Theoretical Studies of a Method toMeasure the Superfluid Fraction of an

Ultracold Atomic Gas

Diplomarbeit

S.T. John

29. November 2012

Fachbereich Physik

Freie Universität Berlin

Betreuer: Prof. Felix von Oppen, PhD

Preface

This thesis is based on work carried out while staying at the Cavendish Labora-

tory in the University of Cambridge, UK. Supervision in Cambridge was given

by Prof. Nigel Cooper (Theory of Condensed Matter group), with support

from Dr. Zoran Hadzibabic (Atomic, Mesoscopic and Optical Physics group).

A condensed version of the work presented in this thesis has been published

in Phys. Rev. A 83, 023610 (2011).

I am very grateful to my supervisor in Berlin, Prof. Felix von Oppen, for

making it possible to work on and write my Diplom thesis at the University of

Cambridge, UK. My thesis topic grew out of collaboration with Prof. Nigel

Cooper. Many thanks to him for suggesting such a rich topic, and being a

wonderful “unofficial” supervisor all-around. Likewise, Dr. Zoran Hadzibabic

also deserves many thanks for lots of engaging discussions and helpful sug-

gestions. Both were of immense help in bringing this project to fruition. Also,

thanks to the Theory of Condensed Matter group in Cambridge for hosting me

throughout the summer of 2010.

Additionally, I appreciate the generous support by the Studienstiftung des

deutschen Volkes, which made my stay in Cambridge so much easier.

Finally, I would like to thank my wonderful family for all their help and

encouragement!

Contents

1 Introduction 1

2 Background 42.1 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Measuring the superfluid density . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Simulating rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Optically dressed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Corrections 18

4 Interactions 234.1 Weak interactions with asymmetric dispersion . . . . . . . . . . . . . . . . 25

4.1.1 Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . . 25

4.1.2 Depletion of the condensate . . . . . . . . . . . . . . . . . . . . . 27

4.2 Popov approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Critical velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Results 315.1 Weakly interacting BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Strongly interacting BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3 Optimal experimental parameters . . . . . . . . . . . . . . . . . . . . . . . 35

6 Summary 38

Appendices

A Derivation of the interacting distribution function 39A.1 Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A.2 Depletion of the condensate . . . . . . . . . . . . . . . . . . . . . . . . . . 44

B Perturbation theory 47

References 49

1 Introduction 1

1 Introduction

Ultracold atomic gases provide an attractive “playground” for fundamental research: their

weak interactions allow for a simple theoretical description, yet they offer wide experimental

flexibility. (For example, the interaction strength can be varied by changing the atomic species

or employing Feshbach resonances [1]. By changing the geometry of the trap holding the gas,

lower dimensionalities can be considered.) In this highly controllable experimental setting, a

variety of many-body quantum phenomena can be studied, often analogous to phenomena in

other systems, in particular solid state materials and liquid helium [2]. Hence ultracold atomic

gases can act as a “quantum simulator”, allowing access to physical regions of theoretical

interest which could not be reached before.

The most well-known quantum phenomenon occurring in ultracold atomic gases is Bose-

Einstein Condensation: when a bosonic gas is cooled to less than its transition temperature,

the ground state suddenly becomes macroscopically occupied. This Bose-Einstein Con-

densate (BEC) behaves like a “giant matter wave”, such that interference between multiple

condensates can be observed [3].

Closely linked to this, there is superfluidity, a set of hydrodynamic phenomena, the most

famous of which is dissipation-free flow. Yet the connection between these two effects is

subtle. For example, a strongly-interacting 4He liquid well below its transition temperature to

superfluidity is almost 100% superfluid while the condensate fraction is only about 10% [4].

In contrast, an ideal non-interacting Bose gas does not exhibit superfluidity at all, and in a

porous medium (as modelled by a random potential) the superfluid fraction of a BEC vanishes

as soon as the condensate has been depleted to just 25% of the total density [5].

To further the understanding of the physics involved in many-body quantum systems, an

interplay between experiment and theory and thus quantitative measurements are important.

However, while the theoretical descriptions of ultracold atomic gases and other systems can

be remarkably similar, the available experimental probes differ widely. In particular, there is

no established method for measuring the superfluid fraction of ultracold atomic gases.

Traditionally (e.g. in liquid helium), superfluidity is measured by the response of the fluid

to rotation. As long as the walls of the container are rotated slower than the critical velocity

for superfluid flow, the superfluid fraction of the fluid remains in its zero angular momentum

state. Only the normal fraction picks up angular momentum and contributes to the moment

of inertia. This is exploited in the Andronikashvili experiment [6], in which the moment of

inertia is measured using a torsional oscillator (see Fig. 1 on the following page, top left). The

proportion of this non-classical moment of inertia to the classically expected value gives the

normal fraction ρn/ρ, and hence the superfluid fraction ρs/ρ = 1−ρn/ρ (see Fig. 1, bottom

left).

Qualitatively, superfluidity in BECs has already been demonstrated by the existence of

vortices forming in rotated gases [7]. However, due to the intricacy of measuring mass flow,

an accurate quantitative measurement of superfluidity in atomic gases has proved difficult.

2 1 Introduction

Figure 1: In Andronikashvili’s classic experiments on the measurement of superfluidity [6], astack of disks is rotated in a fluid using a torsional oscillator (top left). Only the normal fractionof the fluid is dragged along and contributes to the moment of inertia. By relating the measuredangular momentum with the classically expected moment of inertia, the superfluid densityρs can be determined as a function of temperature (bottom left). Cooper and Hadzibabic [8]propose a related measurement method for ultracold atomic gases in which an atomic ringcurrent is induced by laser beams with orbital angular momentum (right). Coupling ofthe light beams with hyperfine states of the atoms then allows a spectroscopic detection ofthe superfluid fraction. [Reprinted with permission from I. Carusotto, Physics 3, 5 (2010).Illustration: Alan Stonebraker. Copyright 2010 by the American Physical Society.]

In Ref. [8], N. R. Cooper and Z. Hadzibabic recently proposed a novel method for

measuring superfluidity in ultracold atomic gases. The key idea of the proposal is inducing

slow rotation by an artificial gauge field. Raman transitions between hyperfine states imprint

an azimuthal vector potential onto the atoms, thereby simulating the switch to a rotating

reference frame (see Fig. 1, right). Importantly, this couples the external (angular momentum)

with the internal (hyperfine state) degrees of freedom. Therefore the occupational difference

between the hyperfine states can be directly related to the average angular momentum. This

allows a spectroscopic measurement of the superfluid fraction, playing to the strengths of

atomic physics. Hence the Cooper-Hadzibabic approach allows a quantitative measurement

of the superfluid fraction of ultracold gases, similar to the Andronikashvili experiment and

closely following the traditional definition [9].

The original proposal [8] assumed a non-interacting gas on a one-dimensional ring at

1 Introduction 3

zero temperature. In my thesis I extend this proposal in several ways.

First, I consider a uniform three-dimensional gas and include in my calculations the

effects of non-zero temperature and interactions. These broaden the distribution of angular

momenta around zero for a metastable superfluid, and around the classical value set by the

imposed rotation for a fully relaxed gas. This allows me to assess the quantitative accuracy of

the proposed measurement. Furthermore, I estimate experimental parameters which lead to a

good compromise between the theoretical accuracy and the experimentally relevant size of

the spectroscopic signal.

Second, I explicitly calculate the expected experimental signal in two important cases. On

the one hand, calculations for a weakly interacting Bose gas show that the spectroscopically

deduced superfluid fraction closely follows the condensate fraction below the critical temper-

ature. This confirms that the method gives the expected results in this well-understood limit.

On the other hand, in the limit of strong interactions the superfluid and condensate fractions

are decidedly different from each other. I demonstrate that the spectroscopic measurement

allows a clear experimental distinction between the two.

The remainder of this thesis is organised as follows. In Chapter 2, the theoretical back-

ground is laid out and the proposed spectroscopic measurement is explained in detail. In

Chapter 3, quantitative theoretical corrections to the mapping between hyperfine populations

and the superfluid fraction are investigated for a normal gas. This leads to suitable experi-

mental parameters which balance signal size (and hence precision) with the accuracy of the

method. Chapter 4 contains extensions and enhancements to the original proposal, including

the effect of interactions in our model. In Chapter 5, numerical results for the expected

experimental signals are analysed for both weakly and strongly interacting gases. Finally,

Chapter 6 gives a brief summary of this work. The appendices contain detailed derivations.

4 2 Background

2 Background

In the first part of this chapter, we give a short overview of Bose-Einstein Condensation (Sec-

tion 2.1) and superfluidity (Section 2.2). The second part explains the method of superfluidity

measurement that will be analysed in the following chapters of this thesis. We begin with

measurements of the superfluid density (Section 2.3), and then describe the general idea of

simulating rotation with artificial gauge fields (Section 2.4). Finally, we go into further detail

for the simulation of rotation using two-photon Raman transitions in a system of atoms with

three hyperfine levels (Section 2.5).

2.1 Bose-Einstein Condensation

Bosonic particles are particles with integral spin and hence with a total wave function which

is symmetric under particle exchange. They obey Bose-Einstein statistics: at a temperature T

the average occupation of a state ν with energy εν is given by the distribution function

nB(εν,µ) =1

exp[(εν−µ)/kT ]−1. (2.1)

The chemical potential µ is implicitly defined by the total number of particles N; it has to be

chosen such that N =∑ν nB(εν,µ). For the distribution function to remain physical (that is, to

avoid divergences of the occupation number), the chemical potential always has to be less

than the ground-state energy ε0. Without loss of generality we can set ε0 = 0, and therefore

we require µ < 0.

In the thermodynamic limit N →∞, V →∞, and the density n = N/V kept constant

(where V is the volume of the system), the states become infinitesimally close in energy. Then

the sum over states can be replaced by an integral,∑ν→

∫ ∞0 dε g(ε), where g(ε) gives the

density of states at energy ε.

For a uniform gas in a three-dimensional box, the density of states is

g(ε) =Vm3/2

21/2π2~3 ε1/2, (2.2)

where m is the mass of the particles.1 The ground state at ε = 0 has zero weight and is not

taken into account by the integral; therefore we have to consider the ground state explicitly.

The total number of particles is given by

N = N0 + Nex, (2.3)

where N0 is the ground-state occupation, and Nex is the number of particles in excited states.

Above the condensation point, the ground state is not macroscopically occupied and can

1In general, the density of states for a uniform gas in d dimensions is g(ε) ∝ εd/2−1; for a d-dimensionalharmonic trap, the density of states is g(ε) ∝ εd−1 [3].

2.1 Bose-Einstein Condensation 5

be neglected. Thus the chemical potential µ has to be chosen such that N = Nex(T,µ), where

Nex(T,µ) =

∫ ∞

0dε g(ε)nB(ε,µ). (2.4)

As the chemical potential has an upper bound µ = 0, the integral in Eq. (2.4) cannot exceed

a temperature-dependent maximal value. As a result, a critical temperature Tc is implicitly

defined by

N = Nex(Tc,µ = 0). (2.5)

At this temperature a phase transition occurs: below the critical temperature Tc, the excited

states can no longer accommodate all N particles.2 Thus the ground state becomes occupied

by a macroscopic number of particles N0. (This means that in the thermodynamic limit, the

ground state has a fractional population N0/N greater than zero.) This is the Bose-Einstein

Condensate.

The solution of Eq. (2.5) for the uniform gas in three dimensions yields [with ζ(3/2) ≈

2.612]

kTc =2π[

ζ(3/2)]2/3

~2n2/3

m≈ 3.31

~2n2/3

m. (2.6)

The range of a particle’s wave function is described by the thermal de Broglie wavelength,

λT =

√2π~2

mkT, (2.7)

with which we can rewrite Eq. (2.6) as

n λ3T

∣∣∣T=Tc

≈ 2.612. (2.8)

This provides an intuitive picture for BECs: Condensation occurs when the average particle

distance (V/N)1/3 becomes comparable to the de Broglie wavelength, that is, when the wave

functions of atoms begin to overlap significantly.

In a real gas, the particles interact both through long-range van der Waals attraction and

short-range repulsion of the electron clouds of two atoms. The total interaction energy is

given by

Utotal =12

∑i, j

U(ri− r j). (2.9)

In weakly interacting, low-density systems, the interaction potential between two particles

can be well described by a δ peak:

Ueff(r1− r2) ≈ U0 δ(r1− r2). (2.10)

2For a uniform gas in less than three dimensions, the integral in Eq. (2.4) diverges for µ = 0; thereforeBose-Einstein Condensation does not occur. In a harmonic trap, the situation is different and lower-dimensionalBECs do exist. Subsequently, we will focus on the uniform three-dimensional gas.

6 2 Background

In the mean-field approximation, the total interaction energy of the system is given by

Utotal =

∫d3r U0n(r) = U0N. (2.11)

For low-energy scattering, the interaction can be described in terms of s-wave scattering. In

this case the interaction strength U0 can be related to the two-body s-wave scattering length a

by

U0 =4π~2a

m. (2.12)

(In the low-energy approximation, atoms usually are repulsive, so this is a positive quantity.)

The scattering cross-section is given by σ = 4πa2 [3]. Therefore a describes an effective

particle radius. The scattering length is the important interaction parameter in the theory of

Bose-Einstein Condensation.

