Inhomogeneous Bose-Einstein condensates

Habilitationsschrift von

Uwe R. Fischer

tt emptoA

Eberhard-Karls-Universitat Tubingen

Fakultat fur Mathematik und Physik

April 2004

Diese Arbeit ist dem Andenken an meine geliebte GroßmutterKathe Fischer (30.3.1906–12.12.2003) gewidmet.

Contents

Preface vii

1 Quasiparticle universes in Bose-Einstein condensates 11.1 The concept of an effective space-time metric . . . . . . . . . 1

1.1.1 Geometrical aspects . . . . . . . . . . . . . . . . . . . 41.1.2 The metric in Bose-Einstein condensates . . . . . . . . 51.1.3 Generality of the effective metric concept . . . . . . . 6

1.2 Nonuniqueness of the quasiparticle content . . . . . . . . . . 71.2.1 Scaling ansatz in expanding Bose-Einstein condensates 91.2.2 “Cosmological” quasiparticle production . . . . . . . . 121.2.3 Gibbons-Hawking effect in de Sitter space-time . . . . 141.2.4 Detecting the thermal de Sitter spectrum . . . . . . . 20

1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Vortices in Bose-Einstein condensates 262.1 Rapidly rotating vortex lattices . . . . . . . . . . . . . . . . . 27

2.1.1 Vortex core compression at high rotation speeds . . . 282.1.2 Vortex lattice collapse to a single giant vortex . . . . . 32

2.2 Quantum vortex dynamics and quantum electrodynamics . . 33

3 Existence of Bose-Einstein condensation for anisotropicallytrapped gases 373.1 Infinitely extended systems . . . . . . . . . . . . . . . . . . . 373.2 Trapped gases . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Bibliography 45

v

Appendix

A Uwe R. Fischer and Matt VisserAnn. Phys. (N.Y.) 304, 22 (2003) 53

B Uwe R. Fischer and Matt VisserPhys. Rev. Lett. 88, 110201 (2002) 72

C Uwe R. Fischer and Matt VisserEurophys. Lett. 62, 1 (2003) 77

D Petr O. Fedichev and Uwe R. FischerPhys. Rev. Lett. 91, 240407 (2003) 85

E Petr O. Fedichev and Uwe R. FischerPhys. Rev. D 69, 064021 (2004) 91

F Petr O. Fedichev and Uwe R. FischerPhys. Rev. A 69, 033602 (2004) 101

G Uwe R. Fischer and Gordon BaymPhys. Rev. Lett. 90, 140402 (2003) 110

H Petr O. Fedichev, Uwe R. Fischer, and Alessio RecatiPhys. Rev. A 68, 011602(R) (2003) 115

I Uwe R. FischerPhys. Rev. Lett. 89, 280402 (2002) 120

vi

Preface

In the present habilitation thesis, my scientific work in the years 2001-2004 is presented. In the three chapters to follow, I will critically discussthe physical context into which this work and the resulting publications areto be placed; for details the reader is referred to the Appendix, where thefull publications are reproduced.

The first chapter consists of a description of the effective space-timemetric concept in condensed matter, and the application of its quantumand classical aspects to Bose-Einstein condensates. The classical aspectswere dealt with in the publications contained in Appendixes A–C. Aspectsof quantum field theory, in particular the observer dependence of the par-ticle content for a quantum field in curved space-time, leading in de Sitterspace-time to the Gibbons-Hawking effect, and the analog of cosmologi-cal particle production in time-dependent metrics are subsequently treated.The corresponding publications are contained in Appendixes D–F. Quan-tized vortices play a pivotal role in the next, second, chapter. It is explainedhow vortex lattices behave if the rotation velocity of the superfluid gas in-creases such that the vortex cores approach each other; this is Appendix G.In Appendix H, an experimental method is proposed to verify if the long-standing conjecture of an analogy of quantum vortex dynamics and quantumelectrodynamics holds true in a dilute two-dimensional superfluid gas. Thefinal chapter refers to Appendix I. It deals with the possible existence oflong-range coherence in an inhomogeneous Bose-Einstein condensate, if wereduce the effective dimensionality of the system by changing the shape ofthe condensate cloud, making its spatial distribution strongly anisotropic.

I am grateful to Tony Leggett, Gordon Baym, Grisha Volovik, and NilsSchopohl for educating me on some of the aspects of theoretical physicsdescribed in what follows, and the work we have done together in Urbana,Helsinki and Tubingen, respectively. In addition, I thank Ralf Schutzholdand Matt Visser for correspondence and their collaboration at various stagesof the work contained herein. Finally, special thanks are due to Petr O.Fedichev for the fun we had with physics (and otherwise) during my stay inInnsbruck.

vii

Chapter 1

Quasiparticle universes in

Bose-Einstein condensates

1.1 The concept of an effective space-time metric

Curved space-times are familiar from Einstein’s theory of gravitation [1],where the metric tensor gµν , describing distances in a curved space-timewith local Lorentz invariance, is determined by the solution of the Einsteinequations. A major problem for an experimental investigation of the (kine-matical as well as dynamical) properties of curved space-times is that gen-erating a significant curvature, equivalent to a (relatively) small curvatureradius, is a close to impossible undertaking in manmade laboratories. For ex-ample, the effect of the gravitation of the whole Earth is to cause a deviationfrom flat space-time on this planet’s surface of only the order of 10−8 (theratio of Schwarzschild and Earth radii). The fact that proper gravitationaleffects are intrinsically small is basically due to the smallness of Newton’sgravitational constant G = 6.67× 10−11 m3kg−1sec−2. Various fundamentalclassical and quantum effects in strong gravitational fields, a few of whichwe discuss below, are thus inaccessible for Earth-based experiments. Therealm of strong gravitational fields (or, equivalently, rapidly accelerating areference frame to simulate gravity according to the equivalence principle),is therefore difficult to reach. However, Earth-based experiments are desir-able, because they have the obvious advantage that they can be preparedand, in particular, repeated under possibly different conditions at will.

The formalism to be described in what follows is aimed at the realizationof effective curved space-time geometries in perfect fluids, which can indeedbe prepared on Earth, and which mimic the effects of gravity inasmuch thekinematical properties of curved space-times are concerned. Among such

1

perfect fluids are Bose-Einstein condensates, i.e., the dilute matter-wave-coherent gases formed if cooled to ultralow temperatures, where the criticaltemperatures are of order Tc ∼ 100 nK · · · 1µK; for a short review of the(relatively) recent status of this rapidly developing field see [2]. In whatfollows, it will be of some importance that Bose-Einstein condensates belongto a special class of quantum perfect fluids, so-called superfluids [4].

The curved space-times we have in mind in the following are experiencedby sound waves propagating on the background of a spatially and temporallyinhomogeneous perfect fluid. Of primary importance is, first of all, to real-ize that the identification of sound waves propagating on an inhomogeneousbackground, which is itself determined by a solution of Euler and continu-ity equations, and photons propagating in a curved space-time, which isdetermined by a solution of the Einstein equations, is of a kinematical na-ture. That is, the space-time metric is fixed externally by a backgroundfield obeying the laws of hydrodynamics (which is prepared by the experi-mentalist), and not self-consistently by a solution of the Einstein equationsGµν = 8πTµν (where G and the speed of light are set to unity). The lat-ter equations relate space-time curvature (represented by the Einstein tensorGµν = Rµν− 1

2gµνR, where Rµν is the Ricci tensor and R = gµνRµν the Ricciscalar) with the energy-momentum content of all other fundamental fields.This energy-momentum content is represented by the classical quantity Tµν ,which is the regularized expectation value of a quantum energy-momentumtensor [3].

As a first introductory step to understand the nature of the proposedkinematical identity, consider the wave equation for the velocity potentialof the sound field Φ, which in a homogeneous medium at rest reads

[

− 1

c2s

∂2

∂t2+ ∆

]

Φ = 0, (1.1)

where cs is the sound speed. It is a constant in space and time for such amedium at rest. This equation has Lorentz invariance, that is, if we replacethe speed of light by the speed of sound, it retains the form shown abovein the new space-time co-ordinates, obtained after Lorentz-transforming toa frame moving at a constant speed less than the sound speed. Just as thelight field in vacuo is a proper relativistic field, sound is a “relativistic” field.∗

The Lorentz invariance can be made more manifest by writing equation(1.1) in the form Φ ≡ ηµν∂µ∂νΦ = 0, where ηµν =diag(−1, 1, 1, 1) is the(contravariant) flat space-time metric (we choose throughout the signature

∗More properly, we should term this form of Lorentz invariance pseudorelativistic in-variance. We will however use for simplicity “relativistic” as a generic term if no confusioncan arise therefrom.

2

of the metric as specified here), determining the fundamental light-cone-like structure of Minkowski space [5]; we employ the summation over equalgreek indices µ, ν, · · · . Assuming, then, the sound speed cs = cs(x, t) to belocal in space and time, and employing the curved space-time version of the3+1D Laplacian [1], one can write down the sound wave equation in aninhomogeneous medium in the generally covariant form [6, 7]:

1√−g∂µ(

√−ggµν∂νΦ) = 0. (1.2)

Here, g = det[gµν ] is the determinant of the (covariant) metric tensor. It isto be emphasized at this point that, because the space and time derivatives∂µ are covariantly transforming objects in (1.2), the primary object in thecondensed-matter identification of space-time metrics via the wave equation(1.2) is the contravariant metric tensor g

µν [18]. In the condensed-matterunderstanding of analog gravity, the quantities g

µν are material-dependentcoefficients. They occur in a dispersion relation of the form g

µνkµkν = 0,where kµ = (ω/cs,k) is the covariant wave vector, with ~k the ordinaryspatial momentum (or quasi-momentum in a crystal).

The contravariant tensor components gµν , for a perfect, irrotational liq-

uid turn out to be [6, 7, 8]

gµν =

1

Acc2s

(

−1 −v

−v c2s1− v ⊗ v

)

, (1.3)

where 1 is the unit matrix and Ac a space and time dependent function, tobe determined from the proper equations of motion for the sound field (seebelow). Inverting this expression according to g

βνgνα = δβ

α, to obtain thecovariant metric gµν , the fundamental tensor of distance reads

gµν = Ac

(

−(c2s − v2) −v

−v 1

)

, (1.4)

where the line element is ds2 = gµνdxµdxν . This form of the metric has been

derived by Unruh for an irrotational perfect fluid described by Euler andcontinuity equations [6]; its properties were later on explored in more detailin particular by M. Visser [7]. We also mention that an earlier derivation ofUnruh’s form of the metric exists, from a somewhat different perspective; itwas performed by Trautman [8].

The conformal factor Ac in (1.4) depends on the spatial dimension ofthe fluid. It may be unambiguously determined by considering the actionof the velocity potential fluctuations above an inhomogeneous background,

3

identifying this action with the action of a minimally coupled scalar field inD + 1-dimensional space-time (cf. Appendixes D and E):

S =

∫

dD+1x1

2g

[

−(

∂

∂tΦ − v · ∇Φ

)2

+ c2s(∇Φ)2

]

≡ 1

2

∫

dD+1x√−gg

µν∂µΦ∂νΦ , (1.5)

where it is assumed that the compressibility 1/g of the (barotropic) fluid,g = d(ln ρ)/dp, where p is the pressure and ρ the mass density of the fluid, isa constant. Using the above canonical identification, it may easily be shownthat the conformal factor is given by Ac = (cs/g)

4−D. It is mentioned herethat the case of one spatial dimension (D = 1) is special, in that the so-calledconformal invariance in two space-time dimensions implies that the classicalequations of motion are invariant (take the same form) for any space andtime dependent choice of the conformal factor Ac.

1.1.1 Geometrical aspects

The line element ds2 = gµνdxµdxν gives us the distances travelled by the

phonons in an effective space-time world in which the scalar field Φ “lives”.In particular, quasiclassical (large momentum) phonons will follow light-like,that is, here, sound-like geodesics in that space-time, according to ds2 = 0.Using that phonons move on geodesics, we discovered in the publication con-tained in Appendix B the phenomenon that a vortex acts on quasiclassicalphonons as an effective gravitational lens, see for an illustration Fig. 2 inAppendixB.

Noteworthy is the simple fact that the constant time slices obtained bysetting dt = 0 in the line element are conformally flat, i.e. the quasiparticleworld looks on constant time slices like the ordinary (Newtonian) lab space,with a simple Euclidean metric in the case of Cartesian spatial co-ordinateswe display. All the intrinsic curvature of the effective space-time is thereforeencoded in the metric tensor elements g00 and g0i. Together with M. Visser,I described this curvature and its properties for an isotachic fluid (i.e., havinga speed of sound independent of space and time). See the Appendix A, Eqs.(12)–(14) and Eqs. (21)–(23) for the Riemann tensor, and Eqs. (36)–(38)for its conformally invariant, traceless part, the Weyl tensor.

In Appendix C, using the fact that there is curvature in any spatiallyinhomogeneous flow (that is, a flow which is not a simple superposition oftranslational motion and rigid body rotation), we have shown that thereexists a sonic analog of the “warp drive” in general relativity permitting

4

superluminal, i.e., here “superphononic” motion [10]. The point made by usis that in the acoustic curved space-times we consider, there is no violationof any condition on the positivity of energy necessary, which is in markedcontrast to the original warp drives, where local energy densities by necessitymust be negative to permit superluminal travel [11]. This is due to thefact that the Einstein equations, relating curvature and energy-momentumcontent of all fields other than gravitational fields, do not need to be imposedin the analogy.

1.1.2 The metric in Bose-Einstein condensates

We assumed in Eq. (1.5) that the compressibility 1/g is a constant. Thisentails that the (barotropic) equation of state reads p = 1

2gρ2. We then

have, in terms of the interaction between the particles (atoms) constitutingthe fluid, a contact interaction (pseudo-)potential, V (x − x′) = gδ(x − x′).This is indeed the case for the dilute atomic gases forming a Bose-Einsteincondensate. Well below the transition temperature, they are well-describedby the Gross-Pitaevskiı mean field equation for the order parameter Ψ ≡〈Ψ〉, the expectation value† of the quantum field operator Ψ [13]:

i~∂

∂tΨ(x, t) =

[

− ~2

2m∆ + Vtrap(x, t) + g|Ψ(x, t)|2

]

Ψ(x, t). (1.6)

The Madelung transformation reads Ψ =√ρ exp[iφ], where ρ yields the con-

densate density and φ is the velocity potential. It allows for an interpretationof quantum theory in terms of hydrodynamics [14]. Namely, identifying realand imaginary parts on left- and right-hand sides of (1.6), respectively, givesus the two equations

−~∂

∂tφ =

1

2mv2 + Vtrap + gρ− ~

2

2m

∆√ρ

√ρ

≡ µ+ pQ, (1.7)

∂

∂tρ+ ∇ · (ρv) = 0. (1.8)

The first of these equations is the Josephson equation for the superfluidphase. This Josephson equation corresponds to the Bernoulli equationof classical hydrodynamics, where the usual velocity potential of irrota-tional hydrodynamics equals the superfluid phase φ times ~/m, such thatv = ~∇φ/m. The latter equation implies that the flow is irrotational save

†Observe that 〈Ψ〉 6= 0 breaks particle number conservation (the global U(1) invari-ance). We will come back to this point in section 3.2.

5

on singular lines, around which the wave function phase φ is defined onlymodulo 2π. Therefore, circulation is quantized [15], and these singular linesare the center lines of quantized vortices (for a detailed discussion of quan-tized vortices see Chapter 2). The usual isothermal chemical potential µ(which we have chosen to incorporate the kinetic energy term 1

2mv2), is

augmented by the “quantum pressure term” pQ ≡ − ~2

2m(∆√ρ)/

√ρ, which is

the one genuine quantum term in (1.7), because pQ ∝ ~2 (observe that the

first order in ~ is incorporated into the velocity potential). The second equa-tion (1.8) is the continuity equation for conservation of particle number, i.e.,atom number in the superfluid gas. The dynamics of the weakly interacting,dilute ensemble of atoms is thus that of a perfect Euler fluid with quantizedcirculation of singular vortex lines, which are the only vortical excitationsin a superfluid. This is true save for regions in which the density rapidlyvaries and the (genuine quantum) pressure term pQ becomes relevant, whichhappens on scales of order the coherence length ξ0 = ~/

√2gmρ0 where ρ0 is

a constant asymptotic density far away from the density-depleted (or possi-bly density-enhanced) region. This is the case in the depleted-density coresof quantized vortices, which we will be discussing in section 2, or at thelow-density boundaries of the system. The quantum pressure is negligibleoutside these domains of rapidly varying or low density.

The whole armoury of space-time metric description of excitations, ex-plained in the last section, and based on the Euler and continuity equations,is then valid for phonon-like excitations of a Bose-Einstein condensate, withthe space-time metric (1.4), as long as we are outside the core of quan-tized vortices, where the flow is irrotational and the quantum pressure isnegligible.

1.1.3 Generality of the effective metric concept

It is important to recognize that the form (1.2) of the wave equation isvalid generally (with a possible additional scalar potential term). That is, agenerally covariant curved space-time wave equation can be formulated notjust for the velocity perturbation potential in an irrotational Euler fluid,for which we have introduced the effective metric concept. If the spectrumof excitation (in the local rest frame) is linear, ω = cpropk, where cprop isthe propagation speed of some collective excitation, the statement that aneffective space-time metric exists is true, provided we only consider wave per-turbations of a single scalar field constituting a fixed classical background.The argument to reach this conclusion is as follows.

Given that the action density L is a functional of φ and its space-timederivatives ∂µφ, i.e. L = L[φ, ∂µφ], we expand the action to quadratic order

6

in the fluctuations around some stationary classical background solution φ0

of the Euler-Lagrange equations. For any Lagrangian of the specified form,the wave equation for perturbations δφ ≡ Φ above the background φ0 is [9]:

∂µ

(

∂2L∂(∂µφ)∂(∂µφ)

∣

∣

∣

∣

φ=φ0

∂νΦ

)

−(

∂2L∂φ∂φ

− ∂µ

∂2L∂(∂µφ)∂φ

)∣

∣

∣

∣

φ=φ0

Φ = 0.

(1.9)The above equation in covariant notation reads

1√−g∂µ(

√−ggµν∂νΦ) − Veff(φ0)Φ = 0, (1.10)

where Veff(φ0) is a background field dependent effective potential (equal tothe second term in round brackets in equation (1.9) divided by

√−g). Thepotential Veff may, for example, in the simplest case contain an effectivemass of the scalar field, such that the wave equation becomes Klein-Gordonlike, i.e. Φ = −m2c4sΦ.

The effective metric coefficients are, up to an (again dimension depen-dent) conformal factor given by:

gµν(φ0) ∝∂2L

∂(∂µφ)∂(∂µφ)

∣

∣

∣

∣

φ=φ0

. (1.11)

The concept of an effective space-time metric therefore applies to everysystem having a wave equation of second order in both space and timederivatives, corresponding to perturbations propagating on a fixed classicalbackground, which in turn itself determines the metric coefficients.

1.2 Nonuniqueness of the quasiparticle content of

a nonstationary Bose-Einstein condensate

Quasiparticles are the fundamental entities used to describe a interactingcondensed matter system in a particle picture. On the microscopic level,if the elementary constituents interact by two-body forces, we are givena second quantized Hamiltonian operator of the form H =

∑

k εka†kak +

Vkk′ a†k′ a†kakak′ , where Vkk′ are the matrix elements for two-particle inter-

action in a plane wave basis, and εk are the bare single particle energiesof the “elementary” bosons or fermions, which are created by the opera-tors a†

kfrom the proper particle vacuum. One then employs a unitary, i.e.

operator-algebra-conserving Bogoliubov transformation [16] to another set

of quasiparticle operators bk, b†k, see Eq. (1.13) below. This gives the Hamil-

tonian the reduced diagonal form Hred =∑

k ω(k)b†kbk +O(b3), where O(b3)

7

represents additional terms of higher order than quadratic, supposed to besmall compared to the leading diagonalized part of the Hamiltonian for thepicture of noninteracting quasiparticles to make sense. These quasiparticlespossess a (possibly spatially anisotropic) dispersion relation ω = ω(k) whichis linear for various important classes of quasiparticles, ω(k) ∝ k. Amongthese classes of collective excitations are, e.g., phonons, antiferromagneticmagnons [17], or the excitations around the gap nodes of the p-wave super-fluid 3He-A [18].

Phonons are the small momentum quanta of the sound field in solids orfluids, with ω(k) = csk. Their classical equation of motion in a perfect fluidis the generally covariant wave equation (1.2), with the effective space-timemetric (1.4). In the simplest case, that is for a homogeneous medium withspace and time independent density, and a constant speed of sound cs, thequantized velocity potential of the phonons reads [19]

Φ(x, t) =∑

k

√

~cs2V ρ0k

[

bke−icskt+ik·x + b†ke

icskt−ik·x]

, (1.12)

where V is the quantization volume of the system, and ρ0 the constantbackground density. That is, the quantum excitation field in a homoge-neous medium may generally be decomposed into plane waves, with theappropriate frequencies as a function of momentum stemming from the dis-persion relation, here ω = csk. Note, in particular, that for this spatiallyand temporally homogeneous case, the statement that we observe positivefrequency (energy) with respect to the laboratory time interval dt is unique,that is, it can be made independent of time and space. In an inhomogeneousfluid, this is (generally) no longer the case, and the notion of an excitationhaving positive energy may depend on where the detector is located in thefluid, if it is at rest relative to the fluid or moves, and what its natural timeinterval is. The latter may be different from that of the laboratory, due tothe particular way the detector couples to the fluid, see section 1.2.3.

Operator basis dependence of quasiparticle content

The number of particles assigned to the quantum field Φ is unique with re-spect to a certain given state |Θ〉 of the quantum field, nk(Θ) = 〈Θ|a†

kak|Θ〉,

provided we decompose Φ in modes associated to the operators ak and theirHermitian conjugates. The number of particles is in particular, zero for thevacuum state with respect to the field operator ak, defined by ak|0〉 = 0.However, it need not be zero with respect to another set of quasiparticleoperators bk, which has a different vacuum |0〉.

8

To demonstrate the basis dependence of quasiparticle number, we use ageneral Bogoliubov transformation for bosons, the class of (quasi-)particleswe will consider in what follows, of the general form

ai =∑

k

αikbk + βikb†k , (1.13)

where k represents a set of quantum numbers, not necessarily the planewaves used in (1.12). The coefficients in the Bogoliubov transformationmust fulfill certain conditions for the transformation to be unitary, i.e., topreserve the bosonic commutation relations for the new operators, [bi, b

†k] =

δik, [bi, bk] = 0, [b†i, b†

k] = 0. As a consequence of this defining unitary

character, the following conditions on the transformation coefficients musthold:

∑

k

αikα∗jk − βikβ

∗jk = δij ,

∑

k

αikβjk − βikαjk = 0. (1.14)

By using the transformation (1.13), it is straightforwardly shown that the

number of ak particles in |0〉 is given by 〈0|a†kak|0〉 =∑

k′ |βk′k|2: The oldoperator ak does not annihilate the new vacuum |0〉 (and vice versa), andwhat looks empty in one quasiparticle vacuum may be full of quasiparticlesin another. Related to this (formal) operator-basis dependence, the actuallydetected number of particles, as opposed to the formally defined quantitynk(Θ), which refers to one particular quasiparticle vacuum, and its associ-ated creation and annihilation operators, is strongly observer dependent. Itdepends, in particular, on how the detector actually couples to the field Φwhose quanta it measures. Various couplings of the detector, for example todifferent powers of the fluid density, will influence the quasiparticle basis inwhich the detector measures, and thus the quasiparticle number detected.

Now, the salient point is that because the phonon field Φ is a relativisticquantum field, we will be able to map the observer dependence just de-scribed, which is general and holds for any sort of quasiparticles, to theobserver dependence experienced by proper relativistic quantum fields incurved or flat space-time [20, 21]. This dependence is of kinematical ratherthan dynamical origin, cf. [22], and therefore fully within the capabilities ofour proposed analogy.

1.2.1 Scaling ansatz in expanding Bose-Einstein condensates

To model the quasiparticle analogue of expanding universes, we will makeuse of expanding Bose-Einstein condensates, which are produced by reducingthe trapping potential strength, i.e. the harmonic trapping frequency with

9

time, or by increasing the interaction coupling constant. Firstly, it is thusappropriate to describe the evolution of density and velocity distribution inthe expanding gas by discussing the so-called scaling procedure establishedin [26, 27]. The scaling procedure introduces a set of generally three scalingvariables, bi = bi(t), which are a function of time only. Using these scal-ing variables, one writes for the (Cartesian) co-ordinate vector componentsxbi = xi/bi; for the scaled co-ordinate vector we use the shorthand notationxb ≡ ∑

i eixi/bi. It may then be shown that the evolution of the Bose-Einstein condensed gas cloud is described, starting from the initial densityand velocity potential distributions ρ = ρinit(x, t = 0), φ = φinit(x, t = 0),by the following density and velocity distributions [26, 27]:

ρ(x, t) ⇒ ρinit(xb)

V , (1.15)

v = ∇φ(x, t) ⇒ v = ∇φ =∑

i

bibixi + ∇φ(xb, t) . (1.16)

This is true provided the scaling parameters bi fulfill the equations ([28] andAppendix C)‡

bi + ω2i (t)bi =

g(t)

g(0)

ω20i

Vbi, (1.17)

where ω0i are the initial trapping frequencies and the dimensionless “scalingvolume” reads V =

∏

i bi; from here on we take ~ = m = 1. The dotsare time derivatives with respect to laboratory time t here and in what fol-lows. We have taken into account in the above equation that the particleinteraction can be varied in lab time, g = g(t), by means of a suitable Fesh-bach resonance [29, 30]; g(0) is the initial coupling constant. We designate“scaling basis” quantities with a tilde. For example, the (stationary) initialdensity distribution ρinit(x) gives us ρinit(xb) if we replace x → xb.

The scaling evolution is exact in the Thomas-Fermi approximation, whichneglects the quantum pressure term pQ and the kinetic energy 1

2mv2. SinceBose-Einstein condensates were experimentally created, the scaling solutionhas routinely been employed to interpret the time-of-flight pictures withwhich they are visualized as well as analyzed [2].

