slope now appears to be independent of the arbitrary details of the two-body model.
We remark that the model independence of our large-momentum dynamics is nontrivial. If
we had chosen each free parameter Lj by matching hri to an estimate for the nearest-neighbor separation, the predicted slope of the contact would vary by almost an order of magnitude, dependingon the chosen initial condition in Eq. (5.11).4 The approximate agreement between the two- andmany-body models demonstrated in Ref. [150] is a result of the near-equivalence of the requirementsthat hri ≡ (4πn/3)−1/3 and |ψ0 (0)|2 ≡ n for normalized Gaussian functions. As explained above,the latter requirement is more physically motivated, and it leads to improved agreement with the
many-body results while unifying the large-momentum predictions of the various exactly-solvable
two-body models. In the remainder of our discussion, we employ this calibration scheme.
Contact dynamics aside, our models also agree on the subleading oscillatory structure of
the large-momentum dynamics, as shown in Fig. 5.3. As discussed in the previous chapter, theseoscillations have phase 2εk t/~, where εk is the kinetic energy of a single particle of momentum k,
and an amplitude that scales as k −5 . Each distribution shows distinct low-momentum behavior
that is determined by the long-range characteristics of the initial conditions. We can infer from
4This discrepancy is most pronounced for a Lorentzian initial condition ψ0 (r) ∼ 1/(r2 + L2 ), for which hri is
infinite and cannot be matched to the particle spacing. If this initial condition is calibrated as we propose, it yields
Eq. (5.14).

73
Eq. (5.11) that these long-range features occur on a length scale that is set by the parameter Lj ,which is of the order of the mean interparticle spacing for the gas. At such momentum scales, we
expect many-body effects to determine the physics, and this limits the approximate validity of our
two-body models to momenta k ≫ kF and times ωF t ≪ 1. These limits are implicit in all of thetwo-body results that follow.

5.3.3

Quenching to Finite Scattering LengthWith our two-body models properly calibrated, we are well equipped to revisit and generalize
the bound-state oscillations addressed in Sec. 5.2. We will see that the simple two-body approach
illustrates the crucial role played by the bound state after a diabatic quench, while quantitatively
describing the evolution of two-body correlations via the dynamical contact.
As a preliminary matter, our two-body approach leads to an intuitive understanding of boundstate oscillations. The basic structure of Eq. (5.6) suggests that the short range of the boundand scattering contributions may be compared to the two legs of a simple interferometer. The
diabatic quench essentially projects the initial condition onto these two legs, and a different phaseis acquired over each leg as time progresses, as evidenced by Eq. (5.6). The measured momentum
distribution is always defined with respect to free-particle (noninteracting) momentum states, rather
than the scattering states of Eq. (5.4); it is for this definition that the k −4 tail is meaningfully
related to short-range density-density correlations via the contact [152, 147]. Hence, the two legs ofthe interferometer are recombined during a measurement of the momentum distribution, thereby
projecting the quantum state onto the free-particle momentum basis as in Eq. (5.7). The phaseevolution of the bound-state component leads to periodically-modulated interference that is mostpronounced at the length scale of the bound state, r . af . As a result, the contact oscillates, alongwith certain other observables such as the condensate fraction (see Fig. 5.2).
As discussed previously, the various initial conditions of Eq. (5.11) lend themselves to analytical, time-dependent solutions for arbitrary initial and final scattering lengths. These formulae are74

C(t)/C
C(t)/C 00

ωBBtt
Figure 5.4: (Color online) Contact dynamics following a diabatic quench from noninteracting to
700a0 , for a BEC of density n = 1012 cm−3 . The circles represent the many-body-variational data
computed with a momentum grid of 32767 evenly spaced points up to a cutoff Λ = 2000πkF . (Thecutoff is chosen to be large to make a better comparison with the zero-range theory assumed in
the two-body model.) The green line represents the formula given in Eq. (5.15), and the red line
represents the linear growth given in Eq. (5.14).

75
quite complicated in general, but, remarkably, they each predict the same behavior of the contact
¯
¶
µ
i´¯¯2
³
hp
¯
ai
2 2 2¯iωB t(5.15)iωB t ¯¯C(t) = 16π n af ¯1 +−1 e
1 + erfaf

if we take the limits na3i ≪ 1 and na3f ≪ 1. Figure 5.4 plots this result against a many-bodysimulation for the diabatic quench considered already in Fig. 5.1. Apart from a slight offset in theoscillation frequency, the agreement is excellent. We believe that this small frequency deviation isdue to the fact that our numerical solution of the many-body model is constrained to a finite (albeit
large) momentum cutoff Λ, whereas our two-body models are truly zero-range. Any experimental
realization of these oscillations would experience such an offset due to the finite range of true
interatomic interactions. This was certainly the case in the Ramsey experiment of Ref. [48]. It isalso possible that mean-field effects could shift the oscillation frequency [33], but this should not besignificant in the small-na3 limit that we are considering. Aside from the bound-state oscillationsof the contact, the momentum distributions at small scattering length look essentially the same as
in Fig. 5.3, including the subleading k −5 behavior mentioned previously.
It is useful to examine the general dynamics given by Eq. (5.15). At short times ωB t ≪ 1,
the contact evolves continuously from its intial value Ci asr2
2 2
C(t) = Ci +32π n ai (ai − af )
ωB t
π
µ
¶2
.
ai
128π
4/3
3/2
− 1 n ωF t + O(t )+(6π 2 )2/3 af(5.16)In the limit of vanishing initial scattering length, the contact first grows linearly according to
Eq. (5.14) for all values of af . This is shown in Fig. 5.4 for the case of a quench to af = 700a0 .
√However, at nonzero initial scattering length, this linear growth is superseded by nonanalytic t
behavior. At later times ωB t ≫ 1, the contact is oscillatory:
"
µ
¶2
ai
2 2 2
C(t) ≈ 16π n af 1+4
−1
af
#.
µ
¶
ai
+4
− 1 cos (ωB t)
af(5.17)For a diabatic quench upward (af > ai ), the time-averaged contact hC(t)it may be up to five times
larger than the Bogoliubov prediction of C0 = 16π 2 n2 a2f , and the oscillation amplitude may be up

76
20

C(t)/C
C(t)/C 00
10

0
0

5

ωBBtt 1015

Figure 5.5: Contact dynamics following a diabatic quench downwards with af = ai /3. For reference, the green line is the equilibrium contact Ci for the BEC at the initial scattering length ai ,and the red line is the contact C0 for a ground state BEC at the final scattering length af .

to four times as large. Of course, in the limit of no quench (af = ai ), the contact is trivially time
independent.
The case of a diabatic quench downward (af < ai ) reveals interesting physics. Depending on
the ratio of initial and final scattering lengths, the time-averaged contact may be much larger thanthe Bogoliubov prediction C0 , and larger even than the initial contact Ci . Figure 5.5 shows these
dynamics for a quench to af = ai /3, in which case hC(t)it is more than an order of magnitudelarger than C0 and almost twice as large as Ci . The peak-to-trough oscillation amplitude is also
larger than both C0 and Ci . This is in stark contrast to the Bogoliubov case, in which the contact
relaxes to C0 over the fast timescale of the diabatic quench. At least in the transient dynamics,
a diabatic reduction in the scattering length can evidently increase local two-body correlations by
up to a factor of four compared to the initial condition, as evidenced by Eq. (5.17). An important
limitation, however, is that these dynamical correlations are most pronounced at and below thelength scale of the bound state.
The heightened short-range correlations contained in hC(t)it , beyond those already at the
Bogoliubov level, come fundamentally from bound-state physics. For example, if we ignore thecontribution of the bound state to the dynamics in Eq. (5.6), we find that the scattering statesdephase in such a way that the dynamical contact asymptotes towards C0 without any oscillations,regardless of the initial scattering length. This is in qualitative agreement with the Bogoliubov

77
prediction. Once the scattering states dephase, however, the bound state is left to dominate theshort-range wavefunction except in the relatively trivial case where af ∼ ai . The excess shortrange correlations, given by the second bracketed term in Eq. (5.17), are determined entirely by
the original projection of the initial condition onto the bound state.
We can estimate the amplitude of the depletion oscillations by simply integrating over the
relevant portion of the momentum distribution. From Eq. (5.17) and the fact that nk (t) ∼ C(t)/k 4
at large k, the part of the momentum distribution oscillating at the bound-state frequency behavesasnk,osc (t) ∼16π 2 n2 a2f
k4

