Thermodynamics in expanding shell-shaped Bose-Einstein condensates

Brendan Rhyno,1, ∗ Nathan Lundblad,2 David C. Aveline,3 Courtney Lannert,4, 5, † and Smitha Vishveshwara1, ‡

1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA2Department of Physics and Astronomy, Bates College, Lewiston, ME, 04240, USA

3Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, 91109, USA4Department of Physics, Smith College, Northampton, Massachusetts 01063, USA

5Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003-9300, USA(Dated: June 3, 2021)

Inspired by investigations of Bose-Einstein condensates (BECs) produced in the Cold Atom Lab-oratory (CAL) aboard the International Space Station, we present a study of thermodynamic prop-erties of shell-shaped BECs. Within the context of a spherically symmetric ‘bubble trap’ potential,we study the evolution of the system from small filled spheres to hollow, large, thin shells via thetuning of trap parameters. We analyze the bubble trap spectrum and states, and track the distinctchanges in spectra between radial and angular modes across the evolution. This separation of theexcitation spectrum provides a basis for quantifying dimensional cross-over to quasi-2D physics at agiven temperature. Using the spectral data, for a range of trap parameters, we compute the criticaltemperature for a fixed number of particles to form a BEC. For a set of initial temperatures, we alsoevaluate the change in temperature that would occur in adiabatic expansion from small filled sphereto large thin shell were the trap to be dynamically tuned. We show that the system cools duringthis expansion but that the decrease in critical temperature occurs more rapidly, thus resulting indepletion of any initial condensate. We contrast our spectral methods with standard semiclassi-cal treatments, which we find must be used with caution in the thin-shell limit. With regards tointeractions, using energetic considerations and corroborated through Bogoliubov treatments, wedemonstrate that they would be less important for thin shells due to reduced density but vortexphysics would become more predominant. Finally, we apply our treatments to traps that realisti-cally model CAL experiments and borrow from the thermodynamic insights found in the idealizedbubble case during adiabatic expansion.

I. INTRODUCTION

The physics of Bose-Einstein condensates that formclosed shell-like geometries is a fascinating and far-reaching topic which spans a range of physical scalesand phenomena. On astronomic scales, for instance, por-tions of neutron star interiors could potentially containsuperfluid spherical-shell structures [1, 2]. Exotic starsknown as “boson stars” have also been hypothesized toform when a complex scalar field couples to gravity [3].On the microscopic length scales of trapped ultracoldatoms, concentric shells of differing phases can be gener-ated in the setting of Bose-Fermi mixtures [4–7] as wellas in bosonic optical lattices, which can preferentiallyfavor superfluid versus Mott insulating phases in differ-ent regions [8–13]. The prospect of realizing isolatedshell-shaped condensates has received a surge of inter-est [14–27]. However, on Earth, gravity renders such arealization challenging by causing trapped gases to poolat the bottom of the trap. Terrestrial experiments donein free-fall can mitigate this problem [25, 28, 29], but theinherently short condensate lifetimes are undesirable. Itwould thus be ideal to probe shell-shaped condensatesin perpetual free-fall. The International Space Station

∗ brhyno2@illinois.edu† clannert@smith.edu‡ smivish@illinois.edu

(ISS) provides precisely these conditions, operating in amicrogravity environment.

Bose-Einstein condensate (BEC) shells in isolationwould provide an arena for studying numerous interest-ing features which could translate to salient properties ofshells in these various settings. The topology of hollowed-out fluid structures shows innate differences from fully-filled structures. The presence of an inner and an outerboundary affects collective mode spectra, vortex physics,and thermodynamics. With regards to geometry, allthese features show unique characteristic properties asa spherical system undergoes an evolution from filled toslightly hollowed out to the thin-shell limit, includingtell-tale signatures of topological change. In principle,this evolution can be realized by forming a condensatein a trap that can produce the standard filled geometry,and then, as exemplified by the ‘bubble trap’ [14, 15],the hollowing out can take place by tuning trap param-eters. A major aspect of BEC shells, which we addresshere, is the thermodynamics behind how a hollow shellcondensate structure can be created at finite tempera-ture. Towards actual realization of such structures, ourtheoretical analyses closely target the experiments beingconducted by two of us (D. A. and N. L.) aboard the ISS.

In 2018 the Cold Atom Laboratory (CAL), devel-oped by the Jet Propulsion Laboratory was successfullylaunched into orbit aboard the ISS. CAL’s design allowsfor remote generation of BECs in microgravity [22, 26]and has been able to produce large millimeter scale ul-tracold bubbles in the 10 − 100 nK temperature range

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[30]. CAL offers wide capabilities for generating ultra-cold mixtures [22], but important to this work is its abil-ity to generate so-called “dressed potentials” [24]. In thismode of operation, an initial non-hollow condensate isprepared using magnetically trapped 87Rb atoms in the|F = 2,mF = 2〉 hyperfine state [24]. A radio frequency(rf) signal is then turned on which modifies the potentialexperienced by the atoms. By modifying this rf “detun-ing” signal, one can produce potentials capable of harbor-ing BEC shells. The first set of experiments reveal a dra-matic expansion and hollowing-out of an initial conden-sate as it undergoes significant thermodynamic changes[30–32].

In this work, complementing the CAL experiments, wetheoretically investigate the thermodynamic process ofa trapped Bose gas, initially forming a compact filledsphere, undergoing adiabatic expansion due to chang-ing trap parameters to form a large, thin spherical shell.Specifically, we consider two aspects of this process – in abubble trap geometry, for fixed number of gas particles,how does the temperature of the system evolve underadiabatic expansion? How does this contrast with thecritical temperature for condensation for a given set oftrap parameters? We also perform an analysis for the ex-perimental traps employed on CAL. Recent work has be-gun to address related issues; aspects of a thin sphericalshell’s thermodynamics have been investigated [33, 34],including the fate of the Berezinskii-Kosterlitz-Thoulesstransition in a closed, thin-shell geometry [35].

Our findings here make for a comprehensive thermo-dynamic study of this unique structure and evolution inand of itself while also informing the related experiments.They predict the manner in which the temperature of theBose-gas would change during the adiabatic hollowing-out expansion. Comparing with the local critical tem-perature throughout the process, we are led to concludethat there is a narrow window of parameters in whicha large thin condensate shell can be created. We go be-yond the semi-classical approximation, whose predictionswe show should be taken with care in the limit of largethin systems. We also include, at first order, the effect ofinteractions and show that they are largely unimportantin the same large thin limit. Our results, while predict-ing that the first set of shell-potential experiments aboardthe ISS are unlikely to retain condensation at the largerradii, point to the range of parameters wherein large con-densate shells occur. They also ascertain the stability ofthe spectacular systems that these first experiments ex-hibit: compared to regular trapped gases of micron-scale,thousand-fold adiabatic expansion gives rise to exquisitedelicate gigantic thermal gas bubbles.