We now define a “scaling time” variable by

dτsdt

=g(t)/g(0)

V , (1.18)

‡Note that by special convention we generally do not sum over latin indices i, j, · · ·,but only over greek indices µ, ν, · · ·.

10

and the τs dependent scaling functions Fi by

Fi(τs) =Vb2i

g(0)

g(τs)=

1

b2i

dt

dτs. (1.19)

In terms of these quantities, the effective second order action for the scalingbasis fluctuations of the phase of the superfluid order parameter δφ(xb, τs) ≡Φ(xb, τs), takes on the particularly simple form (Appendix F)

S(2) =

∫

dτsd3xb

1

2g(0)

[

−(

∂

∂τsδφ

)2

+ c2Fi(∇biδφ)2

]

, (1.20)

where the scaling speed of sound c =√

g(0)ρinit(xb). Because of the factthat this action does not mix spatial and temporal derivatives, the resultingline element in the scaling variables, according to the identification displayedin (1.5) is diagonal (does not possess g0i terms), and reads

ds2 = gµνdxµdxν =

c

g(0)

√

FxFyFz

[

−c2dτ2s + F−1

i dx2bi

]

. (1.21)

We now consider for simplicity the isotropic case, bi ≡ b, which implies

Fi ≡ F = bD−2 g(0)

g. (1.22)

The generalization of the mode expansion in Eq. (1.12) to inhomogeneousexpanding Bose-Einstein condensates then takes in this isotropic case theform (Appendix F):

Φ(xb, τs) =∑

n

√

g(0)

2V εnφn(xb)

[

bnχn(τs) + b†nχ∗n(τs)

]

. (1.23)

The functions φn(xb) are the stationary solutions of the initial Gross-Pita-evskiı equations for excitations above the initial ground state, designatedby the (set of) quantum numbers n with initial energies εn. The initialThomas-Fermi quantization volume is V ; in the hard-walled cubic box limitthe modes are plane waves, φn(xb) → exp[ik · xb].

The temporal mode functions χn(τs) satisfy the second order ordinarydifferential equation

d2

dτ2s

χn + F (τs)ε2nχn = 0. (1.24)

The case when F is a constant (unity) is particular. In this case, the quan-tum state of the quasiparticle excitations remains unchanged for increasing

11

τs, and a given initial quasiparticle vacuum, in the scaling basis with theassociated quasiparticle operators defined by (1.23), remains empty forever.That is, no quasiparticles are created in that basis, although the superfluidmay be in a highly nonstationary motional state, obtained by changing thetrapping ω = ω(t) rapidly with time. However, a detector which measures ina quasiparticle basis different from the scaling basis, for example due to itsparticular coupling to the superfluid, may still detect that quasiparticles are“created”. We will come back to this possibility in section 1.2.3 below, whenwe discuss the purely choice-of-observer related phenomenon of a thermalstate in a quasiparticle basis belonging to one particular space-time, the deSitter space-time of cosmology.

1.2.2 “Cosmological” quasiparticle production

Consider now the general case that the scaling functions F (τs) is a functionof scaling time τs. The fact that F depends on time implies that the state-ment “the excitation is of positive frequency” (a particle) or of “negativefrequency” (antiparticle) for a given propagating wave cannot be held up forall times τs. This frequency mixing implies that quasiparticles are createdfrom the quasiparticle vacuum, because an initially empty scaling basis vac-uum state does not remain empty during the evolution of the system, i.e.initially bn|0(τs = 0)〉 = 0, but at a later stage bn|0(τs)〉 6= 0.

The fact that annihilation and creation operator parts of the initial vac-uum are mixed, as a consequence of (1.13), is physically due to the fact thatquasiparticles are scattered within the course of time at (time dependent)effective potentials. Physically, the τs dependence of F furnishes such aneffective potential, at which excitations are scattered from negative to posi-tive frequency and vice versa: The equation (1.24) is formally equivalent toscattering of a non-relativistic particle with energy εn by a potential in τsspace,

V (τs) = ε2n(1 − F (τs)). (1.25)

At large τs, the WKB scattering solution of (1.24) reads (Appendix F):

χn =1

F 1/4

(

αn exp

[

−iεn∫ τs

−∞dτ ′s√

F (τ ′s)

]

+ βn exp

[

iεn

∫ τs

−∞dτ ′s√

F (τ ′s)

])

(1.26)where αn,βn are transmission and forward scattering amplitudes, respec-tively. The scattering amplitudes are related via the particle flux conser-vation (unitarity) condition: |αn|2 − |βn|2 = 1. The quantity Nn = |βn|2can be interpreted as the number of scaling basis quasiparticles created fromthe initially empty scaling vacuum, due to the time dependent scattering of

12

excitations at the potential V (τs). This scattering is taking place due to thenonstationary motion of the condensate.

In the WKB approximation, the amplitudes αn and βn are connected ina simple way:

βn = exp[−εn/2T0]αn, (1.27)

where the inverse temperature 1/T0 is given by the integral

1

T0= =

[∫

C

√F dτs

]

, (1.28)

and C is the contour in the complex τs-plane enclosing the closest to the realaxis singular point of the function

√

F (τs) [31]. This gives us the numberof particles created in the mode n, using the flux conservation condition|αn|2 − |βn|2 = 1,

Nn = |βn|2 =1

exp[εn/T0] − 1. (1.29)

The distribution of the created quasiparticles follows a thermal bosonic dis-tribution (Planck spectrum), at a temperature T0. The adiabatic evolutionof trapped gases hence leads to “cosmological” quasiparticle creation withthermal occupation numbers in the scaling basis. The temperature T0 oc-curring in the Planck distribution above depends on the details of the scalingevolution, i.e., on the specific superfluid dynamics imposed by the solutionof Eqs. (1.17), that is T0 is a functional of the temporal evolution ωi = ωi(t),see the example treated in section IV of Appendix F.

We use the term “cosmological” in context with the plain condensed-matter fact that quasiparticles are created in the scaling basis. We now jus-tify this by comparing our effective Bose-Einstein condensate metric (1.21)to line elements which constitute cosmological solutions of the Einstein equa-tions. For example, let dτs = dt by properly adjusting g = g(t), thus choos-ing the coupling constant’s time evolution to be given by g(t) = g(0)V(t),cf. Eq. (1.18). We then obtain that (1.21) equals (up to the conformal factorc/g(0)V) an anisotropic version of the metric of the spatially flat Friedmann-Robertson-Walker (FRW) universe [32]:

ds2 =c

g(0)V

[

−c2dt2 +∑

i

b2i dx2bi

]

. (1.30)

In the standard spatially isotropic form of the FRW metric, all bi = b areequal.

To obtain the exact equivalence to a spatially flat FRW metric, we haveto assume in addition that c is spatially independent, which is fulfilled close

13

to the center of the gas cloud, where the parabolic density profile is ap-proximately flat, and c is essentially a constant. In the spatially isotropiccase,

H =b

b(1.31)

is the Hubble “constant”, which obviously is a constant only if the gasexpands exponentially in laboratory time, b ∝ exp[Ht], just as we need ex-ponentially rapid expansion for a constant Hubble parameter in inflationarycosmological models [33, 34].

We therefore come to the remarkable conclusion that the co-ordinatescaling factor b of the Bose-Einstein condensate quasiparticle universe, oc-curring in the equations of motion of a nonrelativistic condensed mattersystem, has exactly the same meaning, in the quasiparticle world, as thecosmological scale factor of the Universe proper.

1.2.3 Gibbons-Hawking effect in de Sitter space-time

We now treat the case that the Fi are constants and do not depend on scalingtime. Therefore, following Eqs. (1.23) and (1.24), no scaling basis quasi-particles are created through negative and positive frequency mixing. Thesuperfluid can still be in highly nonstationary motion: The time evolutionis according to (1.19) prescribed by

b2i (t) = Cig(0)

g(t)V(t) ⇐⇒ bi

∏

k 6=i bk= Ci

g(0)

g(t), (1.32)

cf. Eq. (1.19), where the Ci are constants. However, though the super-fluid is in motion, no dissipation through intrinsic quasiparticle creationtakes place, because there exists the Fock space “scaling” basis, in whichno quasiparticles are created from the scaling vacuum, and the energy ofthat particular superfluid vacuum is conserved. A particular instance is theisotropic 2D case, b1 = b2 = b, where for constant g(t) = g(0) the conditionon F being constant is fulfilled.§

The de Sitter space-time is a cosmological solution of the Einstein equa-tions, in which space is empty and flat (that is, the constant time slicesare Euclidean space), and all of the curvature of space-time is encoded ina nonvanishing cosmological constant Λ. For obvious reasons, the de Sit-ter space-time is very popular in the quantum field theoretical treatment of

§Cf. Ref. [35], where the fact of superfluid vacuum energy conservation is explainedfrom a different (SO(2,1) symmetry) perspective, and [36] for a quality factor measurementof breathing (monopole) oscillations in a cylindrical geometry.

14

cosmological theories [12], because it highlights the crucial cosmological rolethe energy density of all conceivable quantum fields taken together mightplay: The vacuum energy density may constitute the dominant source termfor space-time curvature in the Einstein equations.

The Gibbons-Hawking effect for geodesic observers [21] in such a de Sitterspace-time is the curved space-time analog of the Unruh-Davies effect [3, 20].The latter consists in the fact that a constantly accelerated detector movingin the flat (purportedly “empty”) space-time vacuum, responds as if it wereplaced in a thermal bath of (quasi-)particles with temperature proportionalto its acceleration. Observer-related phenomena are at the heart of quantumfield theory on nontrivial, and generally curved, space-time backgrounds.They tell us that the particle content of a given quantum field depends onthe (motional) state of the detection apparatus, which is verifying that thereare particles by its “clicks”. More technically speaking, the dependence ofthe particle content of quantum fields in curved space-time is rooted inthe non-uniqueness of canonical field quantization in Riemannian spaces[23]. It is of fundamental importance to make these observer-dependenteffects measurable, because such a measurement constitutes, inter alia, aconsistency check for an all-important concept of standard quantum fieldtheory, namely that the quantization of a given field is carried out on a fixedspace-time background.

That the experimental verification is exceedingly difficult with light be-comes apparent if we calculate the Unruh-Davies temperature: The resultis that it equals TUnruh = [~/(2πkBcL)]a = 4K × a[1020g⊕] , where a is theacceleration of the detector in Minkowski space (g⊕ is the gravity acceler-ation on the surface of the Earth), and cL the speed of light. The hugeaccelerations needed to obtain measurable values of TUnruh make it obviousthat an observation of the effect is decidedly a less than trivial undertaking.Although proposals for a measurement with ultraintense short pulses of elec-tromagnetic radiation have been put forward in, e.g., Refs. [24, 25], it is lessthan obvious how the thermal spectra associated to the effect, which stillfurnish tiny contributions to the total energy, should be discernible fromthe background dominated by the ultraintense lasers used to create largeaccelerations of the elementary particles contained in the plasma.

An isotropic de Sitter universe in a harmonic trap

We first discuss the simplest case of an isotropically expanding gas in a har-monic trap. We will see that this case also serves a pedagogical purpose,because it forces us to distinguish between observer-related phenomena withthermal spectrum, which by definition all have Fi(τs) ≡ 1, and “cosmologi-

15

cal” particle creation with a thermal spectrum, as defined in section 1.2.2,where the scaling functions Fi(τs) in (1.19) depend explicitly on scaling time.

One can create de Sitter universes in an expanding gas by letting itexpand exponentially, b(t) ∝ exp[Ht] where H is the Hubble constant ofcosmological expansion, H = Λ/D for the de Sitter universe discussed, whereD is the spatial dimension. The de Sitter metric in terms of “cosmological”time τc reads [32]

ds2 =c0

g(0)V[

−c20dτ2c + e2Hτcdx2

b

]

, (1.33)

where c0 ≡ c(0) is the central scaling (i.e., initial) speed of sound. Inthe present case of exponential expansion, the “cosmological” time intervaloccurring in the metric above is equal to both the laboratory and the scalingtime interval, dτc = dτs = dt.

However, using the isotropic harmonic expansion setup, one has to facethe difficulty that in order to give the FRW metric (1.30) near the centerof the trap (i.e., close to xb = 0) the (conformal) de Sitter form, one hasto increase exponentially the interaction with laboratory time: Because of(1.31), g(t) ∝ bD ∝ exp[DHt] = exp[c0Λt]. Though the central density alsodecreases exponentially (like b−D/2), the exponential increase of the coupling“constant” incurs strong three-body recombination losses [37], whose totalrate (in three spatial dimensions) is proportional to g4ρ2 ∝ b9 = exp[9c0Λt](this latter relation being valid as long as the gas remains dilute). There-fore, within a short time of order 1/H, the Bose-Einstein-condensed gas ofinteracting single atoms will simply no longer be in existence, because theatoms rapidly form bound states. Such an experiment will leave no timeto actually measure phenomena which depend on the fact that an equilib-rium is established; in particular, the thermal equilibrium for the occupationnumbers in the de Sitter quasiparticle basis will not be established on sucha short time scale.

Even more importantly, though one obtains indeed a thermal spec-trum in the de Sitter quasiparticle basis corresponding to the metric (1.33),F (τs) = F (t) = 1/b2 = exp[−2Ht] depends on time, i.e., it is not aconstant in the quasiparticle basis corresponding to the mode functionsχn ∝ exp[−iεnτs]. The thermal spectrum obtained therefore is, by our phys-ical definition, the thermal “cosmological” quasiparticle creation discussedin the previous section. It is not what we want to observe in our effective deSitter space-time, namely the purely choice-of-observer related phenomenonGibbons-Hawking effect, for which no actual quasiparticle “creation” in thescaling quasiparticle basis should take place.

16

The 1+1D de Sitter universe in a cigar-shaped cloud

To circumvent the problem that the interaction coupling needs to be in-creased exponentially with time, to obtain a de Sitter universe in 2+1 or3+1 isotropic space-time dimensions in an isotropic harmonic trap, I andPetr Fedichev have developed the model of a 1+1D de Sitter universe. This1+1D toy model can be realized in a strongly anisotropic, cigar-shapedBose-Einstein condensate, cf. Appendixes D and E. In particular, in theproposed experimental setup, no time variation of the coupling constant atall is necessary, which is thus a true “constant” also in time.

The analysis of the excitation modes in a strongly anisotropic, elongatedBose-Einstein condensate is based on the adiabatic separation ansatz [38]

Φ(r, z, t) =∑

n

φn(r)χn(z, t), (1.34)

where φn(r) is the radial wavefunction characterized by the quantum numbern (we consider only zero angular momentum modes). The above ansatzincorporates the fact that for strongly elongated traps, i.e. traps for whichωz ω⊥, the dynamics of the condensate motion separates into a fastradial motion and a slow axial motion, which are essentially independent.The χn(z, t) are the mode functions for travelling wave solutions in the zdirection (plane waves for a condensate at rest read χn ∝ exp[−iεn,kt+kz]).The radial motion is assumed to be “stiff” such that the radial part iseffectively time independent, because the radial time scale for adjustmentof the density distribution after a perturbation is much less than the axialoscillation time scales of interest. The ansatz (1.34) works independent fromthe ratio of healing length and radial size of the superfluid cigar. In the limitthat the healing length is much less than the radial size, Thomas-Fermi wavefunctions are used, in the opposite limit, a Gaussian ansatz for the radialpart of the wave function φn(r) is appropriate.

The squared oscillation spectrum of the cigar-shaped condensate cloudreads ε2n,k = c20k

2 + 2ω2⊥n(n + 1) + c20k

2, where c0 =√

µ/2 [38]. A deSitter space-time is then obtained from the effective action for the phasefluctuations of the phonon (n = 0) modes near the center of the trap, i.e.near the center of the long axis at z = 0. The 1+1D action is obtainedafter integrating out the transverse, strongly confined directions, and reads(Appendixes D and E):

S0 =

∫

dtdzπb2⊥R

2⊥

2g

[

−(

∂

∂tχ0 − vz∂z

)2

+c20b2⊥b

(∂zχ0)2

]

≡ 1

2

∫

dD+1x√−gg

µν∂µχ0∂νχ0 . (1.35)

17

The scaling parameters b in the axial (ωz) and b⊥ in the perpendicular (ω⊥)directions are functions of time, such that the action fulfills the identificationwith the action of a scalar field minimally coupled to gravity in Eq. (1.5),which is indicated in the second line above, where the metric is the gµν

occurring in Eq. (1.4).The identification of the above phase-fluctuations action with the action

of a scalar minimally coupled to gravity works, that is, the two actions in thefirst and second line of (1.35) are consistent, if we impose the consistencycondition that

πb2⊥R2⊥

gZ2 =

b⊥√b

c0

⇐⇒ b⊥√b

= 8

√

π

2

1

Z2

√

ρma3s

(

ω⊥

µ

)2

≡ B = const., (1.36)

where Z is a renormalization factor according to χ0 = Zχ0, with χ0 therenormalized wave function, and ρm the initial central density. The factorZ does not influence the (classical) equation of motion δS/δχ0 = 0 (it simplydrops out), but does influence the response of a detector. We will come backto this point in section 1.2.4 below, when we discuss the explicit dependenceof the equilibration time scale for the detector stationary state, Eq. (1.48).

In Appendixes D and E, we were using an alternative form of the 1+1Dde Sitter metric (1.33). This alternative form is the one used in the originalGibbons-Hawking paper [21]. It reads

ds2 = −c20(

1 − Λz2)

dτ2 +(

1 − Λz2)−1

dz2 . (1.37)

We leave out the conformal factor, because due to the conformal invari-ance it is irrelevant for classical physics in 1+1D. The time interval dτ inthe above metric is not the “cosmological” time interval dτc in the cosmo-logical comoving frame version of the metric displayed in (1.33). The twometrics may be transformed into each other using the following co-ordinatetransformations:

exp[−2√

Λc0τ ] = exp[−2√

Λc0τc] − Λz2b ,

z = zb exp[Λc0τc], (1.38)

giving us τ = τ(τc, zb), z = z(τc, zb), which transforms (1.37) into (1.33).The advantage of the form (1.37) is that it is plain in this form that thede Sitter space-time has an event horizon, located in our 1+1D case at theconstant values z = zH = ±Λ−1/2.

18

The quantity Λz2 in the de Sitter metric must be independent of time.This leads us to the requirement

√Λz =

vz

c(t)=

bb⊥√bc0

z =Bb

c0z (1.39)

relating the dynamical parameters of our problem to each other (c(t) is theinstantaneous sound velocity at the center of the cloud). These relationsimply that b = const., and thus that b ∝ t. Then, the cosmological constantΛ becomes independent of time, as is necessary for an analog de Sitter space-time to be established.

The experimental procedure now is determined to be as follows: Preparea Thomas-Fermi (i.e., sufficiently large), strongly anisotropically trapped,cigar-shaped condensate (cf. Fig. 1 in Appendix D). Let it expand in theaxial direction linear in lab time by changing the trapping according to thescaling equations (1.17), such that b ∝ t, and simultaneously expand in theperpendicular direction with the square root of lab time, b⊥ ∝

√t, such that

B = b⊥/√b in (1.36) is a constant. Then, a detector “tuned” to the de Sitter

space-time (1.37), i.e., which works in the de Sitter quasiparticle basis, willmeasure a thermal quasiparticle spectrum, with the de Sitter temperature([21] and Appendixes D and E)

TdS =c02π

√Λ =

B

2πb. (1.40)

The fact that a thermal spectrum is obtained can be directly derived fromthe equations of superfluid hydrodynamics, as expounded in our work inAppendixes D and E. That is, it is not simply postulated due to the (kine-matical) analogy with quantum field theory in de Sitter space-time, but is aresult of the quantized hydrodynamic equations determining the evolutionof the quasiparticle content of the quantized sound field in the de Sitterbasis.

The relation between de Sitter time τ and the laboratory time is fixedby dτ = dt/(

√bb⊥) = dt/(b(t)B) = dt/(btB). [Note that dτ and the scaling

time interval dτs defined in (1.18) differ; dτs = dt/B2b2 = dτ/Bb.] Thetransformation law between t and the de Sitter time τ (on a constant zdetector, such that dt = dt), is then given by

t

t0= exp[Bbτ ], (1.41)

where the unit of lab time t0 ∼ ω−1‖ is set by the initial conditions for the

scaling variables b and b⊥.

19

It is important to recognize that an effective exponential “acceleration”of the oscillation frequencies, either because of the exponential dependenceof laboratory time on scaling time, represented by (1.41), or coming fromthe WKB approximation for the scattering amplitudes and the exponentiallysmall mixing of positive and negative frequency parts resulting therefrom,Eq. (1.27), is sufficient for the thermal spectrum to be obtained. Thoughwe have discussed a space-time which possesses a horizon, no pre-existinghorizons in the parent space-time are necessary per se for thermal occupationnumber distributions to be established, as has been pointed out in [39]. Anexplicit example is the original Unruh-Davies effect, where this parent, globalspace-time is simply Minkowski space.

1.2.4 Detecting the thermal de Sitter spectrum

To detect the Gibbons-Hawking effect in de Sitter space-time, one has to setup a detector which measures frequencies in units of the inverse de Sittertime τ , rather than in units of the inverse laboratory time t; this correspondsto measuring in the proper de Sitter quasiparticle vacuum, where, in par-ticular, positive and negative frequency are defined with respect to τ . Onlythen does one detect quasiparticles which are defined with respect to the deSitter quasiparticle basis, and refer to a vacuum corresponding to exactlythat space-time.

In Appendixes D and E, I and Petr Fedichev have provided such a de-tector. We have shown that a “de Sitter basis” detector is realized by an“Atomic Quantum Dot” (AQD).

The AQD can be implemented in a Bose gas of atoms possessing twohyperfine ground states α and β. The atoms in the state α represent theexpanding Bose-Einstein condensate, and are used to model our expandingde Sitter universe (cf. Fig. 1 in Appendix D). The AQD itself is formedby trapping atoms in the state β in a tightly confining optical potentialVopt created by a laser at the center of the cloud (see Fig. 2 in AppendixD). The interaction of atoms in the two internal levels is described by aset of coupling parameters gcd = 4πacd (c, d = α, β), where acd are thes-wave scattering lengths characterizing short-range intra- and inter-speciescollisions; gαα ≡ g, aαα ≡ as, and gαβ ≡ g. The on-site repulsion betweenthe atoms β in the dot is given by the energy level spacing U ∼ gββ/l

3

between states with a occupation difference of one β atom, where l is thecharacteristic size of the ground state wavefunction of atoms β localizedin Vopt. We consider the so-called collisional blockade limit of large U > 0,where only one atom of type β can be trapped in the dot. This limit assumesthat U is much larger than all other relevant frequency scales in the dynamics

20

of both the AQD and the expanding superfluid, and corresponds to a large“Coulomb blockade gap” in electronic quantum dots [40]. As a result of theseassumptions, the collective co-ordinate of the AQD is modeled by a pseudo-spin-1/2 degree of freedom η, with spin-up/spin-down state correspondingto occupation of the AQD by a single atom or no atom in the hyperfine stateβ. A Rabi laser of frequency Ω, with a detuning ∆ from resonance betweenthe two hyperfine levels α and β, couples atoms of the hyperfine species α,constituting the expanding cigar-shaped superfluid, into the AQD. In ourcollisional blockade limit, the maximal occupation of the AQD is one atomof species β (cf. the inset of Fig. 2 in Appendix D).

The full detector Lagrangian reads (Appendix E, also cf. the Hamilto-nian formulation in Appendix D):

LAQD = i

(

d

dtη∗)

η − Ω√

ρ0(0, t)l3 (η + η∗)

−[

−∆ + (g − g)ρ0(0, t) + gδρ+d

dtδφ

]

η∗η . (1.42)

The fact that the detector has the de Sitter basis as its “natural” quasipar-ticle basis, and therefore measures in de Sitter time, is due to the fact thatthe term linear in the detector co-ordinate η in the detector Lagrangiancouples in a certain manner to the superfluid cigar in which it is embed-ded. Namely, the laser with frequency Ω causing transitions between thetwo hyperfine levels α and β couples to the square root of the central mean-field particle density. This particular coupling (represented by the secondterm in the first line of the above Lagrangian) is what we need, because√

ρ0(0, t) =√ρm/bB =

√ρm/(bBt) =

√ρmdτ/dt. The fact that the cou-

pling coefficient is proportional to dτ/dt is required to establish that thedetector can work as a de Sitter detector, because it transforms the detectorequations, which are a priori in laboratory time, into equations in de Sittertime, see Eqs. (1.43) below.

Adjusting the detuning ∆ properly, such that ∆(t) = (g − g)ρ0(0, t) =(g − g)ρm/(b

2B2t2), the first and the second term in the square brackets inthe second line cancel. One then obtains a simple set of coupled equationsfor the occupation amplitudes of the state ψ = ψβ|β〉 + ψα|α〉 of the AQD:

idψβ

dτ=ω0

2ψα + δV ψβ, i

dψα

dτ=ω0

2ψβ , (1.43)

where dτ is the de Sitter time interval. Were it not for the density oscillationsin the cigar, represented by the potential δV , the above equations (1.43)would represent a simple two-level system, with frequency splitting ω0 =

21

2Ω√

ρml3. The density oscillations contained in the perturbation operatorδV (τ) = (g − g)Bb(τ)δρ(τ) cause transitions between the two undisturbeddetector eigenstates |±〉 = (|α〉 ± |β〉)/

√2 of the two-level system, which are

separated by the energy ω0. The density perturbations in the expandinghost superfluid lead to a damping of the Rabi oscillations with frequencyω0 between these two states. This constitutes the effect of the de Sitterthermal bath to be observed, where the damping happens on the time scaledisplayed in (1.48) below.