4

µ

¶
ai
− 1 cos (ωB t) ,af

(5.18)

aside from the time-independent contribution to the k −4 tail. As mentioned previously, theseoscillations occur at the momentum scale of the bound state, where kaf & 1. Integrating Eq. (5.18),
we find that the oscillating part of the depletion fraction is approximatelyZnex,osc (t)d3 k1∼nk,osc (t)
n
n
(2π)3
kaf >1.
µ¶¡ 3 ¢ ai
∼ 32 naf− 1 cos (ωB t)
af(5.19)The oscillation amplitude given here agrees with the many-body data shown in Fig. 5.2 to withina factor of order unity, and we expect it to be a reasonable estimate as long as the dilutenessparameter na3 is small before and after the diabatic quench.
As a final aside, we note that our two-body analysis is able to generalize the short-time
dynamics following a quench to unitarity. If we calibrate our initial wavefunction for ai ≥ 0 andthen quench to unitarity, the initial contact dynamics are
C(t) = Ci − 32π 2 n2 air2~128π 4/3
n ωF tt+πm(6π 2 )2/3

(5.20)

to leading order in na3i and ωF t. This represents a generalization of Eq. (5.14) for diabatic quenchesfrom small initial scattering length. It is interesting that Eq. (5.20) is equal to the af → ∞ limit ofEq. (5.16), despite the fact that Eq. (5.16) was derived for small final scattering length (na3f ≪ 1).

78

5.4

Summary
We have elucidated the important role of the bound state in determining the contact dynamics

of a diabatically-quenched BEC. We first computed these dynamics using a variational manybody model, demonstrating that large-amplitude oscillations of the contact can be observed evenwith existing magnetic-field-ramp technology. Our calculations reinforce the idea that coherent,
short-range physics can lead to measureable signatures even in the BEC fraction. This is thedominant physics of the quenched gas on short time scales, before many-body effects and loss
become important.
We also developed a calibration scheme for two-body models that leads to an unambiguous,quantitative description of BEC contact dynamics following a sudden quench. Our prescription fixes
both the log-derivative and absolute magnitude of the initial short-range, two-body wavefunction
by matching to the many-body problem, and we are able to derive analytic formulae for the shorttime evolution of the contact in the weakly-interacting and unitarity limits. Our dynamics areshown to be independent of the arbitrary features of the models, and they agree with manybody predictions. This two-body picture indicates that bound-state oscillations of the contact are
analogous to interferometry. We expect that one can account for finite ramp speeds by numericallysolving the two-body Schrödinger equation for a properly-calibrated model.
The dynamical contact can be measured using time-resolved RF spectroscopy, as done in
Ref. [4]. Our analysis shows that even the time-averaged contact hC(t)it may be greatly magnified
relative to the Bogoliubov prediction due to bound-state physics, and this could be observed withan RF pulse that is long compared to the bound-state oscillation period. Measuring the oscillations
themselves necessarily requires using shorter pulses, and that may lead to inconvenient broadeningof the central RF peak. In any event, the temporal constraints on time-resolved RF spectroscopy
depend both on the atomic species and on the transition under consideration, and they are beyond
the scope of the present study.
We reiterate that the bound-state dynamics that we have considered are a coherent, transient

79
effect. They encapsulate the response of a short-range wavefunction to an abrupt change in the
scattering length or, equivalently, a log-derivative boundary condition. At longer time scales, we
expect the oscillations to damp out as the system equilibrates. Similar damping was observed in the
Ramsey experiment of Refs. [48, 33], and it was believed to be due to a combination of incoherent
three-body loss and dephasing from magnetic-field inhomogeneities. Still, the coherence of largemomentum dynamics persisted for many oscillation periods before damping became significant. The
engineering of quench apparatus has improved over the years, especially in creating ramps that are
diabatic with respect to the bound state [112]. This opens the door for systematic experimental
studies of bound-state signatures in quenched BECs.

Chapter 6

Quench-Induced Correlation Waves

Results presented in this chapter can be found in our recent preprint on the arXiv [39], which
has been accepted for publication in Physical Review A.

6.1

Background
It is a generic property of wave mechanics that an abrupt change in a system’s boundary

condition generates waves that propagate outward from the boundary. The tap of a mallet excites
phonons in a percussive chime; with a flick of the wrist, a lion tamer snaps his whip; electricpulses in an antenna generate a radio broadcast. If we think of short-range interactions in terms ofa scattering-length-dependent boundary condition for the relative wave function, as in Eq. (2.17),
then we should expect that a sudden change of the scattering length generates waves that propagate
to nonzero particle separations. In contrast to previous chapters, which focused on short-range
correlations via the dynamical contact, we now shift our focus to longer-range correlation waves
that are generated by a quench.
One landmark experiment on this topic was performed recently at the University of Chicago[85]. A quasi-2D BEC was quenched between two different scattering lengths,1 and the nonequilibrium density distribution was subsequently measured in situ after various hold times. The interfering phonons generated by the quench led to oscillatory pair correlations in the system. Thesewaves, known as Sakharov oscillations [140], are similar in their acoustic nature to the observed
1

The quench was adiabatic with respect to the bound state, so the results of the previous chapter do not apply
to this experiment.

81
fluctuations of the cosmic microwave background [79]. Besides its intriguing relevance to cosmology, this experiment demonstrates how correlations can propagate from the short range (where thequench occurs) to the long range (where the fluctuations are measured) in a many-body system.
Dynamical waves that propagate in the pair correlation function in response to an interaction
quench, hereafter referred to as “quench-induced correlation waves,” have been discussed at length
in the context of several many-body models. For the case of 2D and 3D quenched Bose condensates,
these correlations have been calculated in the Bogoliubov approximation [85, 25, 125] and withquantum kinetic theory [93]. Numerical results were presented for quenched 1D Bose gases in Refs.
[73, 124, 178], with analytical results for the Tonks-Girardeau regime given in Ref. [100]. Other
studies have calculated these waves in quenched single-band Hubbard models using matrix-productstate [5] and variational-Monte-Carlo [24] algorithms, where Lieb-Robinson bounds [108] can limit
the spreading of correlations; these studies accompany recent experimental progress in that realm[27].
This chapter takes a different, but complementary, approach to correlation waves. Inspiredby the results of the previous chapter, we reconsider the question: What does an interaction
quench (alternatively, a quenched boundary condition) do to the relative wave function for a pairof particles? This question lies at the root of the many-body quench problem, where interactions
are pairwise and three-body correlations are often negligible. Two-body models offer the advantagethat they can be solved exactly and give direct access to the wave function [22]. Moreover, they
are immediately relevant to few-body systems in optical tweezers [170, 179, 90] and deep opticallattices [148]. In many instances, they have given insight into understanding nonequilibrium manybody phenomena [15, 70, 119, 13, 69]. These models can moreover give a quantitative descriptionof short-time short-range pair correlations in quenched systems, as demonstrated earlier in thisthesis.
In this chapter, we show that two-body models give an intuitive description of the physics behind quench-induced correlation waves. Section 6.2 reviews ballistic expansion from the standpointof a quench. Our phase-space analysis shows that the correlation waves propagate ballistically, i.e.,

82
as if they were free particles. We demonstrate that these waves, which are inherently nonlocal,
can contribute to the k −4 tail of a dynamical momentum distribution. This result is unexpectedconsidering that the ideas surrounding Tan’s contact relate the k −4 tail exclusively to local correlations. In Section 6.3, we discuss the leading-order behavior of the momentum distribution for
arbitrary interaction quenches. We find that there is generally a competition between short-range
and ballistic physics in the large-momentum limit, an effect that is absent in equilibrium scenarios.
Additionally, we find that the amplitude of the correlation wave is determined chiefly by the initialand final scattering lengths, and also by the initial amplitude of the wave function at vanishingparticle separation. Section 6.4 outlines our solution of the two-body quench problem in the presence of an external lattice potential. We show that ballistic correlation waves can propagate evenin deep lattices, and we present a simple semiclassical model that yields accurate estimates for thetransport that occurs. Section 6.5 summarizes our results.