Another relevant parameter of an interacting BEC is the healing length ξ. This describes

the spatial scale over which the condensate recovers its bulk density, e.g. from the zero-density

condition at “hard wall” boundaries (where the condensate wave function must vanish). This

quantity can be estimated by relating the kinetic energy per particle, which is of order

~2/2mξ2, with the interaction energy, given by nU0 [3]:

~2

2mξ2 = nU0, (2.13)

or

ξ2 =~2

2mnU0=

18πna

. (2.14)

2.2 Superfluidity

Instead of being a single property of a condensed-matter system, superfluidity is the name for

a whole set of related phenomena. Common examples are dissipation-free flow, quantisation

of vortices, and vanishing entropy of the superfluid. Superfluids can be well described by

the two-fluid model. In the two-fluid model, the fluid consists of a non-zero “condensate”

fraction in just one single-particle state (as in a BEC), which carries the superfluid properties,

and a “normal” fraction, which behaves like an ordinary fluid.3 We will return to the two-fluid

model in the next section.

Superfluidity breaks down as soon as an obstacle in the flow of the system provides an

energy high enough to create excitations, thereby allowing dissipation. The flow velocity at

which this happens is given by the Landau criterion [3]. It can be derived as follows.

3However, the superfluid density is not the same as the condensate density N0/V . Depending on the systemunder consideration, there can be a significant difference. For example, as mentioned in Chapter 1, superfluid 4Hewill be almost 100% superfluid at sufficiently low temperatures; however, neutron scattering experiments showthat, due to the strong interactions, the ground state will be depleted to a fraction of only 10% of the total numberof particles. In ultracold atomic gases, on the other hand, there is a vast change in the density profile during thetransition to a BEC, while the onset of superfluidity is much harder to detect [4].

2.2 Superfluidity 7

If a system has energy E and momentum p in one frame of reference, then the energy in a

frame moving with a velocity v relative to the first (corresponding to a Galilean transformation)

is given by

E(v) = E− p · v +12

Mv2, (2.15)

where M is the total mass of the system. We first consider a fluid without any excitations. In

its rest frame, it has the ground-state energy E0 and zero momentum. Therefore the energy of

the system in the rest frame of an obstacle, moving with a velocity v relative to the fluid, is

E(v) = E0 +12

Mv2. (2.16)

Let us now add a single excitation of momentum p and energy εp. Then the total energy in

the condensate rest frame is

Eex = E0 + εp. (2.17)

In the reference frame in which the obstacle is at rest, the total energy for a single excitation

is given by

Eex(v) = E0 + εp− p · v +12

Mv2. (2.18)

Thus the excitation energy in the rest frame of the obstacle is εp− p · v. (This amounts to a

tilting of the dispersion relation of excitations.) The energy of the obstacle in its own rest

frame is zero, so excitations can be induced as soon as εp − p · v becomes negative. Hence

there exists a minimal velocity, the critical velocity

vcrit = minp

(εp

|p|

). (2.19)

Below this velocity no excitations can be created, and the flow remains dissipation-free.

In an ideal Bose gas, the dispersion relation is quadratic; the energy of a particle of

momentum p is given by ε(p) = p2/2m. Therefore the critical velocity vanishes; any superfluid

flow immediately disappears. Hence an ideal Bose gas is not considered a true superfluid [10].

We now consider the flow of a superfluid along a torus. Describing the superfluid fraction

by the single-particle state |χ0〉 with the wave function

χ0(r, t) = |χ0(r, t)|eiφ(r,t), (2.20)

we can define a superfluid velocity vs by

vs(r, t) ≡~

m∇φ(r, t). (2.21)

From this we see that the flow of the superfluid fraction is irrotational (unless the phase of the

wave function has a singularity). Because the wave function must be single-valued, the phase

8 2 Background

difference along a closed path going around the torus has to be a multiple of 2π:

∆φ =

∮∇φ ·dr = 2πn. (2.22)

Here n is the integer giving the topological winding number of the phase φ. This means that

the circulation,

κ =

∮vs ·dr =

~

m2πn = n

hm, (2.23)

is quantised in units of h/m. It can be shown that the quantisation of the circulation corre-

sponds to (quantised) vortex flow. It is not possible to smoothly change the wave function

from n = 0 to n = 1, therefore the state with n = 0 is metastable. At sufficiently low speed,

when the rotational energy is smaller than the excitation energy of a single vortex, the

superfluid will not follow the rotation and instead remain inert [3, 10].

2.3 Measuring the superfluid density

The concept of a superfluid density, or superfluid fraction, originates in the two-fluid model

for the hydrodynamics of superfluid 4He, proposed by Tisza [11, 12] and Landau [9]. The

fluid, of total density ρ, is assumed to consist of a superfluid component, of density ρs, which

has vanishing viscosity and flows without dissipation, and a normal component, of density

ρn = ρ−ρs. Landau proposed how to measure these separate components [9]. He envisaged

taking superfluid helium at rest in its container, and slowly rotating the walls at a constant

angular velocity ω. The normal component comes to equilibrium with the rotating walls;

however, the superfluid component is unaffected by the rotation and remains at rest. Since

only the normal component moves, the moment of inertia of the fluid is determined by ρn, and

its ratio to the expected classical moment of inertia (defined by the total density ρ) provides

the normal fraction ρn/ρ and hence the superfluid fraction 1−ρn/ρ. Note that it is necessary

that the trap is not perfectly rotationally symmetric (i.e. the walls must be rough), so that the

normal fluid can relax into the steadily rotating state and come into dynamic equilibrium by

changing its angular momentum.

This method was implemented for superfluid helium in the classic experiments by An-

dronikashvili [6]. In those experiments it was not the container that was rotated, but a stack

of disks embedded in the fluid (see Fig. 1 on p. 2). Still, the disks drag just the normal fluid,

so measurements of the moment of inertia of the disks (using a torsional oscillator) allowed a

determination of the normal and superfluid fractions.

The non-classical moment of inertia arising from the superfluid component provides the

standard definition of the superfluid fraction [4]. To discuss this theoretically, it is customary

to consider the fluid to be contained in a ring-shaped toroidal vessel with a radius R much

larger than its transverse dimensions ∆R, see Fig. 2 on the following page.

In this case, the classical moment of inertia for N atoms of mass m is given by Icl = NmR2.

We shall assume this geometry throughout this thesis, in part for theoretical simplicity, but

2.3 Measuring the superfluid density 9

Aφ

vs=ωeR

R

∆R

Figure 2: Geometry of the vessel considered in this thesis: a torus of radius R with transversedimensions ∆R R. Aϕ is the gauge potential introduced by light fields, corresponding torotation with angular frequency ωeff. This induces a superfluid flow with speed vs in thedirection opposite to Aϕ.

also for practical reasons discussed later. The superfluid fraction is then defined [4] by

the average angular momentum 〈L〉 picked up under rotation of the vessel with an angular

frequency ω:ρs

ρ≡ 1− lim

ω→0

(〈L〉Iclω

), (2.24)

where the fraction 〈L〉/ω gives the measured, non-classical moment of inertia. The limit

ω→ 0 of slow rotation of the vessel is required such that the velocity of the walls of the

container ωR does not exceed the critical velocity of the superfluid: ω < vcrit/R. Since

finite-size effects can be much more important in ultracold atomic gases than in typical

experiments on superfluid helium, it is helpful to expand further on this point. For a very

small rotation rate, namely when ω < ~/2mR2, even an ideal Bose gas shows a non-classical

moment of inertia, and could thus be considered to be superfluid [13]. (For ω < ~/2mR2, the

lowest-energy single-particle state remains the state with vanishing angular momentum, so

an ideal condensate has L = 0.) We define superfluidity in the strongest sense of Ref. [13],

which is the sense that is conventionally applied: the angular momentum is measured with an

imposed rotation frequency in the range ~/2mR2 < ω < vcrit/R. The lower limit excludes the

ideal Bose gas as a superfluid. For a large system, R ~/mvcrit, this constitutes a wide range

of frequencies. For a weakly interacting Bose gas, vcrit ∼ ~/mξ (the healing length ξ will be

further considered in Chapter 4), and so the range of frequencies is wide for R ξ.

This method could, in principle, be applied to ultracold atomic gases, using a rotating

deformation of the trap to represent the rotating walls of the container. However, in practice

this is difficult, as a measurement of the induced angular momentum in ultracold gases is

hard, due to the lack of good probes coupling to the mass flow. While it might be possible to

measure some energy shift depending on the frequency ω, the total energy is very small,

E =12

Iω2 =12

I(~∆`/m∗R2

)2' 0.1µeV.

A signal this small is technically difficult to measure [8].

10 2 Background

Instead, in this thesis we explore a theoretical proposal [8] in which the rotation is

simulated by an optically induced gauge field. A key feature of this method is that it allows

one to measure the angular momentum spectroscopically and hence deduce the superfluid

fraction of an ultracold atomic gas. Since spectroscopic measurements on the atoms in a BEC

are comparatively easy and naturally suited to the strengths of atomic physics, this has the

scope to be a powerful general method.

2.4 Simulating rotation

The basis of the idea is the coupling of light with orbital angular momentum [14] to internal

atomic spin states, thereby creating an azimuthal gauge field Aϕ [15]. An optical gauge

field is analogous to a magnetic gauge field in charged particles, albeit created solely by the

interaction of a particle with a light field; hence optical gauge fields are also applicable to

neutral particles.

The azimuthal gauge field leads to the same effects as rotation, although in a slightly

different manner. In the presence of the gauge field one must make a distinction between

the “canonical” momentum pcan (corresponding to the operator −i~∇) and the “kinetic”

momentum mv = p = pcan− A. In the absence of a gauge field, a superfluid that is at rest in

the toroidal vessel has no winding number of its phase (no vortices) and corresponds to the

case of vanishing canonical momentum pcan = 0. This does not change when the gauge field

is introduced. However, as the gauge field is turned on, the superfluid picks up a non-zero

velocity p/m = −A/m. On the other hand, for a normal fluid, as the gauge field is switched

on, the fluid will always stay at rest with the walls of the container (provided they are rough).

Thus, compared to the rotating container discussed previously, here it is the superfluid that

moves, while the normal fluid stays at rest. The case of the gauge field is, in fact, exactly

equivalent to a rotating vessel, but where one views the system in the rotating frame of

reference and so experiences the trap (and normal fluid) to be at rest.

We will expand on the connection between rotation and gauge fields. The Hamiltonian in

a rotating frame is given by [16]

Hrot = H− L ·ω, (2.25)

where H and L are Hamiltonian and total angular momentum in the laboratory frame. In our

ring geometry, we can write the kinetic energy terms of Eq. (2.25) for a single particle with

L = r× p as

p2

2m− (r× p) ·ω =

p2⊥

2m+

(~`)2

2mR2 −~ω`

=p2⊥

2m+~2

2mR2

(`−

mR2ω

~

)2

−mR2ω2

2. (2.26)

Here we have used p2 = p2⊥ + p2

‖, where p‖ is the momentum in azimuthal direction, p‖ =

|L|/R = ~`/R. (In our ring geometry, p‖ and ` can be used interchangeably with pz, essentially

2.4 Simulating rotation 11

“unrolling” the ring onto a line with periodic boundary conditions. This will be further

discussed in Chapter 4.) Therefore the Hamiltonian in the rotating frame shows a dispersion

relation which is shifted to an energy minimum at an angular momentum `∗ = mR2ω/~. To

simulate rotation, we have to change the dispersion relation such that the azimuthal (angular

momentum) component acquires this shift to a new minimum at `∗.

This is just what happens when the optically induced gauge field is turned on; the atoms

experience an effective dispersion relation which can be written as

E ' E0 +~2

m∗R2

(`2

2− ``∗

)+O(`3), (2.27)

where ` is the angular momentum in units of ~, such that it is quantised to integer values. m∗

is a new effective mass of the atoms, and `∗ is the rotational shift due to the gauge field. (We

will derive this in the next section.) This effective dispersion is equivalent to an azimuthal

gauge field Aϕ = ~`∗/R. It corresponds to being in a frame of reference rotating with an

effective angular velocity

ωeff =~`∗

m∗R2 . (2.28)

For example, a particle with ` = 0 will have an angular group velocity (1/~)dE/d` = −ωeff.

Following the previous discussion, if ωeff is slowly increased from zero, the superfluid will

remain in its original state with 〈L〉 = 0 but will flow with speed vs = ωeffR in the direction

opposite to Aϕ. In contrast, the normal fluid will pick up an average angular momentum of

~`∗ per particle but will retain zero average velocity.