The response of the detector, that is, the transition rates between thedetector states, can be calculated by evaluating a response function [3] whichmakes use of the expectation value of the product of two b(τ)ρ(τ) operators.The probability per unit time for excitation (P+, transition from |+〉 to |−〉)and de-excitation (P−, transition from |−〉 to |+〉) of the detector takes theform ([20] and Appendix E):

dP±

dτ= lim

T →∞

1

T

∫ T∫ T

dτdτ ′〈δV (τ)δV (τ ′)〉e∓iω0(τ−τ ′)

= limT →∞

B2 (g − g)2

T

∫ T∫ T

dτdτ ′〈b(τ)δρ(τ)b(τ ′)δρ(τ ′)〉e∓iω0(τ−τ ′).

(1.44)

The second-quantized solution of the hydrodynamic equations for the den-sity fluctuations above the superfluid ground state in the expanding cigar-shaped Bose-Einstein condensate reads (Appendixes D–F)

δρ =∑

k

i

√

ε0k

4πR2⊥R‖g

∂

∂t

(

ak exp

[

−i∫ tdt′ε0k

Bb2+ ikzb

])

+ H.c. (1.45)

Using this solution, and inserting into (1.44), we have shown in AppendixesD and E that at late times τ the transition probabilities per unit de Sit-ter, i.e., per unit detector time satisfy detailed balance conditions. Theycorrespond to thermodynamic equilibrium at the temperature TdS in (1.40):

dP+/dτ

dP−/dτ=

nB

1 + nB, (1.46)

where the Bose-Planck distribution function takes the form familiar fromthermostatistics:

nB =1

exp[ω0/TdS] − 1]. (1.47)

The frequency ω0 ∝ Ω can be varied by changing the undressed Rabi fre-quency Ω, varying the intensity of the Rabi laser. The detector thus has a

22

changeable and therefore tunable frequency standard, which can be adjustedto scan the above distribution function for a given TdS.

From the relation (1.46), we come to the remarkable conclusion that aproperly designed detector can “see” a thermal equilibrium distribution in itsquasiparticle basis, though it is embedded in a highly nonstationary systemwith respect to the laboratory frame. The rapidly expanding Bose-Einsteincondensate represents this highly nonstationary system, which hosts the deSitter quasiparticle detector atomic quantum dot. I stress here again that(1.46) is an exact result obtained by quantizing hydrodynamic fluctuationsin a nonstationary superfluid, and not just concluded from a mere compar-ison of the phonon kinematics in our expanding superfluid with that of thequantum field theory of photons in de Sitter space-time.

The equilibration time scale of the detector, and thus the time scale onwhich the Rabi oscillations between the detector states are damped out, isset by the detector frequency standard (the level spacing) ω0, and by therenormalization factor Z (Appendix E):

τequil = Z−2ω−10 ∝ (ρma

3s)

−1/2 (µ/ω⊥)2 ω−10 . (1.48)

The renormalization factor Z contained in (1.36) determines the equilibra-tion rapidity because it physically expresses the strength of detector-fieldcoupling. It is related to the initial diluteness parameter Dp(0) ≡ (ρma

3s)

1/2

of the Bose-Einstein condensate and to the ratio µ/ω⊥, which determinesinasmuch the system is effectively one-dimensional, Z 2 ∝ Dp(0)(ω⊥/µ)2. Toobtain sufficiently fast equilibration, the condensate thus has to be initiallynot too dilute as well as close to the quasi-1D regime, for which the trans-verse harmonic oscillator energy scale is of order the energy per particle, i.e.,µ ∼ ω⊥. These two conditions have another important implication. The ra-tio of the instantaneous coherence length ξc(t) = (8πρ0(0, t)as)

−1/2 ∝ tand the location of the horizons z = zH = ±Λ−1/2, which are station-ary in the present setup, has to remain less than unity within the equili-bration time scale.¶ If this is not the case, the coherence length, whichplays the role of the Planck scale, exceeds the length scale of the hori-zon at equilibration, and the concept of “relativistic” phonons propagat-ing on a fixed curved space-time background with local Lorentz symme-try becomes invalid. The ratio ξc(t)/zH at the lab equilibration time scalet = tequil = t0 exp[2π(TdS/ω0) (µ/ω⊥)2D−1

p (0)], expressed in parameters rel-

¶I thank C. Zimmermann for a pertinent question during a talk given by me atTubingen, leading to this observation.

23

evant to the experiment, is given by

ξc(tequil)

zH=πt0T

2dS

ρmasexp

[

2πTdS

ω0

(

µ

ω⊥

)2 1

Dp(0)

]

. (1.49)

We see that this ratio changes exponentially with both the initial dilutenessparameter Dp(0) and the quasi-1D parameter µ/ω⊥. In most currently re-alized Bose-Einstein condensates, the diluteness parameter Dp ∼ 10−2 [13].Here, we initially need Dp(0) ∼ O(1) to have the condition ξc(tequil)/zH < 1fulfilled, assuming a reasonably large value of the de Sitter temperature TdS.Though the condensate has to be initially quite dense, it is to be stressedthat the central density decays like t−2 during expansion. Therefore, the rateof three-body recombination losses quickly decreases during the expansionof the gas, and the initially relatively dense Bose-Einstein condensate, whichwould rapidly decay if left with a Dp close to unity, can live sufficiently long,the total rate of three-body losses decreasing like ρ2

0(0, t) ∝ t−4.

1.3 Summary

The primary statement to be drawn from the present chapter is that phonons,i.e., low-energy linear-dispersion quasiparticles, moving in a spatially andtemporally inhomogeneous Bose-Einstein-condensed superfluid gas, are kine-matically equivalent to photons, the quanta of the electromagnetic field,moving on geodesics in a curved space-time. We have explored the classicalas well as the quantum aspects of this statement.

On the classical side, the analogy helps to provide us with a simple gen-eral means to study quasiparticle propagation in an inhomogeneous mediumin motion. An example of such an application is the gravitational lensing ef-fect exerted by a superfluid vortex, derived in Appendix B. On the quantumfield theoretical side, we can access within the analogy phenomena which areextremely difficult if not impossible to access with light. One of the fun-damentals of quantum field theory, the fact that the particle content of aquantum field depends on the observer, can thus be experimentally veri-fied for the first time. The basic reason that the phenomena in questionare (comparatively) easy to simulate in a condensed matter system is thatthe energy and temperature scales, under which they occur, relative to thetypical energy scales of the system, can be changed at will by the experi-mentalist in a very controlled manner. More particularly, the temperatureof the thermal spectrum of phonons to be measured in the Gibbons-Hawkingeffect can be made relatively large compared to the axial phonon frequenciesand the actual temperature of the gas itself, by expanding the condensate

24

cloud rapidly enough. In the condensed matter analog, the typical energy ofthe quanta produced can in principle be even made to approach the relevant“Planck” scale, i.e., the point in the quasiparticle energy spectrum where itbegins to deviate from being linear.

Finally, on a more adventurous side, one could conceive of carrying outexperiments in “experimental” cosmology, as opposed to the currently ex-isting purely “observational” cosmology. In such an experimental approachto matters cosmological, one would try to reproduce under certain spec-ified and, in particular, well-defined initial conditions large-scale featuresof the cosmos, in the laboratory setting of nonstationary, inhomogeneoussuperfluid gases.

25

Chapter 2

Vortices in Bose-Einstein

condensates

Vorticity can enter a superfluid only in the form of quantized vortices. In thehydrodynamic long-wavelength limit, the vortices are singular lines on whichthe curl of the superfluid particle momentum does not vanish, rotp 6= 0.The momentum p obeys the macroscopic Bohr-Sommerfeld quantizationcondition

∮

p · dx = 2π~nv = mΓnv, (2.1)

where nv is an integer and Γ = 2π~/m the conventional quantum of velocitycirculation, Γ ≡

∮

v · dx. The closed line integral encircles the vortex coreonce, on which the superfluid phase φ, the potential of p according to p =~∇φ, is defined only modulo multiples of 2π, due to its identification withthe phase of the complex order parameter ψ =

√ρ exp[iφ]. Here, ρ is the

superfluid density (equalling at absolute zero the total fluid density). Weexplicitly emphasize that the quantization condition using the momentum isthe invariant characterization of circulation. It is valid in particular also forrelativistic superfluids, in which the “mass of the superfluid particle” m (andthus the velocity circulation) is not well-defined. Still, a scalar, co-ordinateinvariant quantity momentum circulation can be defined via the dynamicalquantity momentum [41].

The quantized vortices defined by (2.1) will be the elementary objectsunder consideration in what follows. In rotating Bose-Einstein-condensates,they are arranged in lattices, see the following section. The possibility thatthey are created in pairs, for condensates rapidly expanding at supersonicspeeds, is investigated in section 2.2.

26

2.1 Rapidly rotating vortex lattices

If a superfluid is set into rotation, upon increasing the rotation velocity Ω,a triangular lattice of ordered vortex lines forms, with their axis along thatof the externally imposed rotation rate [42]. The areal equilibrium densityof vortices in the rotating superfluid is given by nv = 2Ωv/Γ; setting fromhere onwards ~ = m = 1, we have Γ = 2π and nv = Ωv/π. In an infinitesystem, Ωv, giving the vortex density, equals the externally applied rotationrate Ω [42]. In a finite (mesoscopic) system, like the trapped Bose-Einsteincondensates experimentally realized, Ω − Ωv ∼ O(1/R2), where R is theradius of the condensate perpendicular to the axis of rotation. We neglectthe difference between Ω and Ωv, because it is irrelevant in the limit thatthe total number of vortices is large, Nv 1, which is the limit we willconsider in what follows.

Real vortices are not ideal, point-like objects. They possess a core offinite size, in which the fluid density varies; it goes to zero at the centralaxis in the singular vorticity case, for which the fluid velocity scales like|v| ∝ 1/r. In a Bose-Einstein condensate, the size of the core is determinedby the balance of repulsive interaction energy gρ and quantum pressurepQ = −1

2(∆√ρ)/

√ρ, cf. Eq. (1.7). For a vortex in an infinitely extended

Bose-Einstein condensate, this yields the core size ξ0 = (2gρ)−1/2.In the publication of Appendix G, I and Gordon Baym posed the question

of the structure of the vortex lattice and ultimate fate of the superfluid,at rotation speeds so large that a sizable fraction of the fluid is filled bythe vortex cores. It is a well-established fact that in a superconductor,when vortex cores begin to overlap, the system becomes normal, i.e. thatsuperconductivity ceases to exist [43]. However, for elementary bosons, atabsolute zero, it is not immediately apparent what such a normal phaseshould physically be, in analogy to, say, the unpaired electrons formingthe normal Fermi liquid. Our research thus emerged from the question ifanything analogous to the “going normal” of superconductors at the uppercritical magnetic field Hc2 exists in a neutral gas of bosons. Now, thisquestion is of academic value in a strongly correlated superfluid like heliumII, because the rotation rates required to make the atomic-size vortex coresapproach each other are of order 1012 rad sec−1. In the dilute atomic gases,however, Ω0 = 2gn/~ = ~/mξ2

0 ∼ Ωc2 is a reachable rotation rate. Forξ0 ' 0.2µm [2], Ω0 ' 1.8 × 104 rad/sec in 87Rb and ' 6.9 × 104 rad/sec in23Na, well within the capability of present experiments [44, 45, 46, 47].

It was the experiment conducted in [44], which triggered our interest toinvestigate the behavior of vortex matter in rotating Bose gases, because itshowed perfectly arranged triangular Abrikosov-Tkachenko lattices of more

27

than one hundred vortices. The lattices are mapped in time-of-flight pic-tures, in which it is directly seen that the vortex cores indeed fill a quitesizable fraction of the sample; this fraction grew larger in later experimentsat even greater rotation speeds [46].

2.1.1 Vortex core compression at high rotation speeds

To answer the question after a possible analog of Hc2 in a neutral systemof bosons, we considered in the publication of Appendix G the total energyof the superfluid within Gross-Pitaevskiı theory. We were looking for theminimum of the energy, using a variational solution for the particle densitydistribution in a vortex lattice.

In a frame rotating at an angular velocity Ω, the Hamiltonian takes theform

E′ =

∫

d3r

[

1

2m|(−i∇−mΩ× r)ψ|2

+

(

V (r) − 1

2mΩ2r2

)

|ψ|2 +1

2g|ψ|4

]

. (2.2)

To calculate the rotating frame energy, we subdivided the vortex latticeinto cells of radius ` = Ω−1/2 and evaluated the energy within each cell sep-arately, finally summing over all the cells. This amounts to a vortex matterequivalent of the Wigner-Seitz approximation familiar from condensed mat-ter physics [48]. The Wigner-Seitz method has been shown to yield veryaccurate results, as compared to “exact” calculations, if applied to vortexlattices [49].

Now, concentrating on a single vortex in a particular cell, with unitwinding number around the cylindrical axis, the wave function of that vortexreads ψ(r) = f(x, z)eiφ, in polar co-ordinates centered on the cell origin.Within a cell, labelled by i, we took the simple wave function amplitudeansatz

f(x, z) =

(x/ξ)√

ni(z), 0 ≤ x ≤ ξ,√

ni(z), ξ ≤ x ≤ ` = Ω−1/2,(2.3)

where x is the radial co-ordinate measured from the center ri of the cell. Toaccount for the density variation in the system, we allow the mean densityn(ri, z) within a cell centered on ri to depend on the radial position ofthe cell and its height. Ansatz (2.3) is consistent with the exact boundaryconditions, f(x → 0, z) ∼ x, required for the singly quantized vortex, anddf(x, z)/dx|x=` = 0, i.e., that the densities at the boundaries of the cells

28

match smoothly. While the core radius in a cell should generally depend onthe local density, that is, should generally have the functional dependenceξ = ξ[n(ri, z)], for simplicity we took the variational parameter ξ to dependonly on the mean density in the system.

The mean density n(ri, z) in the cell is

n(ri, z) =1

π`2

∫ `

0d2r |ψ|2 = ni(z)(1 − ζ/2), (2.4)

where ζ ≡ (ξ/`)2 = Ωξ2 ≤ 1 is the ratio of the core area to the transversearea of the unit cell of the lattice. The density between the cells ni(z) henceincreases if ζ increases. This fact is related to particle number conserva-tion, the particles having to “go somewhere” if the vortices get more closelyspaced. The mechanism of particle conservation and the ensuing increase ofparticle density between the vortices is the crucial mechanism for the corecompression effect to be described in the following, because it increases theinteraction energy per particle determining the core size, which thus shrinks.

Using the ansatz (2.3), the rotating frame energy turns out to be

E′[n(r), ζ] =

∫

d3r n(r)

[

a(ζ)Ω + V (r) − 1

2Ω2r2 +

1

2gn(r)b(ζ)

]

. (2.5)

The variational functions a(ζ) and b(ζ) have the following functional depen-dence:

a(ζ) =1

n(r)

∫ `

0dxx

[

(

f

x

)2

+

(

∂f

∂x

)2]

=1 − ln

√ζ

1 − ζ/2,

b(ζ) =2

`2n2(r)

∫ `

0dxxf4(x, z) =

1 − 2ζ/3

(1 − ζ/2)2. (2.6)

The quantity a(ζ), which specifies the renormalization of the kinetic en-ergy by the presence of finite vortex cores, has a minimum at ζ ' 0.558,while b(ζ), which indicates the renormalization of the particle interactionenergy, increases slowly with ζ. Obviously, the precise functional form ofthe quantities depends on the form of the function f in ansatz (2.3) [50].

Variation of the energy after the particle density distribution n(r) yieldsthe Thomas-Fermi result,

n(r) =µ+mΩ2r2/2 − V (r)

gb(ζ), (2.7)

where µ is the isothermal chemical potential (including the kinetic energy)and the quantity µ = µ− a(ζ)Ω has the kinetic energy part subtracted. In

29

an elongated trap with transverse frequency ω⊥ and longitudinal frequencyωz, V = 1

2ω2⊥r

2 + 12ω

2zz

2, the density has the usual form of an (inverted)half-paraboloid [51]

n(r) = (µ/gb)(

1 − (r/Rt)2 − (z/Zt)

2)

, (2.8)

with Z2t = (2µ/m)/ω2

z and R2t = (2µ/m)/(ω2

⊥ −Ω2) the (squared) Thomas-Fermi radii in axial (rotation) direction and perpendicular direction. Then,the condition that the number of particles is the fixed N =

∫

n(r)d3r, givesus N = 8πZtR

2t µ/15gb. Therefore, the chemical potential reads

µ

ω⊥=

1

2

[

15Nbas√mω⊥

ωz

ω⊥

(

1 − Ω2

ω2⊥

)]2/5

. (2.9)

We thus obtain that the Bose-Einstein condensate becomes for Ω → ω⊥

increasingly pancake-shaped, with the two Thomas-Fermi lengths growingrespectively decreasing like Rt ∝ (1−Ω2/ω2

⊥)−1/10 and Zt ∝ (1−Ω2/ω2⊥)1/5,

and with the central density consequently decreasing like n(0) ∝ (1 −Ω2/ω2

⊥)2/5.Using the density distribution (2.8), the minimization of the energy (2.5)

gives us the equation

Ω∂a

∂ζ+g〈n〉

2

∂b

∂ζ= 0 (2.10)

which determines the optimal value of ζ, where 〈n〉 =∫

d3rn2(r)/∫

d3rn(r)is a mean of the density in the system.

For various aspect ratios ω⊥/ωz, the variationally optimized vortex corearea as a fraction of the cell size is displayed in the right panel of Fig. 1 in Ap-pendix G. The crucial features are, first of all, that the relative core area hasan extremal value ζmax = 0.558. For moderate values ω⊥/ωz, the increase ofζ is at first very slow, and only close to Ω = ω⊥ is rapidly approaching ζmax.In very strongly elongated traps, ζ(Ω) approaches a self-similar region forincreasing Ω, in which the vortex core size decreases exactly conformal tothe decreasing cell size, with an essentially constant ratio of core and cellarea given by ζmax.

Lowest Landau level results: Beyond Thomas-Fermi

The Hamiltonian (2.2) is equivalent to that of particles in magnetic andelectric fields, with Ω × r the vector potential, and 2Ω playing the roleof magnetic field; the mass equals the “charge” and the scalar potential isthe sum of trapping and centrifugal potential, cf. [52]. Therefore, one mayconstruct quite direct analogies of rapidly rotating Bose gases to electrons

30

in the plane, subjected to strong perpendicular magnetic fields in the Teslarange (with small additional electric fields), leading in very clean samplesto the familiar fractional quantum Hall effect of fermions [53].

For Ω very close to ω⊥, the vortex lattice enters the so-called mean-fieldquantum Hall regime, a term coined by Ho in [54], which refers to the factthat the particles, at very low densities, get confined to their lowest rota-tional Landau level. This is not the Thomas-Fermi regime we describedabove, in which several Landau levels are significantly populated. For com-pleteness of the discussion, I now describe results for this regime.

It should be noted that this “mean-field” notion of “quantum Hall” hasnothing whatsoever to do with the strongly correlated, quantum Hall liquidstates [53]. The latter states were investigated for bosons, e.g., in [55, 56].The mean-field quantum Hall regime has been essentially realized in experi-ment very recently [46], while the strongly correlated, “true” quantum Hallstates are still off by some margin, because they require rotation rates ex-ceedingly close to the point of centrifugal instability Ω = ω⊥. The extremelystrict conditions for the realization of the latter states are essentially beingdue to the simultaneous requirements of very low particle and large vortexdensities. The ratio of particle number N and vortex number Nv equals thefilling factor of the rotational Landau levels of particles, ν = N/Nv. Thefilling factor has to be of order ν ∼ O(10) for the strongly correlated par-ticle quantum Hall state to be realized [56, 62]. Under the condition that1 − Ω/ω⊥ 1, the system is prone to simply fly apart (to be no longertrapped), because the trapping potential is increasingly reduced by the cen-trifugal potential. It is therefore very difficult to realize such a stronglycorrelated, fractional bosonic quantum Hall state, using a purely harmonictrapping potential (in current experiments ν ∼ 103 . . . 104 [46]).∗

Pethick and Baym have carried out an analysis analogous to the onepresented here (which corresponds to many occupied rotational Landau lev-els and the Thomas-Fermi approximation), but effectively including in theiranalysis those parameter and rotation velocity regimes for which the low-est Landau level description of the “mean-field” quantum Hall state applies[50]. They determined the variational function f in (2.3) from a differentialequation, which follows from the Gross-Pitaevskiı equation, and includes allterms possible within this mean-field theory. The experiment [46] has in-deed confirmed the core compression effect first predicted in the publicationwhich Appendix G comprises, in accordance with the more accurate limitingvalue of core and cell area obtained by [50], ζmax = 0.225.

∗In the recent experiments of [63], a weak quartic potential is added to keep the gasconfined for Ω = ω⊥.

31

2.1.2 Vortex lattice collapse to a single giant vortex

In the harmonically trapped case, a vortex lattice always is, within mean-field theory, the energetically preferred state of the dilute Bose gas. However,if we have an admixture of potentials of higher order than quadratic in thetrapping potential V , there exists the possibility that a transition of thevortex lattice to a so-called giant vortex takes place, which is a vortex witha phase factor exp[iνφ], where ν 1. This possibility is physically due tothe fact that for a confining potential higher than harmonic, the fluid pilesup at the container walls by virtue of the centrifugal force, while for the(degenerate) case of exactly harmonic trapping, the centrifugal force justrenormalizes the trapping frequency according to 1

2mω2⊥ → 1

2m(ω2⊥ − Ω2).

That a transition to a giant vortex state takes place starting from a vor-tex lattice state, by increasing the rotation speed, was analytically shown byus in the final section of Appendix G. To make the problem tractable ana-lytically, we considered the extreme limiting case of a hard-walled container,i.e., a cylindrically symmetric trapping potential of the form

V (r, z) =

0 0 ≤ r < R,

∞ r ≥ R.(2.11)

In practice, one can realize potentials which are (much) steeper than har-monic by using Laguerre-Gaussian (doughnut-shaped) laser beams [57].

At the level of the Thomas-Fermi approximation, which is adequate todescribe the giant vortex regime in large systems (mR2Ω 1), a giantvortex, described by order parameter, ψ(r) =

√

nG(r)eiνφ, has an energy inthe rotating frame which reads

E′G =

∫

d3r nG(r)

(

ν2

2mr2+g

2nG(r) − νΩ

)

. (2.12)

This energy needs to be compared with the vortex lattice energy in (2.5).Minimizing with respect to n(r) we find the analog of Eq. (2.7), nG(r) =(µ − ν2/2mr2)/g, where µ = µ + νΩ. The giant vortex thus has a hole inthe center of radius RG = αR, where α = ν/R

√2mµ.

In the limit of large Ω ( Ωh), where α → 1 (i.e., if the hole size isapproaching the container radius), we expanded in powers of (Ω0/mR

2Ω2)1/2

and found that the giant vortex energy equals

E′G

N= −1

2mR2Ω2 +

√2mR2Ω0

3Ω +

Ω0

18+ · · · . (2.13)

32

Comparing this expression with the minimized lattice energy, we found atransition to the giant vortex state at the rotation velocity

ΩG ' Ω0

9(1 + ln[R/ξ0]). (2.14)

This value is in accordance with what Kasamatsu et al. obtained in anindependent numerical study [58], considering a more realistic quadraticplus quartic trapping potential (also cf. [59, 60]). Again, like the corecompression effect discussed in the above, in helium II such a transitionto a giant vortex state is never observable, and the familiar folkore thatmultiply charged vortices are energetically unstable towards vortices of unitcirculation, in the rotation ranges accessible in that system, holds sway.

We finally mention that transitions to giant-vortex-like states, startingfrom vortex lattices, have already been observed in harmonic [61] as wellas harmonic plus quartic potentials [63]. However, these were nonequilib-rium phenomena dealing with transient states, and not yet the transitionbetween two equilibrium states (vortex lattice and giant vortex) that wehave considered in the publication of Appendix G.

2.2 Quantum vortex dynamics and quantum elec-

trodynamics

It is a rather well-established fact that vortices in a film of superfluid can bemapped to charged particles moving in the plane [64]. The vortex “charge”is the circulation Γ, and the “magnetic” and “electric” fields are

E = ρv × ez, B = −ρez, (2.15)

where v and ρ are the superflow velocity and density, respectively; ez is thevector perpendicular to the plane. It is, furthermore, also commonly ac-cepted that vortices can be understood to be quantum objects, their canon-ical quantization relation in the limit of an incompressible superfluid givenby [15, 41]

[X, Y ] = i(2πρ)−1, (2.16)

where X, Y are the quantized vortex center (collective) co-ordinates in theplane.

Less obvious is if the analogy extends to the quantum and relativistic(i.e., electro-dynamical) domain, that is, if in a suitable limit, vortex dynam-ics in a compressible superfluid is isomorphic to quantum electrodynamics.The idea that such an analogy exists in the hydrodynamic limit is due toPopov [65], and was later on investigated, e.g., by Arovas and Freire [66].

33

In the publication contained in Appendix H, we demonstrated that theidea of a possible direct analogy quantum electrodynamics and quantumvortex dynamics is verifiable in a dilute two-dimensional Bose-Einstein con-densate, by observing the decay of monopole (breathing) oscillations of thegas.

The diluteness of the gas is of importance for the possibility that theanalogy exists, because it implies that the Bogoliubov excitation spectrum,in a homogeneous gas of constant density ρ0 at rest, reads

ε2(k) = gρ0k2 +

k4

4m2. (2.17)

Such a spectrum has no minimum, i.e., is monotonously increasing, andsecond, as a consequence, does have an essentially linear spectrum up tothe “Planck” scale we discussed in the previous chapter, kPl =

√gρ0. This

implies, according to the Landau criterion, that no (linear) excitations canbe created above the superfluid ground state save for phonons. This isnot the case for dense, strongly correlated superfluids like helium II, whererotons can be created if the superflow exceeds vR ' 58 m sec−1 [64].