6.2

Ballistic waves
It is instructive to begin with the simplest case in which quench-induced correlation waves

occur: a measurement of the momentum distribution of a strongly interacting ultracold gas. The
general method is to rapidly turn off the external trap and interactions, thereby freezing the momentum distribution of the gas, and then to allow the sample to expand freely before imaging.
After expansion, the image represents the column-integrated momentum distribution of the gas.
Correlation waves are generated by this simple protocol, as we now demonstrate.
The above-described procedure constitutes an interaction quench in the sense that, trap
effects aside, the scattering length is rapidly changed from some initial value (ai ) to some final
value (af ). The effect on the wave function can be seen in the ballistic expansion of a bound pairof interacting particles in 1D. In terms of the particle separation x and coupling constant g1D , theshort-range interaction potential isVint (x) = g1D δ(x).

(6.1)

83
1

0.5

-0.5

0

0.5

Figure 6.1: Ballistic expansion of a bound-state wave function. The black (thin) line represents theinitial wave function. The blue (thick, solid) line represents the wave function at ~t/2µa2i = 0.008,
and the magenta (dashed) line is the wave function at ~t/2µa2i = 0.015.

One can define a 1D scattering length via a = −~2 /µg1D , where µ is the reduced mass for the pair.
The interactions are attractive (repulsive) for a > 0 (a < 0), and they vanish for a = ±∞.2

For

an initially bound pair of atoms, the relative wave function is1
ψ(x, t = 0) = √ e−|x|/ai .
ai

(6.2)

The solution, upon turning off interactions (af → ±∞), is most compactly written in momentum
space. In analogy with Eq. (2.25), we define the 1D Fourier transform asf˜(k) ≡

Zdxe−ikx f (x).

(6.3)

The time-dependent wave function is then given by√
2 ai −iEk t/~ψ̃(k, t) =e,
1 + k 2 a2i(6.4)

where Ek = ~2 k 2 /2µ is the relative kinetic energy. The short-time dynamics of the position-space
wave function is shown in Fig. 6.1. At t = 0, the wave function has a kink at vanishing particle
separation. This kink is absent for t > 0, where we see a correlation wave that propagates to larger
particle separations.
The ballistic expansion dynamics can be easily visualized with a phase-space representation.
2

This is a bit different from what one finds in 3D, although we will see shortly that the 1D scattering length still
relates to a log-derivative boundary condition.

84

a)

b)

Figure 6.2: The Wigner distribution given by Eq. (6.5) for the ballistic expansion of a 1D boundstate wave function at (a) t = 0 and (b) ~t/2µa2i = 0.4. The dashed white line in (b) represents
the formula x = 2~kt/m, which corresponds to the separation of two classical particles that startout on top of each other (x = 0) and then fly apart with momenta ±~k.

85
The Wigner function [173]W (x, k, t) =

Z

³
y ´
y ´ ³
dyeiky ψ ∗ x + , t ψ x − , t
2
2

(6.5)

gives an approximate sense of the phase-space distribution of the instantaneous quantum stateψ(x, t). The position and momentum distributions can be found by integrating:Zdk2|ψ(x, t)| =W (x, k, t)2π.
Z2|ψ̃(k, t)| = dxW (x, k, t)(6.6)

Figure 6.2(a) shows the Wigner function of the bound state at t = 0. Initially, the k −2 tail of the
momentum-space wave function is responsible for the kink in the position-space wave function at
x = 0 (see Fig. 6.1). This is typical for wave functions of 1D systems with short-range interactions.
It is generally understood that any state behaving as Ψ(x) ≈ Ψ(0)(1 − |x|/a) in the short range
should have a contribution2Ψ(0)Ψ̃(k) ∼
+O
ak 2

µ

1
k3

¶

(6.7)

to the large-momentum limit of the momentum-space wave function [129]. This momentum tail isanalogous to what occurs in 3D: a wave function behaving as Ψ(r) ≈ (rΨ)|r→0 1r in the short range
should have a momentum tail that goes as Ψ̃(k) ∼ (rΨ)|r→0 4π
at large k. The connection between
k2
short-range correlations and large-momentum asymptotics has led to the development of universal
contact relations in 1D [129, 6], akin to Tan’s work in 3D [152, 154, 153]. These ideas have also
been extended to 2D systems [151, 35, 160, 172, 171].
Figure 6.2(b) shows that, after the interactions are turned off, the large-momentum components of the wave function propagate outwards to larger particle separations. Although themomentum distribution does not change during the dynamics [cf. Eq. (6.4)], the momentum components eventually separate spatially in a semiclassical sense, with the fastest modes moving thefarthest. (In the figure, we see that the spatial wings of the phase-space distribution agree very
well with the classical problem in which a pair of particles flies apart with momentum ±~k (whitedashed line), similar to the suggestion of Ref. [23].) This mechanism leads to the usual correspondence between the expanded spatial distribution and the initial momentum distribution, as probed

86
by ballistic expansion measurements of interacting systems. We point out that such a mapping
would not occur if the interactions were turned off adiabatically or if they were left unchanged; it
was necessary to quench the system.
It is interesting that ballistic expansion leads to a momentum distribution whose k −4 taildoes not correspond to a kink in the short-range wave function. Rather, this tail is responsible
for the correlation wave that propagates from the short range to the long range, as evidenced inFig. 6.2. One can alternatively view this correlation wave, and hence the k −4 tail in the dynamical
momentum distribution, to be the result of a rapidly disturbed boundary condition. It can be shownthat the interaction potential in Eq. (6.1) enforces a log-derivative boundary condition¯∂x ψ ¯¯1=−¯
ψ x→0+
a(6.8)

for symmetrized wave functions,3 similar to the Bethe-Peierls boundary condition (2.17) in 3D.
The quench from ai > 0 to af = ±∞ changes this boundary condition discontinuously, thereby
generating a correlation wave. We expect intuitively that such a wave should be generated wheneverthe quench is diabatic and af 6= ai . The strength of the wave should depend on the mismatch
between the initial and final boundary conditions. For example, the generated wave should be
weak when af ≈ ai , and it should be strong when a−1 changes drastically. We expect also thatthe large-momentum behavior of the wave function should contain both short-range and ballistic
contributions, generalizing Eq. (6.7).

6.3

Arbitrary Quenches
We demonstrated in Chapter 5 that it is possible to find closed-form solutions to the two-body quench problem in 3D for a broad class of initial wave functions. The short-time, zero-range
dynamics were found to depend on only three parameters: the initial scattering length ai , the final¯
scattering length af , and the initial zero-range behavior of the wave function rψ(r, 0)¯r→0+ . It isnatural to suppose that a similar universality should persist in the large-momentum content of the3This can be proved by integrating the Schrödinger equation in the viscinity of the interaction, as in Eq. (2.23).

87

Figure 6.3: The Wigner distribution given by Eq. (6.5) for a 1D bound-state wave function
quenched to af = 2ai . The Wigner function is evaluated at the same time as in Fig. 6.2(b), andfor the same initial scattering length ai . The dashed white line represents the same classical model
as shown previously.

quench-induced ballistic wave, as this wave originates in the short range and is a direct responseto the change in boundary condition. We indeed find this to be the case in each dimensionality.
The derivation of the large-momentum limit of the 1D dynamical wave function is given
in Appendix A. In short, one must project the initial wave function onto the complete basis ofenergy eigenstates satisfying the appropriate log-derivative boundary condition, Eq. (6.8), and thenpropagate in time.4 We find that the large-momentum limit of the wave function is2ψ(0, t)ψ̃(k, t) =+af k 2µ¶µ ¶af
2ψ(0, 0)1
−iEk t/~
−1
e
+O
2
ai
(k af − i|k|)
k3

(1D)

(6.9)

for t > 0. The first term shown here comes from the dynamical kink that appears in the short-range
wave function for finite values of af , as in Eq. (6.7) and in accordance with our intuition about
the contact [129]. The second term represents the ballistic wave that is generated by the quench,
similar to Eq. (6.4). As a consistency check, it is easy to verify that Eq. (6.9) agrees with Eq. (6.4)
in the af → ±∞ limit. It is also immediately obvious that the ballistic contribution vanishes in
the limit that no quench occurs (i.e., af → ai ).
It is significant that, after the quench, the large-momentum limit of the wave function has¡¢
two distinct components that are both O k −2 . This occurs whenever the final scattering length
4This is the same approach we used to solve the 3D problem in Section 5.3.1.