This paper is organized as follows: we first con-sider bosons subject to a radial “bubble trap” poten-tial [14, 15]. After exploring the spectral properties ofbubble-trapped states, we use this information to cal-culate thermodynamic quantities in the noninteractingcase. In particular, we determine the BEC critical tem-perature as the potential evolves from a filled-to-hollow

sphere geometry along with the temperature of the sys-tem when the expansion process is performed adiabat-ically. Strikingly, we find that adiabatic expansion ofa BEC leads to condensate depletion. We then con-trast our results with the semiclassical approximationand find semiclassical predictions overestimate the crit-ical temperature and hence partially conceal the con-densate depletion phenomena. Next, we consider theeffects of interactions and dimensionality. Contrastingzero-temperature Gross-Pitaevskii numerics with variousapproximation schemes reveals that, for fixed particlenumber systems, the decreasing density of an expand-ing bubble means the noninteracting description is well-suited to describe thin-shell geometries. We then use thisresult to develop a mean-field theory for Bogoliubov col-lective excitations. Consistent with the above discussion,we find thermodynamic predictions for thin shells are rel-atively close to the predictions in the absence of interac-tions. By construction, the Bogoliubov effective theoryassumes spontaneous U(1) symmetry breaking; hence,we address the eventual breakdown of this description asone crosses over into the 2D limit. Finally, we apply thelessons of the bubble trap to compute thermodynamicsin trapping potentials achieved with CAL. We concludewith an outlook on how our work can inform the CALexperiments and discuss possible manifestations of non-equilibrium physics.

II. BUBBLE TRAP SPECTRUM AND STATES

Here we introduce the bubble trap as means to contin-uously tune the system geometry. We then discuss theproperties of states subject to the bubble trap potentialas it deforms from a filled sphere to a hollow thin shell.

A. Trapping potential

Our focus is the thermodynamics of ultracold bosonicgases trapped in potentials that allow geometries rang-ing from a filled sphere to a nearly two-dimensional hol-low shell. Such dimensional crossovers have been investi-gated employing hard-wall spherical potentials and con-finement to spherical surfaces [36, 37]. Here we employtrapping potential forms that can be continuously tunedto span the whole range and can approximate the relatedexperimental setting of shell-shaped BECs aboard CAL[24].

As our starting point, we model the dilute collectionof trapped, interacting bosons in three dimensions usingthe standard description provided by the Hamiltonian:

H =

∫R3

d3x

[ψ†(− ~2

2m∇2 + V

)ψ +

g

2ψ†ψ†ψ ψ

],(1)

where ψ(~x) represents the annihilation operator for a bo-son of mass m, V (~x) describes an external trapping po-

3

tential, and g represents a coarse-grained contact interac-tion between particles. In terms of microscopic physics,g = 4π~2as/m where as is the s-wave scattering length[38–41]. Here, we study the cases of no interactions(g = 0) and repulsive ones (g > 0).

First, we consider an idealized rotationally-symmetric“bubble trap” potential which can tune between 3Dspheres and quasi-2D thin spherical shells [14, 15]:

Vbubble(~x) =1

2mω2

0s2l

√[(|~x|/sl)2 −∆

]2+ (2Ω)2, (2)

where sl =√

~/2mω0 is the oscillator length associatedwith frequency ω0 and ∆ and Ω are dimensionless pa-rameters that control the radius and width of the poten-tial well [42, 43]. By varying these parameters, we canprobe thermodynamics through the range of desired ge-ometries, as shown in Fig. 1. In particular, ramping upthe value of ∆ at fixed Ω deforms the trap to produce athin-shell whose potential minimum is at fixed, nonzeroradius. Physically, ∆ (Ω) corresponds to the rf detuning(Rabi frequency) of a trapped atomic gas [24].

−50 0 500

100

200

300

−50 0 50 −50 0 50

FIG. 1. (Color online) Top (bottom) row: surface plots (spa-tial slices) of the bubble trap at various detuning values. As∆ increases, the location of the radially symmetric potentialminimum, sl

√∆, expands outwards leading to the formation

of shell-shaped structures.

When considering the bubble trap above, we work inunits where sl = ω0 = m = kB = 1, and set all otherparameters to correspond closely with the physically rel-evant case of ongoing CAL experiments. Namely, wetake the number of atoms to be N = 50000, the di-mensionless “width” parameter to be Ω = 250, and (toexpand from a filled-sphere to a hollow shell) considera range of values for the dimensionless “radius” param-eter: ∆ = 0 → 1000. For a trap with bare frequencyω0/2π = 80 Hz, these correspond to a Rabi frequency of5 kHz and a maximum detuning of 10 kHz. In cases wherewe consider nonzero interactions, we use a value appro-priate for both 87Rb and the CAL experiment, settingg = 0.28~ω0

2 s3l which corresponds to a (dimensionless)

interaction strength of 8πNas/sl = 7000 for N = 50000atoms.

B. Spectrum and states

We first find the energy spectrum and eigenstates fora noninteracting system confined by the bubble trap ofEq. (2). Due to its rotational symmetry, eigenstate wave-functions can be written in the form

φklm(~x) =1

rukl(r)Y

ml (θ, φ), (3)

where (r, θ, φ) are spherical coordinates, and Y ml arespherical harmonics having l = 0, 1, . . . ,∞ and m =−l, . . . , l. The quantum number “k” completes the in-dexing by accounting for the radial direction. Obtainingthe eigenstates then reduces to solving the radial compo-nent of the Schrodinger equation:(− ~2

2m

d2

dr2+ V (r) +

~2l(l + 1)

2mr2

)ukl(r) = εklukl(r),(4)

where ukl(0) = ukl(∞) = 0. For fixed angular mo-mentum quantum number l, the index k = 0, 1, . . .∞is chosen to correspond to increasing energy eigenvalues:ε0,l ≤ ε1,l ≤ · · · ≤ ε∞,l.

We employ a simple finite-difference method to dis-cretize and then numerically solve Eq. (4) as the detun-ing parameter ∆ is varied. As ∆ increases (see Fig. 1),the peak of each eigenstate’s probability density expandsradially outward. Hence, a state localized near the originfor small ∆ becomes squeezed into a hollow thin shell atlarge ∆. Fig. 2a shows the evolution of the bubble trapground state during this hollowing-out procedure. Be-cause the probability density decays gradually towardszero away from its peak, determining the precise ∆ valueat which the topological transition from a filled-to-hollowsphere occurs is subtle. Nevertheless, if we say an innerradius is present when the probability density decays toa ‘sufficiently small’ value while moving radially inwardfrom its peak value, we can estimate the location of thehollowing-out transition. We find the hollowing transi-tion occurs around ∆ ≈ 75.

In the case of a thin spherical shell, it is straightfor-ward to approximate the geometry of the ground state.The radius of the shell corresponds closely to the loca-tion of the potential minimum sl

√∆. Also, the poten-

tial in this limit can be modelled by a shifted harmonicoscillator. Using a one-dimensional shifted oscillator tocapture the the radial behaviour of the ground state, weexpect the thickness of the shell to scale as sl(Ω/∆)1/4.This means the approximate volume of the ground statedensity scales like s3

l∆3/4Ω1/4. We see, therefore, that

as the bubble expands with increasing ∆ and fixed Ω,the thickness of the shell decreases but the total volumeincreases.

Turning our attention to the behavior of the radialwavefunctions ukl(r) (Fig. 2b), we note that the quan-tum number k labels the number of nodes in each radialwavefunction (disregarding the zeros due to boundaryconditions). Since this quantum number labels increas-ingly larger energy eigenvalue states, it is natural to as-

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FIG. 2. (Color online) Eigenstates and spectrum of thenoninteracting bubble-trapped system (a) Probability densityfor the ground state wavefunction in the z = 0 plane at variousdetuning values. In each plot, the color is normalized suchthat black (yellow) corresponds to the minimum (maximum)value. (b) Radial wavefunctions of the bubble trap. Onlythe zero angular momentum states for k = 0, 1, 2 are shown.We show these wavefunctions in the two extreme limits of afilled-sphere, ∆ = 0, and a thin-shell, ∆ = 1000. (c) Partialspectrum of the bubble trap with the potential minimum,Ω ~ω0

2, subtracted off. The color is based on the index k, with

k = 0 corresponding to blue, k = 1 corresponding to orange,and so on. As l increases, so does the corresponding energylevel εkl for a given k. Higher angular momentum states havebeen removed from the image for clarity.

sociate large k values with more oscillatory radial wave-functions (increased radial kinetic energy). The wave-functions with k = 0 are of particular importance as theyhave implications for the dimensionality of the system:these states have their support around the spherically-symmetric potential minimum and hence, for large ∆,correspond to a state localized inside a hollow thin-shell.States with k > 0 have an increased number of nodesin the radial direction and hence a larger kinetic energycost. At large ∆, states with k = 0 have no “excitation”in the radial direction and are indicative of a quasi-2Dsystem.