According to [35], a two-dimensional Bose-Einstein condensate has anintrinsic exact symmetry SO(2,1), the Lorentz group in two spatial andone time dimension. As already mentioned in subsection 1.2.3, this hiddensymmetry leads, at zero temperature, to a formally infinite quality factor,i.e. zero damping, of the monopole (breathing) oscillations of the gas. Thefact that the monopole oscillations are undamped at zero temperature, asfar as ordinary phonon excitations above the superfluid ground state areconcerned, led us to the idea (Appendix H) that an analogy of vortex paircreation to the pair creation of electrons and positrons in quantum electrody-namics [67] could be exploited to confirm the analogy of quantum vortex andelectrodynamics in an oscillating two-dimensional Bose-Einstein-condensedgas. Vortices are nonlinear excitations above the ground state, and arethus not protected by the SO(2,1) symmetry of the two-dimensional Bose-Einstein condensate, which holds for the linear phonon excitations only.

The Schwinger process of quantum vacuum breakdown leading to paircreation is a phenomenon occurring whenever the electric field exceeds themagnetic field (in cgs units). This corresponds in the analogy to the in-stability of a supersonic flow with respect to the spontaneous creation ofvortex-antivortex pairs from the superfluid vacuum, i.e., by using the identi-fication (2.15), the “quantum vortex dynamical” condition for the superfluidvacuum to break down is

|E||B| > cs ⇐⇒ Superfluid Vacuum Breakdown.

34

In a simplified model of the 2+1D vacuum pair creation instability, whichexploits directly the analogy to Schwinger pair creation in quantum electro-dynamics by identifying the analogous quantities, the pair production rateΓ per unit area can be written as ([67, 68], Appendix H)

Γ =1

4π2c2sF3/4

∞∑

n=1

(−1)n+1

n3/2exp

(

−πn(E0v)

2

√F

)

, (2.18)

where we have defined F = E2c2s − B2c4s. We set, within logarithmic ac-curacy, the vortex pair size equal to the Thomas-Fermi radius R of thecondensate, in the static vortex self-energy of a widely separated vortex-antivortex pair 2E0

v = 2πρΛ, with Λ = ln(R/ac). The value of the prefactorin front of the exponential in the above expression is subject to changeswhich are due to microscopic details of vortex motion. We display its value,stemming from taking literal the analogy to quantum electrodynamics alsoon the level of quantum fluctuations (to one loop order), for numerical con-creteness. The behavior of Γ for |E|/|B| & cs is, however, dominated bythe hydrodynamical exponent, whose value is independent of microscopicphysics, and more specifically by the n = 1 term in the above sum.

Due to the creation of vortex-antivortex pairs, with a rate per unit area Γ,the oscillation of the gas experiences damping. The scaling equation of mo-tion (1.17) of the isotropically expanding and contracting two-dimensionalcondensate disk thus contains an additional damping term:

b+ ω2fb−

ω2in

b3= − D

ω5in

b7b4 , (2.19)

where ωin and ωf are the initial and final trapping frequencies, respectively;the time scale for switching from ωin ωf is supposed to be much lessthan ω−1

in . The above relation holds for locally supersonic motion, i.e., if|E|/|B| > cs (F > 0). Here, the constant D explicitly reads

D =48

π8

g7/4√gN

(

ln[

4√

gN/π])

11

2

∑

n

(−1)n+1

n8, (2.20)

the fractional exponents like 7/4 and 11/2 occurring therein related to theThomas-Fermi limit being used.

The solution of the pair-creation-modified scaling equations of motion(2.19) yields a nonexponential decay of the maximal value of the scalingfactor bmax(t),

bmax(t) =bmax(0)

(1 + D′b10max(0)ωf t)1/10, (2.21)

35

where D′ = 35512(ωf/ωin)

5D. The envelope bmax(t) of the scaling parameteroscillations is seen to decay very slowly and in a nonexponential fashion,governed by the Thomas-Fermi exponent 1/10 in Eq. (2.21). In addition,the dissipation rate depends strongly on the initial oscillation amplitude,with the tenth power of bmax(0). These features allow one to distinguishuniquely, by experimental means, the amplitude decay of the free conden-sate oscillations, due to vortex pair creation, from other possible dampingmechanisms. Furthermore, the decay rate becomes significant only for suf-ficiently strong coupling, the constant D in (2.20) scaling like g9/4. Strongcouplings can be realized either by tuning an external magnetic field close toFeshbach resonances [29, 30], or by geometric scattering resonances, whichoccur under variations of the (strong) trapping potential perpendicular tothe plane [80].

Observing the oscillation amplitude decay of a two-dimensional Bose-Einstein condensate, at temperatures close to absolute zero, we thus have asensitive tunable instrument at our disposal to verify a possible analogy ofquantum vortex dynamics and quantum electrodynamics.

36

Chapter 3

Existence of Bose-Einstein

condensation for

anisotropically trapped gases

3.1 Infinitely extended systems

The question on the existence and dimension dependence of off-diagonallong-range order, for an infinitely extended system of scalar uncharged bo-sons, was conclusively answered by Hohenberg [69]. He made use of the factthat in the thermodynamic limit of an infinite system of interacting particlesof bare mass m, the relation (we set ~ = kB = 1 in what follows, where kB

is the Boltzmann constant)

Nk ≥ −1

2+mT

k2

n0

n(3.1)

for the occupation numbers Nk =< b†kbk > of plane wave states enumerated

by k (6= 0) holds, where T is the temperature. The angled brackets hereand in what follows indicate a thermal ensemble (quasi-)average [16]; n0

and n are the condensate density, associated to k = 0, and the total density,respectively. The inequality (3.1) is essentially a particular representationof the celebrated Bogoliubov 1/k2 theorem on correlation functions [16].

The relation (3.1) entails that in one and two spatial dimensions, amacroscopic occupation of a single state, the condensate of density n0, isimpossible: The inequality leads to a contradiction in dimension D ≤ 2 dueto the (infrared) divergence of the wave vector integral of (3.1), which deter-mines the number density of non-condensate atoms. We denote the infraredwave vector cutoff by kc = 2π/R, where R is the size of the system, tending

37

to infinity (hence kc → 0), and the ultraviolet cutoff (the microscopic dis-cretization length) we denote kPl. Taking the integral of both sides of (3.1),and dividing by the total number of particles N , we obtain

N −N0

N≥ 1

kDPl

∫ kPl

2π/RdDk

[

mT

k2

n0

n− 1

2

]

R→∞' n0

n×

mT2πkPl

R (1D),

mTk2Pl

ln[RkPl/2π] (2D).(3.2)

The number of excited atoms N −N0, where N0 is the number of particlesin the condensate, has to be finite. Therefore, condensates cannot exist inone or two spatial dimensions at any finite temperature, because n0 = 0necessarily in the thermodynamic limit of R→ ∞, due to the divergence ofthe right-hand side. Physically, long-range thermal fluctuations, the influ-ence of which becomes increasingly dominant the lower the dimension of thesystem, destroy the coherence and long-range order expressed by the exis-tence of the condensate. It is important to note that a finite thickness (2Dcase) or cross section (1D case) of the system does not alter the statementmade by (3.1) and (3.2): The “wave vector” k enumerates (counts) states,and is not a physical wave vector upon which boundary conditions, due tothe finite extension of the system in the perpendicular directions, also needto be imposed [70].

3.2 Trapped gases

The question of the applicability of the Hohenberg theorem, which is rep-resented by Eqs. (3.1) and (3.2), has renewed interest for trapped Bose-Einstein-condensed vapors of reduced dimensionality, which can be realizedusing various techniques; a rather small (compared to the activity in thefield) selection of recent references is [71, 73, 72, 74, 75, 76]. That thereis indeed a difference to the thermodynamics in an infinite box limit stemsfrom various reasons. First of all, the classification of excited non-condensatestates by plane waves is not the suitable one in a trapped gas: The conden-sation, in the limit of large total particle number N , takes place primarilyinto a single particle state in co-ordinate space [77], and condensate andtotal densities have (in principle arbitrary) spatial dependence, n0 → n0(r),n → n(r). Second, there arises the question how interaction influences theexistence of the condensate, given that inhomogeneity of n0(r) and n(r).An interacting gas generally has a behavior increasingly different from theideal gas the lower the dimension of the system [78, 79, 80, 81].

38

In the publication contained in Appendix I, I aimed at obtaining a re-lation which is analogous to (3.1) in as close as possible a manner. It isimportant to stress that a major strength of (3.1) is the explicit interactionindependence; this holds true as long as this interaction is velocity indepen-dent, i.e., depends only on the (relative) position of the particles [69, 82].A major goal thus was to maintain that independence of the inequality onparticle interactions.

Like Hohenberg in his original paper [69], my analysis started from theBogoliubov inequality [16], which reads

1

2

⟨

A, A†⟩

≥T∣

∣

∣

⟨

[C, A]⟩∣

∣

∣

2

⟨[[

C, H]

, C†]⟩ (3.3)

for any two operators A and C. It is valid for any many-body quantum sys-tem, for which the thermal (quasi-)averages indicated by the angled bracketsare well-defined and finite. This inequality was used by Mermin and Wagnerto prove the absence of ferro- and antiferromagnetism in one and two dimen-sions [84] in a fashion analogous to Hohenberg’s proof of the non-existenceof off-diagonal long-range order for scalar particles in one and two spatialdimensions.

The operators in relation (3.3) were then chosen by me to be the smearedexcitation and total density operators

A =

∫

dDr f(r)δΦ(r), C =

∫

dDr g(r)ρ(r), (3.4)

where f(r) and g(r) are complex regularization kernels and the excitationoperator is defined by

δΦ(r) = Φ(r) − b0Φ0(r). (3.5)

The condensate wave function Φ0(r) is normalized to unity. This choice ofoperators is analogous to the original Hohenberg theorem derivation in [69],where instead of the position space basis chosen here, a plane wave basisappropriate for the thermodynamically infinite system in a box is taken. Inposition space, the smearing procedure is necessary, because products of twoquantum field operators diverge if taken at the same point in space.

The next step is to recognize that the so-called mixed response (the“anomalous” commutator) is taking the nonlocal form

⟨[

ρ(r), δΦ(r′)]⟩

=√

N0 Φ0(r)[

−δ(r − r′) + Φ∗0(r)Φ0(r

′)]

. (3.6)

39

Here, after carrying out the commutator, < b0 >=√N0 =< b†0 > has been

used, where this assignment is valid to O(1/√N0). The difference to the

usual Bogoliubov treatment, which would neglect the second term on theright-hand side of the canonical commutator [δΦ(r), δΦ†(r′)] = δ(r − r′) −Φ0(r)Φ∗

0(r′) altogether, is that the second term in the mixed response func-

tion above would disappear in the Bogoliubov approximation. We will arguebelow that neglect of this term in the mixed response, for finite systems,would lead to a contradiction with existing, i.e., experimentally realizedBose-Einstein condensates. The Bogoliubov approach of [b0, b

†0] ≡ 0 vio-

lates particle number conservation to O(f), where f =√

1 −N0/N , whilethe present approach violates it only to O(f 2), and this difference, whichis quite significant for sufficiently large systems, proves indeed crucial forcondensates to exist.∗

I then derived a sum rule for the denominator on the right-hand side of(3.3), analogous to the well-known f -sum rule

∫ +∞−∞ dω ωS(k, ω) = Nk2/m

for the dynamic structure factor [82], but in co-ordinate space. Using that

⟨[[

ρ(r), H(r)]

, ρ(r′)]⟩

= −i⟨[

∇r · (r), ρ(r′)]⟩

=1

2m

⟨

δ(r′ − r)(

Φ†(r′)∆rΦ(r) + h.c.)

−∆rδ(r′ − r)

(

Φ†(r′)Φ(r) + h.c.)⟩

, (3.7)

I obtained that the double commutator in inequality (3.3) equals

⟨[[

C, H]

, C†]⟩

=1

m

∫

dDr (−∆rg(r)) g∗(r)n(r) , (3.8)

where the expectation value of particle density n(r) = 〈ρ(r)〉 = 〈Φ†(r) Φ(r)〉.Here, I used an operator version of the continuity equation for the currentdensity operator = [Φ†∇Φ − (∇Φ†)Φ]/2im, together with the Heisenbergequation of motion for the density operator ρ: [ρ, H] = i∂tρ = −i∇r · . Thislast relation is valid for a Hamiltonian with no explicit velocity dependence[82], which is where the requirement of velocity independence, mentionedin the above, comes into our relation. Using this method of evaluating thedouble commutator, which relies on exact particle number conservation, Itherefore eliminated any (microscopic) details contained in the Hamiltonian,for example on the detailed nature of the particle interactions, and specificassumptions on these details.

∗For a particle-number-conserving approach, that is, one which is not breaking theglobal U(1)-symmetry (to a given order in the quantity f), see [83].

40

The left-hand side of (3.3) I bounded from above using a Cauchy-Schwarzinequality, cf. Eq. (10) in Appendix I. The final inequality, as derived from(3.3), then reads

N −N0 +a

2

≥ mTN0

∣

∣

∫

dDr g(r)f(r)Φ0(r) −∫

dDr∫

dDr′ g(r)f(r′)|Φ0(r)|2Φ0(r′)∣

∣

2

∫

dDr (−∆rg(r)) g∗(r)n(r),

(3.9)

where a ≤ 1 is given by a = 1 −∫

dDr∫

dDr′f(r)f∗(r′)Φ0(r)Φ∗0(r

′) .The choice of the smearing kernels is in general subject to the require-

ment that the right-hand side of the inequality above is maximized for agiven set N0, n(r),Φ0(r). To obtain the maximum value, it is in principlenecessary to do a functional variation of that right-hand side holding thisset of parameters and distributions fixed. In the publication of Appendix I,I chose a simpler option, namely to take kernels such that they lead to aninequality relating quantities with an easily identifiable geometrical mean-ing. This will render (3.9) closest in its implications to the original infinitesystem relation (3.1). The choice of kernels taken thus was

fk(r) =

Ω−1/20 exp[ik · r] (r ∈ D0)

0 (r /∈ D0)

g(r) =

Φ∗0(r) (r ∈ D0)

0 (r /∈ D0)(3.10)

where Ω0 is the volume of the domain D0 in which Φ0 has finite support,i.e., its (suitably defined) finite normalization volume. The plane wavesfk(r) represent a set of functions, in which we vary the “wave vector” k tomaximize the right-hand side of (3.9).

I defined the effective radius of curvature of the condensate to be thecurvature radius of the condensate wave function, weighted by the totaldensity distribution:

Rc ≡√

N

Ω0

(∫

D0

dDrΦ0(r) [−∆rΦ∗0(r)]n(r)

)−1/2

. (3.11)

The relation (3.9) then takes the final form

1 − F

F≥ 1

N

2πR2c

λ2dB

C(k) − 1

2N0

(

1 − |Φ0(k)|2/Ω0

)

. (3.12)

41

Here, the condensate fraction F = N0/N and the de Broglie thermal wave-length λdB =

√

2π/mT . The variational functional C(k) is given by

C(k) =

∣

∣

∣

∣

n0(k) − Φ0(k)

∫

D0

dDrΦ∗0(r)|Φ0(r)|2

∣

∣

∣

∣

2

, (3.13)

where the Fourier transforms of single particle condensate density and wavefunction are defined to be n0(k) =

∫

dDr |Φ0(r)|2 exp[ik · r] and Φ0(k) =∫

dDrΦ0(r) exp[ik · r].There are various notable features of the inequality (3.12). First of all,

it is seen that the two-dimensional, trapped case is marginal: In the 2Dcase, the radius of curvature Rc scales like Rc ∝ N−1/2. Indeed, in thetrapped gas, the logarithmic divergence of the 2D case in (3.2) is cut offby the trapping potential, and Bose-Einstein condensates can exist even ina (suitably defined) thermodynamic limit [85]. Second, the value of C(k)is strongly reduced because of the second term under the square in (3.13);this is a consequence of the fact that we do not make use of the Bogoliubovapproximation and, therefore, the second term in the mixed response (3.6)is included. Third, the relation represents, after the set N0,Φ0(r), n(r)(and thus also F ) has been specified at a given temperature, an inequalityrepresenting primarily a geometric statement: It gives a bound on the pos-sible ratio of mean curvature radius Rc (which is dominated by the weaklyconfining directions), and the de Broglie wavelength λdB.†

One-dimensional Thomas-Fermi condensate

To demonstrate that the inequality (3.12) is indeed useful, I worked out anexplicit example for an experimentally easy-to-access case. That is, I con-sidered a Thomas-Fermi condensate wave function in a quasi-1D harmonic

trap with ωz ω⊥, which reads Φ0(z) =√

n0TF

(

1 − z2/Z2TF

)

/N . It is as-

sumed that F is sufficiently close to unity, N0 ' N , which amounts in effectto taking the limit of both sides of (3.12) to linear order in 1 − F . FromFig. 1 in Appendix I, it is seen that the function C is strongly peaked at itsglobal maximum km ' 3.7Z−1

TF (λm ' 1.7ZTF), where C(km) ' 1.54 × 10−2.It should be noted that, if one were to make the simple “guess” of, say,λm = ZTF, not using a variational ansatz for the right-hand side of (3.12),one would obtain a (much) weaker bound, and would therefore strongly limitthe usefulness of the inequality.

†I mention here that the final term in (3.12), which corresponds to the “−1/2” on theright-hand side of (3.1), while under most circumstances negligible, may be profitably usedto obtain upper bounds on the possible condensate fraction as a function of temperature[86].

42

Using (3.11), we have Rc = 4ZTF/3√

2π, which tell us that, as expected,the effective radius of curvature is of order the Thomas-Fermi length of thestrongly elongated condensate. Neglecting the second term on the right-hand side of (3.12), I obtained at k = km the rigorous inequality

ZTF

λdB≤ 6.0

√

N −N0 . (3.14)

The bound thus obtained on the maximal length of such a thin BEC cigar isconsistent with the experiments carried out so far, cf. the explicit exampleextracted from [71], given under equation (19) in Appendix I. It is importantto stress, however, that this would not be the case if we had neglected,following the Bogoliubov approximation, the second term under the squarein (3.13). Doing so reduces the right hand side of the above inequality byabout an order of magnitude and leads to a contradiction with experiment.Therefore, in this sense, particle number conservation, which is violated bythe standard Bogoliubov prescription, is necessary for the condensate toexist.

The conditions needed to violate the Thomas-Fermi relation (3.14) be-come more transparent if we write ZTF = 3

√

6Nd2zasω⊥/ωz, a relation valid

as long as the 3D scattering length is much less than the oscillator length inthe perpendicular direction, as d⊥ [87]. Then

ω⊥

ωz≤ 36.7

λ3dB

d2zas

N1/2f3. (3.15)

From this relation it is apparent that, leaving ωz (dz), as and N fixed, theinequality is most easily violated (apart from decreasing the temperature),if the aspect ratio is increased by increasing the transverse trapping ω⊥.This route of increasing the aspect ratio to check relation (3.14) should inparticular be possible in the narrow condensate tubes created in 2D opticallattices [76], where the aspect ratios can be increased to 1000 and beyond.

Conclusion

To conclude, I stress again that the particular advantage of relation (3.12)is its independence on any specific model for the strongly anisotropicallyconfined gas. Furthermore, the application of the inequality is not limitedto (real) ground state forms of Φ0. It is also possible to employ the inequalityto examine if certain more exotic excited-state condensates with complex,strongly inhomogeneous Φ0, like single vortices, vortex lattices, or solitonscan in principle exist.

43

By its very nature, as an inequality, it does not (and cannot) provethat a certain condensate exists. For such a proof one still has to resortto an explicit (many-body) calculation, to show that the state with single-particle wave function Φ0 is macroscopically occupied. However, on theother hand, the inequality can rule out parameter ranges for which any givenset N0,Φ0(r), n(r), at a specified finite temperature, is not possible.

44

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51

Appendix A

Uwe R. Fischer and Matt Visser

“On the space-time curvature experienced by quasiparticle

excitations in the Painleve-Gullstrand effective geometry”

Ann. Phys. (N.Y.) 304, 22-39 (2003)

53

On the space-time curvature experienced

by quasiparticle excitations in the

Painlevee–Gullstrand effective geometry

Uwe R. Fischera,* and Matt Visserb,1

a Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street,

Urbana, IL 61801-3080, USAb Physics Department, Washington University, Saint Louis, MO 63130-4899, USA

Received 11 December 2002

Abstract

We consider quasiparticle propagation in constant-speed-of-sound (iso-tachic) and almost

incompressible (iso-pycnal) hydrodynamic flows, using the technical machinery of general rel-

ativity to investigate the ‘‘effective space-time geometry’’ that is probed by the quasiparticles.

This effective geometry, described for the quasiparticles of condensed matter systems by the

Painlevee–Gullstrand metric, generally exhibits curvature (in the sense of Riemann) and many

features of quasiparticle propagation can be re-phrased in terms of null geodesics, Killing vec-

tors, and Jacobi fields. As particular examples of hydrodynamic flow we consider shear flow, a

constant-circulation vortex, flow past an impenetrable cylinder, and rigid rotation.

2003 Elsevier Science (USA). All rights reserved.

1. Introduction

The description of many natural phenomena is most vividly carried out in terms

of hydrodynamics, because the concept of a streaming liquid elucidates and helps to

understand the physical significance and structure of an underlying theory [1]. In its

Annals of Physics 304 (2003) 22–39

www.elsevier.com/locate/aop

*Corresponding author. Present address: Institut f€uur Theoretische Physik, Universit€aat Innsbruck,

Technikerstrasse 25, A-6020 Innsbruck, Austria. Fax: +43-512-507-2919.

E-mail addresses: Uwe.Fischer@uibk.ac.at (U.R. Fischer), matt.visser@mcs.vuw.ac.nz (M. Visser).1 Present address: School of Mathematical and Computing Sciences, Victoria University of

Wellington, P.O. Box 600, Wellington, New Zealand.

0003-4916/03/$ - see front matter 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0003-4916(03)00011-3

classical sense [2,3], hydrodynamics describes the motion of a continuum, character-

ized by a velocity and density distribution, which for a perfect fluid and in the non-rel-

ativistic limit is described by the Euler and continuity equations. It has been

recognized about 20 years ago by Unruh [4], that the propagation of small perturba-

tions on such a hydrodynamic background, which is itself governed by a continuum

version of Newtonian physics, may be cast into the form of a ‘‘relativistic’’ scalar wave

equation

U 1ffiffiffiffiffiffiffigp ol

ffiffiffiffiffiffiffigpglmomU

¼ 0 ð1Þ

for the velocity potential U of the perturbations. The disturbances propagate in an

effective space-time with metric glm, which is in general curved. The metric glm was

later on shown to be of the Painlevee–Gullstrand form [5], originally invented as an

alternative to the Schwarzschild form of the solution of the Einstein equations for a

point mass source. With the advent of effective curved space-time theories, it became

apparent that the Painlevee–Gullstrand representation of the metric appears in a host

of such theories. They comprise, besides the conventional Euler fluid [4,6], superfluid3He–A [7,8], atomic Bose-condensed vapors [9,10], and general dielectric (quantum)

matter [11–13].

An interesting and important feature of the Painlevee–Gullstrand metric is that it

continues to give an appropriate physical description for quasiparticle propagation

even when the effective space-time possesses a horizon [14]. This occurs because

the condensed matter origin of the metric in the Painlevee–Gullstrand form is the

spectrum of elementary excitations (quasiparticles) [15], which is primary. This phys-

ical energy spectrum, from which the metric is obtained using the fact that for mass-

less quasiparticles the energy spectrum is

glmplpm ¼ 0; ð2Þmust be well-defined and, in particular, real everywhere in the system. In contrast,

for the Schwarzschild form of the metric the spectrum reads

E2 ¼ c2 1

rS

r

2

p2r þ c2 1

rS

r

p2?; ð3Þ

where rS is the usual Schwarzschild radius and pr; p? are radial and transverse

components of the quasiparticle momentum, respectively. The velocity c plays the

role of the speed of light and is equal to the sound speed for phonons. This

‘‘Schwarzschild form’’ of the spectrum exhibits imaginary mode frequencies and

consequently leads to instability of the condensed matter system if a horizon is

present, because it has sections of the transverse momentum p? which result in

E2 < 0 inside the horizon. The Painlevee–Gullstrand metric, on the other hand, gives

real frequencies throughout a condensed matter system possessing a quasiparticle

horizon, which can thus be stable.

The non-equivalence of Schwarzschild and Painlevee–Gullstrand form of the met-

ric is related to the fact that the coordinate transformation relating the Schwarz-

schild solution and the Painlevee–Gullstrand representation becomes singular at the

U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39 23

55

horizon [14]. This fact has, inter alia, led to the usage of Painlevee–Gullstrand co-or-

dinates for investigations of Hawking radiation in the ‘‘conventional’’ black hole

context of gravitational theory [16,17], because these co-ordinates are non-singular

through the horizon, making the appropriate vacuum definition there much simpler.

The intrinsic characteristics of a curved space-time are described in a covariant way

by the Riemann tensor [18,19]. Our objective in this paper is to describe the Riemann-

ian curvature of the effective spaces described by the Painlevee–Gullstrand metric, in

the underlying hydrodynamic terms appropriate to a flowing background fluid. We

shall focus on two physical situations: quasiparticles in flows with a constant speed

of sound (iso-tachic flows) and quasiparticles in an almost incompressible (iso-pycnal)

hydrodynamic flow. By ‘‘almost incompressible’’ we mean that we take both the back-

ground density and the quasiparticle propagation speed relative to the medium to be

constants, and concentrate on those effects that are due to motion of the medium, i.e.,

its velocity distribution. In other words, even if a fluid has a constant ‘‘refractive in-

dex,’’ focussing and defocussing effects can be engendered throughmotion of the fluid.

As particularly interesting examples we demonstrate how the tracks of quasipar-

ticles are distorted by propagation through a shear flow, a constant-circulation vor-

tex flow, around an impenetrable cylinder, and how they propagate through a rigidly

rotating fluid. In a more general context we provide a local definition of ‘‘focal

length’’ in terms of the Riemann tensor, and show how the affine and ‘‘natural’’

(using the Newtonian background time) parameterizations of null geodesics can

be related to each other.