88
1

0.5

-0.5

0

0.5

Figure 6.4: Quenching a bound state to af = −ai /2. The black (thin) line represents the initialwave function. The blue (thick, solid) line represents the wave function at ~t/2µa2i = 0.008, andthe magenta (dashed) line is the wave function at ~t/2µa2i = 0.015. Compare with Fig. 6.1, which
depicts a quench to af = ±∞.

is finite. Figure 6.3 shows this behavior for the case in which an initial bound state at a scattering
length ai > 0 is quenched to a final scattering length af = 2ai . Similar to Fig. 6.2(a), we seelarge-momentum content in the short range that is due to the residual kink in the wave function.
Similar to Fig. 6.2(b), we see that the quench generates a ballistic correlation wave that rapidly
propagates to large particle separations. This is in strong contrast to equilibrium systems, where
only the short-range correlations contribute to the large-momentum asymptotics [129]. For this
1D quench problem, the k −4 tail of the one-body momentum distribution (∼ |ψ̃(k, t)|2 ) does notcorrespond perfectly with the zero-range pair probability, indicating that one must exercise carewhen interpreting the 1D contact in a nonequilibrium context.
The amplitude of the ballistic correlation wave shown in Eq. (6.9) can be adjusted by changing
the final scattering length af . Figure 6.4 shows the short-time position-space wave function for abound state that is quenched to af = −ai /2, evaluated at the same times as in Fig. 6.1. One cansee that the quench from attraction (ai > 0) to repulsion (af < 0) has increased the amplitude of
the correlation wave when compared to the ballistic expansion case (af = ±∞). Equation (6.9)
indicates that this enhancement is by approximately a factor of 3.
One may observe from Eq. (6.9) that quenches to the Tonks-Girardeau regime (af → 0− )89generate especially strong ballistic waves. In this limit, the wave function behaves as2ψ(0, 0) −iEk t/~e+Oψ̃(k, t) =
i|k|

µ

1
k2

¶

(6.10)

for large k and t > 0, with the ballistic component dominating the short-range.5

One can intuit

the k −1 tail by observing that the final energy of the system is determined by the expectation valueof the post-quench Hamiltonian in the initial state; this energy must diverge as gf ∼ −1/af → ∞.
For t > 0, the interaction energy must vanish because ψ(0, t) = 0. Conservation of energy therefore
requires that, after the quench, the kinetic energy diverge:Z¯2 ~ 2 k 2
dk ¯¯
¯
→ ∞.
¯ψ̃(k, t)¯
2π
2µ(6.11)

This was first pointed out by the authors of Ref. [100], who calculated analytically the dynamical
density correlations for a many-body system of density n that is quenched from noninteracting to theTonks-Girardeau regime. Our results connect smoothly with theirs in the short-time (~n2 t/m ≪ 1),
short-range (nx ≪ 1) limit. In this limit, their dynamical pair correlations take the form of a√
relative wave function that behaves exactly as in Eq. (6.10) except that ψ(0, 0) → n. If we wereto simulate the many-body problem with a two-body model, as done in the previous chapter, we
would use this same prescription. This prescription also leads to quantitative agreement (at shorttimes) with the numerical calculations of g (2) (0, t) in Ref. [178], which considered a broad range
of af < 0. This reinforces our idea that properly calibrated few-body models can quantitatively
describe short-time short-range correlation phenomena for quenched many-body systems.
The derivation given in Appendix A for quenched one-dimensional systems can be straightforwardly generalized to two and three dimensions. In direct analogy with Eq. (6.9), the results fort > 0 are2πψ̃(k, t) = −³

ψ(ρ,t)ln(ρ/b)

k2

´¯
¯
¯

ρ→0+

+ ln

µ

ai
af

¶ 2π

³

ψ(ρ,0)ln(ρ/b)´¯¯¯ρ→0+ −iEk t/~¢e+O2
k ln(kaf ) − i π2

¡

µ

1
k3

¶

(2D)

(6.12)

where b > 0 is an arbitrary length scale that makes the argument of the logarithm dimensionless,
5

The numerator and denominator of the first term in Eq. (6.9) both vanish in the af → 0− limit. This subtlety
can be treated along the lines presented in Appendix A.

90
and
¶
µ
µ ¶
af 4π (rψ(r, 0))|r→0+ −iEk t/~
4π (rψ(r, t))|r→0+1
ψ̃(k, t) =
+ 1−
e
+O
2
2
k
ai
k (1 + ikaf )
k4

(3D).

(6.13)For the two-dimensional case, we define the scattering length with the convention that the bound2~
6 Both in 2D and in 3D, we see that the ballistic contribution
state has energy EB = − 2µa
2 [114].

(second term) vanishes when af = ai . It can also be verified that both formulas reduce to thefree-particle result when interactions are turned off (af = ∞ in 2D, and af = 0 in 3D).
The ballistic contribution in Eqs. (6.12) and (6.13) is subleading to the short range in thelarge-k limit, but it is nevertheless large compared to what one expects in equilibrium. The sub¡¢leading terms of all equilibrium states are O k −4 for both dimensionalities. In contrast, we see¡¢¡¢that the quench induces a new subleading structure, which is O k −2 ln−1 (k) in 2D and O k −3
in 3D. This subleading behavior in 3D is responsible for the decaying oscillations that we observedin the BEC momentum distribution after quenching to unitarity, as shown in Figs. 4.2-4.4. Indeed,
the cross term between the ballistic and short-range contributions to Eq. (6.13) oscillates withfrequency Ek /~ = 2εk /~ and decays as k −5 at all times, exactly as observed in the many-bodynumerics of Chapter 4. In that case, however, the nonlocal and ballistic origin of the effect was not
obvious.
Despite the subleading nature of the ballistic terms in Eqs. (6.12) and (6.13) for finite af ,¡¢one can generate leading-order O k −2 behavior by turning off interactions. This is along thelines of the ballistic-expansion arguments presented in Sec. 6.2. If we then turn on interactions

before the wave spreads appreciably, the wave function will develop a short-range singularity that¡¢will separately contribute a term of O k −2 to ψ̃(k, t). As was found for a single quench in 1D[Eq. (6.9)], the short-range and ballistic components can therefore occur at the same order in thelarge-k limit of the wave function. Again, we conclude that the considerations that relate themomentum tail exclusively to short-range correlations in equilibrium do not always hold outside of6

We follow the scattering-length convention used in Ref. [114], which is slightly different from that used inRefs. [151, 35, 160, 172, 171]. They are related by a′2D = 2e−γ a2D , where γ is Euler’s constant, a2D is the scattering
length by our convention, and a′2D is the scattering length in the other convention.

91
equilibrium.
We conclude this section by remarking on the limitations of our ballistic analysis. The zerorange approximation, wherein interactions are represented with boundary conditions at vanishingparticle separation, has been enormously successful in describing ultracold quantum gases nearbroad Feshbach resonances [28]. This approximation is only valid for momenta satisfying kr0 ≪ 1,
where r0 is the range of the interaction. In experiments that use tight optical trapping to create
quasi-low-dimensional geometries, the oscillator length of the tight trap represents another scalethat bounds the “range” of the interaction in the reduced dimensionality. The immediate result isthat the momentum tails discussed in the context of zero-range models do not extend out indefinitely
to large k, although the point of breakdown (kr0 ∼ 1) might not be easily observable in typicalsignal to noise by a broad resonance (see Ref. [147]).
Our analysis also invoked the sudden approximation, wherein the scattering length (alternatively, the boundary condition) is assumed to change instantaneously. The consequence is thatballistic modes of arbitrarily large energy are generated by the quench, as shown in Eqs. (6.9)-(6.13).
Any experimental realization of the quench protocol will occur over a finite timescale [32, 85, 112],
and this will lead to an energy cutoff in the ballistic modes that can be generated. However, opticalswitching of interactions was demonstrated to be possible on timescales that are short compared tothose set by the interaction range [30]. It follows that experimentally feasible quench times do not
introduce any intrinsic constraint on the quench protocol beyond that already introduced by therange of interactions. We do expect, however, that slower quenches will produce weaker correlation
waves; in the limit that interactions are changed adiabatically, no correlation waves are generated.