A partial spectrum of the bubble trap is shown inFig. 2c. Note that for a given k value, as the angularmomentum quantum number l increases, so does the cor-responding energy εkl. Furthermore, for a given l value,we find the energy difference between adjacent k bandsbegins to increase with the detuning parameter ∆. Thespacing between the bottom of the k = 0 and k = 1bands indicates the energy scale at which the system willenter a quasi-2D regime.

III. BUBBLE TRAP THERMODYNAMICS(NONINTERACTING)

In this section, we use the above bubble-trapped statesto numerically compute thermodynamic quantities in theabsence of interactions as the trap evolves from a filledsphere to thin shell geometry. We find that adiabaticexpansion of a BEC leads to condensate depletion andhence an initial condensate can be lost during the process.

A. BEC critical temperature

For a noninteracting (NI) system, thermodynamicproperties follow from Eq. (4) (see appendix A for de-tails on the thermodynamic formalism). Of crucial im-portance for shell-shaped condensates is the BEC criticaltemperature, Tc, which is computed by solving the im-plicit equation:

N =∑kl 6=0

(2l + 1)1

e(εkl−ε0)/kBTc − 1. (5)

where N is the number of particles in the system and wedenote the single-particle ground state (k = 0, l = 0, m =0) with subscript “0”. From the spectrum of the bubbletrap, we can determine the BEC critical temperature as afunction of detuning, ∆. Fig. 3 shows that as the systemexpands from a filled sphere to a hollow, thin shell, thecritical temperature drops significantly.

Qualitatively, this can be explained by considering thephase-space density n/nQ(T ) where n = N/Vol is theparticle density, where Vol represents the characteristicvolume of the bubble, and nQ(T ) ≡ (mkBT/2π~2)3/2 isthe quantum density. Condensation should occur when

5

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10

15

20

25

30

35

40

FIG. 3. The BEC critical temperature of N = 50000 bosonsin the bubble trap versus the detuning parameter ∆, whichcontrols the shell thickness. ∆ = 0 → 1000 corresponds tocontinuously deforming the system from a filled-sphere to hol-low thin-shell (Fig. 1). The temperature is given in units of~ω02kB

where ω0 is the oscillator frequency in Eq. (2).

the phase-space density is on the order of unity. As thesystem expands from a filled sphere into a thin shell, thevolume of the shell increases while the particle numberis fixed, hence the particle density decreases. Therefore,one must lower the temperature to obtain a phase-spacedensity on the order of unity.

In addition to the reduction of Tc at large detuning, thefunction Tc(∆) decreases monotonically with ∆ and alsopossesses an inflection point at a small detuning valueof ∆ ≈ 38. We note this inflection point occurs slightlybefore the hollowing-out transition discussed earlier.

In contrast to a recent calculation by Tononi et al [33]of the BEC critical temperature of a Bose gas in a similargeometry, which relied on the semiclassical approxima-tion, here we obtain our result using the (numeric) spec-trum of the bubble trap. Shortly, we will directly com-pare the semiclassical method with the spectral methodand find that the Tc obtained from the spectrum is lowerthan that obtained semiclassically.

B. Adiabatic expansion and cooling

To model the creation of large bubbles on CAL, weconsider an initially harmonic trap (and filled, spheri-cal system) being expanded outwards by increasing thedetuning parameter ∆. We assume this process is per-formed adiabatically (i.e. without generating heat) andat fixed particle number. Here, we enforce adiabaticityby considering an isentropic (constant entropy) processand compute the temperature during adiabatic expansionby solving the implicit equations for net particle number

N and entropy S:

N=∑kl

(2l + 1)fkl, (6a)

S= kB∑kl

(2l + 1) [(1 + fkl) ln(1 + fkl)− fkl ln fkl] , (6b)

where fkl = 1/(eβ(εkl−µ) − 1) is the Bose-Einstein dis-tribution function at chemical potential µ and inversetemperature β(= 1/kBT ). We model the adiabatic ex-pansion process as follows: we fix a starting temperature(prior to expansion, at ∆ = 0) and then solve Eq. (6) tofind the associated entropy. Next, we model expandingthe gas by increasing ∆. At each stage of the expansion,we determine the new temperature by solving Eq. (6)such that both the particle number N and the entropyS are fixed. Note that when T > Tc one must solvefor both temperature and chemical potential, but whenT < Tc and µ → ε−0 , one must instead solve for thetemperature and number of condensed particles, N0.

We note that one expects adiabatic expansion of thebubble cools the system. For instance, during an isen-tropic process, a free or harmonically trapped BEC obeysthe characteristic relation VolT ν = const. where the ex-ponent ν is positive; hence, increasing the volume of thegas results in a decrease in temperature.

C. Cooling by adiabatic expansion depletes thecondensate

We next present numerical solutions for the temper-ature of a bubble-trapped system during adiabatic ex-pansion at various starting temperatures; the results aresummarized in Fig. 4a. We note that the BEC criticaltemperature decreases faster than the temperature of thegas as it adiabatically expands. This means that if onedoes not sufficiently cool the system before beginning toexpand it into a bubble shape, an initially condensedsystem transitions during expansion into a thermal gas.An explicit example of this can be seen for the adiabaticexpansion temperature profile with initial temperaturekBTi = 30~ω0

2 (which corresponds roughly to a temper-ature of 60 nK for a 80 Hz trap). Although the systembefore expansion is in the condensed phase, it eventuallyfinds itself above the condensate transition temperatureas the parameter ∆ increases.

Even if the initial temperature is low enough to remainin the condensed phase for the entire expansion, we findthat the condensate fraction decreases as the system ex-pands. In other words, adiabatic expansion in this caseleads to a loss of phase-space density; this is a negativeversion of the long-known phase-space density increasesat constant entropy exploited in various experiments [44–47]. Figure. 4b explicitly shows the depletion upon adi-abatic expansion for a few different initial temperaturesby plotting the condensate fraction N0/N . This effect ispronounced when comparing the local condensate den-sity with the excited state density at various stages in

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(c)

FIG. 4. (Color online) (a) BEC critical temperature (blackline) of N = 50000 particles in the bubble trap along with var-ious adiabatic expansion temperature profiles (colored lines).Each adiabatic expansion corresponds to a different initialtemperature when ∆ = 0: kBTi/

~ω02

= 10, 20, 30, . . . . Notethat Tc decreases faster than any adiabatic temperature pro-file; thus, expanding the gas isentropically leads to a decreas-ing condensate fraction. (b) Plot of the condensate fraction,N0/N , during the adiabatic expansions shown in the previ-ous image. (c) Comparison of the local condensate density(colored-dashed lines) and excited state density (black-solidlines) at various stages of the kBTi = 20 ~ω0

2adiabatic expan-

sion (shown along the z-axis). Above each density curve isthe associated value of ∆ (marked at the potential minimum

location sl√

∆).

the expansion process. If we consider a condensed gasbeing expanded adiabatically whose initial temperatureis kBTi = 20~ω0

2 , we show in Fig. 4c that there is clearcondensate depletion at large ∆.