2. Painleve–Gullstrand curvature in 3 þ 1 dimensions

In the following discussion the quasiparticle spectrum is assumed to be linear in the

fluid rest frame for ‘‘small’’ quasiparticle momenta, E ¼ cjpj corresponding to (2), anddeviating from linearity for momenta approaching the ‘‘Planck scale’’ of the system at

hand. In general the ð3þ 1Þ-dimensional Painlevee–Gullstrand metric [5] reads

gtt ¼ q

c½c2 v2; gti ¼ q

cvi; gij ¼

q

cdij: ð4Þ

That is, the metric has space-time interval

ds2 ¼ q

c

c2dt2 þ dijðdxi vidtÞðdxj vjdtÞ

: ð5Þ

By special convention, the indices on the 3-velocity are always raised and lowered

using the flat 3-dimensional Cartesian metric so that vi ¼ vi.

In the case of irrotational fluid flow (for instance in a superfluid outside the cores

of the (singular) quantized vortices), the dAlembertian equation (1) can be derived

directly from a linearization procedure based on the Euler and continuity equations

[4,6]; the existence and relevance of the Painlevee–Gullstrand effective metric then

follows as a rigorous theorem. If distributed vorticity is present, the situation is

more subtle [20]: In hydrodynamics with distributed vorticity one obtains a rather

complicated system of coupled differential equations, one of which contains the

24 U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39

56

dAlembertian operator (and therefore also contains the effective metric) as a sub-

sidiary quantity [20]. Thus for hydrodynamics with distributed vorticity, the effec-

tive metric is not the whole story—but certainly an important part of the story.

In particular, if one appeals to the eikonal approximation (in this context identical

to the WKB approximation) one can derive Pierces approximate wave equation

[21]. In this approximation one can write down the quasiparticle spectrum directly

in terms of the effective metric [20].

Note that the constant-time hypersurfaces are conformal to ordinary flat Carte-

sian space. As long as we are interested in quasiparticles that propagate along the

null cones of this effective metric (that is, quasiparticles moving at the speed c relative

to the medium), it is permissible to neglect the overall conformal factor of q=c andconsider the simplified metric

gtt ¼ ½c2 v2; gti ¼ vi; gij ¼ dij: ð6Þ(This is simply the statement that conformal transformations leave null curves and,

in particular, null geodesics, invariant.) The inverse of this simplified metric is

gtt ¼ 1

c2; gti ¼ vi

c2; gij ¼ dij vivj

c2: ð7Þ

Note that the Newtonian time parameter t provides a preferred foliation of the

spacetime into space + time, and that this preferred foliation will prove very useful.

Suppose now that the speed of sound is iso-tachic, independent of position and

time. Then we can choose coordinates to set the speed c of linear quasiparticle dis-

persion equal to unity, a convention adopted in the formulae below. The ð3þ 1Þ-di-mensional Painlevee–Gullstrand metric [5] then reads

gtt ¼ 1þ v2; gti ¼ vi; gij ¼ dij: ð8ÞIn general relativistic language the lapse function in the ADM formulation [19] is

now unity and all the space-time curvature is encoded in the shift function—which

here describes the physical velocity of the fluid. The inverse metric is

gtt ¼ 1; gti ¼ vi; gij ¼ dij vivj: ð9ÞTurning to the computation of curvature, the 24 independent connection coefficients

read (cf. [22])

Ctij ¼ Dij;

Cttt ¼ vivkDik ¼1

2ðv rÞv2;

Ctti ¼ vjDij;

Cijk ¼ viDjk;

Citt ¼ otvi vkoivk þ vivlvkDlk ¼ otvi 1

2dij

vivj

ojv2;

Citj ¼ vivkDjk þ Xij:

ð10Þ

Here we have defined the deformation rate and angular velocity tensors by

U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39 25

57

Dij ¼1

2oivj

þ ojvi

¼ oðivjÞ ¼ Dji;

TrD ¼ divv;

Xij ¼1

2oivj

ojvi

¼ o½ivj ¼ Xji:

ð11Þ

The deformation rate is in general relativistic language the extrinsic curvature of the

constant-time hypersurfaces, while the angular velocity tensor is in fluid mechanics

language equivalent to the vorticity vector defined via xi ¼ ijkXjk. The above tensors

result in the unique decomposition of oivj ¼ ðr vÞij ¼ Dij þ Xij into a symmetric

and an antisymmetric tensor.

The components of the Riemann curvature tensor afford the basic symmetries

R½lm½qk ¼ R½qk½lm, which are supplemented by R½lmqk ¼ 0 and Rl½mqk ¼ 0 [19]. The

Riemann components that need to be calculated are thus Rtitj, Rijkl, and Rtijk,

the rest follow by the (anti-)symmetry properties. A tedious but straightforward

computation (which follows a variant of the Gauss–Codazzi decomposition)

yields

Rijkl ¼ DikDjl DilDjk; ð12Þ

Rtijk ¼ oiXjk þ vl DklDij

DjlDik

; ð13Þ

Rtitj ¼ otDij þ DXð þXDÞij ðD2Þij vkvk;ij þ vkvl DklDij

DjkDil

: ð14Þ

Here we have defined ðDXþXDÞij DikXkj þ XikDkj and similarly ðD2Þij DikDkj.

The appearance and interpretation of the Riemann components may be greatly

simplified if we consider them in an orthonormal, locally Minkowskian tetrad frame

fealg. Greek indices denote the usual space-time indices, Roman letters from the be-

ginning of the alphabet indicate tetrad indices, while Roman letters from the middle

of the alphabet denote space indices. Whenever there is any chance of confusion, car-

ets on indices are used to indicate that the components are given in the tetrad frame.

The tetrad frame fealg is defined byglm ¼ gabe

alebm: ð15Þ

In the simplest gauge it is given by

ettt ¼ 1; etti ¼ 0; eııt ¼ vi; e||i ¼ d||i: ð16Þ

The inverse basis satisfies

glm ¼ gabe la e

mb : ð17Þ

Note the use of index placement to distinguish eal from its inverse e la . Hence

ealelb ¼ dab as well as e

la e

am ¼ dlm. In a time plus space decomposition

e ttt ¼ 1; e t

ıı ¼ 0; e itt ¼ vi; e

j

ıı ¼ dj

ıı : ð18ÞThus, for any given vector X , the components in the various frames are related

by

26 U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39

58

Xa e la Xl ðXtt;XııÞ ¼ ðXt þ vjXj;XiÞ ð19Þ

and

X a ealXl ðX tt;X ııÞ ¼ ðX t;X i viX tÞ: ð20Þ

These index conventions greatly simplify the formulae below. Calculating the Rie-

mann tensor in the tetrad frame gives

Rıı||kkll ¼ DikDjl DilDjk; ð21Þ

Rttıı||kk ¼ oiXjk; ð22Þ

Rttııtt|| ¼ d

dtDij ðD2Þij þ ðDXþXDÞij; ð23Þ

where

d

dt¼ ot þ v r ð24Þ

is the usual convective derivative. The tetrad components Rabcd tell us how a La-

grangian observer moving with the fluid perceives the curvature of the effective

space-time described by the Painlevee–Gullstrand metric (8).

The components in the tetrad and co-ordinate frames are related by

Rabcd ¼ eaaebbecceddRabcd : ð25Þ

In the tetrad frame, the Ricci tensor

Rab ¼ Rcacb ¼ Rttattb þ Rkkakkb ð26Þhas the components

Rtttt ¼ Rkkttkktt ¼ Rttkkttkk ¼ d

dtTrD Tr ðD2Þ; ð27Þ

Rttıı ¼ Rttkkkkıı ¼ okXki ¼1

2Dvi

1

2oiðTrDÞ ¼ 1

2ðr xÞi; ð28Þ

Rıı|| ¼ Rttııtt|| þ Rkkııkk|| ¼d

dtDij ðDXþXDÞij þ DijTrD; ð29Þ

where we remind the reader that we have defined the vorticity vector

xi ¼ xi ¼ ijkXjk ¼ ðrotvÞi ¼ ðr vÞi: ð30Þ

The curvature scalar thus becomes

R ¼ Rabgab ¼ Rtttt þ Rkkkk ¼ 2

d

dtTrDþ ðTrDÞ2 þ Tr ðD2Þ ð31Þ

and contains the trace of the deformation tensor and the trace of its square, but not

the vorticity. Finally, the Einstein tensor takes the form

U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39 27

59

Gtttt ¼ Rtttt þ1

2R ¼ 1

2ðTrDÞ2 1

2TrðD2Þ; ð32Þ

Gttıı ¼ Rttıı ¼ 12ðr xÞi; ð33Þ

Gıı|| ¼ Rıı|| 1

2dıı||R ¼ d

dtDij

dijTrD

þ TrD Dij

1

2dijTrD

1

2dijTrðD2Þ ðDXþXDÞij: ð34Þ

We emphasise that although the Ricci and Einstein tensors are non-trivial, and cer-

tainly objects of physical interest, there is at this level no need for or justification for

imposing Einstein equations—though these Ricci and Einstein tensors are properties

of the flow, they are not directly related to the stress-energy tensor generating that flow

and thus the effective space-time curvature experienced by the quasiparticles. In su-

perfluids, for example, the ‘‘Einstein action’’ proportional to the curvature scalar (31)

is smaller than the simple kinetic energy of the superflow by the factor a2=l2, where a isthe atomic scale and l the scale on which the velocity field varies [7], so that the

‘‘Einstein action’’ is subdominant in determining the velocity field.

It is sometimes convenient to work with the conformally invariant, traceless part

of curvature. This is given by the Weyl tensor [23]

Cabcd ¼ Rabcd þ ga½dRcb þ gb½cRda þ1

3Rga½cgdb; ð35Þ

where the brackets indicate anti-symmetrization on the indices they enclose. This

gives

Cıı||kkll ¼ Rıı||kkll þ dıı½llRkk|| þ d||½kkRllıı þ1

3Rdıı½kkdll||; ð36Þ

Cttıı||kk ¼ oiXjk 1

2di½jðr xÞk; ð37Þ

Cttııtt|| ¼ 1

2

d

dtDij

1

3dijTrðDÞ

ðD2Þij þ1

3dijTrðD2Þ

þ 1

2TrðDÞ Dij

1

3dijðTrDÞ

þ 1

2ðDXþXDÞij: ð38Þ

3. Examples

3.1. General iso-pycnal flows

Suppose now that the flow is not only iso-tachic (constant speed of sound) but

also iso-pycnal (constant background density). This corresponds to an ‘‘almost in-

compressible’’ fluid such as water. The major change from the previous section is

the simplification that comes from the continuity equation:

28 U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39

60

dq

dt¼ 0 ) r v ¼ 0 ) TrD ¼ 0: ð39Þ

The form of the Riemann tensor is not affected, though for the Ricci tensor we now

have

Rtttt ¼ TrðD2Þ; ð40Þ

Rttıı ¼1

2Dvi; ð41Þ

Rıı|| ¼d

dtDij ðDXþXDÞij: ð42Þ

The Ricci scalar simplifies to

R ¼ TrðD2Þ: ð43Þ

Thus the Ricci curvature scalar is positive semidefinite for iso-pycnal flows, and

vanishes if and only if the deformation D is zero.

The Einstein tensor is now

Gtttt ¼ 12TrðD2Þ; ð44Þ

Gttıı ¼1

2Dvi; ð45Þ

Gıı|| ¼d

dtDij

1

2dijTrðD2Þ DXð þXDÞij: ð46Þ

Finally, the Weyl tensor for iso-pycnal flows reduces to

Cıı||kkll ¼ Rıı||kkll þ dıı½llRkk|| þ d||½kkRllıı þ1

3Rdıı½kkdll||; ð47Þ

Cttıı||kk ¼ oiXjk þ di½jDvk; ð48Þ

Cttııtt|| ¼ 12

d

dtDij þ

1

2ðDXþXDÞij ðD2Þij þ

1

3dijTrðD2Þ: ð49Þ

3.2. Shear flow

As a first simple example of a non-trivial incompressible flow (TrD ¼ 0), consider

the flow with constant shear

v ¼ x0ð0; x; 0Þ ð50Þwhich has both constant deformation Dxy ¼ Dyx ¼ ð1=2Þx0 and constant vorticity

xz ¼ x0 ¼ 2Xxy ¼ 2Xyx (all other components vanishing) [24]. The Riemann cur-

vature components are

U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39 29

61

Rttııtt|| ¼ 14x20Pij;

Rttıı||kk ¼ 0;

Rıı||kkll ¼1

4x20ðhikhjl hilhjkÞ;

ð51Þ

where hik ¼ hki is unity if ðikÞ ¼ ðxyÞ and zero otherwise. The projection operatorPij dij ninj ð52Þ

where n ¼ ð0; 0; 1Þ is a unit vector in z-direction ensures that the curvature has non-zero components only in the x- and y-directions.

For the Ricci and Einstein tensors

Rttıı ¼ Rıı|| ¼ 0;

Rtttt ¼ 12x20 ¼ TrðD2Þ;

R ¼ 1

2x20;

Gtttt ¼ 14x20;

Gıı|| ¼ 14x20dij;

Gttıı ¼ 0:

ð53Þ

Thus the quasiparticles are seen in their effective space-time to be moving on a

ð3þ 1Þ-dimensional manifold of constant scalar curvature, with radius of curvatureinversely proportional to the shearing rate x0.

3.3. Vortex flow of constant circulation

A somewhat more interesting case is the constant-circulation flow in the x–y plane

vy ¼cx

x2 þ y2; vx ¼ cy

x2 þ y2ð54Þ

appropriate to a vortex flow well outside the central core, where the circulation isH

v ds ¼ 2pc. In this case you would not want to trust the geometry for r < rc ¼ c

because at r ¼ rc the flow goes supersonic. This flow has

Dxx ¼2cxy

r4¼ Dyy ;

Dxy ¼cðy2 x2Þ

r4¼ Dyx;

Diz ¼ Dzi ¼ 0;

Xij ¼ 0:

ð55Þ

Note the ‘‘duality’’ between the vortex core and the far field. In the core the de-

formation rate is zero and the vorticity is non-zero, while in the far field it is the

30 U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39

62

vorticity that is zero and deformation that is non-zero. The Riemann curvature

tensor takes the form:

Rxxyyxxyy ¼ det D ¼ c2

r4;

Rttıı||kk ¼ 0;

Rttııtt|| ¼ ðv rÞDij ðD2Þij ¼ ðv rÞDij c2

r4Pij:

ð56Þ

More explicitly

Rttxxttxx ¼c2

r6ðy2 3x2Þ;

Rttyyttyy ¼c2

r6ðx2 3y2Þ;

Rttxxttyy ¼ 4c2xy

r6;

Rttııtt|| ¼ c2

r64xixj

dijr2

:

ð57Þ

Therefore the Ricci tensor, curvature scalar, and Einstein tensor read

Rtttt ¼ 2c2

r4; Rttıı ¼ Rıı|| ¼ 0; R ¼ 2c2

r4; ð58Þ

Gtttt ¼ c2

r4; Gıı|| ¼ dij

c2

r4; Gttıı ¼ 0: ð59Þ

It is mildly amusing to note that the vortex geometry is uniquely determined by the

cylindrical symmetry plus the equation Gab / dab (not gab).

3.4. Streaming motion past a cylinder

The most complex flow we discuss here is provided by the 2-dimensional stream-

ing motion from right to left past a cylinder of radius a. According to the circle the-

orem [3], the complex velocity potential of such a flow is given by

w ¼ U Z

þ a2

Z

; ð60Þ

where Z ¼ xþ iy and U is the velocity at infinity in negative x-direction. This results

in the flow

vx ¼ U 1

þ a2y2 x2

r4

; vy ¼ 2Uxya2

r4: ð61Þ

The velocity at infinity is restricted to be U < 1=2, for the maximal velocity on thecylinder surface to be less than the speed of sound. The formulae for deformation

and vorticity (which is identically zero for this flow) read

U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39 31

63

Dxx ¼2Ua2

r6xð3y2 x2Þ ¼ Dyy ;

Dxy ¼2Ua2

r6yðy2 3x2Þ ¼ Dyx;

Diz ¼ Dzi ¼ 0;

Xij ¼ 0:

ð62Þ

The Riemann components show that the flow past a cylinder, due to its reduced

symmetry, yields a more complicated space-time geometry for quasiparticles than the

vortex flow:

Rxxyyxxyy ¼ det D ¼ 12TrðD2Þ ¼ 4U

2a4

r6;

Rttıı||kk ¼ 0;

Rttııtt|| ¼ ðv rÞDij ðD2Þij ¼ ðv rÞDij þPij detD;

ð63Þ

where the last line reads more explicitly

Rttxxttxx ¼2U 2a2

r8a2ðy2

5x2Þ þ 3ðx4 6x2y2 þ y4Þ

;

Rttyyttyy ¼2U 2a2

r8a2ðx2

5y2Þ 3ðx4 6x2y2 þ y4Þ

;

Rttxxttyy ¼ 12U2a2

r8xyða2 2x2 þ 2y2Þ:

ð64Þ

These latter components show that the ‘‘circulation’’ Ua2 is not the only relevant

parameter of the flow, in contrast to the constant-circulation vortex case, as we may

expect from the reduced symmetry of the flow past the cylinder.

The curvature scalar

R ¼ 8U 2a4

r6ð65Þ

decays much more quickly with distance from the cylindrical object than the cur-

vature of the vortex flow, Eq. (58).

3.5. Rigid rotation

The simplest example of a non-trivial incompressible flow (TrD ¼ 0) is pure rota-

tion v ¼ Xðy; x; 0Þ, which has zero deformation Dij ¼ 0, and constant vorticity xz ¼x0 ¼ 2Xxy ¼ 2Xyx ¼ 2X (all other components vanishing). This flow is appropriate

for instance deep inside the core of a vortex where the fluid effectively rotates as a

‘‘rigid’’ body. (In ordinary fluids this happens because viscosity dominates in the

core; in superfluids there is a more dramatic effect in that the superfluid goes normal

close enough to the core.) Also note that the core has a maximum size given by

jvj ¼ 1, that is, rc ¼ 2=x0.

32 U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39

64

For the rigid rotation flow it is easy to see that the Riemann curvature tensor is

identically zero, either (1) by brute force application of the above formulae, or more

subtly (2) by going to a rotating frame (of angular velocity X ¼ 2x0) in which the

velocity is identically zero, evaluating the Riemann tensor there (where it is blatantly

zero), and transforming back to the rotating frame. Although the Riemann tensor is

identically zero, there is interesting physics going on: The fact that pure rotation

leads to zero Riemann curvature is ultimately responsible for the fact that Eqs.

(12) and (21) do not contain any terms quadratic in X, a result that otherwise has

to be simply asserted based on explicit calculation.

Additionally, we emphasise that even though the Riemann tensor is zero, the

Christoffel symbols are definitely not zero. Indeed

Citt ¼ X2rrri; ð66Þ

Citj ¼ Xij ¼1

2ijkx

k: ð67Þ

These two portions of the Christoffel symbols are of course simply representing the

centrifugal and Coriolis pseudo-forces. All other components are zero.

A further (approximate) example of such a flow is encountered if one considers

the coarse-grained flow induced by a lattice of vortices [25]. An (infinite) lattice ro-

tates as if it were a solid body, with a vortex density nv ¼ X=pc prescribed by the ro-tation velocity X and the circulation 2pc, assumed to be equal for each individual

vortex. For the vortex lattice, it follows from the vanishing of the Riemann curvature

that a collimated quasiparticle beam can pass a (sufficiently dilute) lattice without

(on average) being deflected.

4. Geodesic deviation

An invariant measure of the strength of a flow pattern as regards its influence on

quasiparticle motion may be defined to be the value of the curvature scalar R / sj

at a certain given distance s from the flow-generating object (cf. Fig. 1, illustrating

the generic situation of flow past an object placed in a homogeneous stream). Among

the flows discussed in the previous section the shear flow is strongest in that sense

(because the ‘‘flow generating object’’ is covering all space, j ¼ 0), followed by the

vortex flow (j ¼ 4) and the flow past the cylinder (j ¼ 6). Finally rigid rotation,

which has zero R and is ‘‘flat’’ (j ¼ 1). It is the simplest conceivable non-trivial

(i.e., inhomogeneous) flow with the property of having all Rabcd equal to zero.

A non-vanishing Riemann tensor leads to tidal (relative) acceleration of nearby

geodesics, described by the Jacobi equation of geodesic deviation for quasiparticles

D2na

dk2þ Ra

bcdubncud ¼ 0: ð68Þ

The above relation gives the covariant relative acceleration of two nearby geodesics,

with null tangent vectors u separated by the displacement vector n, and with the

U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39 33

65

geodesics affinely parametrized by k. (At this stage all we need to know is that use of

an ‘‘affine parameter’’ simplifies many formulae; in the following section we will

derive a relationship between the affine parameter and physical Newtonian time t.)

The fact that the constant time slices of the metric (5) are conformally identical to

flat Cartesian space in three dimensions, entails that the space-time curvature of the

quasiparticle world is reflected in a relative acceleration of quasiparticle rays in the

Newtonian lab world of non-relativistic hydrodynamic flow.

Consider a family of geodesics in the x-direction, with tangent vector u ¼ ðutt;utt; 0; 0Þ and a purely space-like separation in the y-direction n ¼ ð0; 0; dy; 0Þ. We thenhave

D2½dydk2

þ Ryyttyytt

n

þ Ryyxxyyxx

ðuttÞ2o

½dy ¼ 0: ð69Þ

This can be viewed as a parametrically driven harmonic oscillator (driven in the

affine parameter k), with ‘‘frequency’’

XðkÞ ¼ uttffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ryyttyytt þ Ryyxxyyxxp

: ð70ÞPhysically this means that by looking at the components of the Riemann tensor we

can see if the effective geometry locally acts as a focussing lens [corresponding to

XðkÞ real] or as a diverging lens [corresponding to XðkÞ imaginary]. Since (in thefocussing case, and assuming a reasonably uniform medium) two initially parallel

geodesics will focus down to a point after an elapse of affine parameter dk ¼ p=XðkÞ,the corresponding local focal length is (in physical distance units) given by

f local ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kRttyyttyy þ Rxxyyxxyykp : ð71Þ

Note the strengths and weaknesses of this concept—it provides a local position and

orientation dependent notion of focal length appropriate for nearly parallel geode-

sics (nearly parallel quasiparticles; so one is automatically working ‘‘on axis’’ and

ignoring ‘‘spherical abberation’’), but this definition of f local does in general not

provide significant global information. If the Riemann tensor is strongly inhomo-

geneous, varying on length scales significantly smaller than f local, then this concept of

Fig. 1. The quasiparticle geodesic deviation at a distance vector s caused by an object placed in a flow with

velocity v1 at infinity (the generic case of the situation in Section 3.4).

34 U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39

66

local focal length is not particularly useful. In particular, in the vortex geometry of

[26], with flow (54), the focussing effect we had in mind was a global effect due to

quasiparticles passing by opposite sides of the vortex core, with impact parameter

b—this is not a situation that can be described by the Jacobi equation. The global

result obtained there for f ¼ f global ¼ ð2b3=3pr2cÞ½1þOðrc=bÞ is not the local f localdefined above. Indeed two initially parallel quasiparticles passing by on the same side

of the vortex core will be driven apart from each other by geodesic deviation—it is

this effect that leads to the ‘‘cylindrical abberation’’ of the lens discussed in [26].

A case where the local focal length does acquire global meaning is the shear flow

(50), for which the focal length (71) becomes a constant

f shear ¼ffiffiffi

2p

p

x0

: ð72Þ

The focal length is in this case bounded by the atomic length scale itself, simply due

to the requirement that the concept of hydrodynamics makes sense. This further

strengthens the notion of the shear flow being the strongest possible flow as regards

its influence on quasiparticle motion, because any other flow has more stringent

bounds on the global f .

One useful refinement of the local focal length concept introduced in Eq. (71) is to

consider null geodesics (quasiparticle paths) propagating in an arbitrary unit direc-

tion uu and then use indices M and N to denote the two spatial directions perpendic-

ular to uu. Then the local focal length can be generalized to a 2 2 matrix

f localMN ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kRttMttN þ RııM ||N uuııuu||kp : ð73Þ

The square root and inverse is to be taken in the matrix sense, and the two eigen-

values of fMN are the two principal focal lengths along the direction uu. If these ei-

genvalues differ it is a signal of astigmatism.

5. Non-affine parameterization of null geodesics

While the use of affine parameters for null geodesics is standard in general relativ-

ity, it should be borne in mind that in the present Painlevee–Gullstrand context there

is a preferred temporal foliation provided by the Newtonian time parameter t. It is

worth the technical bother of using the non-affine parameterization in terms of t here

in order to make aspects of the physics clearer.

In general, we know that along any null geodesic there will be some relationship

between affine parameter k and Newtonian time t. For instance we can assert

dk ¼ exp½fðtÞ dt: ð74Þ

In the affine parameterization the geodesic equation for a null curve is just

ulrlum ¼ 0; um dxm

dk:

U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39 35

67

If we choose a non-affine parameterization

uulrluum ¼ _ffðtÞuul; uum dxm

dt:

The geodesic equation becomes

d2xl

dt2¼ C

lab

dxa

dt

dxb

dtþ _ffðtÞ dx

l

dt: ð75Þ

In this form it is clear that the physical acceleration of the quasiparticle is related to

gradients in the Painlevee–Gullstrand metric. It is extremely useful to derive an ex-

plicit relationship between the affine parameter k and the physical Newtonian time t.

To do this let us start with the notion of a stationary geometry (technically: there

exists a time-like Killing vector; colloquially: a time-independent geometry). The

time-like Killing vector takes the form

Kl ¼ ð1; 0Þ; Kl ¼ ð½1 v2;vÞ: ð76Þ

The tangent vector to the null geodesic is denoted

ul ¼ dxl

dk¼ dt

dk1;dx

dt

: ð77Þ

It is a standard theorem that the 3þ 1 inner product between a geodesic tangent

vector and a Killing vector is conserved, as long as the geodesic is affinely param-

eterized. Thus

glmKlum ¼ dt

dk1

v2 þ v dxdt

¼ constant: ð78Þ

On the other hand, because ul is a null vector

1 v2 þ 2v dxdt

dx

dt

2

¼ 0: ð79Þ

Eliminating between these two equations, we can normalize in such a way that

dt

dk¼ exp½fðtÞ ¼ 1

"

v2 þ dx

dt

2#1

: ð80Þ

That is

fðtÞ ¼ ln 1

"

v2 þ dx

dt

2#

: ð81Þ

If the fluid is not moving, then v ¼ 0 and jdx=dtj ¼ 1 so t / k. If the fluid is moving

we simply have to live with this position-dependent factor relating the affine pa-

rameter k (in terms of which the geodesic equations are most easily written down) to

36 U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39

68

the Newtonian time parameter t (in terms of which the physical acceleration is most

easily calculated).