6.4

Lattice Transport
It makes sense to suppose that the ballistic nature of quench-induced correlation waves should

allow for transport over potential-energy barriers. Semiclassically speaking, some part of the k −2
ballistic tail in Eq. (6.9) always has enough energy to cross a barrier of finite height, as suggested inRef. [47]. We expect that the amount of transport can be tuned by adjusting the amplitude of the

92
wave and, therefore, the strength of the quench. In this section, we investigate the quench-induced
dynamics that occurs for a pair of particles on a single site of a 1D optical lattice. We find that asemiclassical adaptation of our quantum description gives a good measure of the quench-induced
transport.
Interaction-quench effects in an optical lattice were recently discussed in the numerical resultsof Refs. [121, 122]. There, the authors used the multi-layer multi-configuration time-dependent
Hartree method for bosons (ML-MCTDHB) to investigate the dynamics of several interacting
bosons in a few lattice sites. They found that a quench can trigger rapid transport between wells,
as well as breathing and cradle modes within a given well. Such higher-band effects are ignored intypical Hubbard models that only include the lowest Wannier state in the formalism. Bound states,
strong interactions, and/or strong quenches may distort the wave function considerably from theWannier description, thereby necessitating models that encompass higher bands. The inclusion ofhigher bands, either in the ML-MCTDHB sense or in the spirit of a multi-band Hubbard model[53], makes it difficult to obtain numerically converged results for many-body systems on a latticewith strong interactions and strong quenches. Our two-body calculation should provide a usefulbenchmark in quantitatively understanding the rapid transport that takes place after an interaction
quench.
The relative and center-of-mass coordinates do not separate for the case of an interacting pairof atoms in an optical lattice. We therefore resort to numerics to investigate the exact quantum
dynamics. Without loss of generality, we consider identical bosons of mass m. The time-dependent
Schrödinger equation for this system can then be written asi~∂Ψ~2 ∂ 2 Ψ~2 ∂ 2 Ψ−+ Vlat (x1 )Ψ + Vlat (x2 )Ψ + Vint (x1 − x2 )Ψ
=−2∂t2m ∂x12m ∂x22(6.14)

where the interaction potential is given by Eq. (6.1), and the optical lattice potential of spacing ℓand depth V0 is given by
Vlat (xj ) = V0 sin2

³ πx ´jℓ.

(6.15)
2h
It is customary to measure lattice depth in units of the recoil energy ER = 8mℓ
2 . Inasmuch as

93
the interaction quench directly excites relative momenta, it is convenient to work with the relative
coordinate x = x1 −x2 and the center-of-mass coordinate X = (x1 +x2 )/2. After changing variables
and using a trig identity, one finds thatµµ¶³ πx ´¶∂Ψ~2 ∂ 2 Ψ~2 ∂ 2 Ψ
2πXi~Ψ + Vint (x)Ψ .
cos=−−+ V0 1 − cos∂tm ∂x2
4m ∂X 2ℓℓ

(6.16)The energy eigenstates corresponding to Eqs. (6.14)-(6.16) were found numerically in Ref. [164](see also Ref. [166] for the 3D analogue). Here, we instead solve for the dynamical wave function
by time-evolving an initial condition with the widely-used split-operator method. This method is
described in detail in Ref. [107].
In our numerics, we exploit bosonic symmetry [Ψ(x, X) = Ψ(−x, X)] by discretizing only
for x ≥ 0 and taking spectral transforms along this variable with the discrete cosine transform.
We discretize the center-of-mass coordinate X for both positive and negative values, and we take
spectral transforms along that variable using the fast Fourier transform. Our grid consists of 213
equally spaced points for x ∈ [0, 64ℓ] meshed with 27 equally spaced points for X ∈ [−4ℓ, 4ℓ]. We
model short-range interactions on the spatial grid by employing a potential that has support onlyat grid points where x = 0. We have found that representing δ(x) → δx,0 /∆x, where ∆x is thegrid spacing along the x direction, leads to the correct log-derivative boundary condition Eq. (6.8)
in the limit that ∆x ≪ |a|. This condition is satisfied by all simulations discussed in this section.
Our study focuses on quenched systems for which the induced transport is expected to be
the most significant. As indicated in Eq. (6.9) and alluded to in Ref. [47], the amplitude of theballistic wave is proportional to the initial probability amplitude that the atoms are in the same
position, ψ(0, 0). This quantity is largest, in equilibrium, when the system is in a bound state. We
therefore choose the initial condition for the transport problem Eq. (6.16) to be a bound state ina single lattice site. This configuration represents a subsystem of the state described by Ref. [163],
which reported observing a single molecule per lattice site. For a deep lattice, the bottom of the√well can be approximated as a harmonic-oscillator potential of frequency ω = 2 V0 ER /~ and width

94

a)

b)

Figure 6.5: Quenching a bound state from ai = 0.2ℓ to af = −ai on a single lattice site. The lattice
is assumed to have a depth of 10ER . a) The initial two-body probability density ℓ2 |Ψ(x1 , x2 )|2 fora bound state on a lattice site (t = 0). b) The same quantity calculated at ~t/mℓ2 = 0.01. Notethe logarithmic color scale, whose lower limit is a cutoff.

95
aho =

q

~
mω . One can write the approximate initial condition asΨ(x, X) = ψ0 (x)φ0 (X),
where φ0 (X) =

³

2
πa2ho

´1/4

2

(6.17)

2e−X /aho describes the center-of-mass degree of freedom, and ψ0 (x) is themolecular state dressed by the oscillator [22]. For a ≪ aho , one can show that ψ0 approaches theordinary bound state given by Eq. (6.2). For our simulations, we will consider an initial bound stateof scattering length ai = 0.2ℓ in a lattice of depth V0 = 10ER . The two-body probability density
associated with this initial condition is shown in separate-particle coordinates in Fig. 6.5(a).
As discussed previously, we expect that an interaction quench will generate an energetic
correlation wave that propagates over the potential barriers that separate individual lattice sites.
This transport is shown in Fig. 6.5(b) a short time after quenching to af = −ai . The wave has thesame general structure as in Fig. 6.4, with spatially decaying oscillations and a cusp of reduced
probability when both particles come together. Even after such a short time, we see that the wave
already extends a couple of lattice sites in each direction.
It is instructive to quantify the amount of quench-induced transport that takes place. We
can define a dynamical probability for the likelihood that both atoms remain in the central well:Z

PCC (t) =Z

dx1|x1 |< 2ℓ

dx2 |Ψ(x1 , x2 , t)|2 .

(6.18)|x2 |< 2ℓ

In like manner, we also define the probability that both atoms have tunneled,Z

PT T (t) =|x1 |> 2ℓ

Z

dx1

dx2 |Ψ(x1 , x2 , t)|2 ,(6.19)

dx2 |Ψ(x1 , x2 , t)|2 .

(6.20)|x2 |> 2ℓ

and the probability that a single atom has tunneled,
PT C (t) = 2

Z|x1 |> 2ℓdx1

Z|x2 |< 2ℓHere, we have exploited the symmetry of the bosonic wave function. The complementarity of theintegration regions results in the identity PCC + PT T + PT C = 1 at all times. These probabilities
are plotted for af = ±∞ and af = −ai in Figs. 6.6(a) and 6.6(b), respectively. In both cases,

96
the atoms begin in the central well [PCC (0) ≈ 1]. After the quench, the transport probabilities
smoothly saturate to values that depend on af . We note that the transport is substantial eventhough the lattice depth is of the order required for a typical Mott-insulating state in 1D [87, 180].
The ballistic description of the previous section leads to an intuitive, semiclassical model oftransport. We can estimate the saturated values of PCC , PT T , and PT C by considering the following
question: What fraction of the momentum distribution describes ballistic atoms that are energetic
enough to make it over the barrier?
The simplest analysis can be made for the case in which the interactions are turned off(af = ±∞). Short-range physics then does not contribute to the momentum distribution, andwe can consider ballistic effects to stem entirely from the momentum-space version of the initialcondition Eq. (6.17), similar to our analysis in Sec. 6.2. One can find the initial two-body wave
function Ψ̃(k1 , k2 , 0) from Eq. (6.17) via
Ψ̃(k1 , k2 , 0) = ψ̃0 (k)φ̃0 (K)= ψ̃0 (k1 − k2 )φ̃0µ

k1 + k2
2(6.21)¶

where we have changed to separate-particle momentum coordinates k1 and k2 from the relative andcenter-of-mass coordinates k and K. In a semiclassical sense, we expect that atoms with kinetic
energy εki < V0 don’t make it over the barrier. Hence, we estimate that the probability for both
atoms to stay in the central lattice site is given by
Z

PCC →

εk1 <V0

dk1
2π

Z

dk2
|Ψ̃(k1 , k2 , 0)|2 .
2π(6.22)

εk2 <V0

Similarly, we estimate the other transport probabilities to be
Z
Z
dk2
dk1
|Ψ̃(k1 , k2 , 0)|2
PT T →
2π
2π
εk1 >V0

PT C → 2Z

εk1 >V0

εk2 >V0

dk1
2π

Z

dk2
|Ψ̃(k1 , k2 , 0)|22π.