A key feature of the proposal [8] is that since the gauge field is generated by mixing

internal hyperfine states of the atoms, this provides a natural coupling between internal and

external degrees of freedom. Spectroscopically measuring the population of the different

internal spin states allows the average angular momentum per atom ~〈`〉 to be deduced. We

will focus on the population difference of the hyperfine states |+1〉 and |−1〉 in a three-level

system, but this could be adapted to other internal structures. (The three-level system will be

described in further detail in the next section.) For a single atom with angular momentum `

we define the population difference as

∆p` ≡ |ψ−1(`)|2− |ψ+1(`)|2 . (2.29)

Here ψ±1 refer to the amplitudes of the original, uncoupled hyperfine states |0〉, |+1〉 and |−1〉

in the dressed state |ψ〉 when the light is turned on:

|ψ〉 = ψ−1(`) |−1〉+ψ0(`) |0〉+ψ+1(`) |+1〉 . (2.30)

A measurement of the populations N+1 and N−1 for a gas of such atoms leads to the fractional

12 2 Background

population difference ∆p, which may be expressed in terms of ∆p` as

∆p ≡N−1−N+1

N=

∑`〈n`〉

[|ψ−1|

2− |ψ+1|2]∑

`〈n`〉=

∑`〈n`〉∆p`∑`〈n`〉

. (2.31)

Here 〈n`〉 gives the average occupation of states with angular momentum `. Within the

assumption that ∆p` can be expanded to first order,

∆p` ' ∆p0 +∆p′`+O(`2), (2.32)

one can deduce the angular momentum expectation value

〈L〉N~≡ 〈`〉 ≡

∑`〈n`〉`∑`〈n`〉

'∆p−∆p0

∆p′. (2.33)

Putting this back into Eq. (2.24), one gets

ρs

ρ' 1− lim

`∗→0

∆p−∆p0

∆p′`∗, (2.34)

where the appropriate moment of inertia Icl = Nm∗R2 has been used. For a perfect superfluid

we would expect to find ∆p ≡ ∆p0, whereas for a normal fluid we would expect to find

∆p ≡ ∆p0 +∆p′`∗, thus allowing us to distinguish between the two.

The main goal of this thesis will be to analyse the quantitative accuracy of Eq. (2.34),

allowing for corrections that arise from the higher-order terms that are neglected in Eqs. (2.27)

and (2.32). However, for now, note that the spectroscopic technique should show a distinct

qualitative signature of superfluidity. For a normal fluid (or the normal fraction), it does

not matter in which order one increases `∗ and cools the gas to its final temperature: that

is, these two operations “commute”. However, for the superfluid fraction these operations

do not commute: if one first cools at `∗ = 0 and then imposes non-zero `∗, the superfluid

is pushed into a (metastable) state in which it is moving with respect to the walls of the

container; if one first imposes non-zero `∗ and then cools, the superfluid will be formed

at rest with respect to the walls. Thus, depending on the order, the system is led either to

the non-relaxed metastable superfluid condensed in a state of vanishing (canonical) angular

momentum (`c = 0), which we label “SF”, or to the relaxed superfluid, condensed in the

ground state (`c = `∗), which we label “RSF”. The two cases will have different fractional

hyperfine populations, ∆pRSF , ∆pSF, so the change in population allows a clear qualitative

signature of metastable superfluid flow.

2.5 Optically dressed states

There are several well-established theoretical proposals for how optical fields can be used

to create fictitious gauge fields in neutral atomic gases [17]. The scheme we follow here

is closely related to that implemented in the experiments of the Spielman group at NIST

2.5 Optically dressed states 13

[18, 19]. However, it is adapted to generate an azimuthal vector potential, by using optical

beams with orbital angular momentum [15]. As described previously, throughout this work

we assume that the gas is confined in a toroidal trap (cf. Fig. 2) with radius R large compared

to its transverse dimensions ∆R. This simplifies the experimental implementation of the

azimuthal gauge field, as it will be sufficient to require that the optical fields are uniform only

over the range ∆R.

We consider atoms with three hyperfine levels [18] in their electronic ground state (e.g.23Na with F = 1). The degeneracy of the three hyperfine states MF = 0,±1 is lifted by

applying a weak external magnetic field B, thereby inducing a Zeeman shift4 ∆E = ZMF

with an energy gap Z = gFµBB between the hyperfine states. These states are coupled by two

co-propagating Laguerre-Gauss beams with frequencies ω1, ω2 and orbital angular momenta

`1, `2. (Laguerre-Gauss beams have helical phase fronts, hence an azimuthal component

appears in their Poynting vector. This creates an orbital angular momentum along the beam

axis, distinct from the inherent photon spin; often both are combined into a single quantity,

the optical angular momentum [14].)

The frequencies are chosen such that they are detuned from any actual electronic transition,

so single-photon transitions are suppressed. Instead, the hyperfine states are coupled by two-

photon Raman transitions (see Fig. 3). In such a transition an atom absorbs a photon from

one beam, is raised to a “virtual” intermediate state5, and, by stimulated emission of a second

photon, immediately falls down into another hyperfine level of the electronic ground state.

Because of conservation of energy and angular momentum, under absorption of the first

photon, the atom transitions from a state with energy E and angular momentum ` to E +~ω1

and `+ `1; after emission of the second photon, the final state has energy E + ~(ω1 −ω2)

and angular momentum `+ `1 − `2. For every two-photon transition the atom experiences

a net change in its centre-of-mass angular momentum of ∆` = `1− `2, while the change in

linear momentum can be neglected. Hence there exists a coupling between |MF = −1, `−∆`〉

and |MF = 0, `〉 as well as between |MF = 0, `〉 and |MF = +1, `+ ∆`〉. Note that the two

light beams are slightly detuned from the Raman two-photon resonance, with detuning

δ ≡ (ω1−ω2)−Z/~.

In principle, we would also have to include in our description the electronic excited state

which the laser beams couple with the hyperfine-split electronic ground state. However, when

considering two-photon Raman transitions with a large detuning ∆L = (E1 − E0)/~−ω1,2

from the actual energy difference between electronic states, E1−E0, the electronic excited

state becomes a “dark state” and can be integrated out. Hence we only need to consider the

three-level ground state.

4For simplicity, we neglect any quadratic Zeeman effect, assuming that the magnetic field is small.5In two-photon transitions, this is allowed by the energy-time uncertainty principle, ∆E∆t & ~/2, where ∆t is

the difference in the photon arrival times and ∆E is the detuning from the electronic transition energy.

14 2 Background

ħδ

Z

Z

MF=−1

MF=0

MF=+1ħω1

ħω2

virtual level

ℓ−∆ℓ

ℓ

ℓ+∆ℓ

ħδ

E

ħΔL

Figure 3: Sketch of a two-photon Raman transition between hyperfine levels, adapted fromSpielman et al. [18, 19].

The Hamiltonian for the Zeeman splitting of the electronic ground state is given by

Hgs =

−Z 0 0

0 0 0

0 0 +Z

. (2.35)

A general atom-photon interaction is given by

Hint(t) = −d · E(t),

where the electric field strength is

E(t) = E1 cos(ω1t) +E2 cos(ω2t),

and d is the operator for the transition dipole moment [20]. After integrating out the dark

state, the two-photon Raman interaction is given by

Hint(t) = ~ΩR cos[(ω1−ω2)t]

0 1 0

1 0 1

0 1 0

. (2.36)

Here ΩR is the two-photon Rabi frequency which describes the coupling between hyperfine

states. Its strength is given by

~ΩR =E1 · d1∗E2 · d2∗

~∆L, (2.37)

2.5 Optically dressed states 15

where d1∗ and d2∗ are the transition dipole moments between the electronic excited state

(denoted “∗”) and the two hyperfine states involved in the transition. (We assume that all

three hyperfine levels have approximately the same transition dipole moment.)

We apply a Rotating Wave Approximation [21], in which we neglect nonresonant pro-

cesses (in which the atom falls from a higher to a lower level by absorbing a photon, or

rises from a lower to a higher level by emitting a photon). This is achieved by writing

cos(ωt) = [exp(iωt) + exp(−iωt)]/2 and subsequently neglecting the rapidly oscillating expo-

nential terms. We are lead to the following Hamiltonian for the two-photon interaction:

Hint(t) =~ΩR

2

0 exp[−i(ω1−ω2)t] 0

exp[i(ω1−ω2)t] 0 exp[−i(ω1−ω2)t]

0 exp[i(ω1−ω2)t] 0

. (2.38)

To remove the time dependence, we switch to the Interaction picture. We find that the

Hamiltonian for the electronic ground state is given by

ˆHgs = ~

−δ 0 0

0 0 0

0 0 +δ

, (2.39)

where δ = (ω1−ω2)−Z/~ is the effect of the Zeeman splitting. The resulting interaction with

the light field is

HI = ~

0 ΩR/2 0

ΩR/2 0 ΩR/2

0 ΩR/2 0

. (2.40)

We have not yet taken into account that the Laguerre-Gauss beams couple Raman transitions

with changes in angular momentum: the kinetic energy links with the hyperfine states

according to

Hkin =~2

2mR2

(`+∆`)2 0 0

0 `2 0

0 0 (`−∆`)2

. (2.41)

The full Hamiltonian is given by H = Hkin + ˆHgs + HI , and we arrive at

H(`) = ~

~

2mR2 (`+∆`)2−δ ΩR/2 0

ΩR/2 ~2mR2 `

2 ΩR/2

0 ΩR/2 ~2mR2 (`−∆`)2 +δ

. (2.42)

The ` dependence has been made explicit; each atomic state is defined by the amplitudes

of the three hyperfine states and its angular momentum `. For each ` there are three energy

eigenvalues of Eq. (2.42), corresponding to three dressed energy bands.

In Fig. 4, we show the results of a numerical diagonalisation of the Hamiltonian (2.42).

16 2 Background

-2 -1 0 1 2

0

2

4

E/[h2(∆

`)2/M

R2]

ΩR = 2h(∆`)2/MR2

ΩR = 0

`∗

`/∆`

Figure 4: Coupled (solid lines) and uncoupled (dashed lines) energy bands of a three-levelsystem, for δ= 0.5~(∆`)2/mR2 (from [8]). The smooth curves interpolate between the allowedinteger values of `.

The three uncoupled energy levels for ΩR = 0 are shown as dotted lines. When the light is on,

ΩR , 0, these levels are mixed and lead to the energy levels of the dressed states (solid lines).

For non-zero detuning δ the minimum of the lowest band is displaced to a non-zero angular

momentum `∗. Provided that all atoms are restricted to states in the lowest-energy dressed

band, the atoms experience an effective dispersion relation of the form (2.27) in which the

non-zero `∗ plays the role of an azimuthal gauge field.

Throughout this thesis, we assume that only this lowest dressed band is occupied. This

is justified if the chemical potential µ and temperature kT are small compared to the band

splitting, which is of order ~ΩR. Hence we require that ΩR be sufficiently large. The lowest

band will be referred to as ε0‖(`) in Chapter 4. To obtain the results shown in Chapter 5, we

determine ε0‖(`) by the numerical solution of Eq. (2.42). However, to allow an understanding

of the general trends, we derive some analytic expressions which are valid for large ΩR, where

the bands are far apart from each other and the lowest band is nearly parabolic. We carry out

perturbation theory in 1/ΩR, for which the unperturbed Hamiltonian is given by Eq. (2.40),

H0 = HI , and the perturbation is given by H′ = ˆHgs + Hkin. The perturbative analysis is

described in Appendix B, and shows [8] that the minimum of the dispersion relation is shifted

from ` = 0 to

`∗ ' −√

2δ

ΩR∆`+O(1/Ω2

R) , (2.43)

and the bare mass is increased to an effective mass m∗, with

m∗ ' m1 +

√2~(∆`)2

mR2ΩR

+O(1/Ω2R). (2.44)

Similarly, perturbative calculations of |ψ−1(`)|2 and |ψ+1(`)|2 show that they have equal and

opposite contributions linear in `. The difference |ψ−1(`)|2− |ψ+1(`)|2 can indeed be written

2.5 Optically dressed states 17

as a series in `, as in Eq. (2.32), with

∆p0 'δ

ΩR

[√

2−~(∆`)2

mR2ΩR

]+O(1/Ω3

R), (2.45)

∆p′ ' −√

2~∆`

mR2ΩR+O(1/Ω2

R). (2.46)

As parameters representative of experiments, throughout this thesis we consider sodium with

m = 23mp (where mp is the proton mass) in a trap of radius R = 10µm. A typical value of

the effective mass is m∗ ≈ 1.15m (at a two-photon Rabi frequency ΩR = 2π×100kHz and for

∆` = 50).

With this the stage has been set and we can now consider the questions motivating this

thesis: How applicable is this proposal—what is the range of validity of the presented method?

How does the inclusion of non-zero temperature and finite interactions change the results?

18 3 Corrections

3 Corrections

In the preceding chapter, we have summarised the theoretical proposal of Ref. [8]. This

showed how to relate the superfluid fraction to a spectroscopically determined hyperfine

population imbalance [see Eq. (2.34)]. The quantitative accuracy of this relation relies on

the validity of the termination of the Taylor expansions in Eqs. (2.27) and (2.32) at quadratic

and linear orders, respectively. If these (terminated) expansions were exact, the superfluid

fraction would be perfectly determined by Eq. (2.34). In practice, higher-order corrections do

exist, that is,

E = Eparabolic + c`3 + · · · (3.1)

∆p` = (∆p`)linear + c′`2 + · · · (3.2)

where Eparabolic and (∆p`)linear correspond to the lower-order expansions (2.27) and (2.32).

The higher-order corrections have two major implications.

First, corrections to ∆p` of quadratic or higher order in ` lead to a deviation of the

spectroscopic measurement [the right-hand side of Eq. (2.33)] from the actual average angular

momentum 〈`〉.