In a similar manner, the Jacobi equation can be rewritten as

D2na

dt2 _ffðtÞDn

a

dtþ Ra

bcduubncuud ¼ 0: ð82Þ

While this looks somewhat messier than the affinely parameterized Jacobi equation

(68), the physics is the same. In particular if we start with two initially parallel null

geodesics (Dn=dt ¼ 0 at t ¼ 0), and assume a locally homogeneous medium, we are

led to the same notion of local focal length as discussed in the previous section.

6. Discussion

We have shown how the generation of curved Riemannian space-time geometries

for quasiparticles is possible based purely on the velocity pattern of a non-relativistic

flow. Conversely, one might conceive of solving for a flow field from a given space-

time geometry. This is a highly nonlinear problem, as becomes obvious from the re-

lations (21)–(23). It is, however, certainly no more nonlinear or complicated than

solving the Einstein equations of general relativity themselves. While the Painlevee–

Gullstrand geometry discussed here does not provide us with the most generic case

(remember that the constant time surfaces are (conformally) flat; for generalizations

allowing for more general space-time metrics see [10]), it shows that the underlying

kinematical structure of a curved space-time can in principle be perfectly non-relativ-

istic. The dynamical identification of this effective geometry with general relativity,

i.e., imposing the Einstein equations, is a more advanced step [7], but is possible

in principle as well.

There are several generalizations of the current analysis that would be of interest:

(1) If the quasiparticle propagation speed (c, local speed with respect to the back-

ground medium) is varying then the geometry exhibits ‘‘index gradient’’ effects in ad-

dition to effects generated by the motion of the medium. While technically

straightforward, the relevant calculations of the Riemann tensor are computation-

ally messy and the physical interpretation is not so clear (unless the medium is com-

pletely at rest; in which case one recovers standard ‘‘index gradient’’ physics). (2) If

the density varies from place to place, then it is necessary to distinguish the ‘‘geomet-

rical quasiparticle’’ regime (the analogue of geometrical optics) from the ‘‘wave qua-

siparticle regime’’ (the analogue of wave optics). In the geometrical approximation

the results of the present paper can be carried over; in the wave regime one needs

to carry out an analysis in terms of Green functions and wave equations; the entire

armoury of quasiparticle trajectories as null geodesics of the effective metric breaks

down and must be replaced by a more fundamental wave description.

In summary: The use of pseudo–Riemannian geometry has important applica-

tions well beyond the confines of general relativity. In particular quasiparticle prop-

agation in condensed matter systems can often be characterized in terms of an

‘‘effective’’ space-time geometry; most easily described in Painlevee–Gullstrand form.

U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39 37

69

If the background medium is a fluid, then the Riemann curvature (and Christoffel

symbols, etc.) can be calculated in terms of shear (deformation) and vorticity of

the fluid. Ultimately this analysis relates the focussing and deflection of quasiparti-

cles to the properties of the fluid flow.

Acknowledgments

URF acknowledges support by the Deutsche Forschungsgemeinschaft (FI 690/2-1)

and the ESF Programme ‘‘Cosmology in the Laboratory.’’ MV was supported by the

US Department of Energy.

References

[1] E. Madelung, Z. Phys. 40 (1927) 322.

[2] Sir Horace Lamb, Hydrodynamics, Republication of the sixth edition 1932, Dover, New York, 1945.

[3] L.M. Milne-Thomson, Theoretical Hydrodynamics, fifth ed., Macmillan, New York, 1968.

[4] W.G. Unruh, Phys. Rev. Lett. 46 (1981) 1351.

[5] P. Painlevee, C.R. Hebd. Acad. Sci. (Paris) 173 (1921) 677–680;

A. Gullstrand, Arkiv. Mat. Astron. Fys. 16 (1922) 1–15;

A generalization of the Painlevee–Gullstrand form of the Schwarzschild metric to the Kerr case may be

found in C. Doran, Phys. Rev. D 61 (2001) 067503;

A pedagogical discussion, in particular a comparison to Eddington–Finkelstein and Kruskal–Szekeres

co-ordinates is contained in K. Martel, E. Poisson, Am. J. Phys. 69 (2001) 476–480.

[6] M. Visser, Class. Quantum Grav. 15 (1998) 1767–1791;

M. Visser, Phys. Rev. Lett. 80 (1998) 3436.

[7] G.E. Volovik, Phys. Rep. 351 (2001) 195–348.

[8] G.E. Volovik, JETP Lett. 69 (1999) 705–713 [Pisma Zh. EEksp. Teor. Fiz. 69 (1999) 662–668].

[9] L.J. Garay, J.R. Anglin, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 85 (2000) 4643–4647;

L.J. Garay, J.R. Anglin, J.I. Cirac, P. Zoller, Phys. Rev. A 63 (2001) 023611.

[10] C. Barceloo, S. Liberati, M. Visser, Class. Quantum Grav. 18 (2001) 1137–1156.

[11] W. Gordon, Ann. Phys. (Leipzig) 72 (1923) 421–456.

[12] U. Leonhardt, Phys. Rev. A 62 (2000) 012111.

[13] R. Sch€uutzhold, G. Plunien, G. Soff, Phys. Rev. Lett. 88 (2002) 061101.

[14] U.R. Fischer, G.E. Volovik, Int. J. Mod. Phys. D 10 (2001) 57–88 (gr-qc/0003017).

[15] Commonly the two notions of ‘‘quasiparticle’’ (pole in the Green function) and ‘‘elementary

excitation’’ (giving energy vs. momentum curves) are used interchangeably in the literature, so also

here. For a careful discussion of their differences see E.H. Lieb, Phys. Rev. 130 (1963) 1616–1624.

[16] M.K. Parikh, F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042.

[17] R. Sch€uutzhold, Phys. Rev. D 64 (2001) 024029.

[18] Georg Friedrich Bernhard Riemann: €UUber die Hypothesen, welche der Geometrie zu Grunde liegen,

Springer, Berlin 1923, 3rd ed. (Inaugural lecture, G€oottingen, June 10th, 1854, with a commentary by

H. Weyl).

[19] C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, New York, 1973.

[20] S.E. Perez Bergliaffa, K. Hibberd, M. Stone, M. Visser, Wave Equation for Sound in Fluids with

Vorticity, cond-mat/0106255.

[21] A.D. Pierce, J. Acoust. Soc. Am. 87 (1990) 2292–2299.

[22] M. Stone, Phys. Rev. E 62 (2000) 1341–1350.

[23] S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press,

Cambridge, 1973.

38 U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39

70

[24] A more general shear flow and its influence on quasiparticle geodesics in dielectric media has been

considered by P. Ben-Abdallah, J. Quant. Spectrosc. Rad. Trans. 73 (2002) 1–11.

[25] V.K. Tkachenko, Zh. EEksp. Teor. Fiz. 49 (1965) 1875 [Sov. Phys. JETP 22 (1966) 1282].

[26] U.R. Fischer, M. Visser, Phys. Rev. Lett. 88 (2002) 110201.

U.R. Fischer, M. Visser / Annals of Physics 304 (2003) 22–39 39

71

Appendix B

Uwe R. Fischer and Matt Visser

“Riemannian Geometry of Irrotational Vortex Acoustics”

Phys. Rev. Lett. 88, 110201 (2002)

72

VOLUME 88, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 18 MARCH 2002

Riemannian Geometry of Irrotational Vortex Acoustics

Uwe R. Fischer*Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080

Matt Visser†

Physics Department, Washington University, Saint Louis, Missouri 63130-4899(Received 10 October 2001; revised manuscript received 4 February 2002; published 4 March 2002)

We consider acoustic propagation in an irrotational vortex, using the technical machinery of differen-tial geometry to investigate the “acoustic geometry” that is probed by the sound waves. The acousticspace-time curvature of a constant circulation hydrodynamical vortex leads to deflection of phonons atappreciable distances from the vortex core. The scattering angle for phonon rays is shown to be quadraticin the small quantity G2pcb, where G is the vortex circulation, c the speed of sound, and b the impactparameter.

DOI: 10.1103/PhysRevLett.88.110201 PACS numbers: 02.40.Ky, 43.20.+g

Introduction.—The last few years have seen consider-able interest in the development of analog models of andfor general relativity [1–3]. These analog models providea two-way street: sometimes they illuminate aspects ofgeneral relativity and sometimes the machinery of differ-ential geometry can be used to illuminate aspects of theanalog model. In this article, we use mathematical meth-ods developed in the framework of differential geometryto study acoustic propagation in an irrotational vortex. Weanalyze the “acoustic” space-time geometry created by thevortex in terms of its metric tensor, Killing vectors, andgeodesics. The medium is assumed to be “almost incom-pressible,” by which we mean that we take the backgrounddensity and the speed of sound relative to the medium tobe constant, and focus on the effects due to motion of themedium. If the flow and its perturbations are irrotationalthere is a rigorous theorem to the effect that sound can bedescribed by a curved-space scalar wave equation usingan effective acoustic geometry, this theorem being derivedby using the Euler and continuity equations of classical hy-drodynamics [1,2]. (Even if distributed vorticity is present,which is not the case considered here, then in the eikonalapproximation there are rigorous theorems to the effectthat sound ray propagation can be described by curved-space geodesics in the same “effective acoustic geome-try” [4].)

Using the acoustic space-time approach, we shall showthat the quasiclassical scattering process of phonons bythe vortex leads to a scattering angle quadratic in the smallquantity G2pcb, where G is the vortex circulation, c thespeed of sound, and b the impact parameter of the phonontrajectory relative to the vortex. Because of the fact that thelowest order scattering angle is quadratic in G2pcb andnot linear, there is (up to third order) no net modificationof the transverse force exerted by a phonon beam directedtowards the vortex. The effect of the acoustic space-timecurvature is that the vortex acts as a converging lens onquasiclassically moving phonons.

Riemannian geometry of vortex acoustics.—For an ir-rotational vortex, the velocity profile is given in terms ofthe circulation G

Hy ? d x 2pg by

y G

2pru

g

ru . (1)

If we adopt cylindrical coordinates r, u, and z, so that thespatial metric is

gij

2

4

1 0 00 r2 0

0 0 1

3

5 , (2)

the covariant and contravariant components of the velocityare yu g; yu gr2. Now go to a four-dimensionaldescription in terms of the coordinates t, r, u, and z. Then,in terms of the speed of sound c, the 3 1 1-dimensionalacoustic geometry of the vortex is described by the space-time metric [1,2]

gvortexmn

2

6664

2c2 1 gr2 0 2g 0

0 1 0 0

2g 0 r2 0

0 0 0 1

3

7775

. (3)

The scalar velocity potential of phonons lives in the curvedspace-time world given by gvortex

mn . This form of the metricis often referred to as the Painlevé-Gullstrand form [2,5].It is important to note that this vortex metric is not iden-tical to the metric of a massless spinning cosmic string, asolution of the Einstein equations with cylindrical symme-try. That metric corresponds to

gstringmn

2

6664

2c2 0 2g 0

0 1 0 02g 0 r2 2 g2c2 0

0 0 0 1

3

7775

. (4)

The two metrics (3) and (4), vortex and cosmic string,agree asymptotically at large r,

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VOLUME 88, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 18 MARCH 2002

gvortexmn gstring

mn 1 1 Or2c r2 . (5)

Here we have defined the “core radius”

rc jgjc

. (6)

This denotes the radius at which the flow goes supersonic.The assumption of incompressibility as well as the pro-file (1) certainly break down at this radius, but the actualdimensions of the vortex core (for a superfluid vortex be-ing set by the coherence length j0) are often larger thanrc. The two metrics are markedly different at intermedi-ate values of r. In particular, we shall see below that theydiffer significantly well outside the vortex core radius rc.The metric of the spinning cosmic string is everywhereflat (the space-time curvature is identically zero). In con-trast, even at intermediate distances, the acoustic geometryof the vortex is not flat (the space-time Riemann tensor isnot zero), and there are significant effects on the propa-gation of null geodesics (sound rays) at values of r welloutside the vortex core. That there is a region of vanish-ing classical deflection outside a vortex or a magnetic fluxtube, as conventionally assumed in the standard formula-tion of the Aharonov-Bohm problem (see, e.g., [6,7]), isthus only asymptotically true for a hydrodynamical vortex,even if the generalized vorticity (hydrodynamical vorticityand/or magnetic flux) vanishes everywhere outside the vor-tex core. It is hence not a priori clear that computationsof the Iordanskiı force based on the spinning string met-ric (the analog-gravitational Aharonov-Bohm effect [8,9]),give the full force exerted by a phonon beam on an actualhydrodynamical vortex.

It is straightforward to compute the Ricci curvaturescalar corresponding to the metric (3),

Rvortex

2g2

c2r4

2r2c

r4. (7)

The curvature of the space-time experienced by the quasi-particles is thus significant at distances which can be welloutside the core domain. That the flow be considered welloutside the core domain is required by the relation (6),which states that at a distance rc from the rotation axis thevelocity around the core equals the speed of sound, so thatcompressibility is no longer negligible.

A simple criterion for the curvature of the effectivespace-time to be significant for the motion of the phononaround the vortex at a given distance r is how the phononwave vector magnitude k compares with the inverse space-time curvature radius at that distance. The wave vectormagnitude is then to be compared with

kcr 2pp

R 2p

p2

rc

1

rrc2. (8)

Well outside the core, kcrc ø 1. The phonon “sees” thecurvature of the space-time if k exceeds kcr, and behaves

as a particle moving on a geodesic in that space-time. Viceversa, the topological structure of the acoustic space-timerelated to the Iordanskiı force (the Aharonov-Bohm effectin the space-time of the spinning string) dominates thebehavior of the phonons if k ø kcr.

For a specific physical example of an irrotational vortexgeometry, in the dilute limit of a Bose-condensed atomicvapor (BEC) [10], we can relate the circulation G, thecoherence length j0, and the speed of sound by usingthe relations j2

0 h22mgn and c p

gnm. Theycombine to give the relation

2prcc 2pp

2 cj0 jGj 2pjgj BEC (9)

for a singly quantized vortex with G hm, where g isthe strength of interaction related to the s-wave scatter-ing length a by g 4p h2am. The “acoustic” core sizeis thus in this example of the same order as the actual(quantum-mechanical) core size of varying density. Thehydrodynamic circulation, calculated with the speed ofsound along a core circumference with radius rc

p2 j0,

equals the quantum of circulation in the dilute gas limit.This relation qualitatively also holds in the dense, stronglycorrelated superfluid helium II (superfluid 4He), where j0

is of order the atomic size. The relation (9) yields for thecurvature scalar in (7)

Rvortex

4j20

r4. BEC (10)

Curvature of space-time implies that particle worldlineson geodesics in that space-time deviate from being ini-tially parallel after some proper distance of travel alongthe geodesic. From the general space-time interval of thePainlevé-Gullstrand metric in the form [2]

ds2 2dt2 1 dijdxi 2 yidt dxj 2 yjdt , (11)

it is apparent that a constant time slice of the effectivespace-time of quasiparticles is just ordinary Euclideanspace. Hence spatial distances measured on constant timeslices in the effective space-time are identical to distancesin the Newtonian lab world, and a real force is actingupon the phonon. It is, however, to be stressed that theactual motion of the phonon in the lab world is describedcorrectly by the motion in the full effective space-time,that is, the phonon is not just a (Lagrangian) particledragged along by the flow, if that flow is inhomogeneous.

Phonon motion in acoustic geometry.—To explicitlysee how the vortex flow affects acoustic propagation, andthereby get a handle on how acoustic influences can affectthe vortex, we use the eikonal approximation and considerphonons instead of sound waves. Phonons then follow nullgeodesics in the acoustic geometry [2]. Associated withspace-time symmetries are so-called Killing vectors [11],along which the metric is invariant. For both geometries(vortex and spinning string) there are three such Killingvectors, corresponding to translations in the t, u, and z

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directions:

K1m 1, 0, 0, 0; K2m

0, 0, 1, 0 ;

and K3m 0, 0, 0, 1 . (12)

This leads to three conserved quantities along eachgeodesic [11]

KAmgmndxn

dl 2kA . (13)

Here l is some arbitrary affine parameter for the geodesic[11]. We are interested in null geodesics which representthe paths of sound rays in the acoustic geometry. Fromthese three conservation laws we see that for the vortexgeometry

∑

2c2 1

µg

r

∂2∏dt

dl2 g

du

dl 2k1 ; (14)

2gdt

dl1 r2 du

dl 2k2 ; (15)

and

dz

dl 2k3 . (16)

By elimination between the first two equations

dt

dl

µ

k1 1k2g

r2

∂

c22. (17)

We are furthermore particularly interested in sound raysthat come in from infinity, so without loss of generalityit is possible to rescale l to choose k1 c2, and to thendefine k2 c2k2 and k3 2k3. Using the new affineparameter, a brief computation yields

dz

dt

k3

1 1 k2gr2(18)

and

du

dt

g

r22

k2c2r2

1 1 k2gr2. (19)

To now calculate drdt we use the fact that the sound raysare null curves so that

gmndxm

dt

dxn

dt 0 . (20)

Therefore

2c2 1

µg

r

∂2

2 2gdu

dt

1

µdr

dt

∂2

1 r2

µdu

dt

∂2

1

µdz

dt

∂2

0 , (21)

leading to, by substituting (18) and (19),

dr

dt

s

c2 2

µk2c2r

1 1 k2gr2

∂2

2

µk3

1 1 k2gr2

∂2

.

(22)

The three equations (18), (19), and (22) completely specifythe path of the sound ray in terms of the time parameter t

and the two nontrivial constants of motion k2 and k3. Thesehave the physical interpretation that k3 yz

` bc , c,while in terms of the impact parameter b

k2 g 2 cb

p

1 2 b2

c2. (23)

The radial motion is more usefully recast as an “energyequation”

1

2

µ

dr

dt

∂2

1 V r 1

2c2, (24)

with the “potential”

V r 1

2c2 k2cr2 1 b2

1 1 k2gr22. (25)

The form of this potential is sketched in Fig. 1 (assum-ing k2g . 0 and setting b 0). Note that at large dis-tances it has the standard centrifugal barrier proportionalto 1r2, while at short distances (however, already insidethe “acoustic” core) it falls quadratically to zero.

The general deflection angle of the ray as a function ofk2 and k3 is obtained by integrating

Du 2Z `

rturn

du

drdr , (26)

where the turning point rturn is determined by solvingdrdu 0. For simplicity, we set b k3 0 in the fol-lowing. To lowest order in the small parameter jgjcb

rcb

rturn jbj∑

1 2

µrc

b

∂2

1 O

µr3

c

b3

∂∏

. (27)

For b ! `, at fixed g and c (fixed core radius rc), thedeflection then evaluates from integrating

du

dr

b1 2 r2c r2

q

r2 2 r2turn r2 2 r2

c 1 Or3

cb3 (28)

FIG. 1. The effective potential of phonon motion in the vor-tex acoustic space-time, with b 2rc and k3 y`

z 0, rc

c 1.

110201-3 110201-3

75

VOLUME 88, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 18 MARCH 2002

along the ray, according to Eq. (26). The result is toquadratic order in rcb

Du p sgnb

∑

1 13

4

r2c

b21 Or3

c b3

∏

. (29)

The limit of zero classical force corresponds to the limitof vanishing core size rc c ! ` and/or g ! 0), as maybe inferred from the vanishing of the curvature scalar (7).This is directly seen in the vanishing, to quadratic orderin rc, of the classical scattering of the ray away from astraight line caused by the acoustic space-time curvature.The action of the vortex on the phonons is schematicallydepicted in Fig. 2.

From the deflection angle (29), one can easily obtain thefocal length as a function of impact parameter. Indeed, twoinitially parallel phonon beams that pass by opposite sidesof the vortex with impact parameter b will intersect at adistance 2f beyond the vortex, where

f 2

3p

b3

r2c

∑

1 1 O

µrc

b

∂∏

. (30)

The fact that the focal length depends on impact parametershows that the vortex exhibits what is known in opticsas “spherical abberation,” which in the present context ismore correctly termed “cylindrical abberation.”

In an experiment, one should take into account that thetransverse size of a phonon beam is limited by the wave-length l, which has to be less than or equal to the impactparameter b for the quasiclassical phonon scattering pic-ture to make sense. Thus the minimum focal length isgiven by

fmin 2

3p

l3

r2c

∑

1 1 O

µrc

l

∂∏

. (31)

In superfluid 4He, where rc 0.6 Å, the roton minimumoccurs at lroton 3 Å. The phonon wavelength has tobe about three times the roton wavelength, for an ap-proximately linear phonon dispersion to apply, so thatl * 15rc, which results in fmin 700rc. Accordingly,the focal length is bounded by f * 40 nm.

In dilute Bose-Einstein condensates, due to theBogoliubov-type spectrum of these systems, which doesnot display a roton minimum, the phonon dispersion is toa good approximation linear up to l 2pj0

p2 prc,

hence fmin 20rc. With rc 0.3 mm, this results inf * 6 mm for Bose-Einstein condensates. The latter esti-

Phonon rayb

rc

FIG. 2. Deflection of phonons by the vortex, which acts as aconverging lens.

mate indicates that vortical focusing effects are potentiallywithin the realm of experimental feasibility.

Discussion.—An incoming phonon of finite momen-tum k passing a singular vortex is deflected by a classi-cal force acting upon it. This force is equivalent to anacoustic space-time curvature induced in the vicinity ofthe vortex by that flow. From (29), it follows that thevortex acts as a converging lens. The fact that around ahydrodynamical vortex there is a classical force field atappreciable distances outside the core, entails that calcu-lations based on the assumption that there is no force inthe vorticity free region outside the core are potentiallymisleading. In particular, there are finite distance effectsassociated with a hydrodynamical vortex over and abovethe familiar Aharonov-Bohm effect.

U. R. F. acknowledges support by the Deutsche For-

schungsgemeinschaft (FI 690/2-1) and the ESF Pro-gramme “Cosmology in the Laboratory.” M. V. wassupported by the U.S. Department of Energy.

*Electronic address: ufischer@uiuc.edu†Electronic address: visser@kiwi.wustl.edu

[1] W. G. Unruh, Phys. Rev. Lett. 46, 1351–1353 (1981).[2] M. Visser, Classical Quantum Gravity 15, 1767–1791

(1998); Phys. Rev. Lett. 80, 3436–3439 (1998).[3] G. E. Volovik, Phys. Rep. 351, 195–348 (2001).[4] C. Barceló, S. Liberati, and M. Visser, Classical Quantum

Gravity 18, 1137–1156 (2001).[5] P. Painlevé, C. R. Hebd. Acad. Sci. (Paris) 173, 677– 680

(1921); A. Gullstrand, Ark. Mat. Astron. Fys. 16, 1–15(1922).

[6] A. L. Shelankov, Europhys. Lett. 43, 623– 628 (1998).[7] M. V. Berry, J. Phys. A 32, 5627–5641 (1999).[8] G. E. Volovik, JETP Lett. 67, 881– 887 (1998).[9] M. Stone, Phys. Rev. B 61, 11 780–11786 (2000).

[10] A. J. Leggett, Rev. Mod. Phys. 73, 307– 356 (2001).[11] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation

(Freeman, San Francisco, 1973).

110201-4 110201-4

76

Appendix C

Uwe R. Fischer and Matt Visser

“Warped space-time for phonons moving in a perfect

nonrelativistic fluid”

Europhys. Lett. 62, 1-7 (2003)

77

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84

Appendix D

Petr O. Fedichev and Uwe R. Fischer

“Gibbons-Hawking effect in the sonic de Sitter space-time

of an expanding Bose-Einstein-condensed gas”

Phys. Rev. Lett. 91, 240407 (2003)

85

Gibbons-Hawking Effect in the Sonic de Sitter Space-Timeof an Expanding Bose-Einstein-Condensed Gas

Petr O. Fedichev1,2 and Uwe R. Fischer1

1Leopold-Franzens-Universitat Innsbruck, Institut fur Theoretische Physik, Technikerstrasse 25, A-6020 Innsbruck, Austria2Russian Research Center Kurchatov Institute, Kurchatov Square, 123182 Moscow, Russia

(Received 17 April 2003; published 11 December 2003; corrected 8 January 2004)

We propose an experimental scheme to observe the Gibbons-Hawking effect in the acoustic analog ofa 1 1-dimensional de Sitter universe, produced in an expanding, cigar-shaped Bose-Einsteincondensate. It is shown that a two-level system created at the center of the trap, an atomic quantumdot interacting with phonons, observes a thermal Bose distribution at the de Sitter temperature.

DOI: 10.1103/PhysRevLett.91.240407 PACS numbers: 03.75.Kk, 04.70.Dy

Cosmology in the early universe is a branch of physicswhich is, for all too obvious reasons, removed far fromexperiment. To use a concise formulation of the primarydilemma which cosmology faces, there exists nothing likethe possibility of ‘‘reproducing’’ experiments, becausethere has in fact ‘‘only ever been one experiment, stillrunning, and we are latecomers watching from the back’’[1]. While there can be no truly experimental cosmology,as regards, in particular, the reproduction of the (pre-sumed) extreme conditions which prevailed in the earlystages of the universe, one might look for analogousphenomena in condensed matter experiments, which are

indeed reproducible, and can be done at energy scalessmaller by many orders of magnitude [2].