(6.23)

εk2 <V0

These probabilities are plotted as the horizontal dashed lines in Fig. 6.6(a), and they agree reasonably well with the saturation observed in the dynamics.

97

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

0.1

a)

0.2

0

0.1

0.2

b)

Figure 6.6: Lattice transport for a bound state that is quenched from ai = 0.2ℓ to (a) af = ±∞ and(b) af = −ai . The solid blue line denotes the dynamical probability of both atoms occupying thecentral well of the lattice, PCC (t); the solid cyan line is the probability of a single atom occupying
the central well PT C (t); the solid red line is the probability that no atoms occupy the central
lattice site, PT T (t). The horizontal dashed lines correspond to the semiclassical estimates for theseprobabilities.

98
When af is finite, the transport estimates should include only the ballistic contribution to
the momentum distribution. This much is clear from the fact that, in the absence of a quench, thek −4 tail of the momentum distribution contributes to the short range instead of to transport. Moregenerally, the momentum distribution has a mixture of short-range and ballistic effects, as shownin Eq. (6.9) to leading order. We have found that, in most cases, an accurate estimate of transport√
probabilities requires going beyond leading order so as to suitably include momenta k ∼ 2mV0 /~.
(S)

In using Eqs. (6.21) and (6.23), we replace ψ̃0 (k) with the full ballistic wave function ψ̃bal (k, t)derived in Appendix A and given by Eq. (A.14). For Eq. (6.22), we make this same replacement
and also add in the probability that the atoms remain bound after the quench, since bound atoms
will remain in the central well during the short timeframe of quench-induced transport.7

The

resulting estimates for the case of af = −ai are plotted as dashed lines in Fig. 6.6(b). The
increased transport that occurs for this quench is well described by the semiclassical estimate. Thisagreement owes itself to the fact that a wave of energy εk incident on a potential barrier of height
V0 has near unity transmission for εk ≫ V0 . These waves dominate the integrals in Eq. (6.23) whenthe quench is strong.
It is interesting that the saturation timescale in Figs. 6.6(a)-(b) does not appear to dependon the final scattering length of the quench. We have found that the saturation time is well√approximated by the time it takes an atom of momentum k = 2mV0 /~ to travel one lattice spacing.
This supports our semiclassical description of quench-induced transport. For the lattice depth usedin our simulations, the saturation timescale is smaller than the lowest-band tunneling time by morethan two orders of magnitude. The higher-band physics at play in this transport process comesfrom our use of strong interactions [95, 46] as well as from the quench itself [121, 122, 141].

6.5

Summary
In this chapter, we have taken a wave-function-based approach to describe the correlation

waves induced by an interaction quench. Our calculations made use of the zero-range approxima7

Recall that this binding probability vanishes when the post-quench interactions are repulsive (af < 0).

99tion for particle-particle interactions, represented here with a scattering-length-dependent boundary condition at vanishing particle separation. Within this approximation, the interaction quench
disturbs the boundary and generates a wave that propagates ballistically to nonzero particle separations. We have derived the leading-order behavior of this wave in momentum space for one, two, andthree spatial dimensions. These results are intuitive in that the amplitude of the correlation wave
depends only on the initial amplitude at the boundary and the scattering length before and after
the quench. In each dimensionality, the ballistic contribution to the wave function dominates thenext-to-leading-order terms that occur in equilibrium systems. Particularly interesting is the fact
that, in one dimension, the k −2 tail of the momentum-space wave function is generally determinedby both short-range and ballistic effects. Similar results can occur in two and three dimensions,depending on the quench sequence. It is significant that a protocol as simple as a quench can surprise the intuition that usually associates large-momentum behavior exclusively with short-range
physics. On this account, our two-body calculations indicate that one must exercise care wheninterpreting the contact out of equilibrium. We note that the contact dynamics we computed in
the previous two chapters are well defined, as ballistic effects were subleading to the short range inthose cases.
Our simulations reveal that quench-induced correlation waves can cause considerable transport in a 1D optical lattice. The amount of transport that takes place is readily tunable by altering
the initial short-range pair probability of the state, as well as the strength of the quench. Our
analytic two-body calculation makes possible a semiclassical framework within which both thetransport and the saturation time can be estimated with surprising accuracy. We expect thatsimilar results hold for optical-lattice systems in higher dimensionalities whose numerical calculations are more challenging. It would be interesting to see what role these ballistic dynamics mightplay in a quenched many-body system. For example, the system described in Ref. [163], which
was essentially a Mott insulator of molecules in a lattice, might have phase coherence partiallyrestored by the colliding ballistic waves that a quench might generate. One can expect generallythat ballistic waves should be damped by collisions in a many-body system. This damping is dif-100
ficult to model quantitatively without introducing a certain amount of arbitrariness to the theory[137, 138, 92, 94, 93]. At the same time, it is the crux of the question of how isolated quantum
many-body systems equilibrate despite the high level of excitation provided by a quench. It may
be possible to shed light on the matter by investigating how ballistic waves collide even at the
few-body level. This remains for future work.

Chapter 7

Conclusion

The recurring theme of this thesis is that dynamically-tuned interactions can lead to interesting nonequilibrium phenomena. Our studies have focused on interaction quenches, where
the scattering length a is suddenly changed and the system is allowed to evolve with the newinteractions. When applied to Bose-condensed systems, this procedure has been used to createanti-clustered attractive states in 1D [76], generate time-dependent correlation waves in 2D [85],and probe the resonant limit (a → ∞) of a Bose-Einstein condensate (BEC) in 3D [112]. Theseground-breaking experiments, while advancing our understanding of quantum many-body physics,
inspire further investigations of what an interaction quench does to a quantum state.
Our first results dealt with the short-time dynamics of a BEC that is quenched to unitarity.
We studied this system using a time-dependent, many-body variational approach that encodes
pair correlations. Although we incorporated the resonant interactions with several short-range
model potentials, the time evolution of the momentum distribution was found to be essentiallyindependent of the model, and it scaled universally with density-derived units. We also studied theevolution of short-range pair correlations via the dynamical contact, which grows and saturates ina way that is quantitatively consistent with the largest momentum modes measured in the 85 Rb
experiment [112].
We then shifted gears to study the contact dynamics of BECs that are quenched to small
scattering length. Although one might expect time-dependent Bogoliubov theory to be adequate inthis regime, we found that this is not always the case. Fast quenches allow pairs of atoms to project102
nontrivially onto the Feshbach-molecular bound state, and this can lead to short-range dynamics
that are absent in Bogoliubov theory. Our many-body simulations revealed that the contact exhibits
high-contrast oscillations at the frequency of the bound state, and this translates into oscillations of
the condensate fraction. We then set up an intuitive two-body model that yielded a quantitatively
accurate formula for the oscillatory contact, while highlighting the important role played by thebound state in the dynamics.
Our two-body framework was then applied to investigate the nonlocal correlation waves
that are generated by a sudden quench. We were able to represent these waves analytically inmomentum space for arbitrary initial and final scattering length. Our phase-space analysis revealed
that they travel ballistically, ie. free-particle-like. One unexpected result was that these long-range
correlation waves can contribute to the k −4 tail of the momentum distribution, depending on thequench protocol. This finding is at odds with the ideas surrounding Tan’s contact, which relate thelarge-k asymptotics exclusively to short-range pair correlations. Our result, which came from anexact solution to the Schrödinger equation, provides a valuable demonstration of how equilibriumbased intuition about wave functions can break down in a nonequilibrium context. Our analytic
calculation of correlation waves additionally lends itself to quantitative predictions for quenchinduced transport over potential-energy barriers. A possible extension of this work would be toinvestigate how ballistic waves collide with each other at the few-body level, as such collisions are
expected to occur in many-body systems and constitute the first steps towards equilibration.
It would be worthwhile to generalize the ideas of this thesis to include three-body effects
related to Efimov physics. Several decades ago, Efimov showed that there is a ladder of three-bodybound states that accumulates near the dissociation threshhold at large scattering length [54, 55].
This effect has been shown to alter the Tan relations by introducing a three-body contact parameterto account for short-range triplet probabilities [18]. Attempts to measure the three-body contact ina weakly-interacting BEC yielded a value that was consistent with zero [174]. At large scattering
length, however, one might expect the ladder of Efimov states to play a more significant role in
the physics. In the same sense that we observed growth of the two-body contact after quenching

103
to unitarity, the three-body contact might also grow appreciably, albeit over a longer timescale. It
should be possible to describe this growth at short times within a calibrated three-body calculation,
similar to our approach in Chapter 5. Although it is unlikely that an analytical solution will befound for such a calculation, the problem could still be solved numerically by the methods describedin Ref. [150].