Second, corrections to the effective kinetic energy E of cubic or higher order in ` mean

that even the very definition of the superfluid fraction, Eq. (2.24), breaks down. A basic

assumption of this definition is that the rotational properties of a (normal) gas are entirely

characterised by its moment of inertia. This is correct provided the kinetic energy is a

quadratic function of the angular momentum (or, equivalently, of velocity).

To expand on this, let us consider a transformation to a frame rotating at angular frequency

ω. The energy in the laboratory frame consists of a kinetic and an interaction term:

E =∑

i

12

mv2i +

12

∑i, j

U(ri− r j). (3.3)

Importantly, the interaction energy only depends on the distance |ri− r j| between two particles.

Unlike position, distance (in the non-relativistic case assumed throughout this thesis) is frame-

independent. Hence the interaction energy has the same form regardless of the frame of

reference. The velocity vi of particles in the laboratory frame is related to the velocity v′i in

the frame of reference which is rotating with the fluid by [22]

vi→ v′i = vi−ω× ri. (3.4)

The energy in the rotating frame is given by

E′ ≡∑

i

12

mv′2i +12

∑i, j

U(r′i − r′j). (3.5)

3 Corrections 19

In terms of laboratory coordinates, this is

E′ =∑

i

12

m (vi−ω× ri)2 +12

∑i, j

U(r′i − r′j)

=∑

i

12

mv2i −

∑i

mvi · (ω× ri) +∑

i

12

m(ω× ri)2 +12

∑i, j

U(ri− r j). (3.6)

Combining the first and last term gives the laboratory-frame energy E. Within the assumption

that ω and ri can be considered perpendicular to each other, we find

E′ = E−ω ·∑

i

mri× vi +12

∑i

mr2i ω

2. (3.7)

Using L =∑

i mri× vi and Icl =∑

i mr2i , the total energy transforms according to

E→ E′ = E−ω · L +12

Iclω2. (3.8)

Thus, for a state of a given total angular momentum L, the only material property character-

ising the net energy change is the classical moment of inertia Icl. For non-parabolic kinetic

energy, however, the kinetic energy does not transform in any simple way, but depends on the

population of the individual angular momentum states and hence also on how interactions

and/or temperature have distributed particles among these levels. The moment of inertia of

even a normal gas cannot be assumed simply to be a constant Icl. As we will show later, in

the limit ΩR →∞ the band becomes perfectly parabolic, so the definition Eq. (2.24) does

become well-defined. We will revisit the issue of parabolicity of the dispersion relation at the

end of this chapter.

We now turn to estimate the quantitative effects of these two forms of correction.

First, we consider the effects of the higher-order corrections to ∆p`, Eq. (3.2). Because of

the departure of ∆p` from its linear expansion, we find ∆p′`∗ , ∆p`∗ −∆p0. This difference

is illustrated for different combinations of ΩR and ∆` in Fig. 5, where the dashed lines show

∆p′`∗ and the circles show the actual difference ∆p`∗ −∆p0. As a consequence, Eq. (2.34)

would give a systematically incorrect result even in the T → 0 limit. In practice, this can be

corrected by replacing the denominator in Eq. (2.34) by ∆p`∗ −∆p0.

A more serious problem arises from the fact that non-zero temperature and/or interac-

tions populate a range of ` states. Due to this broadening of the distribution function, the

higher-order corrections to ∆p` in Eq. (3.2) lead to the situation that for a normal gas (or a

relaxed superfluid) with average momentum 〈`〉 = `∗, the spectroscopic signal ∆p is not just

determined by the signal of the state at `∗, but depends on the overall distribution function, so

∆p , ∆p`∗ . Similarly, for the (metastable) superfluid with 〈`〉 = 0, one has ∆p , ∆p0. Hence

one would incorrectly determine the superfluid fraction. To estimate the size of this error,

here we consider the effect of temperature using the example of an ideal Boltzmann gas.

(Interactions will be considered in Chapters 4 and 5.) We populate the different ` states in

20 3 Corrections

the lowest band (see Fig. 4) according to a Boltzmann distribution n` ∝ exp[−ε0‖(`)/kT ]. (For

the purposes of the superfluidity measurement, the difference between a normal fluid and a

relaxed superfluid, both possessing the same (average) angular momentum, is negligible.)

The results are shown for a range of temperatures by the solid lines in Fig. 5 on the following

page. If the effects of thermal broadening were negligible, all these curves would agree with

∆p`∗ −∆p0 (the circles in Fig. 5).

Increasing ΩR decreases the relevance of higher-order corrections, but also diminishes

the signal size ∆p′`∗ and thus leads to a bigger experimental uncertainty. Increasing ∆` has

the opposite effect. One goal of this thesis is to find parameters for which a good compromise

can be reached, such that there is both a large experimental signal and small systematic

inaccuracies.

For δ/ΩR 1, the influence of ΩR and ∆` on signal strength and higher-order corrections

can be seen in the expressions from perturbation theory. Using the perturbative expansions in

1/ΩR (cf. Section 2.5), the signal size is given by

∆p′`∗ ' 2δ

ΩR

~

mR2ΩR(∆`)2 +O(1/Ω3

R), (3.9)

and considering the corrections to ∆p` in Eq. (3.2), the second-order coefficient c′ is given by

c′ ' −3√

2δ

ΩR

(~

mR2ΩR

)2

(∆`)2 +O(1/Ω4R). (3.10)

These expressions show that increasing ∆` is an effective way to increase the signal size,

but also has an adverse effect on the accuracy. Accuracy is improved by increasing ΩR,

but at a reduction in signal size. As a trade-off one would thus try to make ∆` as high as

possible, and then increase ΩR as long as the signal size remains big enough. As a reasonable,

and experimentally feasible, compromise we choose ∆` = 50 and ΩR = 2π×100kHz [as in

Fig. 5 (c)]. These are the values we will use throughout the remainder of this thesis.

Second, and finally, we return to the definition of superfluidity in Eq. (2.24), and estimate

its adequacy. Measuring the relative deviation (〈`〉− `∗)/`∗ is a way to assess the departure

from parabolicity due to the terms of E which are odd in `; the even terms like `2, `4, . . .

cancel in the calculation of the expectation value (where N =∑∞`=−∞ n` is the normalisation

constant)

〈`〉 =1N

∞∑`=−∞

n`` =1N

∞∑`=1

(n` −n−`)`. (3.11)

For the Boltzmann gas of Fig. 5, the relative deviation is maximal at `∗ ∼ δ→ 0 (with a

maximum value ∝ 1/Ω2R) and decreases with higher δ. At T = 1000Hz6, the maximum

value of the deviation at δ→ 0 is about 0.3%. To further quantify the size of non-parabolic

6For easy comparison with other parameters, such as the Rabi frequency, we express temperatures in frequencyunits, such that 1 kHz = 48 nK.

3 Corrections 21

0.0 0.5 1.0 1.5 2.0δ/ΩR

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

∆p−

∆p

0

(a) ΩR = 2π× 5 kHz, ∆`=10

∆p′` ∗T=200HzT=500HzT=1000HzT=2500Hz

` ∗ =−1

` ∗ =−2

` ∗ =−3

` ∗ =−4

` ∗ =−5

0.0 0.5 1.0 1.5 2.0δ/ΩR

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

∆p−

∆p

0

(b) ΩR = 2π× 50 kHz, ∆`=10

∆p′` ∗T=200HzT=500HzT=1000HzT=2500Hz

` ∗ =−1

` ∗ =−2

` ∗ =−3

` ∗ =−4

` ∗ =−5

0.0 0.5 1.0 1.5 2.0δ/ΩR

0.00

0.01

0.02

0.03

0.04

0.05

0.06

∆p−

∆p

0

(c) ΩR = 2π× 100 kHz, ∆`=50

∆p′` ∗

T=200Hz

T=500Hz

T=1000Hz

T=2500Hz

` ∗ =−1

` ∗ =−5

` ∗ =−10

` ∗ =−15

` ∗ =−20

` ∗ =−25

` ∗ =−30

Figure 5: Results for a normal gas: ideal signal size ∆p′`∗ and temperature-dependentdeviation from this as a function of the detuning δ/ΩR. Data are for a classical Boltzmanngas at several values of ΩR and ∆`. Circles denote the zero-temperature limit (∆p`∗ −∆p0) atdifferent values of `∗. The curves are labelled from top to bottom, and `∗ is labelled from leftto right.

22 3 Corrections

corrections to the dispersion relation, we carry out perturbation theory up to fourth order in

1/ΩR [23]. (For a longer description, see Appendix B.) The next terms in the series in ` are

E ' Eparabolic−√

2δ

ΩR

~4(∆`)3

(mR2)3Ω2R

`3 +1

2√

2

~5(∆`)4

(mR2)4Ω3R

`4 +O(`5), (3.12)

where both coefficients have higher-order contributions of O(1/Ω4R). The contributions of

the cubic and quartic terms relative to the parabolic term (~2`2/2m∗R2) are of order 10−4`

and 10−7`2, respectively. This is on the same order as the 0.3% estimated previously: for

T = 1000Hz and `∗ = 1, the root mean square angular momentum is√〈`2〉 ∼ 16, so we would

expect the relative correction due to the cubic term to be about 0.2%. This deviation of the

dispersion relation from parabolicity sets an upper bound to the accuracy with which one can

determine the superfluid fraction.

4 Interactions 23

4 Interactions

In addition to the effects of non-zero temperature on a normal fluid, we want to know what

happens to ∆p due to interactions in a superfluid. It is important to note that a gas is only

superfluid because of interactions; in the limit of vanishing interaction strength, the critical

velocity (see Section 4.3) goes to zero, so a metastable superfluid flow cannot be maintained.

Here we do not attempt a full model of the interacting gas in a trap. Rather, we aim to use

a method that includes interactions in the simplest way. We therefore consider an interacting

gas with uniform density. The density inhomogeneity near the walls is on the scale of the

healing length ξ = (8πna)−1/2, which is the distance over which the condensate recovers its

bulk value from zero density at the walls. We assume that ξ ∆R and that we can thus

neglect the density inhomogeneity. (Typical values in experiments would be n ∼ 1014 cm−3

and a ∼ 10nm, corresponding to a healing length ξ ∼ 200nm, which can be neglected in

comparison with trap sizes of ∆R ∼ 1µm.) We therefore model the gas in a ring-like trap by

considering a torus of radius R and with a rectangular cross-sectional area A = L× L (see

Fig. 6).

pz

L

L

R

Figure 6: The system geometry employed in this chapter, a toroidal trap with hard walls anda cross-sectional area L×L. For the radius R L, angular momentum ` can be unrolled ontothe pz direction, pz = ~`/R, which then has effective periodic boundary conditions. Thus `and pz will be used interchangeably.

Since we neglect density inhomogeneities, we consider the system to be uniform and impose

periodic boundary conditions in the transverse directions. For L ∼ ∆R R we can unroll the

angular momentum from the circumferential direction of the torus onto a line with periodic

boundary conditions, equivalent to a linear momentum pz = (~/R)`, where ` is the angular

momentum (in units of ~) defined previously. With this understanding we will use ` and pz

interchangeably.

We are only interested in the azimuthal motion, pz or `, in which we would like to measure

superfluidity. Therefore we will integrate out the two transverse directions p⊥. These have a

simple kinetic energy dispersion, p2⊥/2m. In the azimuthal direction we allow for a general

dispersion relation ε0‖(pz), set by the energy of the lowest band of the Hamiltonian (2.42).

The complete dispersion relation is given by

ε0(p) =1

2mp2⊥+ ε0

‖(pz). (4.1)

24 4 Interactions

Later on we will compare the superfluid fraction with the condensate fraction, so it is

instructive to determine the number of excited particles Nex. Here we will determine this for

an ideal non-interacting gas, as many of the features will carry over to the interacting case

discussed later. In a non-interacting gas the number of excited particles is given by

Nex =∑p,0

nB(ε0(p)

), (4.2)

where nB(ε) = [exp(ε/kT )−1]−1 is the Bose-Einstein distribution in the condensed system

(for which the chemical potential vanishes, µ = 0). We split the sum,

Nex = N⊥ex(pz = 0) +∑pz,0

N⊥ex(pz), (4.3)

defining

N⊥ex(pz) ≡∑p⊥,0

nB(p2⊥/2m + ε0

‖(pz)

). (4.4)

[We assume without loss of generality that the energy minimum is at ε0‖(pz = 0) = 0.] We

switch to an integral representation according to∑

p →L

2π~

∫dp. The term N⊥ex(pz) with

pz , 0 does not pose any problems. However, when evaluating the pz = 0 contribution, we

find

N⊥ex(pz = 0) =A

(2π~)2

∫d2 p⊥ nB

(p2⊥/2m

)=

2πA(2π~)2

∫ ∞

0dp p

1exp[(p2/2m)/kT ]−1

, (4.5)

which is infrared divergent. At the lower limit, p→ 0, the integrand is approximately given

byp

exp[(p2/2m)/kT ]−1'

pp2/2mkT

=2mkT

p.

Thus the indefinite integral at small momenta is a logarithm. Note that due to the uncertainty

principle there exists a low-momentum cut-off pmin = ~/L. Fortunately, in the thermodynamic

limit, L→∞, the resulting logarithmic divergence is unproblematic. To see this, we make

the integral in Eq. (4.5) dimensionless by substituting p ≡ p/√

2mkT . In terms of the thermal

de Broglie wavelength λT =√

2π~2/mkT , this yields

N⊥ex =2πA

(2π~)2

∫ ∞

~/Ldp p

1exp(p2/2mkT )−1

= 2Aλ2

T

∫ ∞

λT /√

4πLdp

pexp(p2)−1

. (4.6)

Hence for the low-momentum divergence we find

N⊥ex(pz = 0) ∼ (A/λ2T ) log(L/λT ).