An archetypical model of cosmology is the de Sitteruniverse, in which space is essentially empty, and thecurvature of space-time is due to a nonvanishing cosmo-logical constant [3]. The de Sitter space-time is used ininflationary models of the cosmos [4], displays a cosmo-logical horizon, and is associated with a thermal spec-trum of particles at a temperature TdS /

p

[5].Atomic Bose-Einstein condensates (BECs) [6] have

emerged as one of the most suitable condensed mattersystems for the simulation of quantum phenomena ineffective space-times [7–12]. In the present study, we in-vestigate an effective 1 1-dimensional [1 1D]de Sitter universe and the associated thermal phononspectrum of the Gibbons-Hawking type [5]. We examinethe propagation of axial low-energy modes in expanding,strongly elongated condensates, and find that the spec-trum consists of one massless (phonon) mode and asequence of massive excitations, moving in an effectivecurved 1 1D space-time. It is demonstrated that a1 1D version of the de Sitter metric [3] can be real-ized by the massless phonon mode in a linearly expandingBose-Einstein condensate with constant particle interac-tion. We show that an atomic quantum dot (AQD) [13],placed at the center of the cloud, can be used to measurethe Gibbons-Hawking quantum process. The detector’scoupling to the superfluid is constructed in such a waythat the natural time interval of the detector is equal to

the time interval in the de Sitter metric. Therefore, thedetector inside the expanding condensate is capable ofmeasuring a thermal state at the de Sitter temperature.The Gibbons-Hawking effect is a curved space-time gen-eralization of the Unruh-Davies effect, in which a con-stantly accelerated detector in vacuum responds as if itwere placed in a thermal bath with temperature propor-tional to its acceleration [14,15]. Therefore, our proposalrepresents a means to confirm the observer dependenceof the particle content of a quantum field in curved space-time [16].

Our approach is based on the by now well establishedidentity of the action of a massless scalar field propagat-ing on a curved space-time background in D 1 dimen-sions and the action of the phase fluctuations in amoving inhomogeneous superfluid [17–19] (we set h m 1, where m is the mass of a superfluid particle):

S Z

dD1x1

2g _ r2 c2r2

1

2

Z

dD1xg

pg@@: (1)

Here, x; t is the superfluid background velocity,cx; t

g0p

is the velocity of sound, 0x; t is thebackground density, and g 4as is the coupling con-stant describing the short-range interaction between theparticles in the superfluid; as is the s-wave scatteringlength. In the second line of (1), the conventionalhydrodynamic action is identified with the action of aminimally coupled scalar field in an effective curvedspace-time. In the following, we derive the effective1 1D action of the form (1) for the lowest axialexcitations of an elongated condensate from the full 3Ddynamics of hydrodynamic excitations of the cigar-shaped condensate, and identify the metric tensor g.

We consider a large condensate confined in a stronglyanisotropic harmonic trap, characterized by the frequen-cies !k and !? in the axial and radial directions, respec-tively (!? !k). The initial condensate density isgiven by the usual Thomas-Fermi (TF) expression

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jTFj2 m1 r2=R2? z2=R2

k: Here, m is the maxi-mum initial density and the squared TF radii are R2

k 2=!2

k and R2? 2=!2

?, where mg is the (initial)chemical potential. According to [20], the temporal evo-lution of a Bose-Einstein-condensed atom cloud undervariation of the trapping frequencies is described by thescaling condensate wave function

r=b?; z=b 0 1=2 expi0 : (2)

Here, 0 R

g00; tdt _bbz2=2b _bb?r2=2b? is the

background phase, where b?t and bt are the scalingparameters describing the condensate expansion in theradial (rr) and axial (zz) directions; cf. Fig. 1. The mean-field condensate density 0 jTFj2r=b?; z=b=b2

?bcontains the scaling volume b2

?b, and are the fluctua-tions of the density around mean field.

The elementary excitations above the ground state ofthe BEC in the limit !?=!k ! 1 were first studied in[21]. The description of the modes is based on an adiabaticseparation for the axial and longitudinal variables of thephase fluctuation field:

r; z; t X

n

nrnz; t; (3)

where nr is the radial wave function characterized bythe quantum number n (we consider only zero angularmomentum modes). For long wavelength axial excita-tions, the dispersion relation reads [21]

2nk 2!2

?nn 1 c20k

2; (4)

where c0

=2p

and k is the axial wave number(kR? 1). In the action (1), we use Eq. (3), with therescaled radial wave function n nr=b?, and inte-grate over the radial coordinate. We then find the follow-ing effective action for the axial modes of a given radialquantum number n:

Sn Z

dtdzb2?Cn

2g _n vz@zn2 cc2

n@zn2

M2n

2n: (5)

Here, the common factor Cnz is given by b2?Cnz

R

r<rmd2r2

n, the averaged speed of sound is cc2nz; t

gC1n b2

?R

r<rmd2r0

2n, and the effective mass term

M2nz; t gC1

n b2?

R

r<rmd2r0@rn2; the radial cigar

size is given by r2m R2

?b2?1 z2=R2

kb2. The phonon

branch of the excitations corresponds to the n 0 solu-tion of Eq. (4), for which the radial wave function 0 const [21]; hence the mass term M0 0 and C00 R2

?, cc0; t ccn00; t c0=b?b1=2.The action (5) can be identified with the action of a

minimally coupled scalar field in 1 1D, close to thecenter of the condensate. The line element reads [17,18]

ds2 Acc2 v2zdt2 2vzdtdz dz2; (6)

where Ac is some arbitrary (space and time dependent)conformal factor and we set c ccz 0; t. The actions(1) and (5) can be made consistent if we renormalize thephase field according to Z ~ and require thatZ2b2

?C00=g 1= cc0, leading to

B b?b1=2

8

2

r1

Z2ma

3s1=2

!?

2

const: (7)

The constant factor Z in the above relations does notinfluence the equation of motion, and corresponds to arenormalization of the field amplitude, Z ~, whichdetermines the coupling of ~ to a detector [22].

We require that the space-time metric is, in appropri-ately chosen coordinates, identical to that of a 1 1Dde Sitter universe. We first apply the transformationc0d~tt ctdt, connecting the laboratory time t to thetime variable ~tt. Defining vz=c

p

z B _bb=c0z (notethat the dot on b and other quantities always refers toordinary laboratory time), this results, up to the confor-mal factor Ac, in the line element ds2 c2

01 z2d~tt2 2c0z

p

d~ttdz dz2. We then employ a secondtransformation c0d( c0d~tt z

p

dz=1 z2, with atime independent , to obtain the 1 1D de Sittermetric [3,5]

ds2 c201 z2d(2 1 z21dz2: (8)

The quantity B2 _bb2=c20 is a ‘‘cosmological constant,’’

provided _bb does not depend on t. The transformationbetween t and the de Sitter time (, at a given coordinatez, then involves the exponential ‘‘acceleration,’’ t=t0 expB _bb(, where t0 2=!k is set by the initialconditions. The (conformally invariant) temperature as-sociated with the effective metric (8) is the Gibbons-Hawking temperature [5]

TdS c0

2

p

B

2_bb; (9)

and the horizon(s) is located at z zH Rk!2

k=21=2. Combining the latter relation with(9), we see that zH=Rk 1, if TdS !k=4. We arethus justified in neglecting the z dependence in C0 andcc to describe the physics inside the horizon surfaces.

FIG. 1 (color online). Expansion of a cigar-shaped Bose-Einstein condensate. The stationary horizon surfaces are lo-cated at zH, respectively. The thick dark lines represent laserscreating an optical potential well in the center.

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According to Eq. (5), the equation of motionS0=0 0 is given, at constant B, by

B2b2d

dt

b2d

dt0

1

C0zb@zb cc2zbC0zb@zb0 0;

(10)

where zb z=b is the scaling coordinate. Apart fromthe factor C0zb, stemming from averaging over theperpendicular direction, this equation corresponds tothe hydrodynamic equation of phase fluctuations in in-homogeneous superfluids [23].

At t ! 1, the condensate is in equilibrium and thequantum vacuum phase fluctuations close to the center ofthe condensate can be written in the following form:

0 X

k

g

4C00Rk0k

s

aakei0ktikz H:c:; (11)

where aak; aayk are the annihilation and creation operators of

a phonon. The initial quantum state of phonons, which wecall the ‘‘adiabatic vacuum,’’ is the ground state of thesuperfluid and is annihilated by the operators aak (thisdefinition of vacuum and excitations corresponds to the‘‘adiabatic basis’’ of [15]). Assuming that the expansionis switched on adiabatically, the density fluctuation op-erator, which is conjugate to 0, then reads

X

k

0k

4C00Rkg

s

d

dt

(

aak exp

"

iZ t dt00k

Bb2 ikzb

#)

H:c: (12)

The above equation completely characterizes the n 0evolution of the condensate density fluctuations.

The particle content of a quantum field state in curvedspace-time depends on the observer [16]. We now showthat the de Sitter time interval d( dt=bB dt=b1=2b?can be effectively measured by an atomic quantum dot[13]. Here, the AQD is an effective two-level system witha time-dependent level splitting (see Fig. 2). It can bemade in a gas of atoms possessing two hyperfine groundstates * and +. The atoms in the state * represent thesuperfluid cigar, and are used to model the expandinguniverse. The AQD itself is formed by trapping atoms inthe state + in a tightly confining optical potential Vopt.The interaction of atoms in the two internal levels isdescribed by a set of coupling parameters gcd 4acd(c; d f*;+g), where acd are the s-wave scatteringlengths characterizing short-range intra- and interspeciescollisions; g** g, a** as. The on-site repulsion be-tween the atoms + in the dot is U g++=l

3, where l is thecharacteristic size of the ground state wave function ofatoms + localized in Vopt. In the following we considerthe collisional blockade limit of large U > 0, where onlyone atom of type + can be trapped in the dot. Thisassumes that U is larger than all other relevant frequency

scales in the dynamics of both the AQD and the expand-ing superfluid. (The value of g++ may be increased usingFeshbach resonances [24].) As a result, the appropriatecollective coordinate of the AQD is modeled by apseudo-spin-1=2, with spin-up/spin-down state corre-sponding to occupation by a single/no atom in state +.

Conversion of atoms between states * and + is causedby a laser-driven transition characterized by Rabi fre-quency and detuning . The Hamiltonian of theAQD interacting with the superfluid is

HAQD f g*+00; t 0; tg 1 1z2

00; tl31=2 expi 01 H:c:;

(13)

where we use Pauli matrix notation, i.e., 1 1x i1y.We introduce the wave function of the AQD in the form + expi 0j+i *j*i. Using the scalingexpression (2) for the condensate wave function, we find

id +

d( fbB 0g*+ g Vg + !0

2 *;

id *

d( !0

2 +;

(14)

where ( is the de Sitter time, and !0 2ml31=2 is

independent of (. The perturbation potential

V( g*+ gBb(0; ( (15)

provides the coupling between the AQD and the expand-ing superfluid [if g*+ ’ g, higher order terms in thedensity fluctuations have to be taken into account in theRabi term of (13)]. We suggest to operate the detector att g*+ g00; t g*+ gm= _bb2B2t2, sothat in zeroth order in V we obtain an effective two-level system with the level splitting !0, which plays therole of a frequency standard of the detector. By adjustingthe laser intensity contained in the Rabi frequency , onecan change !0, and thus probe the response of the detec-tor for various frequencies.

FIG. 2 (color online). Level scheme of the atomic quantumdot, which is embedded in the superfluid cigar, and created byan optical well for atoms of a hyperfine species different fromthat of the condensate. Double occupation of the dot is pre-vented by a collisional blockade mechanism.

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The detector response can be calculated using the two-point correlation function of the coupling operator VV.The probabilities for the detector to ‘‘click,’’ correspond-ing to excitation (P) and deexcitation (P) of the de-tector, per unit of de Sitter time, are given by [14,15]

dPd(

limT!1

1

T

Z T Z T

d(d(0hVV(VV(0iei!0((0:

(16)

Using the explicit form of the detector-to-field coupling(15), we obtain

hVV(VV(0i / ma3s1=2

!?

2B2g*+=g 12T2

dS

sinh2TdS( (0 :

(17)

This correlation function is proportional to one character-izing a thermal phonon state at a temperature TdS in acondensate at rest. This means that our de Sitter detectorat the center of the condensate responds to the scalingvacuum as if it were placed in a thermal bath at a tem-perature TdS. Indeed, substituting the correlator (17) intoEq. (16) we find that the quantities dP=d( at late timesare time independent, and satisfy detailed balance con-ditions corresponding to thermal equilibrium at thede Sitter temperature given in Eq. (9):

dP=d(

dP=d( nB

1 nB

; (18)

where nB exp!0=TdS 11 is the Bose distributionfunction. We stress that the AQD observes a thermalspectrum, even though the quantum state of the fluctua-tions is adiabatically connected to the initial vacuumstate, i.e., no excitations in the adiabatic basis character-ized by the aak are created (note that the time interval ofthe adiabatic vacuum is dt=Bb2 d(=b). The latter pro-cess corresponds to ‘‘cosmological’’ quasiparticle produc-tion [11,12], which can be neglected in our situation.

Our experimental proposal consists in the followingsteps: (i) preparation of a large TF condensate at very lowtemperatures; (ii) introduction of an AQD at the center(respectively an array of AQDs to increase the signal),which can be used to determine the initial temperature ofthe cloud; (iii) linear axial expansion of the condensateaccording to (7). The Rabi oscillations of the AQD can beused to monitor the thermal distribution nB in (18). SinceZ2 / ma

3s1=2!?=2 and P / Z2 [22], the initial

condensate has to be quite dense and close to the quasi-1D regime. The particle density throughout the clouddecreases like t2. Therefore, the rate of three-body lossesquickly decreases during expansion, and comparativelylong observation times are feasible.

The AQD as a detector is suitable to measure thede Sitter time interval, because it couples linearly to thesquare root of the density of the superfluid gas. In prin-ciple, one can construct detectors coupling to differentpowers of density or superfluid velocity, and then more

generally experimentally study the nonuniqueness of theparticle content of various quantum states in curvedspace-time [16]. For example, outcoupling pairs of atomsby photoassociation is a means to set up a detector whichhas d(=dt / 0. Finally, we note that while in this Letterwe have studied only the n 0 massless axial phononmodes, strongly elongated condensates can also be used tostudy the evolution of massive bosonic excitations.Together with a natural Planck scale EPlanck , thisprovides the opportunity to investigate, on a laboratoryscale, the influence of finite quasiparticle mass and thetrans-Planckian spectrum on the propagation of relativ-istic quantum fields in curved space-time.

We thank E. A. Cornell for an inspiring discussion onthe experimental feasibility of our theoretical ideas. Weacknowledge helpful discussions with R. Parentani,R. Schutzhold, G. E. Volovik, P. Zoller, and A. Recati.P. O. F. has been supported by the Austrian FWF andthe Russian RFRR, and U. R. F. by the FWF.We gratefullyacknowledge support from the ESF Programme‘‘Cosmology in the Laboratory.’’

[1] G. R. Pickett et al., Nature (London) 383, 570 (1996).[2] G. E. Volovik, The Universe in a Helium Droplet (Oxford

University Press, Oxford, 2003).[3] W. de Sitter, Mon. Not. R. Astron. Soc. 78, 3 (1917).[4] A. Linde, hep-th/0211048.[5] G.W. Gibbons and S.W. Hawking, Phys. Rev. D 15, 2738

(1977).[6] J. R. Anglin and W. Ketterle, Nature (London) 416, 211

(2002).[7] W. G. Unruh and R. Schutzhold, Phys. Rev. D 68, 024008

(2003).[8] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys.

Rev. Lett. 85, 4643 (2000).[9] C. Barcelo, S. Liberati, and M. Visser, Classical Quantum

Gravity 18, 1137 (2001).[10] U. Leonhardt, T. Kiss, and P. Ohberg, J. Opt. B 5, S42

(2003).[11] P. O. Fedichev and U. R. Fischer, cond-mat/0303063.[12] C. Barcelo, S. Liberati, and M. Visser, gr-qc/0305061.[13] A. Recati et al., cond-mat/0212413.[14] W. G. Unruh, Phys. Rev. D 14, 870 (1976).[15] N. D. Birrell and P. C.W. Davies, Quantum Fields in

Curved Space (Cambridge University Press,Cambridge, England, 1984).

[16] S. A. Fulling, Phys. Rev. D 7, 2850 (1973).[17] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981).[18] M. Visser, Classical Quantum Gravity 15, 1767 (1998).[19] U. R. Fischer and M. Visser, Ann. Phys. (N.Y.) 304, 22

(2003); Phys. Rev. Lett. 88, 110201 (2002).[20] Yu. Kagan, E. L. Surkov, and G.V. Shlyapnikov, Phys.

Rev. A 54, R1753 (1996); Y. Castin and R. Dum, Phys.Rev. Lett. 77, 5315 (1996).

[21] E. Zaremba, Phys. Rev. A 57, 518 (1998).[22] P. O. Fedichev and U. R. Fischer, cond-mat/0307200.[23] S. Stringari, Phys. Rev. Lett. 77, 2360 (1996).[24] A. Marte et al., Phys. Rev. Lett. 89, 283202 (2002).

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Publisher’s Note: Gibbons-Hawking Effect in the Sonic de Sitter Space-Timeof an Expanding Bose-Einstein-Condensed Gas

[Phys. Rev. Lett. 91, 240407 (2003)]

Petr O. Fedichev and Uwe R. Fischer(Received 16 January 2004; published 27 January 2004)

DOI: 10.1103/PhysRevLett.92.049901 PACS numbers: 03.75.Kk, 04.70.Dy, 99.10.Fg

This Letter was published online on 11 December 2003 with typographical errors in the second line of Eq. (1) and thelast line of the paragraph containing the equation. In both instances, the g’s representing the metric tensor elements andits determinant were inadvertently changed from sans serif to italic which made them indistinguishable from theinteraction constant g. To distinguish the two characters, the metric tensor elements have been changed back to theoriginal sans serif g, and the determinant of the metric tensor to g, as of 8 January 2004. The font style is not correctin the printed version of the journal.

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Appendix E

Petr O. Fedichev and Uwe R. Fischer

“Observer dependence for the phonon content of the sound

field living on the effective curved space-time background

of a Bose-Einstein condensate”

Phys. Rev. D 69, 064021 (2004)

91

Observer dependence for the phonon content of the sound field living on the effective curvedspace-time background of a Bose-Einstein condensate

Petr O. FedichevLeopold-Franzens-Universitat Innsbruck, Institut fur Theoretische Physik, Technikerstrasse 25, A-6020 Innsbruck, Austria

and Russian Research Center Kurchatov Institute, Kurchatov Square, 123182 Moscow, Russia

Uwe R. FischerLeopold-Franzens-Universitat Innsbruck, Institut fur Theoretische Physik, Technikerstrasse 25, A-6020 Innsbruck, Austria

and Eberhard-Karls-Universitat Tubingen, Institut fur Theoretische Physik, Auf der Morgenstelle 14, D-72076 Tubingen, Germany~Received 3 October 2003; published 22 March 2004!

We demonstrate that the ambiguity of the particle content for quantum fields in a generally curved space-time can be experimentally investigated in an ultracold gas of atoms forming a Bose-Einstein condensate. Weexplicitly evaluate the response of a suitable condensed matter detector, an ‘‘atomic quantum dot,’’ which canbe tuned to measure time intervals associated with different effective acoustic space-times. It is found that thedetector response related to laboratory, ‘‘adiabatic,’’ and de Sitter time intervals is finite in time and nonsta-tionary, vanishing, and thermal, respectively.

DOI: 10.1103/PhysRevD.69.064021 PACS number~s!: 04.62.1v, 03.75.Kk, 04.70.Dy

I. INTRODUCTION

The feat of reproducing certain features of the physics ofrelativistic quantum fields on curved space-time backgrounds@1,2#, in the laboratory, now seems closer than ever before.Because of the realization of the fact that phonons or gener-ally ‘‘relativistic’’ quasiparticles, propagating in a nonrelativ-istic background fluid, experience an effective curved space-time @3–7#, various exotic phenomena of the physics ofclassical as well as quantum fields in the curved space-timesof gravity are getting within the reach of laboratory scaleexperiments @8#.

The particle content of a quantum field state in curved orflat space-time depends on the motional state of the observer.A manifestation of this observer dependence is the Unruh-Davies effect, which consists in the fact that a constantlyaccelerated detector in the Minkowski vacuum responds as ifit were placed in a thermal bath with temperature propor-tional to its acceleration @1,2#. This effect has eluded obser-vation so far: The value of the Unruh temperature TUnruh5@\/(2pkBcL)#a54 K3a@1020g % # , where a is the accel-eration of the detector in Minkowski space (g % is the gravityacceleration on the surface of the Earth! and cL the speed oflight, makes it obvious that an observation of the effect is aless than trivial undertaking. Proposals for a measurementwith ultraintense short pulses of electromagnetic radiationhave been put forward in, e.g., Refs. @9,10#.

In the following, we shall argue that the observer depen-dence of the particle content of a quantum field state incurved space-time, related to the familiar nonuniqueness ofcanonical field quantization in Riemannian spaces @11#, canbe experimentally demonstrated in the readily available ul-tralow temperature condensed matter system Bose-Einsteincondensate ~BEC!. More specifically, we argue that theGibbons-Hawking effect @12#, a curved space-time generali-zation of the Unruh-Davies effect, can be observed in anexpanding BEC. To explicitly show that particle detection

depends on the detector setting, we construct a condensedmatter detector tuned to time intervals in effective laboratoryand de Sitter space-times, respectively @13#. We show thatthe detector response is strongly different in the two cases,and associated with the corresponding effective space-time.Furthermore, we describe a system of space-time coordinatesin which there is no particle detection whatsoever takingplace, which we will call the ‘‘adiabatic’’ basis, and whichsimply corresponds to a detector at rest in the ~conformal!Minkowski vacuum.

II. HYDRODYNAMICS IN AN EXPANDINGCIGAR-SHAPED BEC

It recently became apparent that among the most suitablesystems for the simulation of quantum phenomena in effec-tive space-times are Bose-Einstein condensates ~BECs! @14–18#. They offer the primary advantages of dissipation-freesuperflow, high controllability, with atomic precision, of thephysical parameters involved, and the accessibility of ul-tralow temperatures @19,20#. Even more importantly in thecontext under consideration here, the theory of phononicquasiparticles in the spatially and temporally inhomogeneousBEC is kinematically identical to that of a massless scalarfield propagating on the background of curved space-time inD11 dimensions.

It was shown by Unruh that the action of the phase fluc-tuations F in a moving inhomogeneous superfluid may bewritten in the form @3,4# ~we set \5m51, where m is themass of a superfluid constituent particle!:

S5E dD11x1

2g F2S ]

]tF2v•“F D 2

1c2~“F !2G[

12E dD11xA2ggmn]mF]nF . ~1!

Here, v(x,t) is the superfluid background velocity, c(x,t)5Agr0(x,t) is the velocity of sound, where g is a constant

PHYSICAL REVIEW D 69, 064021 ~2004!

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describing the interaction between the constituent particles inthe superfluid (1/g is the compressibility of the fluid!, andr0(x,t) is the background density. In the second line of ~1!,the conventional hydrodynamic action is identified with theaction of a minimally coupled scalar field in an effectivecurved space-time. Furthermore, the velocity potential of thesound perturbations in the BEC satisfies the canonical com-mutation relations of a relativistic scalar field @14#. We there-fore have the exact mapping, on the level of kinematics, ofthe equation of motion for phononic quasiparticles in a non-relativistic superfluid, to quantized massless scalar fieldspropagating on a curved space-time background with localLorentz invariance.

To model curved space-times, we will consider the evo-lution of the BEC if we change the harmonic trapping fre-quencies with time. For a description of the expansion ~orcontraction! of a BEC, the so-called scaling solution ap-proach is conventionally used @21#. One starts from a cigar-shaped BEC containing a large number of constituent par-ticles, i.e., which is in the so-called Thomas-Fermi ~TF! limit@22#. According to Ref. @21#, the evolution of a Bose-condensed atom cloud under temporal variation of the trap-ping frequencies v i(t) and v'(t) ~in the axial and radialdirections, respectively! can then be described by the follow-ing solution for the condensate wave function

C5

CTF

b'Ab

expF2iE gr0~x50,t !dt1ibz2

2b1i

b'r2

2b'

G .

~2!

Here, b' and b are the scaling parameters describing thecondensate evolution in the radial ( r) and axial ( z) direc-tions ~cf. Fig. 1!. The initial (b5b'51) mean-field conden-sate density is given by the usual TF expression

uCTFu25rTF~r ,z !5rmS 12

r2

R'

2 2

z2

R i2D . ~3!

Here, rm is the maximum density ~in the center of the cloud!

and the squared initial TF radii are R i252m/v i

2 and R'

2

52m/v'

2 . The initial chemical potential m5rmg , where g54pas , and as is the scattering length characterizing

atomic collisions in the ~dilute! BEC. In our cylindrical 3Dtrap, we have for the initial central density

rm5S 6Nv'

2 v i

A8pg3/2 D 2/5

.

The condition that the TF approximation be valid impliesthat m@v i ,v' . The solution ~2! of the Gross-Pitaevskimean-field equations becomes exact in this TF limit, inde-pendent of the ratio v i /v' . However, the solution becomesexact also in the limit that v' /v i→0, independent of thevalidity of the TF limit, the system then acquiring an effec-tively two-dimensional character @23,24#. We will see belowthat in the opposite limit of v i /v'→0 there is an ‘‘adiabaticbasis’’ in which no axial excitations are created during theexpansion, i.e., with respect to that basis there are, in particu-lar, no unstable solutions possible, implying the stability ofthe expanding gas against perturbations.

According to ~2!, the condensate density evolves as

r0~r ,z ,t !5

rTF~r2/b'

2 ,z2/b2!

b'

2 b, ~4!

and the superfluid velocity

v5

b'

b'

rer1bb

zez ~5!

is the gradient of the condensate phase in Eq. ~2!. It increaseslinearly to the axial and radial boundaries of the condensate.