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Appendix A

Two-Body Solution after a Quench

Several results from Chapters 5 and 6 rely on an analytic solution to the two-body quench
problem in momentum space. Dimensionality does not play a significant role in the strategy ofthe derivation. The general idea, already outlined at the beginning of Section 5.3.1, is to expand
the wave function in the eigenstates of the zero-range model. One can always write down energynormalized scattering states, and there is also a discrete bound-state solution for a > 0. The difficult
part of the problem involves evaluating the integral that superposes the scattering solutions, but
we will see that some methods from complex analysis can simplify the calculation. These ideas
do not depend directly on the dimensionality of the problem; hence, without loss of generality, we
will work through the details for the case of a quench in 1D. At the end of this Appendix, we will
describe what alterations are necessary to solve the problem in 3D and 2D. We simplify expressions
2~
by scaling distances by an arbitrary length scale ξ and energies by 2µξ
2 , where µ is the reduced

mass for the pair.

A.1

Solving the Problem in 1D
In free space, the time-dependent Schrödinger equation for the relative wave function ψ(x, t)isi∂ψ(x, t)∂ 2 ψ(x, t)2=−− δ(x)ψ(x, t)
2
∂t
∂x
af

(A.1)

where af is the 1D scattering length after the quench. Without loss of generality, we will consider
symmetrized initial conditions, such that ψ(x, 0) = ψ(−x, 0). This symmetry is preserved by the

116
Schrödinger equation whether or not the two particles are identical bosons. The overall effect of
the interaction is to enforce the log-derivative boundary condition shown in Eq. (6.8), which is¯∂x ψ ¯¯1=− .
¯ψ x→0+a(A.2)

This can be proved by integrating Eq. (A.1) over a vanishingly small interval [−ǫ, ǫ] and assuming
even symmetry.
We can propagate a given initial condition ψ(x, 0) in time by expanding in the energy eigenstates that satisfy the post-quench log-derivative boundary condition. The scattering states are£¤(S)ψk′ (x) = Ak′ sin(k ′ |x|) − k ′ af cos(k ′ x) ,Ek′ = k ′2 ,

(A.3)

where
1
Ak′ = q2πk ′ (1 + k ′2 a2f )(A.4)

is a constant that enforces energy normalization. These states are uniquely defined for k ′ > 0. For
af > 0, the bound state solution is1ψB (x) = √ e−|x|/af ,af

1
EB = − 2 .
af

(A.5)

It will be helpful to decompose the time-dependent wave function onto its scattering and boundcontributions, which we will treat separately. Thus, we write [117]ψ(x, t) = ψ (S) (x, t) + ψ (B) (x, t),

(A.6)

Z∞

(A.7)

where
ψ
and

(S)

(x, t) ≡

0

¯
E
D
¯
(S)
(S)
dEk′ e−iEk′ t ψk′ (x) ψk′ (x′ )¯ ψ(x′ , 0)¯­®
ψ (B) (x, t) ≡ Θ(af )e−iEB t ψB (x) ψB (x′ )¯ ψ(x′ , 0) ,(A.8)

and where h·|·i denotes a projection integral. The Heaviside function Θ(af ) determines whether or
not the bound state should be included in the dynamics. We will assume that ψ(x, 0) is normalizable
and smooth everywhere except possibly for a nontrivial log-derivative at x = 0.

117
It is most convenient to solve for the momentum-space wave function. This requires taking
the Fourier transform of the energy eigenstates. For af > 0, the Fourier transform of the boundstate can be inferred from Eq. (6.4)√
2 af.
ψ̃B (k) =1 + k 2 a2f(A.9)

The Fourier transform of the scattering states takes a more complicated form, but it can be written
as
·
¢
¡
(S)
ψ̃k′ (k) = Ak′ −2k ′ π i + k ′ af δ(k ′2 − k 2 ) +2k ′k ′2 − k 2 − iǫ¸,

(A.10)

where we use the convention ǫ → 0+ . This formula can be verified by taking the inverse Fouriertransform. The first term can be easily integrated due to the Dirac delta function, and the second
term can be integrated by using the residue theorem [2]. One must close the contour in the upper
half plane for x > 0, and the lower half plane for x < 0. The pole structure in (A.10) leads to theabsolute-value dependence in the position-space scattering state (A.3). With these ingredients, we
can rewrite the time-dependent wave function (A.6)-(A.8) in momentum space:ψ̃(k, t) = ψ̃ (S) (k, t) + ψ̃ (B) (k, t),(A.11)

Z∞(A.12)

where
ψ̃

(S)

(k, t) =

0

and

¯
D
E
¯
(S)
(S)
dEk′ e−iEk′ t ψ̃k′ (k) ψk′ (x′ )¯ ψ(x′ , 0)¯®
­ψ̃ (B) (k, t) = Θ(af )e−iEB t ψ̃B (k) ψB (x′ )¯ ψ(x′ , 0) .

(A.13)

Having subdivided the scattering eigenstate as in (A.10), there are thus two parts that compose the scattering contribution ψ̃ (S) (k, t) in Eq. (A.12). The first part can be evaluated triviallyin momentum space by exploiting the delta function in Eq. (A.10). We write it as follows:
¯
E
D
¯
(S)
(S)
ψ̃bal (k, t) ≡ −2π|k|A|k| (i + |k|af ) ψ|k| (x′ )¯ ψ(x′ , 0) e−iEk t .

(A.14)

118

Figure A.1: Integration contour in the complex k ′ plane. The crosses denote the poles of thebracketed factor in Eq. (A.15). The final leg of the integral is along arg[k ′ ] = −π/4.

We call this the “ballistic” contribution to the wave function due to its free-particle-like f (k)e−iEk tbehavior, similar to Eq. (6.4). The second contribution can be written as(S)(k, t) =
ψ̃sr

Z∞

′

′ −ik′2 t

2k dk e

0

·

¸D
¯
E
2k ′ Ak′(S) ′ ¯
′
ψ
(x
)
ψ(x
,
0)
¯
,
′
k
k ′2 − k 2 − iǫ(A.15)

and we will see that it generally contributes to the short-range part of the wave function. In sum,
we can write(S)(S)ψ̃(k, t) = ψ̃bal (k, t) + ψ̃sr(k, t) + ψ̃ (B) (k, t).

(A.16)

for the full momentum-space wave function.
One can make progress with Eq. (A.15) by exploiting the residue theorem. The integration,
as written, is along the positive real k ′ axis. If we close the contour as shown in Fig. A.1, thecontribution from the |k ′ | → ∞ arc vanishes. The only poles that can contribute residues mustcome from the analytic continuation of the scattering projection inside the integration loop. We
have found empirically that, if ψ(x, 0) decays smoothly and without oscillation (such as for a boundstate, a ground-state Busch wave function [22], or forms analogous to Eq. (5.11)), the integrand isanalytic inside the closed contour and the integral vanishes.1 The two straight legs of the integral
1(S)

More generally, the residues from any poles inside the contour can be absorbed into the definition of ψ̃bal (k, t).
This occurs, for example, if ψ(x, 0) already has a ballistic component. We will ignore such exotic initial conditions
in our analysis.