4.1 Weak interactions with asymmetric dispersion 25

The other contributions are extensive,

N⊥ex(pz , 0) ∼ (V/λ3T ),

soN⊥ex(pz = 0)N⊥ex(pz , 0)

∼λT

Rlog(L/λT ), (4.7)

and N⊥ex(pz = 0) can be neglected in the limit λT R. The same reasoning applies when

taking interactions into account, so from now on we will drop the pz = 0 contribution to Nex

and replace∑

p,0 by∑

pz,0∑

p⊥,0.

4.1 Weak interactions with asymmetric dispersion

In the following we consider an interacting gas, with a general, possibly asymmetric, non-

interacting dispersion relation ε0(p). The fundamental Hamiltonian is given by

H =∑i,pε0

i (p)a†i,pai,p +1

2V

∑i jkl

∑p,p′,q

Ui jkl(q)a†i,p+qa†j,p′−qak,p′ al,p, (4.8)

where the indices i, j, k and l stand for internal states (e.g. one of the three hyperfine bands).

[Ui jkl(q) is the Fourier transform of the pairwise interaction function U i jkleff

(r− r′), describing

two-body scattering from states the internal states k, l to i, j.] Since we study atoms in the

ultracold limit, all interactions can be treated as contact interactions, Ueff(r− r′) = U0δ(r− r′).The interaction strength U0 can be related to the scattering length a by U0 = 4π~2a/m,

assuming for simplicity that the scattering length is independent of the internal state. The

Hamiltonian in the momentum basis is

H =∑i,pε0

i (p)a†i,pai,p +1

2VU0

∑i jkl

∑p,p′,q

a†i,p+qa†j,p′−qak,p′ al,p. (4.9)

As we assume that only the lowest band is occupied (cf. Section 2.5), we can ignore inter-band

mixing and simplify Eq. (4.9) to the Hamiltonian we will consider hereafter:

H =∑

pε0(p)a†pap +

U0

2V

∑p,p′,q

a†p+qa†p′−qap′ ap. (4.10)

(A more detailed derivation of the following can be found in Appendix A.)

4.1.1 Bogoliubov transformation

We assume that the condensate has only one macroscopically occupied state pc = 0. Should

the condensate be in a state pc , 0, we can shift all momenta by relabelling the states to

p′ ≡ p− pc with energy ε0′(p′) = ε0(pc + p′). We note that, for a superfluid, the state pc is not

necessarily the ground state: for example, as in the protocol described previously, a superfluid

26 4 Interactions

may remain condensed in ` = 0 even after a gauge field is applied such that the lowest-energy

single-particle state has `∗ , 0. Our discussion in this section also applies to these metastable

superfluid states.

The creation and annihilation operators of the ground state are a†0 and a0, respectively.

We have 〈a†0a0〉 = N0, where N0 is the number of particles in the condensate. For N0 1,

we can neglect the commutator [a0, a†

0]− = 1 compared to the operator a†0a0, and replace the

ground-state operators by a c-number: a0 ' a†0 '√

N0. Expanding the interaction term and

only keeping terms of order O(N0) or higher (i.e. assuming the only relevant interactions are

with the condensate state), one finds

H 'Nε0(0) +U0N2

2V+

∑p,0

[ε0(p)− ε0(0) + U0n0

]a†pap +

U0n0

2

∑p,0

(a†pa†−p + apa−p

)=∑′

p,0

[ε0(p)− ε0(0) + ε1] a†pap + [ε0(−p)− ε0(0) + ε1] a†−pa−p

+

∑′

p,0

ε1(a†pa†−p + apa−p

)+ Eoffset, (4.11)

where the prime on the sum denotes that we only sum over (an arbitrary) half of momentum

space. Here n0 ≡N0/V is the condensate density, and we have defined ε1 ≡U0n0 = 4π~2an0/m,

which sets the chemical potential, µ = ε1 + ε0. (This is in contrast to the non-interacting case,

where µ = ε0.) The constant terms have been absorbed into Eoffset.

To disentangle the mixed term∑′

p,0 ε1(a†pa†−p + apa−p

), we diagonalise the Hamiltonian

using a Bogoliubov transformation. We introduce new operators αp and βp,

αp := uap + va†−p,

α−p ≡ βp := ua−p + va†p.(4.12)

The coefficients u and v will be chosen below. The inverse transformation is given by

ap := uαp− vα†−p,

a−p ≡ bp := uα−p− vα†p.(4.13)

We require that our new operators still follow Bosonic commutation relations, therefore we

impose

[α, α†]− = [β, β†]− = u2− v2 = 1. (4.14)

The Hamiltonian (4.11) is diagonalised by choosing

u2 =12

(ε

ε+ 1

)and v2 =

12

(ε

ε−1

), (4.15)

leading to

H = Eoffset +∑p,0

12

(ε − ε) +∑p,0

(ε + γ)︸ ︷︷ ︸εexc

α†pαp, (4.16)

4.1 Weak interactions with asymmetric dispersion 27

where εexc(p) = ε(p) + γ(p) is the energy of Bogoliubov excitations. We have defined7

γ(p) =12

[ε0(p) + ε0(−p)

]− ε0(0),

γ(p) =12

[ε0(p)− ε0(−p)

],

ε(p) = γ(p) + ε1,

ε(p) =

√ε(p)2− ε2

1 =√γ(p)[γ(p) + 2ε1].

(4.17)

γ(p) denotes the symmetric part of the non-interacting dispersion relation, shifted such that

γ(pc) = 0. ε(p) is the symmetric part of the Bogoliubov excitation energy. γ(p) denotes the

asymmetric part of the excitation energy, and is the same for both the non-interacting and

interacting cases. As opposed to the standard case of a symmetric dispersion relation, we find

γ , 0 and a dependence of the energy εexc of Bogoliubov excitations on `∗. We shall use this

in Section 4.3 to determine the critical velocity at which the metastable superfluid becomes

unstable.

4.1.2 Depletion of the condensate

We can write the operator for the total number of particles as

N = N0 +∑p,0

a†pap. (4.18)

When we apply the Bogoliubov transformation, we find

N ' N0 +∑p,0

12

(ε

ε−1

)+

∑p,0

ε

ε

⟨α†pαp

⟩, (4.19)

where we have already taken the expectation value. At zero temperature, there are no

Bogoliubov excitations present, and

N(T = 0) = N0 +∑p,0

12

(ε

ε−1

). (4.20)

After changing the sum to an integral, we can analytically integrate out the transverse

directions. In terms of the number density n = N/V we get

n(T = 0) = n0 +n0a2πR

∑pz,0

[1 +

γ

ε1

(1−

√1 + 2

ε1

γ

)], (4.21)

where γ = γ(pz; p⊥ = 0) is evaluated at p⊥ = 0. Each term of the sum gives the corresponding

`-state occupation n`(T = 0).

7Since we assumed that the condensate is in pc = 0, the last term of γ(p) in (4.17) refers to the condensatestate; in general, the argument to ε0 has to be changed appropriately: γ(p) = 1

2 [ε0(pc + p) + ε0(pc − p)]− ε0(pc).

28 4 Interactions

To find the temperature-dependent depletion, we calculate the expected excitation number:

〈α†pαp〉 = nB[εexc(p)

]. The energy of such thermal excitations is given by εexc = ε + γ. Chang-

ing the integration over transverse directions into one over the dimensionless x ≡ p2⊥/2mε1

leads to

n(T )−n(T = 0) =n0aπR

∑pz,0

∫ ∞

0dx

x + 1 +γ/ε1√(x + 1 +γ/ε1)2−1

×1

exp[(ε1

√(x + 1 +γ/ε1)2−1 + γ

)/kT

]−1

. (4.22)

Here n0 = n0(T = 0) is the zero-temperature condensate fraction. Again we set γ = γ(pz; p⊥ =

0). In general, this integral has to be evaluated numerically. Similarly to previously, the terms

of this sum give the finite-temperature part of the distribution function, n`(T )−n`(T = 0).

4.2 Popov approximation

To extend the validity of our approximation to higher temperatures, we employ the Popov

approximation. Compared to a standard Hartree-Fock treatment, this additionally includes

terms which are linear in 〈a†a†〉 and 〈aa〉, accounting for low-energy excitations: the inter-

action mixes particle-like and hole-like excitations [24]. In the homogeneous system, the

Popov approximation leads to the same Hamiltonian as in the Bogoliubov theory, except for a

simple excitation-independent energy offset. Thus the distribution function formally stays the

same; the only difference is that in the Popov approximation we need to determine n0 = n0(T )

self-consistently, both at zero and at non-zero temperatures [3, 25].

As the zero-temperature distribution (4.21) is symmetric around ` = 0, at T = 0 the average

angular momentum is always zero, 〈`〉 ≡ 0, for any interaction strength. This recovers the

expectation that ρs/ρ = 1 even if the condensate is depleted by interactions. We calculate the

zero-temperature depletion for a parabolic dispersion relation ε0‖

= ~2`2

2mR2 = γ. We approximate

the sum in Eq. (4.21) by an integral,

n−n0

n0=

a2πR

2πR2π~

∫ ∞

0dpz

[1 +

γ

ε1

(1−

√1 + 2

ε1

γ

)]. (4.23)

This can be evaluated analytically, giving

n−n0

n0=

a√

2m2π~

√4π~2an0

m

(+

43

√2)

=8

3√π

(n0a3

)1/2, (4.24)

where we substituted ε1 = 4π~2an0/m. The resulting expression for the condensate fraction,

n0

n'

[1 +

83√π

(n0a3

)1/2]−1

, (4.25)

is not in a closed form; we have to find a self-consistent solution for n0 numerically. At small

4.3 Critical velocity 29

depletions nex/n 1, Eq. (4.25) simplifies to the Bogoliubov result [3],

n0

n' 1−

83√π

(na3

)1/2. (4.26)

According to this theory, one would simply occupy the lowest band as stated in the

interacting distribution function given by Eq. (4.22) until convergence is achieved. For a

parabolic dispersion relation, even the zero-temperature distribution function [cf. Eq. (4.21)]

extends to very high `: At large momenta, ` 1, the square root in Eq. (4.21) can be expanded

to quadratic order in ε1/γ, and we find an asymptotic behaviour

n` 'n0a2πR

ε1

γ∝

1`2 . (4.27)

However, the physical distribution of atoms does not extend beyond `max ∼ R/Re, where

Re is the range of the interaction potential (in practical terms, this is of the same order

as the scattering length a). Thus we do not trust the expression for n`, Eq. (4.22), for

`/R & 1/Re ∼ 1/a, and so it is pointless to do the whole sum. In any case, all the important

physics characterising the superfluid response is contained in p . ~/ξ (equivalent to ` . R/ξ).

Hence we introduce a cut-off by ignoring states with ` > `cut. We choose `cut ∼ 10×2πR/ξ,

for which the relative error in ∆pRSF−∆pSF due to the cut-off with respect to `cut =∞ is on

the order of 10−4.

4.3 Critical velocity

In order that the system is (meta)stable, the minimum of the excitation energy εexc = ε + γ has

to be positive: min` εexc(`) > 0. This leads to a critical angular momentum shift `∗crit above

which, for |`∗| > `∗crit, the superfluid flow is unstable. [This is the Landau criterion for stable

superfluid flow, equivalent to Eq. (2.19).] The determination of the critical `∗ in the general

case requires a full (numerical) determination of the dispersion relation of the lowest dressed

band, ε0‖(pz). In the limit in which the non-interacting dispersion relation can be taken to be

parabolic [cf. Eq. (2.27)], the symmetric and antisymmetric parts of ε0‖

are given by

γ(`) =~2

2m∗R2 `2 and γ(`) = −

~2`∗

m∗R2 `, (4.28)

respectively. Then the lowest-energy Bogoliubov excitation for a given ` is

εexc(p⊥ = 0;`) = ε + γ '

√~2ε1

m∗R2 |`| −~2`∗

m∗R2 ` , (4.29)

assuming ~2`2/2m∗R2 ε1. The excitation energy is positive for all ` only if

~2|`∗|/m∗R2 <√~2ε1/m∗R2 .

30 4 Interactions

Substituting ε1 = 4π~2an0/m, this corresponds to

|`∗| < `∗crit =

√4πR2an0

m∗

m. (4.30)

If |`∗| < `∗crit, then εexc(`) > 0 for all `. On the other hand, if |`∗| > `∗crit, then there always is

some ` such that εexc(`) < 0 and hence the system is thermodynamically unstable. It will relax

due to the introduction of vortices. Equation (4.30) shows that the metastable superfluid will

become unstable when the condensate density is too low (i.e. when the temperature is too

high), when interactions are too weak, or when `∗ is too large.