It is important to note that although condensate deple-tion during adiabatic expansion presents an experimen-tal hindrance to observing large BEC shells, Fig. 4b alsoshows that the degree of severity depends crucially on theinitial temperature prior to expansion. A system whichstarts off colder experiences depletion at a far slower ratewhen compared to one which starts off warmer.

D. Validity of the semiclassical approximation

It is instructive to see how results obtained using stan-dard semiclassical methods (see appendix A) comparewith those obtained using the (numeric) bubble trapspectrum (Fig. 5). Importantly, we find that the pre-dictions of the semiclassical approximation become lessaccurate as the shell is expanded (in particular, note thedifference in predicted BEC critical temperature at large∆). The semiclassical approximation assumes the tem-perature is much larger than the energy level spacingsover the energy range in which it is used. Discrepanciesbetween semiclassical and spectral calculations indicatethat this condition is not being satisfied. As the bub-ble expands, the nature of energy levels change (recallFig. 2c) which, in turn, lowers both Tc and the tempera-ture during adiabatic evolution. Although level spacingsat the bottom of each k band decrease during expansion,the effects of higher angular momentum states must alsobe considered. From Fig. 5, we conclude that the com-bined effect of a decreasing temperature with the modi-fied level spacings means semiclassical predictions shouldbe taken with some level of caution for thin-shell systemsor fully-expanded bubbles.

Notably, the semiclassical prediction for Tc in the thin-shell becomes considerably higher than that obtained us-ing the spectrum. At the highest detuning shown, thedifference is about 6.5~ω0

2 (roughly 12.5 nK for a 80 Hztrap). Hence, for some range of initial temperatures be-fore expansion, the semiclassical approximation will in-correctly predict that an expanding gas will remain con-densed when in fact it will transition to the normal phase.

Finally, we note that when T & Tc, the temperaturepredictions during adiabatic expansion don’t vary greatlybetween the spectral and semiclassical methods. Fur-thermore, even below Tc, the difference between the twomethods is far less severe than that seen in the predictionof the BEC critical temperature.

IV. EFFECTS OF INTERACTIONS ANDDIMENSIONALITY

Below, the inclusion of interactions is discussed. AtT = 0, if the particle number is held fixed, we find a

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FIG. 5. (Color online) Comparison of our previous results us-ing the spectrum of the bubble trap with the semiclassical ap-proximation. Here, we show the BEC critical temperature forN = 50000 particles along with various adiabatic expansiontemperature profiles. [The undashed (dashed) black (blue)line is Tc using the spectral (semiclassical) approach. Theundashed (dashed) grey (magenta) lines are adiabatic expan-sion profiles using the spectral (semiclassical) approach.]

noninteracting description to be a good approximationfor thin shells. We then use this result to develop aneffective low temperature theory in a spontaneous U(1)symmetry broken phase and find, in the thin-shell regime,that interactions do little to modify the results obtainedpreviously. We conclude by considering the breakdown ofthis mean-field theory approach as one crosses over from3D to 2D physics.

A. Validity of the noninteracting description

Having described the thermodynamics of a bubbletrapped gas in the NI (g = 0) limit, we now addressthe question of where the noninteracting description ac-curately captures salient features even when interactionsare present. Working at zero temperature, we show herethat for thin shells at fixed particle number, the NI pic-ture indeed constitutes a reasonable description. Wejustify this assertion in two ways: 1) we analyze nu-meric solutions of the Gross-Pitaevskii equation for anexperimentally-relevant interaction strength and contrastthis data with both weakly- and strongly-interacting lim-iting cases, 2) we perform a variational calculation to gaininsight into the role of interactions in the thin-shell limit.

Comparing condensate wavefunctions: Themany-body ground state of Eq. (1) is characterized bya condensate wavefunction ψc(~x) which obeys the time-independent Gross-Pitaevskii (GP) equation:

0 =

(− ~2

2m∇2 + V (~x)− µ+ g|ψc(~x)|2

)ψc(~x). (7)

Because the GP equation is nonlinear, analytic solutionsare typically unfeasible. Fortunately, in cases whereg → 0 or (~2/2m)∇2ψc → 0, which correspond phys-ically to non- and strongly-interacting systems, respec-tively, analytic solutions can be found.

In the absence of interactions, g = 0, we denote thesolution as the noninteracting condensate wavefunction:ψNI(~x) =

√N0φ0(~x) where the subscript “0” denotes

the single-particle ground state. In the opposite limit ofstrong interactions, we use the Thomas-Fermi (TF) ap-proximation and treat the kinetic contribution to Eq. (7)as negligibly small [38, 39]. Dropping the kinetic termallows one to solve the GP equation to obtain the TFcondensate wavefunction: |ψTF| =

√(µ− V )/g when

V (~x) < µ and ψTF = 0 otherwise [38, 39].Numerical solutions for the ground state of Eq. (7) (at

a range of ∆ values for the bubble trap potential, Eq. (2))were found using an imaginary-time algorithm [48] andtaking the experimentally-relevant value of dimensionlessinteraction strength, 8πNas/sl = 7000.

We now calculate and compare the kinetic energy,∫d3xψ∗c (− ~2

2m∇2)ψc, trap potential energy,

∫d3xV |ψc|2,

and interaction energy,∫d3x g2 |ψc|

4, at zero-temperatureusing the numerically-solved (GP), noninteracting (NI),and strongly-interacting (TF) condensate wavefunctions.Fig. 6a shows the energy fractions as ∆ is increased (ex-panding the system from a filled-sphere to a thin-shell)at T = 0, with fixed particle number, using the threecondensate wavefunctions.

From the numerically-solved GP results, one can seeclearly that the energy stored in interactions is the dom-inant contribution for small ∆ (filled spheres), but be-comes the least significant contribution for large ∆ (thinshells). For parameter regimes in which the interactionenergy fraction is sufficiently suppressed, one expects theNI case to give a reasonable description of the system.

Let us now compare results between the various con-densate wavefunctions. For small ∆, the TF approxi-mation does reasonably well compared to the numericalsolution (GP) whereas the NI approximation does poorly;hence, interactions are expected to be important in thedescription of a bubble-trapped system in this param-eter regime. Conversely, as ∆ increases, and the shellexpands, the TF approximation becomes a poor approx-imation of the GP numerics, whereas the NI approxi-mation performs reasonably well. In particular, the TFenergy fractions become nearly constant, whereas the GPinteraction energy fraction falls drastically and begins toconverge with the NI wavefunction results for ∆ & 300.Thus, for thin shells, a NI description is a good approxi-mation to the full solution with interactions. The break-down of the TF approximation as ∆ increases is due toits failure to capture the rising kinetic energy fraction ofthe GP condensate in this regime. We also note that, al-though the NI and GP interaction energy fractions beginto converge, the NI wavefunction systematically overesti-mates the kinetic and underestimates the trap potentialenergy fractions. To further contextualize these conclu-

8

0 500 10000.0

0.2

0.4

0.6

0.8

1.0GP

NI

TF

0 500 1000 0 500 1000

(a)

−40−20 0 20 40−40−20

0

20

40

(b)

FIG. 6. (Color online) Comparison between the zero-temperature GP, NI, and TF condensate wavefunctions atvarious detunings for N = 50000 atoms in the bubble trap.(a) From left to right, the fraction of energy stored as ki-netic, trap potential, and interaction energy. Note: thebubble trap reference energy has been subtracted off fromthe potential. We also treated quantum depletion as nearlynegligible for the NI condensate wavefunction [38] (we setN0 = 0.99N for T = 0). (b) Local condensate density atvarious detunings. The density is plotted in the z = 0 plane:nc(x, y, 0) = |ψc(x, y, 0)|2. In each plot, the color is normal-ized such that black (yellow) corresponds to the minimum(maximum) density value.

sions, we direct the reader to Fig. 6b which shows the(T = 0) local condensate density at various detuning val-ues. Here, one can see clearly that for filled spheres (thinshells) the TF (NI) condensate wavefunction captures thesalient spatial features of the GP results.