119πthen cancel, and we can rewrite Eq. (A.15) as an integral along k ′ = ze−i 4 for real, nonnegative z:
π
(S)
ψ̃sr
(k, t) = −4ie−i 4

=

−i π4

4ie
k2

Z∞

2

dze−z t

0

Z∞

dz

¯
h
Ei
D
z2
(S) ′ ¯
′
′
A
ψ(x
,
0)
ψ
(x
)
¯
′
π
k
k
−iz 2 − k 2
k′ 7→ze−i 4

2
e−z t

1 + iz 2 /k 20hz 2 Ak′

D

¯
Ei
¯
(S)
ψk′ (x′ )¯ ψ(x′ , 0)′

(A.17)π

k 7→ze−i 4where the factor in brackets has been analytically continued. Mathematica can evaluate this quantity in closed form for several interesting cases, including where ψ(x, 0) takes a form similar toEq. (5.11) or is a bound state. Unfortunately, the expressions are usually too lengthy to usefully
write down.
(S)

The physical significance of ψ̃sr (k, t) can be seen if one compares it with the zero-range
contribution to the scattered wave function, ψ (S) (0, t). Using Eq. (A.3), we can write the result asψ(S)(0, t) =

Z∞
0

¯
E
D
¯
(S)
(S)
dEk′ e−iEk′ t ψk′ (0) ψk′ (x′ )¯ ψ(x′ , 0)

= −af

Z∞
0

¯
E
D
¯
(S)
dEk′ e−iEk′ t k ′ Ak′ ψk′ (x′ )¯ ψ(x′ , 0)−i π4

= 2iaf e

Z∞
0(A.18)

¯
h
Ei
D
2
¯
(S)
dze−z t z 2 Ak′ ψk′ (x′ )¯ ψ(x′ , 0)′π

k 7→ze−i 4where we have again exploited the integration contour in Fig. A.1. A direct comparison of Eq. (A.17)
and Eq. (A.18) indicates that
2ψ
(S)
ψ̃sr
(k, t) =(S) (0, t)

af k 2

+O

µ

1
k4

¶

(A.19)(S)

for large k satisfying k 2 t ≫ 1. This verifies our claim that ψ̃sr generally encodes the short-range
behavior of the scattered wave. The Gaussian suppression in Eqs. (A.17)-(A.18) indicates that
this contribution to the wave function vanishes in the t → ∞ limit. This is as expected for an
unconfined wave packet composed entirely of scattering states, which must spread out in space astime passes. With significantly less work, one can combine (A.5) and (A.9) to show thatψ̃(B)

2ψ (B) (0, t)(k, t) =+O
af k 2µ1
k4¶(A.20)

120
for the bound-state contribution to the dynamical wave function.
We now examine the large-momentum behavior of the ballistic contribution to the wave
function, given by Eq. (A.14). This requires finding the asymptotics of the projection integralZ∞−∞dx [sin (k|x|) − kaf cos (kx)] ψ(x, 0)(A.21)

for a symmetric wave function whose short range behaves as ψ(x, 0) ≈ ψ(0, 0)(1−|x|/ai ), and whoselong range is regular and smooth. The second part of this integral is reminiscent of Eq. (6.7), andevaluates to
−kaf

Z∞

−∞

dx cos (kx) ψ(x, 0) = −kaf
= −kafZ∞

dxeikx ψ(x, 0)(A.22)−∞·

2ψ(0, 0)+O
ai k 2

µ

1
k3

¶¸

as discussed in Ref. [129]. The first part of the integral can be understood as we integrate by partsZ∞dx sin (k|x|) ψ(x, 0) = 2−∞

Z∞dx sin (kx) ψ(x, 0)

0

¯∞
µ
¶
¶
Z∞ µ
¯
cos (kx)cos (kx)=2 −ψ(x, 0)¯¯ − 2 dx −
ψ ′ (x, 0)
k
k
0
,
0
Z∞
2ψ(0, 0) 2+=
dx cos (kx) ψ ′ (x, 0)
k
k
0
µ ¶
1
2ψ(0, 0)+O=k
k2

(A.23)

where we have assumed that ψ(x, 0) vanishes in the x → ∞ limit, and that ψ ′ (x, 0) is smooth anddecays at large x. Combining (A.22) and (A.23), we find thatZ∞−∞

2ψ(0, 0)dx [sin (k|x|) − kaf cos (kx)] ψ(x, 0) =
k

µ
¶
µ ¶
af
1
1−
+O
ai
k2

(A.24)

for the projection asymptotics. This leading-order behavior encodes the mismatch between theinitial and final boundary conditions of the quench. Inserting this result into Eq. (A.14), we findthat(S)ψ̃bal (k, t) =

µ

af
−1
ai

¶

2ψ(0, 0)e−iEk t + O
2(k af − i|k|)

µ

1
k3

¶

after some algebra. Combining Eqs. (A.19), (A.20), and (A.25), we arrive at Eq. (6.9).

(A.25)

121

A.2

Generalizing to 3D
The solution described in the previous section generalizes naturally to three dimensions. The

reason for this is that the 1D wave function ψ1D (x, t) and the 3D wave function rψ3D (r, t) satisfy
the same equations of motion and log-derivative boundary conditions. One can compare the 3D
eigenstates in Eqs. (5.4)-(5.5) to those in 1D, given by Eqs. (A.3)-(A.5). The main difference is in
the definition of the Fourier transform in 3D, which leads to slightly different expressions for thebound and scattering states in momentum space:ψ̃B (k) =(S)ψ̃k′ (k) = Bk′

·

q
8πa3f

(A.26a)1 + k 2 a2f

4πk ′ af
2π 2
(1 − ikaf ) δ(k − k ′ ) + ′2kk − k 2 − iǫ¸

(A.26b)where the normalization constant is given by1
Bk′ = r
³
´.
2
2
′
2
4π k 1 + k af

(A.27)

Otherwise, the solution proceeds as outlined in Section A.1. The results of Chapter 5 can be foundby following the steps of the derivation up to Eq. (A.17) and performing the necessary integrals with
the initial conditions given in Eq. (5.11). To derive the large-k asymptotics shown in Eq. (6.13),
one finds from the contour integration that
¡ (S)
¢¯
µ ¶
¯
4π
rψ
(r,
t)
1
(S)
r→0+
ψ̃sr (k, t) =
+O
2
k
k4
¡
¢¯
µ ¶
4π rψ (B) (r, t) ¯r→0+1(B)ψ̃ (k, t) =
+O
2
k
k4

(A.28a)(A.28b)in analogy with Eqs. (A.19)-(A.20).

A.3

Generalizing to 2D
The generalization to two dimensions is more subtle. First, the scattering and bound wave

functions are defined in terms of Bessel functions, so there is no immediate functional connection

122
between the 2D and 1D eigenstates. As an added complication, there are a couple different conventions in the literature for defining the 2D scattering length for zero-range interactions. We choose
2~
to follow the convention of Ref. [114], in which the bound state has energy EB = − 2µa
2 . Withinfthis framework, the bound state can be written as
1

K0ψB (ρ) = q
πa2f

µ

ρ
af

¶

,

(A.29)and the scattering states can be written as(S)ψk′ (ρ) = Dk′

where

·

¸
π/2
′
N0 (k ρ) ,J0 (k ρ) −ln (k ′ af )′(A.30)¸2 !#−1/2
π/2
Dk′ = 4π 1 +ln (kaf )"Ã·(A.31)

enforces energy normalization.2 We write them asq
4πa2fψ̃B (k) =1 + k 2 a2f
2π(S)ψ̃k′ (k) = Dk′
ln (k ′ af )·π´1δ(k − k ′ ) ³ ¡ ′ ¢ln k af + i− ′2
k2
k − k 2 − iǫ(A.32a)¸(A.32b)in momentum space. These transforms can be verified with the aid of Mathematica. The solutionof the quench problem then proceeds as in Section A.1. To derive the large-k asymptotics shownin Eq. (6.12), the contour integration in Fig. A.1 reveals that³ (S)´¯ψ (ρ,t) ¯µ ¶2π ln(ρ/b) ¯
1
ρ→0+
(S)
ψ̃sr (k, t) = −
+O
2
k
k4
³ (B)
´¯
(ρ,t) ¯µ ¶2π ψln(ρ/b)
¯
1
ρ→0+
(B)
ψ̃ (k, t) = −
+O
2
k
k4

(A.33a)(A.33b)where b > 0 is an arbitrary length scale that makes the argument of the logarithm dimensionless.
These relations are analogous to Eqs. (A.19)-(A.20).

2

For small ρ, these states behave as ψ(ρ) ∝ ln

h

ρeγ
2af

i

+ O(ρ). The logarithms that appear in 2D constrain the

scattering length to be strictly nonnegative. See, for example, Ref. [133].