5 Results 31

5 Results

To present the results of our calculations for an interacting gas, we consider two scenarios. In

the first one, we study a Bose-Einstein condensate (BEC) with weak interactions, Section 5.1,

for which we show that the proposed measurement technique gives the expected feature that

the superfluid density and condensate density are almost the same. In the second scenario,

Section 5.2, we turn to the more interesting case of a BEC with strong interactions, for

which the condensate fraction and superfluid fraction are significantly different from each

other. We show that the measurement protocol can distinguish the superfluid fraction from

the condensate fraction. Finally, we will analyse the trade-off between signal strength and

accuracy in Section 5.3.

As described previously, in our studies the condensate is assumed to be uniform and

boundary effects are neglected. When discussing temperature dependences, we shall also

assume that the density n is kept constant. For a harmonically trapped gas, with falling

temperature the peak density quickly rises by more than an order of magnitude from near

the transition point to when most of the atoms are in the condensate. We ignore this effect,

since we are primarily interested in the low temperature properties of the BEC and less in

the behaviour around the transition point. In the results we present, we choose the following

parameters: m = 23mp, R = 10µm, ΩR = 2π×100kHz, ∆` = 50, and n = 1014 cm−3. These

are typical parameters achievable with current technology.

5.1 Weakly interacting BEC

For a new experimental technique, it is important to establish that it works in a situation

where the result is known. To that end, it will be valuable to see the results for a weakly

interacting BEC, which can be well described by mean-field theories. Here we present the

signal that one can expect to measure in such an experiment, for a weakly interacting BEC

with na3 = 10−4. The condensate fraction at T = 0 is approximately 99%, showing negligible

depletion.

As described in Section 2.3, the appearance of superfluidity displays a clear qualitative

signal in this measurement technique. The processes of cooling and of imposing a non-

zero gauge field `∗ do not “commute” for a superfluid, so its response is hysteretic. First

introducing `∗ (by increasing the Raman detuning δ) and then cooling the gas leads to a

condensate in the new ground state, `c = `∗. This is the “relaxed superfluid” (RSF), with

spectroscopic signal ∆pRSF. On the other hand, cooling to below the transition point at

`∗ = δ = 0 creates the superfluid in `c = 0. Subsequently increasing δ leads to a metastable

(non-relaxed) superfluid (SF), with spectroscopic signal ∆pSF. Hence it is possible to measure

two separate curves, ∆pRSF and ∆pSF. In contrast, for a normal fluid, cooling and increasing

δ do commute.

In Fig. 7 (a), we show the spectroscopic signatures of the metastable superfluid (∆pSF)

and the relaxed superfluid (∆pRSF) at temperatures below the BEC transition temperature Tc.

32 5 Results

(The raw signal has been scaled by `∗ so that the numbers are of the same order.) One can see

that the curves of the metastable superfluid (∆pSF) terminate at certain temperatures. This

termination happens at a temperature (below the transition temperature) when the condensate

becomes sufficiently depleted so that |`∗| > `∗crit. Thus curves with higher |`∗| break off at

lower temperatures. The difference in the signals ∆pRSF , ∆pSF below this temperature is a

clear signature of the metastable superfluid flow.

To extract a quantitative measure of the superfluid fraction from these results, one could

take the curves for the metastable superfluid ∆pSF and use these in Eq. (2.34). Application

of Eq. (2.34) requires knowledge of ∆p0 and ∆p′. As we have discussed, the result will

also include some quantitative corrections from higher-order terms in the Taylor expansions

(3.1) and (3.2) of E` and ∆p`. In view of these facts, it is helpful to take the difference

∆pRSF−∆pSF. This removes the dependence on ∆p0 and also removes additive systematic

errors. The differences ∆pRSF −∆pSF are shown in Fig. 7 (b). Finally, we scale these

differences by the value of the same quantity that is obtained at T = 0 for a weakly interacting

gas,

∆pRSF(T = 0,na3 ≈ 0)−∆pSF(T = 0,na3 ≈ 0) .

[Since here we consider na3 = 10−4, this is almost identical to scaling by the T = 0 limit

of the curves in Fig. 7 (b).] This value is for a situation (negligible interactions) in which

we know the system should be completely condensed and perfectly superfluid at T = 0. It

provides a direct measurement of the quantity ∆p′`∗, while also removing some multiplicative

systematic errors.

The final scaled curves are shown in Fig. 7 (c). These show what the spectroscopic

measurement would give for the superfluid fraction of the weakly interacting BEC as a

function of temperature. The different values of `∗ correspond to different effective rotation

rates. For comparison, the dashed line shows the numerically determined condensate fraction

n0/n.8 In this case, of a weakly interacting BEC, it is expected that the condensate and

superfluid fractions should coincide. Figure 7 (c) shows that this result is recovered to a very

good accuracy by the spectroscopic measurement of the superfluid fraction. Furthermore,

in a practical experimental measurement with a harmonically trapped gas we expect to find

even better agreement between the spectroscopic measurement and ρs/ρ: here we chose

n = 1014 cm−3, which is a representative value for the density in the BEC, but this leads to

a transition temperature which is several times higher than that found in experiments. In a

typical experiment, the density would vary such that Tc is lower, so kT/~ΩR would be smaller

and quantitative corrections should have less of an effect.

8The condensate fraction n0/n deviates from the 1− (T/Tc)3/2 behaviour expected in an ideal Bose gas due toweak interactions as considered within the Popov approximation.

5.1 Weakly interacting BEC 33

0.0 0.2 0.4 0.6 0.8 1.0T/Tc

0.017

0.018

0.019

0.020

∆p/|`∗|

(a)

` ∗ =−20, RSF` ∗ =−10, RSF` ∗ =−1, RSF` ∗ =−20, SF` ∗ =−10, SF` ∗ =−1, SF

0.0 0.2 0.4 0.6 0.8 1.0T/Tc

0.00

0.01

0.02

0.03

0.04

0.05

∆p(RSF)−

∆p(SF)

(b) ` ∗ =−20

` ∗ =−10

` ∗ =−1

0.0 0.2 0.4 0.6 0.8 1.0T/Tc

0.0

0.2

0.4

0.6

0.8

1.0

∆p(RSF)−

∆p(SF)

(norm

aliz

ed t

o 1

at T

=0) (c)

n0/n (` ∗ =−1, RSF)

` ∗ =−1

` ∗ =−10

` ∗ =−20

Figure 7: Qualitative behaviour of the proposed superfluidity measurement as a functionof temperature (in terms of the ideal Bose gas transition temperature Tc ≈ 31.1kHz) atna3 = 10−4: (a) the raw ∆p signal scaled by |`∗|; (b) difference signal ∆pRSF −∆pSF; (c)difference signal normalised to 1 at T = 0, and comparison with the condensate fraction n0/n.Curves break off due to reaching |`∗| > `∗crit '

√4πR2an0. The curves are labelled from top to

bottom.

34 5 Results

5.2 Strongly interacting BEC

We will now consider how the spectroscopic method can be used to provide an accurate

quantitative measurement of the superfluid fraction in a regime where the condensate fraction

n0/n and the superfluid fraction are significantly different. To this end, we shall consider a

relatively strongly interacting BEC, with an interaction parameter up to na3 = 0.1. We cannot

trust Popov theory to be an accurate quantitative theory of an atomic gas in regimes of strong

interactions. However, we can still use the results of Popov theory to assess the accuracy of

the spectroscopic method of measuring the superfluid fraction. Specifically, we know that,

even for a strongly interacting gas, in the low temperature limit the superfluid fraction should

be ρs/ρ = 1, while the condensate fraction can be significantly depleted. Here we wish to

demonstrate that the difference between condensate fraction and superfluid fraction can be

observed using the spectroscopic measurement method.

As described in Chapter 3, while considering an accurate quantitative measurement of

ρs/ρ, we face problems coming from higher-order corrections, which even cause the usual

definition of superfluidity to break down. For this reason, in order to proceed we follow the

protocol that was outlined in Section 5.1. We propose to define a spectroscopically measured

superfluid fraction by

ρspecs

ρ≡

∆pRSF(T,na3)−∆pSF(T,na3)∆pRSF(T = 0,na3 ≈ 0)−∆pSF(T = 0,na3 ≈ 0)

. (5.1)

The denominator gives the reference signal at zero temperature and with negligible interac-

tions, where the system is completely condensed into the ground state and 100% superfluid.

(The interactions cannot be exactly zero, as one requires |`∗| < `∗crit '√

4πR2an0 for metasta-

bility of the superfluid flow.) In principle, all four ∆p values in Eq. (5.1) can be measured

in one experimental system by using a Feshbach resonance [1] to tune a. All measurements

need to be taken at the same values for ΩR, ∆` and `∗. One should take the limit `∗ → 0,

analogous to ω→ 0 in the definition (2.24).

Furthermore, from the same chapter, all higher-order corrections vanish for ΩR→∞, so

in this limit the definition (2.24) coincides with the spectroscopic definition (5.1). Thus, the

superfluid fraction is obtained from the spectroscopic signal (5.1) by taking the limit

ρs

ρ≡ lim

ΩR→∞

(ρ

specs

ρ

). (5.2)

Evaluating Eq. (5.1) at different values of ΩR gives a way to assess the influence of non-

parabolic corrections. Furthermore, this offers the possibility of improving the measurement

by extrapolating to ΩR→∞.

To analyse the quantitative accuracy of this method, we at first keep T = 0, that is, we

consider the accuracy of the method for the case of a perfect superfluid where we know that

〈`〉 = 0 (or `∗ in the relaxed SF). We wish to show that at an interaction strength where the

5.3 Optimal experimental parameters 35

10-6 10-5 10-4 10-3 10-2

na3

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45∆p

(a)

` ∗ = −20, RSF

` ∗ = −20, SF

` ∗ = −10, RSF

` ∗ = −10, SF

` ∗ = −1, RSF

` ∗ = −1, SF

10-6 10-5 10-4 10-3 10-2

na3

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

ρsp

ecs

/ρ

(b)

` ∗ = −40

` ∗ = −35

` ∗ = −30

` ∗ = −20

` ∗ = −10

` ∗ = −1

n0/n (` ∗ = −1, SF)

Figure 8: Quantitative analysis of interaction effects at zero temperature: (a) raw ∆p signal; (b)spectroscopically determined superfluid fraction ρspec

s /ρ, and comparison with the condensatefraction n0/n. Note that the condensate depletion is significantly higher than the deviation ofρ

specs /ρ from 1. The curves are labelled from top to bottom.

BEC is reasonably depleted, e.g. nex/n ∼ 0.1, the 100% superfluidity is still recovered by the

spectroscopic method. The numerical results are shown in Fig. 8, as a function of increasing

dimensionless interaction strength na3 while keeping the density n fixed. Figure 8 (a) shows

the raw ∆p results for the relaxed (∆pRSF) and metastable (∆pSF) superfluids. Figure 8 (b)

shows the resulting spectroscopically determined superfluid fraction, from Eq. (5.1), using

na3 = 10−7 to represent na3 ≈ 0. For comparison, in Fig. 8 (b) the dashed line shows

the numerically determined condensate fraction n0/n for the SF.9 The departure of the

spectroscopic measurement of the superfluid fraction from 1 [the solid lines in Fig. 8 (b)]

shows that the error in determining the superfluid fraction by the spectroscopic method

increases with interaction strength. However, this error is much smaller than the excited

fraction nex/n. Therefore, the spectroscopic method allows one to distinguish clearly between

the condensate fraction and the superfluid fraction in a strongly interacting BEC.

5.3 Optimal experimental parameters

In the preceding sections we have shown that the spectroscopic method is capable of providing

both a qualitative signature of superfluidity and a quantitatively accurate way to measure the

superfluid fraction. We now turn to discuss the experimental parameters to use to optimise

this technique as a quantitative measurement of superfluid fraction.

As was discussed in Chapter 3, there is a trade-off between the signal size and the

quantitative accuracy of the method. In particular, we showed that it is advantageous to have

as large a value of ∆` as possible, and then to increase ΩR as much as possible to improve

9For the parameters used, this is indistinguishable from the condensate density determined by self-consistentlysolving Eq. (4.25) for n0, showing that non-parabolic corrections are irrelevant when considering the condensatefraction.

36 5 Results

0 5 10 15 20 25 30 35 40−` ∗

0.02

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

∆p′ ` ∗

1−ρ specs /ρ

Figure 9: Trade-off between signal size and accuracy at zero temperature. Signal size isassumed to be given by ∆p′`∗; compare Fig. 5 (c). Accuracy is determined by the deviationof the spectroscopic measurement (5.1) from the expected unity. This deviation has beencalculated at na3 = 10−2 (corresponding to a condensate depletion nex/n ≈ 12%), amountingto a vertical slice through Fig. 8 (b) at na3 = 10−2.

accuracy, but not so much as to make the signal ∆p too small. From these considerations we

were led to choose ∆` = 50 (from practical considerations of achieving beams of high angular

momentum), and we settled on a compromise value of ΩR = 2π×100kHz. However, there

still remains the question of what is the best detuning δ (and therefore value of `∗) to use.

Formally, following Eq. (2.34) we should consider the limit `∗→ 0, such that the superfluid

remains metastable even very close to Tc. But in this limit the signal becomes very small.

What is a reasonable value of `∗ to use?

To explore this question, in Fig. 9 we present the results of calculations at zero temperature

and na3 = 10−2, corresponding to a condensate depletion of about 12%.