Variational calculation: To further understand theabove results, let us look for an origin to the decreasinginteraction energy fraction during the zero-temperatureexpansion. This result might be perplexing – one mightsuspect interactions to play a prominent role when theshell becomes thin and atoms get squeezed into a spaceof small width. However, one should keep in mind thatthe particle density is going down as the system becomesthinner: the particle number is held fixed, while the con-densate volume, which we argued earlier should scale as

s3l∆

3/4Ω1/4 in the NI case, is increasing as the bubbleexpands.

To see this more concretely, we perform a variationalcalculation [38] with a trial condensate wavefunction:

ψtrial(~x) = AF

(|~x| − ab

)eiϕ(~x), (8)

where the variational parameters A, a, and b are allnonnegative real constants, ϕ(~x) is the phase of thecondensate wavefunction, and F (·) is a (nonnegative)smooth real function which decays to zero for large ar-guments [49]. For example, we could use a Gaussian,

F (u) = e−u2/2 [38, 49], from which one can see the length

scales have a natural interpretation: a ≈ (Rout + Rin)/2describes the average shell radius and b ≈ Rout−Rin de-scribes the shell thickness where Rout (Rin) is the outer(inner) radius of the condensate shell. In this context,we can define a thin-shell as one for which a/b 1.

We now insert the trial wavefunction into H[ψ∗c , ψc]−µN [ψ∗c , ψc] where H[·] and N [·] are the Hamiltonian andparticle number functionals respectively. Considering nosuperflow, ~

m∇ϕ = 0, we extremize the functional andobtain the following forms for the kinetic, trap poten-tial, and interaction energy terms in the thin shell limit,b/a→ 0:

Etrialkin ≡

∫R3

d3xψ∗trial

(− ~2

2m∇2

)ψtrial ∼ C1

N

b2, (9a)

Etrialpot ≡

∫R3

d3xV (~x)|ψtrial|2 ∼ C2Na2b2, (9b)

Etrialint ≡

∫R3

d3xg

2|ψtrial|4 ∼ C3

N2

a2b, (9c)

where we have used the relationship N ∼ C0A2a2b as

b/a → 0 to replace the (amplitude) variational parame-ter A with the particle number N (which normalizes thetrial wavefunction), and the numbers C0, . . . , C3 are in-dependent of the variational parameters A, a, and b. Toreach Eq. (9b), we took the thin-shell limit, b/a → 0,such that Ω remained fixed and assumed the variationalparameter a was close to the location of the bubble trap’spotential minimum sl

√∆. For clarity, in Eq. (9b) we also

subtracted off the contribution due to the bubble trapreference energy Ω~ω0

2 .From these asymptotic results, one finds that the frac-

tion of energy stored in interactions relative to the totalenergy becomes vanishingly small in the thin-shell limit.This occurs because the ratio of interaction energy overkinetic energy is proportional to Nb/a2. Therefore, forfixed particle number, the zero-temperature interactionenergy fraction tends toward zero for increasingly thinshells.

B. Do interactions help preserve the condensate?

In the NI case, we found earlier that adiabatically ex-panding a condensate into a thin shell led to a decreasing

9

condensate fraction. A natural question is how interac-tions modify this picture. However, based on the argu-ments of the previous section, we should not expect dra-matic changes in the thin-shell limit where the NI case isexpected to be a good approximation. In general, com-puting thermodynamics in the presence of interactionsis far more involved than in their absence [40, 41]; herewe proceed using a standard Bogoliubov quasiparticle de-scription (see appendix A). In particular, at temperaturesfar below the U(1) symmetry breaking transition, we de-velop a mean-field theory by expanding the boson fieldabout the NI condensate wavefunction ψc(~x) ≈ ψNI(~x).This formalism assumes that the NI condensate wave-function is a reasonable approximation to the true solu-tion of the GP equation, which we have shown above tobe correct for a thin shell. Thermodynamics then followfrom solutions to the Bogoliubov equations (see appendixB for the case of the bubble trap) which we solve numer-ically using a finite-difference method.

As a means to probe the effect of interactions on thethermodynamics of an expanding system, we set a con-densate fraction, N0/N , and then model expanding thegas by increasing ∆. Instead of doing the expansion adi-abatically, here we simply ask: what temperature is re-quired to maintain the given condensate fraction? This ismotivated by the fact that the NI condensate wavefunc-tion is a poor approximation of the true condensate wave-function at low-detunings. To evolve an initial (∆ = 0)temperature isentropically, we need to determine and fixthe entropy at the start of the expansion, but because oureffective theory assumes the NI condensate wavefunctionis a reasonable reference state, any thermodynamic pre-dictions we make at such small ∆ are suspect.

In Fig. 7 we compare the temperatures required for an80% condensate fraction in the NI and Bogoliubov for-malisms. In the regime of applicability of our Bogoliubovdescription using the NI condensate wavefunction, we seethat there is a range of geometries over which interactionsdo increase the temperature required to obtain the givencondensate fraction, but that this effect is very small. Wenote that the Bogoliubov temperature briefly drops be-low its NI counterpart, but then rises and remains slightlyabove for the rest of the expansion. Consistent with thearguments of the the previous section, we thus find thatinteractions do little to change the thermodynamics ofthin shells.

C. Dimensional crossover

While the previous thermodynamic calculations de-pend on mean-field considerations, there are several lim-its in which these approaches would not suffice. First,the finite size of the sphere results in deviations fromthe thermodynamic limit. In principle, first order correc-tions could be taken into account using finite-size scalinganalyses [50].

As we consider thinner shells, a more drastic effect

0 200 400 600 800 10000

5

10

15

20

25

30

35

40

FIG. 7. (Color online) Bubble trap expansions of N = 50000atoms done at a fixed-condensate fraction N0/N . The blackline shows the NI BEC critical temperature, whereas the col-ored lines show the temperature required for an 80% conden-sate fraction. The solid lines correspond to NI data whereasthe markers correspond to Bogoliubov collective excitations.

comes from reduction in effective dimensionality. Thesystem enters the two-dimensional regime at tempera-tures below which excitations along the radial directionbecome energetically unfavorable. More precisely, ap-pealing to the energy spectrum in Fig. 2c, energy levelsεk,l are characterized by excitations along the radial (k)and angular (l) directions. For temperatures low enoughto excite only the k = 0 band, we have the conditionkBT < ε1,0 − ε0,0. At the highest detuning shown in

Fig. 2c, we find ε1,0−ε0,0 ≈ 4~ω0

2 which corresponds to atemperature scale on the order of 10 nK for a 80 Hz trap.Below this temperature, and at the largest detunings, weexpect the emergence of two-dimensional physics; thisregime clearly requires treatments that go beyond mean-field.