In Fig. 9 the diamonds show the signal strength, as given by ∆p′`∗. This is the ideal

difference between the signals of a normal fluid and a superfluid at T = 0, neglecting higher-

order corrections to ∆p`. As in Fig. 5 (c), this is a good approximation to the actual signal.

These results show that, for the parameters chosen, the fractional imbalance ∆p must be

measured to an accuracy of about 0.01 to 0.05. While this is a small fractional imbalance, it

should be feasible to detect signals of this size by averaging over many shots.

The circles in Fig. 9 display the inaccuracy in the spectroscopic measurement of the

superfluid fraction, i.e. the relative deviation of ρspecs /ρ from the expected value of 1 at T = 0.

Hence the circles in Fig. 9 correspond to a vertical slice through Fig. 8 (b) at na3 = 10−2.

Here, an inaccuracy of 0.03 means that a 100% superfluid would be misinterpreted as only

having 97% superfluid fraction. In contrast, recall that the condensate fraction is about 88%.

So this still allows a clear distinction between the superfluid and condensate fraction.

The results in Fig. 9 would suggest picking |`∗| ≈ 35, where the inaccuracy passes

through zero. However, one should note that these results only show the inaccuracy at zero

5.3 Optimal experimental parameters 37

0.0 0.2 0.4 0.6 0.8 1.0T/Tc

0.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04

ρsp

ecs

/ρs

` ∗ =−40

` ∗ =−35

` ∗ =−30

` ∗ =−25

` ∗ =−20

` ∗ =−10

` ∗ =−1

Figure 10: Comparison of ρspecs /ρ and ρs/ρ = 1− 〈`〉/`∗ at na3 = 10−2 as a function of

temperature (in terms of the ideal Bose gas transition temperature Tc ≈ 31.1kHz). As inFig. 7, the curves break off for |`∗| > `∗crit '

√4πR2an0. At the left-hand edge of the graph,

each curve starts at the y value corresponding to the one in Fig. 8(b) at na3 = 10−2. The curvesare labelled in this order from top to bottom.

temperature. In Fig. 10 we plot the ratio ρspecs /ρs as a function of temperature. Here ρspec

s is

the spectroscopically measured superfluid density and ρs is the expected superfluid density, as

computed from the usual definition of superfluidity (2.24), ρs/ρ = 1−〈`〉/`∗. The ratio stays

close to 1 over a wide temperature range, showing that the spectroscopic method provides a

good measure of the superfluid fraction.

Combining the issue of signal size and accuracy over a range of temperatures, we find that

(for the parameters studied) a good choice is |`∗| between 20 and 30. In this range, the signal

size is approximately 0.05; the spectroscopically measured superfluid fraction is accurate to

about 0.02 (much less than the depleted fraction of 0.12); the accuracy remains at this level

up to about 0.9Tc.

38 6 Summary

6 Summary

I studied a novel proposal [8] to measure the superfluid fraction of atomic BECs and assessed

its feasibility. This technique uses optically induced gauge fields to simulate rotation. The

superfluid response then appears in the spectroscopic signal. The proposal considered a

non-interacting Bose gas on a one-dimensional ring at zero temperature.

In my work I calculated the expected spectroscopic signatures in a three-dimensional

BEC with uniform density, including a broadening of the angular momentum distribution

function due to both non-zero temperature and interactions between atoms. In conclusion, I

demonstrated that this technique can be used to obtain a qualitative experimental signature

of the superfluid response. The signature is given by the different spectroscopic signals

depending on whether the artificial gauge field is turned on before or after the gas has been

cooled below the transition temperature. In addition, I showed that the spectroscopic signal

can be made large enough for experimental detection.

Most importantly, I showed that this technique also allows a quantitative measurement

of the superfluid density. The accuracy of the measurement is improved as the signal size

is decreased. However, with realistic values for experimental parameters, I found that a

compromise is possible in which there is both a sufficiently large signal (which can be

reasonably measured in experiments) and the superfluid density can be determined with

sufficient accuracy to obtain useful information.

Furthermore, I showed that this allows a clear experimental measurement of the distinction

between condensate fraction and superfluid fraction in a strongly interacting Bose gas.

In summary, my work demonstrated the usefulness of the Cooper-Hadzibabic approach

and provided guidance for parameters in future experimental implementations.

The results of this thesis, as published in Ref. [26], may serve as a stepping stone both

towards experimental realisations, which can benefit from a computational model that can

quickly accommodate emerging questions, and towards deeper theoretical understanding of

superfluidity in BECs.

39

Appendices

A Derivation of the interacting distribution function

A.1 Bogoliubov transformation

We start from the Hamiltonian (4.10),

H =∑

pε0(p)a†pap +

U0

2V

∑p,p′,q

a†p+qa†p′−qap′ ap.

For the ground state occupied by N0 atoms, |Ψ0(N0)〉, the creation and annihilation operators

yield

a†0 |Ψ0(N0)〉 =√

N0 + 1 |Ψ0(N0 + 1)〉 ,

a0 |Ψ0(N0)〉 =√

N0 |Ψ0(N0−1)〉 .(A.1)

For macroscopical occupation, N0 1, we can approximate the ground-state operators by a

c-number: a0 ' a†0 '√

N0. Only keeping terms which are at least of order O(N0), the sum in

the interaction term reduces according to∑p,p′,q

a†p+qa†p′−qap′ ap ' N20 + N0

∑p,0

(a†pa†−p + apa−p + 4a†pap). (A.2)

[Terms of order O(N3/20 ) vanish due to momentum conservation.] Then the Hamiltonian up to

quadratic order in a and a† is

H 'N0ε0(0) +

∑p,0

ε0(p)a†pap +U0N2

0

2V+

U0N0

2V

∑p,0

(a†pa†−p + apa−p + 4a†pap)

=N0ε0(0) +

U0N20

2V+

∑p,0

[ε0(p) + 2U0n0

]a†pap +

U0n0

2

∑p,0

(a†pa†−p + apa−p), (A.3)

where n0 = N0/V . Because we want the ground-state population to come out of the calculation,

we express N0 in terms of the operator for the total number of particles,

N =∑

pa†pap = N0 +

∑p,0

a†pap. (A.4)

Fluctuations in the total number of particles are negligible, hence we can replace N by its

expectation value N. Again only keeping terms of order O(N) or higher, we can give the

ground state occupation in terms of the total number of particles as

N20 ' N2−2N

∑p,0

a†pap. (A.5)

40 A Derivation of the interacting distribution function

Plugging this into the constant terms of Eq. (A.3), we find

N0ε0(0) +

U0N20

2V' Nε0(0) +

U0N2

2V+

[−ε0(0)−

U0NV

]∑p,0

a†pap. (A.6)

Therefore the term ∑p,0

[ε0(p) + 2U0n0

]a†pap

acquires another summand (−ε0(0)−U0N/V) inside the brackets. The first part (−ε0(0))

cancels any constant energy offset in the non-interacting dispersion relation ε0(p). Because

the sum over states already contributes another order of N, in the sum we can approximate

the second part (−U0N/V) by −U0n0, and so we arrive at

H ' Nε0(0) +U0N2

2V+

∑p,0

[ε0(p)− ε0(0) + U0n0

]a†pap +

U0n0

2

∑p,0

(a†pa†−p + apa−p). (A.7)

To disentangle the interaction term (a†pa†−p + apa−p), we will apply a Bogoliubov transfor-

mation. However, due to the term ε0‖(pz) in the dispersion relation, in general we find

ε0(p) , ε0(−p). Thus we explicitly write

H = Eoffset +∑′

p,0

[ε+

0 a†pap + ε−0 a†−pa−p + ε1(a†pa†−p + apa−p)], (A.8)

where we have defined

ε1 ≡ U0n0,

ε+0 ≡ ε

0(p)− ε0(0) + ε1,

ε−0 ≡ ε0(−p)− ε0(0) + ε1.

(A.9)

The constants are taken up by

Eoffset ≡ Nε0(0) +U0N2

2V. (A.10)

The prime on the sum in Eq. (A.8) signifies that we have restricted the sum to only half of

momentum space. (It is arbitrary which half we choose.)

For the next step, we look at just one term of the sum over p. To make the following

equations clearer, we temporarily change our notation as follows:

εa ≡ ε+0 , εb ≡ ε−0 , a(†) ≡ a(†)

p , b(†) ≡ a(†)−p. (A.11)

In this notation, a single term of the sum is given by

h = εaa†a + εbb†b + ε1(a†b†+ ab). (A.12)

A.1 Bogoliubov transformation 41

We perform the Bogoliubov transformation

α = ua + vb†,

β = ub + va†.(A.13)

To ensure that α(†) and β(†) still represent bosonic operators, we have to set[α, α†

]−

=[ua + vb†,ub + va†

]−

= u2− v2 != 1. (A.14)

The same holds for [β, β†]−. Hence the inverse transformation is given by

a = uα− vβ†,

b = uβ− vα†.(A.15)

We can write combinations of a(†) and b(†) in terms of α(†) and β(†) as follows:

a†a =(uα†− vβ

) (uα− vβ†

)= u2α†α+ v2ββ†−uv

(α†β†+ βα

)= v2 + u2α†α+ v2β†β−uv

(α†β†+ αβ

), (A.16)

b†b =(uβ†− vα

) (uβ− vα†

)= u2β†β+ v2αα†−uv

(β†α†+ αβ

)= v2 + u2β†β+ v2α†α−uv

(α†β†+ αβ

), (A.17)

a†b† =(uα†− vβ

) (uβ†− vα

)= u2α†β†+ v2αβ−uv

(α†α+ ββ†

)= −uv + u2α†β†+ v2αβ−uv

(α†α+ β†β

), (A.18)

ab =(uα− vβ†

) (uβ− vα†

)= u2αβ+ v2α†β†−uv

(αα†+ β†β

)= −uv + u2αβ+ v2α†β†−uv

(α†α+ β†β

). (A.19)

Plugging this into Eq. (A.12), we get

h =εaa†a + εbb†b + ε1(a†b†+ ab

)=(εa + εb

)v2 + εa

(u2α†α+ v2β†β

)+ εb

(u2β†β+ v2α†α

)−

(εa + εb

)uv

(α†β†+ αβ

)+ ε1

[−2uv +

(u2 + v2

) (αβ+ α†β†

)−2uv

(α†α+ β†β

)]=(εa + εb

)v2−2ε1uv

+(εau2 + εbv2−2ε1uv

)α†α

+(εav2 + εbu2−2ε1uv

)β†β

+[ε1

(u2 + v2

)−

(εa + εb

)uv

] (α†β†+ αβ

). (A.20)

To get rid of the cross-term (α†β†+ αβ), we have to choose u and v such that

ε1(u2 + v2

)−

(εa + εb

)uv = 0. (A.21)

42 A Derivation of the interacting distribution function

The condition u2− v2 = 1 [Eq. (A.14)] is automatically enforced by writing

u = cosh t, v = sinh t. (A.22)

Then Eq. (A.21) reads

ε1(cosh2 t + sinh2 t

)−

(εa + εb

)sinh t cosh t !

= 0. (A.23)

With the identities

2sinh xcosh x = sinh(2x) and cosh2 x + sinh2 x = cosh(2x), (A.24)

this reduces to

tanh(2t) =2ε1

εa + εb . (A.25)

Therefore t is given by

t =12

arc tanh2ε1

εa + εb =14

ln(εa + εb + 2ε1

εa + εb−2ε1

). (A.26)

This leads to

u2 = 1 + v2 =12

εa + εb√(εa + εb)2

−4ε21

+12. (A.27)

Introducing the shorthand notations

ε ≡12

(εa + εb

)and ε ≡

√ε2− ε2

1 , (A.28)

the Bogoliubov coefficients can be written in a simple way:

u2 =12

(ε

ε+ 1

), v2 =

12

(ε

ε−1

). (A.29)

Now we only need to find an expression for uv:

uv =√

u2v2 =12

√ε2

ε2 −1 =12

√√√ε2−

(ε2− ε2

1

)ε2− ε2

1

=12

√ε2

1

ε2

=12ε1

ε. (A.30)

We have to take the positive root for consistency, so that the cross-term actually vanishes.

A.1 Bogoliubov transformation 43

With this we arrive at

h =(εa + εb

)v2−2ε1uv +

(εau2 + εbv2−2ε1uv

)α†α+

(εav2 + εbu2−2ε1uv

)β†β

=ε(ε

ε−1

)− ε1

ε1

ε+

[εa 1

2

(ε

ε+ 1

)+ εb 1

2

(ε

ε−1

)− ε1

ε1

ε

]α†α

+

[εa 1

2

(ε

ε−1

)+ εb 1

2

(ε

ε+ 1

)− ε1

ε1

ε

]β†β

=ε − ε +

[εε

ε− ε1

ε1

ε+

12

(εa− εb

)]α†α+

[εε

ε− ε1

ε1

ε−

12

(εa− εb

)]β†β

=ε − ε +

[ε +

12

(εa− εb

)]α†α+

[ε −

12

(εa− εb

)]β†β. (A.31)

We further define γ ≡ 12

(εa− εb

). Then

h = ε − ε + (ε + γ) α†α+ (ε − γ) β†β. (A.32)

Going back to the full Hamiltonian, we find

H = Eoffset +∑′

p,0

[ε+

0 a†pap + ε−0 a†pap + ε1(a†pa†−p + apa−p

)]= Eoffset +

∑′

p,0

[ε − ε + (ε + γ) α†pαp + (ε − γ) α†−pα−p

], (A.33)

where we introduced the following definitions:

γ(p) =12

[ε0(p)− ε0(−p)

],

γ(p) =12

[ε0(p) + ε0(−p)

]− ε0(0),

ε(p) = γ(p) + ε1,

ε(p) =

√ε(p)2− ε2

1 =√γ(p)[γ(p) + 2ε1].

(A.34)

Because ε(p) = ε(−p) is symmetric and γ(−p) = −γ(p) is antisymmetric, the sum over p can

be extended to cover all of momentum space again:

H = Eoffset +∑p,0

12

(ε − ε) +∑p,0

(ε + γ)︸ ︷︷ ︸εexc

α†pαp. (A.35)

εexc(p) = ε(p) + γ(p) is the energy of Bogoliubov excitations with momentum p.

44 A Derivation of the interacting distribution function

A.2 Depletion of the condensate

We can write the operator for the total number of particles as

N = N0 +∑p,0

a†pap. (A.36)

It is not directly obvious that it is equivalent whether we look at “positive” or “negative” p;

there is no antisymmetric prefactor as in Eq. (A.35). The expression a†pap is only valid for

the “positive” half of momentum space. Hence we have to consider

N = N0 +∑′

p,0

(a†pap + a†−pa−p

). (A.37)

According to the Bogoliubov transformation, we have

a†pap = v2 + u2α†pαp + v2α†−pα−p−uv(α†pα

†−p + αpα−p

),

a†−pa−p = v2 + u2α†−pα−p + v2α†pαp−uv(α†pα

†−p + αpα−p

).

(A.38)

Thus

N = N0 +∑′

p,0

[2v2 + (u2 + v2)α†pαp + (u2 + v2)α†−pα−p

]−

∑′

p,0

2uv(α†pα

†−p + αpα−p

)= N0 +

∑p,0

v2 +∑p,0

(u2 + v2)α†pαp−∑p,0

uv(α†pα

†−p + αpα−p

). (A.39)

In eigenstates of the Hamiltonian (A.35), the cross-term vanishes when taking the expectation

value:⟨α†pα

†−p + αpα−p

⟩= 0. Inserting the expressions for u2 and v2 from Eq. (A.29), this

leads to

N ' N0 +∑p,0

12

(ε

ε−1

)+

∑p,0

ε

εα†pαp. (A.40)

At zero temperature the system is in its ground state, 〈α†pαp〉 = 0, therefore

N(T = 0) = N0 +∑p,0

12

(ε

ε−1

). (A.41)

Using

ε(p) =p2⊥

2m+γ(pz) + ε1, (A.42)

where

γ(pz) =12

(ε0‖(pz) + ε0

‖(−pz)

)− ε0‖(0), (A.43)

A.2 Depletion of the condensate 45

and switching to an integral representation according to∑

p→L

2π~

∫dp, we find

∑p,0

12

(ε

ε−1

)'

∑pz,0

A(2π~)2

∫d2 p⊥

12

(ε

ε−1

)=

A(2π~)2

∑pz,0

2π∫ ∞

0dp p

12

(ε

ε−1

). (A.44)

This integral can be solved exactly:

∑p,0

12

(ε

ε−1

)=

A4π~2

∑pz,0

∫ ∞

0dp p

p2

2m +γ(pz) + ε1√[p2

2m +γ(pz) + ε1

]2− ε2

1

−1

=

A4π~2

∑pz,0

12

[−2mγ(pz)− p2 +

√[2mγ(pz) + p2] [2mγ(pz) + p2 + 4mε1

]]∣∣∣∣∣∣∞0

=A

4π~2

∑pz,0

[12

2mε1−

(−2mγ(pz) + 2m

√γ(pz)

[γ(pz) + 2ε1

])]=

mA4π~2

∑pz,0

(ε1 +γ(pz)−

√γ(pz)

[γ(pz) + 2ε1

]). (A.45)

Following the discussion in Chapter 4, the term with pz = 0 and p⊥ , 0 has been omitted. In

terms of the density n = N/V (with V = 2πRA), we find

n(T = 0) = n0 +mε1

8π2~2R

∑pz,0

[1 +γ/ε1

(1−

√1 + 2ε1/γ

)]. (A.46)

With ε1 = U0n0 = 4π~2n0a/m, the prefactor can be simplified:

mε1

8π2~2R=

n0a2πR

.

We arrive at our final expression for n(T = 0):

n(T = 0)−n0 =n0a2πR

∑pz,0

[1 +

γ

ε1

(1−

√1 + 2

ε1

γ

)]. (A.47)

In the Popov approximation, this has to be solved for n0 in a self-consistent way.

To evaluate the effects of finite temperature on the ground-state depletion, we consider

N(T )−N(T = 0) =∑p,0

ε

εfp, (A.48)

where fp is the number of excitations with momentum p, given by

fp = nB(ε + γ) =1

exp[(ε + γ)/kT ]−1. (A.49)

46 A Derivation of the interacting distribution function

Again, we switch to an integral representation:

∑p,0

ε

εfp '

∑pz,0

A(2π~)2

∫d2 p⊥

ε

εfp =

A(2π~)2

∑pz,0

2π∫ ∞

0dp p

ε

εfp (A.50)

Explicitly, this is

∑p,0

ε

εfp '

A2π~2

∑pz,0

∫ ∞

0dp p

p2⊥

2m +γ(pz) + ε1√[p2⊥

2m +γ(pz) + ε1

]2− ε2

1

×1

exp

√[p2⊥

2m +γ(pz) + ε1

]2− ε2

1 + γ(pz)

/kT

−1

. (A.51)

We make the integral dimensionless by substituting

x ≡p2

2m1ε1. (A.52)

Again considering the density, this leads to

1V

∑p,0

ε

εfp =

mε1

(2π~)2R

∑pz,0

∫ ∞

0dx

ε1(x +γ(pz)/ε1 + 1)√ε2

1

([x +γ(pz)/ε1 + 1

]2−1

) fp

=mε1

4π2~2R

∑pz,0

∫ ∞

0dx

x + 1 +γ/ε1√(x + 1 +γ/ε1)2−1

×1

exp[(ε1

√(x + 1 +γ/ε1)2−1 + γ

)/kT

]−1

. (A.53)

As previously, the prefactor can be simplified by substituting ε1 = 4π~2n0a/m:

14π2

m~2R

4π~2

mn0a =

n0aπR

.

Finally, we arrive at

n(T )−n(T = 0) =n0aπR

∑pz,0

∫ ∞

0dx

x + 1 +γ/ε1√(x + 1 +γ/ε1)2−1

×1

exp[(ε1

√(x + 1 +γ/ε1)2−1 + γ

)/kT

]−1

, (A.54)

where the integral has to be solved numerically. Again, n0 must be determined self-

consistently. The integration is well-defined as long as our system is stable. If the argument

of the Bose distribution function is less than zero, εexc < 0, this signifies an instability,

invalidating our assumptions.

B Perturbation theory 47

B Perturbation theory

Because higher-order perturbation theory becomes increasingly prone to careless mistakes,

we calculate the coefficients to Taylor expansions in ` and 1/ΩR of E and ∆p` in Mathematica,

using the following recursive algorithm [23]. We want to find the solutions to

H |ψn〉 = En |ψn〉 . (B.1)

We split the Hamiltonian into H = H0 +λH′, assuming that the unperturbed Hamiltonian H0

has the known solutions

H0 |n〉 = En,0 |n〉 . (B.2)

We expand the solutions of Eq. (B.1) in Taylor series about λ = 0:

En =

∞∑s=0

En,sλs, (B.3)

|ψn〉 =

∞∑s=0

|ψn,s〉λs, (B.4)

where

|ψn,s〉 =∑

m

Cmn,s |m〉 , (B.5)

and Cmn,0 = δmn. The index s counts the order of the perturbation. Plugging these series into

Eq. (B.1) leads to the recursion relations

En,s = 〈n| H′ |ψn,s−1〉−

s−1∑j=1

En, jCnn,s− j, (B.6)

Cmn,s =1

En,0−Em,0

∑k

〈m| H′ |k〉Ckn,s−1−

s∑j=1

En, jCmn,s− j

(for m , n). (B.7)

Finally, we have

Cnn,1 = 0; Cnn,s = −12

s−1∑j=1

∑m

Cmn, jCmn,s− j, s > 1. (B.8)

For large ΩR, the dressed bands are far apart from each other and the lowest band is nearly

parabolic. To find the non-parabolic corrections, we apply these recursion relations to the

unperturbed Hamiltonian

H0 =~ΩR

2

0 1 0

1 0 1

0 1 0

. (B.9)

48 B Perturbation theory

This Hamiltonian has the exact eigenvalues

E0 = −~ΩR√

2, E1 = 0, E2 = +

~ΩR√

2,

with corresponding eigenstates

|0〉 =

1

−√

2

1

, |1〉 =

−1

0

1

, |2〉 =

1√

2

1

.Corrections to the unperturbed eigensystem due to the perturbation

H′(`) = ~

~

2mR2 (`+∆`)2−δ 0 0

0 ~2mR2 `

2 0

0 0 ~2mR2 (`−∆`)2 +δ

(B.10)

are determined by solving the recursive Eqs. (B.6)–(B.8) with Mathematica. This results in a

series expansion in 1/ΩR and ` for E(`) and |ψ(`)〉. Keeping terms up to the fourth order in

1/ΩR yields the higher-order terms for ∆p` [Eq. (3.10)] and energy [Eq. (3.12)].

REFERENCES 49

References

[1] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga. “Feshbach resonances in ultracold

gases”. Rev. Mod. Phys. 82, 1225 (2010).

[2] I. Bloch, J. Dalibard, and W. Zwerger. “Many-body physics with ultracold gases”. Rev.

Mod. Phys. 80, 885 (2008).

[3] C. J. Pethick and H. Smith. Bose-Einstein Condensation in Dilute Gases; 2nd ed.

Cambridge University Press, Cambridge, 2008.

[4] A. J. Leggett. “Superfluidity”. Rev. Mod. Phys. 71, S318 (1999).

[5] A. Griffin, D. W. Snoke, and S. Stringari (Eds.). Bose-Einstein Condensation. Cambridge

University Press, Cambridge, 1996.

[6] E. Andronikashvili. J. Phys. USSR 10, 201 (1946).

[7] A. L. Fetter. “Rotating trapped bose-einstein condensates”. Rev. Mod. Phys. 81, 647

(2009).

[8] N. R. Cooper and Z. Hadzibabic. “Measuring the superfluid fraction of an ultracold

atomic gas”. Phys. Rev. Lett. 104, 030401 (2010).

[9] L. D. Landau. J. Phys. USSR 5, 71 (1941).

[10] J. F. Annett. Superconductivity, Superfluids and Condensates (Oxford Master Series in

Condensed Matter Physics). Oxford University Press, Oxford, 2004.

[11] L. Tisza. Nature 141, 913 (1938).

[12] L. Tisza. “The theory of liquid helium”. Phys. Rev. 72, 838 (1947).

[13] J. M. Blatt and S. T. Butler. “Superfluidity of an ideal bose-einstein gas”. Phys. Rev.

100, 476 (1955).

[14] S. Franke-Arnold, L. Allen, and M. Padgett. “Advances in optical angular momentum”.

Laser & Photonics Reviews 2, 299 (2008).

[15] G. Juzeliunas, P. Öhberg, J. Ruseckas, and A. Klein. “Effective magnetic fields in

degenerate atomic gases induced by light beams with orbital angular momenta”. Phys.

Rev. A 71, 053614 (2005).

[16] L. D. Landau and E. M. Lifshitz. Statistical Physics, Part 1. Butterworth Heinemann,

Oxford, 1981.

[17] J. Dalibard, F. Gerbier, G. Juzeliunas, and P. Öhberg. “Artificial gauge potentials for

neutral atoms”. ArXiv e-prints (2010).

50 REFERENCES

[18] I. B. Spielman. “Raman processes and effective gauge potentials”. Phys. Rev. A 79,

063613 (2009).

[19] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman.

“Bose-einstein condensate in a uniform light-induced vector potential”. Phys. Rev. Lett.

102, 130401 (2009).

[20] B. W. Shore. The Theory of Coherent Atomic Excitation. Wiley-Interscience, New York,

1990.

[21] C. Cohen-Tannoudji. Atom-Photon Interactions. Wiley-Interscience, New York, 1992.

[22] P. Nozieres and D. Pines. The Theory of Quantum Liquids, Volume 2. Addison-Wesley,

New York, 1990.

[23] F. M. Fernandez. Introduction to perturbation theory in quantum mechanics. CRC

Press, Boca Raton, FL, 2001.

[24] L. Pitaevskii and S. Stringari. Bose-Einstein Condensation (International Series of

Monographs on Physics). Oxford University Press, Oxford, 2003.

[25] V. N. Popov. Functional Integrals and Collective Excitations (Cambridge Monographs

on Mathematical Physics). Cambridge University Press, Cambridge, 1988.

[26] S. T. John, Z. Hadzibabic, and N. R. Cooper. “Spectroscopic method to measure the

superfluid fraction of an ultracold atomic gas”. Phys. Rev. A 83, 023610 (2011).