As with planar thin films, defects such as vor-tices form the natural source of excitations that com-plement the mean-field hydrodynamic long-wavelength,low-frequency excitations [51]. In previous work bytwo of the current authors and collaborators, weevaluated the energy for creating a vortex-antivortexpair at opposite sides of a spherical condensateshell of finite thickness [52]: Ethin-shell

vortex /Ethin-shell ≈2π~2

3gm δ(

ln Rout

ξ0+ ln δ

ξ0+ 4.597

)where Rout is the outer

radius, δ is the shell thickness, ξ0 is the coherencelength, and g is the 3D interatomic interaction strength.Thus, depending on the parameters, even before enteringthe two-dimensional limit, we may well access a finite-thickness regime in which vortex-antivortex pair excita-tions become favorable above a certain temperature, thusdestroying condensation in the system.

In the two dimensional limit, the proliferation of suchvortex-antivortex pairs renders the critical temperatureto take the Berezinskii-Kosterlitz-Thouless (BKT) form

10

[53, 54]. The effect of curvature on such physics is highlyinteresting in and of itself [55–58]. In recent work, Tononiand co-workers demonstrated that in the precise shell set-ting considered here and of relevance to CAL, the BKTtransition finds its finite-size manifestation [35].

V. APPLICATIONS TO THE CAL TRAP

Having found the change in temperature and conden-sate transition temperature of a system undergoing adi-abatic expansion from a filled sphere to a thin shell in anidealized bubble trap, we now consider the expansion of asystem in a realistic CAL trap: V (~x) = VCAL(~x). In themicrogravity environment aboard the ISS, CAL is ableto produce exceptionally large mm-scale bubbles in bothfilled and hollow regimes. Figure. 8 shows a small sampleof the CAL dressed potentials experienced by magneti-cally trapped 87Rb atoms. As discussed in [24], theseremarkable dressed potentials are generated using atom-chip current configurations. By increasing the rf detun-ing signal, one can deform the trap geometry to produceshell-shaped potential-minimum surfaces. The CAL trapis thus broadly similar to the bubble trap. As a result,much of the bubble trap physics discussed above shouldbe applicable to the CAL experiment.

0

50

100

0

50

100

0

50

100

−100 0 1000

50

100

−50 0 50 −50 0 50

FIG. 8. (Color online) Spatial slices and surface plots of theCAL trap at various detuning frequencies. In contrast to thebubble trap, the CAL trap is anisotropic, hence the imagesare shown along different spatial axes. Each consecutive rowof images corresponds to a higher detuning frequency, withthe first row corresponding to the initial or “bare” trap.

There are crucial differences in the details though; un-like the bubble trap, the CAL trap is spatially inho-mogeneous, breaking rotational symmetry. This meansquantum-mechanical modelling of CAL trap thermody-namics is significantly more difficult, requiring solutionsof three-dimensional partial differential equations as op-posed to one-dimensional radial equations. Fortunately,some of the experimentally-relevant adiabatic expansionsoccur at temperatures above Tc and hence numerically-expensive diagonalization can be avoided through use of

the semiclassical approximation.Here we use the semiclassical approach to obtain ther-

modynamics during system expansion in the CAL trap.To carry out our calculations, CAL atom-chip currentdata is used as input to generate a spatial grid of dressedpotentials, VCAL(~x), appropriate for 87Rb atoms in the|F = 2,mF = 2〉 hyperfine state [24]. Using thesemiclassical approximation for NI atoms, we computeboth the BEC critical temperature and the tempera-ture of the system during adiabatic expansions for afew experimentally-relevant initial temperatures (Ti =97, 289, 387, 597 nK) with the results displayed in Fig. 9.We see both the BEC critical temperature and the tem-perature during adiabatic expansions decrease with ap-plied detuning frequency. However, Tc does not decreasenoticeably faster than the adiabatic curve for the caseof the initially-condensed gas. This same phenomenonwas observed in the semiclassical treatment of the bub-ble trap (see Fig. 5). In the case of the bubble trap, wesaw that for temperatures at or below Tc, the semiclassi-cal approach became less accurate as detuning increasedand the bubble became thin. Most notably, we saw thatthe semiclassical model overestimates the BEC criticaltemperature for large bubbles. Hence, we find it is likelythat, at these experimentally-relevant starting tempera-tures, that adiabatic expansions in the traps on CAL willproduce thermal clouds rather than condensates.

−40 −30 −20 −10 0 10 20 30101

102

103

Tc

97

289

387

597

FIG. 9. (Color online) BEC critical temperature (black line)of N = 50000 noninteracting 87Rb atoms in the CAL trapalong with various adiabatic expansion temperature profiles(colored lines) calculated semiclassically.

VI. OUTLOOK

In summary, we have presented a thorough study of thethermodynamic properties of BECs in shell-shaped ge-ometries while working in parameter regimes applicableto CAL experiments. We began by charting out the spec-

11

trum and eigenstates of a bubble trap potential throughthe evolution of trap parameters for a condensate froma filled sphere to hollow thin shell. Based on this analy-sis, we numerically computed thermodynamic quantitiesfor a noninteracting system and found that adiabatic ex-pansion leads to condensate depletion. Even in the pres-ence of interactions, we argued that our conclusions holdfor fixed particle systems in the thin-shell limit. Finally,using semiclassical methods to calculate thermodynamicproperties of 87Rb atoms in CAL experimental traps, wemade the connection between this realistic system andthe bubble trap more concrete.

While several immediate avenues open up with regardsto BECs in shell-shaped geometries, we first note: theCAL experiments, corroborated by theory, show some-thing remarkable. In the microgravity environment,theshell inflation mechanism provided by the bubble trapallows for macroscopically large, suspended and con-tained gas structures to emerge. While macroscopicquantum coherence across these structures are likely totake further efforts, having topologically non-trivial ther-mal gases that span linear dimensions of the order ofmillimeters is a feat in and of itself. Our studies showhere that it is not impossible to retain condensationduring adiabatic expansion into these remarkable bub-bles. The next-generation CAL experiments promise toachieve such structures, providing fertile ground for stud-ies of multiple aspects of condensate shells.

With regards to equilibrium and linear response fea-tures, previous works have elucidated several prospects.These thermodynamic analyses would require further de-velopments to connect with experiments, such as moresophisticated techniques for handling interactions, highercomputational power for handling realistic trap geome-tries, and accounting for the strong asymmetry presentin actual traps. Thermodynamic studies would also servewell to interface with other aspects of condensate shellsthat are being studied, including collective modes andvortex physics. Connecting the thermodynamic studiespresented here with other settings for shell-shaped con-densates, such as in stellar bodies, optical lattices, andBose-Fermi mixtures, would offer new insights.

An additional realm poised for investigation involvesnon-equilibrium dynamics. In this work, we assumed adi-abaticity and that the dynamic shell evolution involveda quasi-static process. Generally, there can be a varietyof dynamical situations where adiabaticity breaks down;tracking the rate of entropy growth would be one steptowards quantifying such deviations. Perhaps the mostdrastic deviation from adiabaticity concerns regimes inwhich the system is dynamically tuned across the BECtransition. Our results do indeed suggest that suchtransitions during expansion are likely. The expansionwould then lead to non-equilibrium critical phenomenonof Kibble-Zurek physics [59–63]. This mechanism for pro-ducing defects, in this case, vortices [60, 64–66] and morecomplex defects in spinful condensates [67–71], wouldnot only require important consideration in attempts to

achieve quantum coherence, micro-gravity and the shell-shaped geometry could perhaps conspire to realize themuch sought-after universal scaling behavior associatedwith this phenomenon.

These form but a few considerations in the fascinat-ing and diverse realms that host these unique shell-shaped condensate structures from stellar bodies to sys-tems of co-existent phases on Earth to the on-going stud-ies aboard the International Space Station. A commonunderlying thread is the extreme conditions and dynamictuning offered by Nature at the astronomical realms andby advances in the ultracold experiments. One mightspeculate on how much of the thermodynamics of ex-panding shells discussed here would be relevant in stellarevolution and formation of neutron stars. In the mean-while, while we have demonstrated that retaining a con-densate through expansion can be a delicate matter, wehave shown consistent with hints from CAL, that in theultracold experimental realm, it is entirely possible tocreate remarkable, gigantic, diaphanous thermal bubbles.

ACKNOWLEDGMENTS

We are grateful to Karmela Padavic and Kuei Sun forinvolved conversations and laying part of the backgroundwork in previous collaborations. We thank Bryan Clarkfor illuminating conversations that informed numerics forrealistic CAL settings. This work made use of the IllinoisCampus Cluster supported by the University of Illinoisat Urbana-Champaign. This work was supported by theNational Aeronautics and Space Administration througha contract with the Jet Propulsion Laboratory, CaliforniaInstitute of Technology.

Appendix A: Thermodynamics of dilute bosons

Consider the Hamiltonian H for a collection of diluteinteracting bosons in R3, Eq. (1). We are interested inthermodynamics, hence compute thermal averages using

〈· · · 〉 = Tr( 1Z e−β(H−µN) · · · ), where the trace is over

Fock space, β(= 1/kBT ) is the inverse temperature, µ

is the chemical potential, N is the number operator, andZ is the grand partition function:

Z = Tr(e−β(H−µN)

)= N

∫[Dψ∗Dψ] e−S[ψ∗,ψ]/~.

(A1)

Above, we introduced the coherent state path integral inthe usual way [72]: N is an unimportant constant pref-actor, [Dψ∗Dψ] is the integration measure of a complex-valued field ψ(τ, ~x) with “time” coordinate τ ∈ [0, β~](the field obeys periodic boundary conditions in time),

12

and the action is

S =

∫dτd3x

[ψ∗(~∂τ −

~2

2m∇2 + V − µ

)ψ +

g

2|ψ|4

].

(A2)

As T → 0+ (β → ∞), only field configurations whichminimize the action contribute significant weight to thepartition function. We denote such configurations as“condensate wavefunctions” ψc [38, 39] — they are staticand obey the time-independent Gross-Pitaevskii equa-tion, Eq. (7).

1. Thermodynamics of noninteracting bosons

With interactions turned off (g = 0), thermodynamicsare obtained from the spectrum and wavefunctions of thesingle-particle Schrodinger equation [38, 39]. In this sec-tion, let “m” (εm) denote the eigenstates (eigenvalues)of the Schrodinger equation:(

− ~2

2m∇2 + V (~x)

)φm(~x) = εmφm(~x). (A3)

Further, let m = 0 denote the ground state of this equa-tion (which we assume is non-degenerate) and ε0 denotethe ground state energy. Expanding the field operator as

ψ(~x) =∑m φm(~x)bm in the operator formalism or the

field as ψ(τ, ~x) =∑nm e

−iωnτφm(~x)ψnm, where ωn arebosonic Matsubara frequencies [72], in the path integralformalism gives

Z =∏m

1

1− e−β(εm−µ), (A4)

where µ < ε0 in order to keep the trace bounded. Fromderivatives of the grand free energy, − 1

β lnZ, various

thermal expectation values can be computed:

N=∑m

fm, (A5a)

E=∑m

fmεm, (A5b)

S= kB∑m

[(1 + fm) ln(1 + fm)− fm ln fm] , (A5c)

where we denote the system’s particle number, energy,and entropy by N , E, and S respectively, and fm =1/(eβ(εm−µ) − 1) is the Bose-Einstein distribution func-tion. If we move onto spatially resolved quantities, suchas the one-body density matrix,

〈ψ†(~x)ψ(~x′)〉 =∑m

φ∗m(~x)φm(~x′)fm, (A6)

we see these require information on the wavefunctions.Thus, at least formally, solving Eq. (A3) gets us the ther-modynamics.

2. The semiclassical approximation

When solutions to Eq. (A3) are intractable, a usefulapproximation exists. Provided the temperature is muchlarger than the spacing between single-particle energylevels,

kBT ∆E, (A7)

where ∆E is a characteristic energy scale for the levelspacings, we can approximate thermodynamic quanti-ties by replacing sums over eigenstates with integrals[38, 39]. Making this replacement still requires a no-tion of what the states in Eq. (A3) are; fortunately, thiscondition is consistent with using the classical relationε~p(~x) = 1

2m |~p|2 + V (~x) [38, 39]. Provided the condition

in Eq. (A7) is met, after integrating over momentum, onefinds in the normal phase, T > Tc:

N= nQ

∫R3

d3xLi 32(e−β(V−µ)), (A8a)

E= nQ

∫R3

d3x

[3

2kBT Li 5

2(e−β(V−µ))

+V Li 32(e−β(V−µ))

], (A8b)

S= kBnQ

∫R3

d3x

[5

2Li 5

2(e−β(V−µ))

+β(V − µ) Li 32(e−β(V−µ))

], (A8c)

where nQ(T ) ≡ (mkBT/2π~2)3/2 is the quantum densityand Lis(z) =

∑∞n=1 z

n/ns is the polylogarithm or, ascommonly referred to in this context, the Bose function[73].

3. Thermodynamics of (weakly) interacting bosonsat low-temperatures

With interactions turned on (g > 0), computing thethermodynamics is no longer trivial [40, 41]. In general,path integral Monte Carlo methods offer a powerful wayto proceed [74–76]. Green function methods have foundsuccess in analytically addressing questions such as theshift in Tc from its NI value [77–79]. At temperatures wellbelow (or commensurate to) the BEC critical tempera-ture, it is common to employ some variation of the mean-field theory introduced by Bogoliubov [37–39, 80, 81].Because our interests lie in ultracold dilute atomic gases,we will restrict ourselves to temperatures far below theBEC transition, T Tc, and use a Bogoliubov quasipar-ticle description.

At sufficiently low temperatures, provided the dimen-sion of space is larger than the lower critical dimension(which is 2 from the Mermin-Wagner–Hohenberg [82, 83]or Coleman [84] theorem), the system can spontaneouslybreak the U(1) symmetry of the Hamiltonian. This is sig-

nified by a nonvanishing field expectation value: 〈ψ〉 ∼

13

ψc as T → 0 where the ground state field configuration,ψc, is a solution to Eq. 7. At these low-temperatures, weperform a saddle-point analysis to reach an effective low-temperature theory for fluctuations out of the conden-sate. Namely, we use the “Bogoliubov shift”: ψ = ψc+δψ[37, 81]. In the operator formalism:

H − µN= −Ecint

+

∫R3

d3x δψ†(− ~2

2m∇2 + V − µ+ 2g|ψc|2

)δψ

+

∫R3

d3xg

2

(ψ2cδψ

†δψ† + h.c.)

+O(δψ)3, (A9)

where Ecint ≡g2

∫d3x|ψc|4 is the interaction energy of the

condensate wavefunction and the fluctuations, δψ, obeybosonic commutation relations. The quadratic portionof the operator in Eq. (A9) can be brought into diagonalform by use of a Bogoliubov transformation [38, 39, 85]:

H−µN = −Ecint

+Epair +1

2MP 2z +

∑λ

(Eλ>0)

Eλβ†λβλ +O(δψ)3. (A10)

In the equation above, the energies, Eλ, of the (bosonic)

quasiparticles, βλ, are given by solving the Bogoliubovequations [38, 39]:

Eλuλ(~x)=

(− ~2

2m∇2 + V − µ+ 2g|ψc|2

)uλ(~x)

+gψ2cvλ(~x), (A11a)

−Eλvλ(~x)=

(− ~2

2m∇2 + V − µ+ 2g|ψc|2

)vλ(~x)

+gψ∗2c uλ(~x), (A11b)

where we introduced the wavefunctions uλ(~x) and vλ(~x).For positive energy solutions, the wavefunctions can bechosen to obey the orthonormality conditions [85]:

δλλ′ =

∫R3

d3x (u∗λuλ′ − v∗λvλ′) , (A12a)

0 =

∫R3

d3x (uλvλ′ − vλuλ′) , (A12b)

The Bogoliubov collective excitations in Eq. (A10) arecreated by:

β†λ ≡∫R3

d3x(uλδψ

† − vλδψ), (A13)

and thus uλ(~x) and vλ(~x) have the interpretation of a“particle” and “hole” wavefunction respectively. We alsosee the emergence of a “momentum operator” [38, 85],

Pz ≡∫R3

d3x(ψc δψ

† + h.c.), (A14)

associated with a zero-mode solution of the Bogoliubovequations [85, 86]. The “mass” in the Hamiltonian,

M ≡ ∂µ∫d3x|ψc|2, is a (positive) inverse energy scale

introduced for convenience [85]. Lastly, there is a shift inthe reference energy that arises due to the pairing termsin Eq. (A9). This energy shift, which is nonpositive, isgiven by:

Epair ≡ −∫R3

d3x

(1

2M|ψc|2 +

∑λ

(Eλ>0)

Eλ|vλ|2). (A15)

We can define a “position operator”, Qz, which obeysthe canonical commutation relation [Qz, Pz] = i as [38,86]

Qz ≡∫R3

d3x

(− i∂µψc

Mδψ† + h.c.

). (A16)

Note: these zero-mode operators commute with the βλin Eq. (A10). From the new operators, the original fluc-tuation field operator can be written as [86]

δψ =∂µψcM

Pz − iψc Qz +∑λ

(Eλ>0)

(uλβλ + v∗λβ†λ). (A17)

For temperatures far below the U(1) symmetry break-ing transition, we assume that, when calculating observ-ables, it is permissible to ignore the third and fourthorder fluctuation terms in Eq. (A10). For consistency,this requires that the “condensate depletion” is small:

〈∫d3xδψ†δψ〉/N 1. Thus, we reach an effective

Hamiltonian for the fluctuations:

Heff ≡ −Ecint + Epair +1

2MP 2z +

∑λ

(Eλ>0)

Eλβ†λβλ. (A18)

The eigenstates of Eq. (A18) are simply those of themomentum and quasiparticle number operators:

|Pz, Nλ〉 =e−

12P

2z+i√

2Pz β†z+ 1

2 (β†z)2

π1/4

∏λ

(Eλ>0)

(β†λ)Nλ√Nλ!

|vac〉,

(A19)

where Pz ∈ R, Nλ ∈ Z≥0, βz is a ladder operator for

the zero-mode [86], βz ≡ (Qz + iPz)/√

2, and |vac〉 is the

vacuum state of the βz and βλ operators.We should now be in a position to approximate ther-

mal averages in the broken U(1) phase using 〈· · · 〉 ≈Tr( 1Z e−βHeff · · · ), but as discussed in [86] doing so naively

leads to inconsistent results. Because the eigenstates ofthe effective Hamiltonian have definite zero-mode mo-mentum, i.e. Pz is a good quantum number, we neces-sarily find 〈Q2

z〉 diverges as a result of the Heisenberg

uncertainty principle. Because 〈δψ†δψ〉 contains a termproportional the variance in the zero-mode position op-erator, we then violate the condition that the conden-sate depletion be small [86]. Thus, when using the ef-fective Hamiltonian to compute thermal expectation val-ues, one cannot simultaneously trace over the zero-mode

14

momentum states and demand the depletion be small.Here, we avoid this issue in a pragmatic way throughour choice of the condensate wavefunction. We takethe condensate wavefunction to be the NI (g = 0) so-lution ψNI(~x) =

√N0φ0(~x). Although this may seem

like a poor choice, we find in the main text that us-ing the NI condensate wavefunction can be justified inthe limit of thin-shells at fixed particle number. Tak-ing ψc(~x) ∼

√N0φ0(~x) leads to both Pz and Qz van-

ishing if we neglect fluctuations in the m = 0 mode:

δb0 ≡ b0 −√N0 → 0. Treating b0 as a c-number is well-

justified for T Tc where N0 is macroscopically large;thus, this approximation is standard in the Bogoliubovtreatment of the weakly interacting Bose gas [38, 39, 80].Therefore, using the NI condensate wavefunction meansour effective Hamiltonian is still given by Eq. (A18), butwith the zero-mode momentum term removed.

We now calculate thermodynamic quantities in thepresence of (weak) interactions. First, the requirement

that 〈ψ〉 ∼ ψc as T → 0 is consistent with the effectivetheory: the field obtains a nonzero expectation value,

〈ψ(~x)〉 =√N0φ0(~x), spontaneously breaking the U(1)

symmetry of the Hamiltonian. This also states the con-densate wavefunction is the mean-field in the broken sym-metry phase for all temperatures at which the effectiveHamiltonian remains valid. Next, the one-body densitymatrix becomes

〈ψ†(~x)ψ(~x′)〉 = N0φ∗0(~x)φ0(~x′)

+∑λ

(Eλ>0)

[fλu∗λ(~x)uλ(~x′) + (1 + fλ)vλ(~x)v∗λ(~x′)] , (A20)

where fλ = 1/(eβEλ − 1) is the Bose-Einstein distribu-tion function for the Bogoliubov quasiparticles. It is thenstraightforward to show the number of particles is:

N = N0(T )

+∑λ

(Eλ>0)

∫R3

d3x[fλ|uλ|2 + (1 + fλ)|vλ|2

]. (A21)

Notice that to ensure self-consistency at temperature T ,

the number of particles in the NI ground state, N0(T ),must be chosen such that Eq. (A21) is satisfied.

Appendix B: Bogoliubov equations in the bubbletrap

Within the mean-field theory discussed in appendixA, thermodynamics of the interacting system followfrom solutions to the Bogoliubov equations, Eq. (A11).Due to rotational invariance of the bubble trap andthe associated NI condensate wavefunction, ψNI(r) =√N0 u

NI0 (r)/

√4π r, where uNI

0 (r) is the ground state so-lution of the radial equation, Eq. (4), we can write theBogoliubov “particle” and “hole” wavefunctions respec-tively as

uklm(~x)=1

rukl(r)Y

ml (θ, φ), (B1a)

vklm(~x)=1

rvkl(r)Y

ml (θ, φ), (B1b)

where the notation is the same as that used in Eq. (3).Here the radial wavefunctions ukl(r) and vkl(r) (whichobey open boundary conditions) satisfy the Bogoliubovanalog of the radial equation:

Eklukl=

(− ~2

2m∂2r + V +

~2l(l + 1)

2mr2− µ+ 2g|ψNI|2

)ukl

+gψ2NI vkl, (B2a)

−Eklvkl=(− ~2

2m∂2r + V +

~2l(l + 1)

2mr2− µ+ 2g|ψNI|2

)vkl

+gψ∗NI2 ukl. (B2b)

The number of particles in the U(1) symmetry brokenphase is then

N = N0(T )

+∑kl

(Ekl>0)

(2l + 1)

∫ ∞0

dr

[fkl|ukl|2 + (1 + fkl)|vkl|2

],(B3)

where fkl = 1/(eβEkl − 1).

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