The excitations in the limit v i /v'→0 were first studiedin Ref. @25#. The description of the modes is based on anadiabatic separation for the axial and longitudinal variablesof the phase fluctuation field:

F~r ,z ,t !5(n

fn~r !xn~z ,t !, ~6!

where fn(r) is the radial wave function characterized by thequantum number n ~we consider only zero angular momen-tum modes!. The above ansatz incorporates the fact that forstrongly elongated traps the dynamics of the condensate mo-tion separates into a fast radial motion and a slow axial mo-tion, which are essentially independent. The xn(z ,t) are themode functions for traveling wave solutions in the z direction~plane waves for a condensate at rest read xnexp@2ien,kt1kz#). The radial motion is assumed to be ‘‘stiff’’ such thatthe radial part is effectively time independent, because theradial time scale for adjustment of the density distributionafter a perturbation is much less than the axial oscillationtime scales of interest. The ansatz ~6! works independent ofthe ratio of healing length and radial size of the BEC cigar.In the limit that the healing length is much less than theradial size, TF wave functions are used, in the opposite limit,a Gaussian ansatz for the radial part of the wave functionfn(r) is appropriate.

For axial excitations characterized by a wavelength l52p/k exceeding the radial size R' of the condensate, wehave kR'!1, and the dispersion relation reads, in the TFlimit for the radial wave function @25#

FIG. 1. Expansion of a cigar-shaped Bose-Einstein condensate.The stationary horizon surfaces are located at 6zH , respectively.The thick dark lines represent lasers creating an optical potentialwell in the center.

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en ,k2

52v'

2 n~n11 !1

v'

2

4~kR'!2

52v'

2 n~n11 !1c02k2, ~7!

where c05Am/2. Observe that the central speed of sound c0is reduced by a factor A2 from the well-established valueAgrm5Am for an infinitely extended liquid @25#.

Equations ~6! and ~7! can be generalized for an expandingcondensate. Substituting in Eq. ~6! the rescaled radial wavefunction fn[fn(r/b'), and inserting the result into the ac-tion ~1!, integrating over the radial coordinates, we find thefollowing effective action for the axial modes of a givenradial quantum number n:

Sn5E dtdzb'

2 Cn~z !

2g@2~ xn2vz]z!

21 cn

2~z !~]zxn!2

1M n2~z !xn

2# , ~8!

where the common ‘‘conformal’’ factor Cn(z) is given by

b'

2 Cn~z !5Er,rm

d2rfn2 . ~9!

The integration limits are fixed by the z dependent radial sizeof the cigar rm

25R'

2 b'

2 (12z2/R i2b2). The averaged speed of

sound reads

cn2~z ,t !5

g

Cnb'

2 Er,rm

d2rr0fn2 ~10!

and the ~space and time dependent! effective mass term is,for a given radial mode, obtained to be

M n2~z ,t !5

g

Cnb'

2 Er,rm

d2rr0@]rfn#2. ~11!

The phonon branch of the excitations corresponds to the gap-less n50 solution of Eq. ~7!. In this case the radial wavefunction f0 does not depend on the radial variable r @25#,and the mass term vanishes, M 050. We then obtain thefollowing expressions,

C0~z !5pR'

2 S 12

z2

b2R i2D , ~12!

and for the z dependent speed of sound ( c[ cn50):

c2~z ,t !5

c02

b'

2 bS 12

z2

b2R i2D . ~13!

We will see below that we need these expressions in the limitz→0 only, because only in this limit we obtain the exactmapping of the phonon field to a quantum field propagatingin a 111D curved space-time.

III. THE 1¿1D DE SITTER METRIC IN THECONDENSATE CENTER

We identify the action ~8! with the action of a minimallycoupled scalar field in 111D, according to Eq. ~1!. Remark-ably, such an identification is possible only close to the cen-ter z50. The contravariant 111D metric may generally bewritten as @4#

gmn5

1

Acc2 S 21 2vz

2vz c22vz

2D , ~14!

where Ac is some arbitrary ~space and time dependent! factorand c5 c(z50). Inverting this expression to get the covari-ant metric, we obtain

gmn5AcS 2~c22vz

2! 2vz

2vz 1 D . ~15!

The term A2ggmn contained in the action ~1! gives the fa-miliar conformal invariance in a 111D space-time, i.e., theconformal factor Ac drops out from the action and thus doesnot influence the classical equations of motion. We thereforeleave out Ac in the formulas to follow, but it needs to beborne in mind that the metric elements are defined always upto the factor Ac .

The actions ~1! and ~8! can be made consistent if werenormalize the phase field according to F5ZF and requirethat

b'

2 C0~0 !

gZ2

5

1

c~16!

holds. The factor Z does not influence the equation of mo-tion, but does influence the response of a detector ~see Sec.IV A below!. In other words, it renormalizes the coupling ofour ‘‘relativistic field’’ F to the laboratory frame detector.More explicitly, Eq. ~16! leads to

b'Ab 58Ap

2

1

Z2Armas

3S v'

mD 2

[B5const. ~17!

According to the above relation, we have to impose that theexpansion of the cigar in the perpendicular direction pro-ceeds like the square root of the expansion in the axial di-rection. The constant quantity B can be fixed externally ~bythe experimentalist!, choosing the expansion of the cloudappropriately by adjusting the time dependence of the trap-ping frequencies v i(t) and v'(t), according to the scalingequations @21,26#

b1v i2~ t !b5

v i2

b'

2 b25

v i2

B2b3,

~18!

b'1v'

2 ~ t !b'5

v'

2

b'

3 b5

v'

2

B3b5/2.

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Since both C0 and c depend on z, an effective space-timemetric for the axial phonons can be obtained only close tothe center of the cigar-shaped condensate cloud. This is re-lated to our averaging over the physical perpendicular direc-tion, and does not arise if the excitations are considered inthe full D-dimensional situation, where this identification ispossible globally. Also note that the action ~8! does not con-tain a curvature scalar contribution of the formxn

2A2g R@gmn(x)# , i.e., that it only possesses trivial con-formal invariance @27#.

We now impose, in addition, the requirement that the met-ric is identical to that of a 111D universe, with a metric ofthe form of the de Sitter metric in 311D @12,28#. We firstapply the transformation c0d t 5c(t)dt to the line elementdefined by ~15!, connecting the laboratory time t to the timevariable t . Defining vz /c5ALz5(Bb/c0)z ~note that thedot on b and other quantities always refers to ordinary labo-ratory time!, this results, up to the conformal factor Ac , inthe line element

ds252c0

2~12Lz2!d t 222c0zALd t dz1dz2. ~19!

We then apply a second transformation c0dt5c0d t1zALdz/(12Lz2), with a constant L . We are thus led tothe 111D de Sitter metric in the form @12#

ds252c0

2~12Lz2!dt21~12Lz2!21dz2. ~20!

The transformation between t and the de Sitter time t ~on aconstant z detector, such that d t 5dt), is given by

tt0

5exp@Bbt# , ~21!

where the unit of laboratory time t0;v i21 is set by the initial

conditions for the scaling variables b and b' . The tempera-ture associated with the effective metric ~20! is the Gibbons-Hawking temperature @12#

TdS5

c0

2pAL5

B2p

b . ~22!

The ‘‘surface gravity’’ on the horizon has the value aH

5c02AL5c0Bb , and the stationary horizon~s! are located at

the constant values of the z coordinate

z56zH56R iA v i2

2mL. ~23!

Combining ~22! and ~23!, we see that zH /R i is small ifv i /TdS!4p . Therefore, the de Sitter temperature needs tobe at least of the order of v i for the horizon location~s! to bewell inside the cloud. The latter condition then justifies ne-glecting the z dependence in C0 and c in Eq. ~16!. Thoughthere is no metric ‘‘behind’’ the horizon, i.e., at large z, thisshould not affect the low-energy behavior of the quantumvacuum ‘‘outside’’ the horizon.

IV. DETERMINING THE PARTICLE CONTENT OF THEQUANTUM FIELD

The particle content of a quantum field state depends onthe observer @1,2,11#. To detect the Gibbons-Hawking effectin de Sitter space, one has to set up a detector that measuresfrequencies in units of the inverse de Sitter time t , ratherthan in units of the inverse laboratory time t. We will showthat the de Sitter time interval dt5dt/bB5dt/Abb' can beeffectively measured by an atomic quantum dot ~AQD!@29,30#. The measured quanta can then, and only then, beaccurately interpreted to be particles coming from aGibbons-Hawking type process with a constant de Sittertemperature ~22!. We then use the tunability for other timeintervals feasible with our detector scheme, and contrast thisde Sitter result with what the detector ‘‘sees’’ if tuned tolaboratory and ‘‘adiabatic’’ time intervals.

The AQD can be made in a gas of atoms possessing twohyperfine ground states a and b . The atoms in state a rep-resent the superfluid cigar, and are used to model the expand-ing universe. The AQD itself is formed by trapping atoms instate b in a tightly confining optical potential Vopt . The in-teraction of atoms in the two internal levels is described by aset of coupling parameters gcd54pacd (c ,d5$a ,b%),where acd are the s-wave scattering lengths characterizingshort-range intra- and interspecies collisions; gaa[g , aaa

[as . The on-site repulsion between the atoms b in the dotis U;gbb /l3, where l is the characteristic size of the groundstate wave function of atoms b localized in Vopt . In thefollowing, we consider the collisional blockade limit of largeU.0, where only one atom of type b can be trapped in thedot. This assumes that U is larger than all other relevantfrequency scales in the dynamics of both the AQD and theexpanding superfluid. As a result, the collective coordinate ofthe AQD is modeled by a pseudo-spin-1/2, with the spin-up/spin-down state corresponding to occupation by a singleatom/no atom in state b .

We first describe the AQD response to the condensatefluctuations in the Lagrangian formalism, most familiar in afield theoretical context. The detector Lagrangian takes theform

LAQD5iS d

dth*Dh2@2D1gab~r0~0,t !1dr !#h*h

2VAr0~0,t !l3S expF2iE0

tgr0~0,t8!dt81idfGh*

1expF iE0

tgr0~0,t8!dt82idfGh D . ~24!

Here, D is the detuning of the laser light from resonance,r0(z50,t) is the central mean-field part of the bath density,and l is the size of the AQD ground state wave function. Thedetector variable h is an anticommuting Grassmann variablerepresenting the effective spin degree of freedom of theAQD. The second and third lines represent the coupling ofthe AQD to the surrounding superfluid, where df and dr arethe fluctuating parts of the condensate phase and density at

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z50, respectively. The laser intensity and the effective tran-sition matrix element combine into the Rabi frequency V;below we will make use of the fact that V can easily experi-mentally be changed as a function of laboratory time t, bychanging the laser intensity with t .

To simplify ~24!, we use the canonical transformation

h→h expF2iE0

tgr0~0,t8!dt81idfG . ~25!

The above transformation amounts to absorbing the superflu-id’s chemical potential and the fluctuating phase df into thewave function of the AQD, and does not change the occupa-tion numbers of the two AQD states. The transformation ~25!gives the detector Lagrangian the form

LAQD5iS d

dth*D h2VAr0~0,t !l3~ h1h*!

2F2D1~gab2g !r0~0,t !1gabdr1

d

dtdfG h*h .

~26!

The laser coupling ~second term in the first line! scales asb21/2b'

21 , and hence like the de Sitter time interval in unitsof the laboratory time interval, dt/dt . We suggest that thedetector be operated at the time dependent detuning D(t)5(gab2g)r0(0,t)5(gab2g)rm /( b2B2t2), which thenleads to a vanishing of the first two terms in the squarebrackets of ~26!.

We now reintroduce the wave function of the AQD stem-ming from a Hamiltonian formulation, c5cbub&1caua&.An ‘‘effective Rabi frequency’’ may be defined to be v0

52VArml3; at the detuning compensated point, we thenobtain a simple set of coupled equations for the AQD ampli-tudes

idcb

dt5

v0

2ca1dVcb , i

dca

dt5

v0

2cb , ~27!

where t is the de Sitter time.We have thus shown that the detector equations ~27! are

natural evolution equations in de Sitter time t , if the Rabifrequency V is chosen to be a constant, independent of labo-ratory time t. We will see in Sec. IV B that, adjusting V in acertain time dependent manner, within the same detectorscheme, we can reproduce time intervals associated to vari-ous other effective space-times.

The coupling of the AQD to fluctuations in the superfluidis described by the potential

dV~t !5~gab2g !Bb~t !dr~t !. ~28!

Neglecting the fluctuations in the superfluid, the level sepa-ration implied by ~27! is v0, and the eigenfunctions of thedressed two level system are u6&5(ua&6ub&)/A2. Thequantity v0 therefore plays the role of a frequency standardof the detector. By adjusting the value of the laser intensity,

one can change v0, and therefore probe the response of thedetector for various phonon frequencies. Note that if gab isvery close to g, to obtain the correct perturbation potential,higher order terms in the density fluctuations have to betaken into account in the Rabi term of ~26!.

To describe the detector response, we first have to solvethe equations of motion ~8! for the phase fluctuations, andthen evaluate the conjugate density fluctuations. The equa-tion of motion dS0 /dx050 is, for time independent B, givenby

B2b2ddt

~b2x0!2

1C0~zb!

]zb@ c2~zb!C0~zb!]zb

x0#50,

~29!

where zb5z/b is the scaling coordinate. Apart from the fac-tor C0(zb), stemming from averaging over the perpendiculardirection, this equation corresponds to the hydrodynamicequation of phase fluctuations in inhomogeneous superfluids@31#. At t→2` , the condensate is in equilibrium and thequantum vacuum phase fluctuations close to the center of thecondensate can be written in the following form:

x05A g4C0~0 !R ie0,k

akexp@2ie0,kt1ikz#1H.c.,

~30!

where ak , ak† are the annihilation and creation operators of a

phonon. The initial quantum state of phonons is the groundstate of the superfluid and is annihilated by the operators ak .With these initial conditions, the solution of ~29! is

x05A g4C0~0 !R ie0,k

akexpF2iE0

t dt8e0,k

Bb21ikzbG1H.c.

~31!

The solution for the canonically conjugate density fluctua-tions, consequently, is

dr5iA e0,k

4C0~0 !R ig]

]t S akH expF2iE0

t dt8e0,k

Bb21ikzbG J D

1H.c. ~32!

Equations ~31! and ~32! completely characterize the n50evolution of the condensate fluctuations. Observe that theevolution proceeds without frequency mixing in the adiabatictime interval defined by dta5dt/Bb2 ~the ‘‘scaling time’’interval dt/B2b2 defined in Ref. @26# is proportional to thisadiabatic time interval!. Therefore, in the ‘‘adiabatic basis,’’no frequency mixing occurs and thus no quasiparticle exci-tations are created. This hints at a hidden ~low-energy! sym-metry, in analogy to the ~exact! 211D Lorentz groupSO~2,1! for an isotropically expanding BEC disk, discussedin Ref. @23#.

A. Detection in de Sitter time

The coupling operator dV causes transitions between thedressed detector states u1& and u2& and thus can be used toeffectively measure the quantum state of the phonons. Weconsider the detector response to fluctuations of C , by goingbeyond mean field and using a perturbation theory in dV .

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There are two physically different situations. The detector iseither at t50 in its ground state, (ua&1ub&)/A2, or in itsexcited state, (ua&2ub&)/A2. We define P1 and P2 to bethe probabilities that at late times t the detector is excitedrespectively deexcited. Using second order perturbationtheory in dV , we find that the transition probabilities for thedetector may be written

P65(k

ge0,k

4R iC0~0 !S gab

g21 D 2

B2uT6u2, ~33!

where the absolute square of the transition matrix element isgiven by

uT6u25UE

0

` dt

b~t !expF6ie0,kE

0

t dt8

b~t8!1iv0tGU2

.

~34!

Calculating the integrals, we obtain

P65JS gab

g21 D 2

B2gp

2BbR iC0~0 !3H nB

11nB ,~35!

where the ~formally divergent! sum

J5(k

v0

e0,k, ~36!

and the factors

nB5

1exp@v0 /TdS#21

~37!

are Bose distribution functions at the de Sitter temperature~22!. We conclude that an expansion of the condensate in zdirection, with a constant rate faster than the harmonic traposcillation frequency in that direction, gives an effectivespace-time characterized by the de Sitter temperature TdS .

We now show that J is proportional to the total de Sittertime of observation, so that the probability per unit time is afinite quantity @32#. At late times, the detector measures pho-non quanta coming, relative to its space-time perspective,from close to the horizon, at a distance dz5zH2z!zH5L21/2. The trajectory of such a phonon in the coordinatesof the de Sitter metric ~20!, at late times t , is given by ~cf.Fig. 2!

lnF zH

dzG52ALc0t . ~38!

This implies that the central AQD detector measures quantathat originated at the horizon with large shifted frequency

e0,k5

v0

A2L1/4dz1/25

v0

A2exp@c0tAL# . ~39!

Making use of the above equation, we rewrite the summationover k in ~36! as an integral over detector time:

J5(k

v0

e0,k5

R iv0

pc0E de0,k

e0,k5

R iv0

pc0ALc0E dt

5

R iv0

pc0Bbt . ~40!

Therefore, the probabilities per unit detector time ~de Sittertime! read, where upper/lower entries refer to P1/P2 , re-spectively:

dP6

dt5S gab

g21 D 2

B2gv0

2C0~0 !c03H nB

11nB .~41!

They are finite quantities in the limit that t→` . In labora-tory time, the transition probabilities evolve according to

P6~ t !5P0v0

TdSlnF t

t0G3H nB

11nB ,~42!

where, from relation ~16!, P05Z2@(gab /g21)B#2/2. Wesee that the detector response is, as it should be, proportionalto Z2, the square of the renormalization factor of the phasefluctuation field.

The absorption and emission coefficients dP6 /dt satisfyEinsteinian relations. Therefore, the detector approachesthermal equilibrium at a temperature TdS on a time scaleproportional to Z22v0

21. Our de Sitter AQD detector thusmeasures a stationary thermal spectrum, even though its con-densed matter background, with laboratory time t, is in ahighly nonstationary motional state. Since Z2

Armas3(v' /m)2, not-too-dilute condensates with v';m

~i.e., close to the quasi-1D regime @33#! are most suitable forobserving the Gibbons-Hawking effect.

The verification of the fact that a thermal detector statehas been established proceeds by the fact that the two hyper-

FIG. 2. Typical trajectories of phonons (ds250) in the de Sitter

metric ~20!. The path taken by phonon I, which is at early timespropagating near the horizon, is described in the text. The pathtaken by phonon II, which approaches the horizon surface 2zH atlate times t , does not lead to an excitation of the de Sitter detectorplaced at z50.

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fine states a and b are spectroscopically different states ofthe same atom, easily detectable by modern quantum opticaltechnology. When the optical potential is switched on, theatoms are in the empty a state originally, which is an equal-weight superposition of u1&5(ua&1ub&)/A2 and u2&5(ua&2ub&)/A2. The thermalization due to the Gibbons-Hawking effect takes place in the dressed state basis consist-ing of the two detector states, i.e. of the states u1& andu2&, on a time scale given by the quantities P6 in Eq. ~42!.For the laboratory observer, the Gibbons-Hawking thermalstate will thus appear to cause damping of the Rabi oscilla-tions on the thermalization time scale, i.e., friction on thecoherent oscillating motion between the two detector statesoccurs, due to the thermal phonon bath perceived by thedetector. The occupation of the detector states can be mea-sured directly using atomic interferometry: A p/2 pulsebrings one of them into the filled (b) and the other into theempty (a) state. To increase the signal to noise ratio, onecould conceive of manufacturing a small array of AQDs in asufficiently large cigar-shaped host superfluid, and monitorthe total population of b atoms in this array.

B. Detection in laboratory time

We contrast the above calculation with the response theAQD detector would see if tuned to laboratory time. This canbe realized if we allow Vt , such that VAr0(0,t)5VArm/(Bbt)5const, in the Rabi term on the right-handside of ~26!, to be time independent. The detector has, there-fore, dt as its natural time interval in this setting. ThePainleve-Gullstrand metric ~15! in pure laboratory framevariables, assuming B2b2

@v i2/b2 as in the derivation of the

de Sitter metric ~20!, reads

ds252

c02

B2b2t2~12Lz2!dt2

2

2zt

dzdt1dz2. ~43!

The metric ~43! is asymptotically, for large t, becoming thatof Galilei invariant ordinary 1D laboratory space, i.e. it isjust measuring length along the z direction, because thespeed of sound in the ever more dilute gas decreases like 1/tand the ‘‘phonon ether’’ becomes increasingly less stiff.

The transition probabilities for absorption respectivelyemission are now given by P65@ge0,k/4R iC0(0)#(gab /g21)2uT6u2, where the matrix elements are, cf. Eq. ~34!,

uT6u25UE

2`

` dt

Bb2expF6ie0,kE

2`

` dt8

Bb21iv0tGU2

.

~44!

Substituting the adiabatic time interval dta5dt/(Bb2t2)leads for large t to

ta5t0s21/~Bb2t !, ~45!

where t0a5*2`1`dt/(Bb2t2). The transformation to adiabatic

time maps tP@2` ,1`# onto taP@2` ,t0a# and, by furthersubstituting y5e0,k(ta2t0a), we have

uT6u25

1

e0k2 UE

0

`

dy expF iS y7

v0e0k

Bb2

1y D GU

2

. ~46!

The integral is a linear combination of Bessel functions. Totest its convergence properties, we are specifically interestedin the large e0k limit. Performing a stationary phase approxi-mation for large A56v0e0k /Bb2, we have for positive A~absorption! that the integral above becomes J(A)5(pAA)1/2exp@22AA# and for negative A ~emission! J(A)5(pAuAu)1/2. The final result then is

P65S gab

g21 D 2

A2pArmas3S v'

mD 2

3E0

EPlde0kA v0

e0kBb23H exp@24Av0e0k /Bb2#

1

5S gab

g21 D 2

A2prmas3S v'

m D 2

35.

12

A4EPlv0

Bb2

,

~47!

where EPl;m is the ultraviolet cutoff in the integral for theemission probability P2 , the ‘‘Planck’’ scale of the super-fluid. Because of the convergence of the absorption integralfor P1 , the total number of particles detected remains finite,and there are no particles detected by the effective laboratoryframe detector at late times. This is in contrast to the deSitter detector, which according to ~41! still detects particles,in a stationary thermal state.

There is a detector setting that corresponds to a detector atrest in the Minkowski vacuum. This setting is represented bythe adiabatic basis, with time interval defined by dta5dt/Bb2, realizable with the AQD by setting the Rabi fre-quency V1/t . Then, no particles whatsoever are detected,i.e., no frequency mixing of the positive and negative fre-quency parts of ~32! does take place. The associated space-time interval

ds25b2@2c0

2dta21dzb

2# ~48!

is simply that of ~conformally! flat Minkowski space inthe spatial scaling coordinate zb and adiabatic time coordi-nate ta .

V. SUMMARY AND CONCLUSIONS

We summarize the effective space-times considered inthis article, and the associated time intervals in Table I. Themajor observation of the present investigation is that thephysical nature of the effective space-time considered in thecondensed matter system reflects itself directly in the quasi-particle ~phonon! content measured by a detector that has anatural time interval equal to the time interval of this particu-lar effective space-time. We have demonstrated that the no-

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tion of observer dependence can be made experimentallymanifest by an atomic quantum dot placed at the center of alinearly expanding cigar-shaped Bose-Einstein condensate,which has a tunable effective time interval. Thus a possibil-ity opens up to confirm experimentally, in an effectivecurved space-time setting, that, indeed, ‘‘a particle detectorwill react to states which have positive frequency with re-spect to the detector’s proper time, not with respect to anyuniversal time’’ @1#.

The Gibbons-Hawking effect in the BEC is intrinsicallyquantum: The signal contains a ‘‘dimensionless Planck con-stant,’’ i.e. the gaseous ~loop expansion! parameter Armas

3.This implies that a reasonable signal-to-noise ratio can beachieved only by using initially dense clouds with stronginterparticle interactions. On the other hand, the phononicquasiparticles of the superfluid can be regarded as noninter-acting only in a first approximation in Armas

3!1. The ef-

fects of self-interaction between the phonons, induced bylarger values of the gaseous parameter, can lead to decoher-ence and the relaxation of the phonon subsystem. The sameline of reasoning applies to the evolution of quantum fieldsin the expanding universe. The interactions between quasi-particle excitations and their connection to decoherence pro-cesses in cosmological models of quantum field propagationand particle production are, therefore, important topics forfuture work.

ACKNOWLEDGMENTS

We acknowledge helpful discussions with R. Parentaniand R. Schutzhold. P. O. F. has been supported by the Aus-trian Science Foundation FWF and the Russian Foundationfor Basic Research RFRR, and U. R. F. by the FWF. Theyboth gratefully acknowledge support from the ESF Program‘‘Cosmology in the Laboratory.’’

@1# W.G. Unruh, Phys. Rev. D 14, 870 ~1976!.@2# N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved

Space ~Cambridge University Press, Cambridge, England,1984!.

@3# W.G. Unruh, Phys. Rev. Lett. 46, 1351 ~1981!.@4# M. Visser, Class. Quantum Grav. 15, 1767 ~1998!.@5# R. Schutzhold and W.G. Unruh, Phys. Rev. D 66, 044019

~2002!.@6# U.R. Fischer and M. Visser, Ann. Phys. ~N.Y.! 304, 22 ~2003!;

Phys. Rev. Lett. 88, 110201 ~2002!.@7# P. Nurowski, E. Schucking, and A. Trautman, in On Einstein’s

Path, Essays in Honor of Engelbert Schucking, edited by A.Harvey ~Springer, New York, 1999!, Chap. 23.

@8# G. E. Volovik, The Universe in a Helium Droplet ~Oxford Uni-versity Press, Oxford, 2003!; for a broad overview on effectivespace-times in condensed matter systems.

@9# E. Yablonovitch, Phys. Rev. Lett. 62, 1742 ~1989!.@10# P. Chen and T. Tajima, Phys. Rev. Lett. 83, 256 ~1999!.@11# S.A. Fulling, Phys. Rev. D 7, 2850 ~1973!.@12# G.W. Gibbons and S.W. Hawking, Phys. Rev. D 15, 2738

~1977!.@13# We will use the notions of particle and quasiparticle inter-

changeably, as far as excitations of quantum fields obeyingrelativistic respectively pseudorelativistic wave equations areconcerned. The particles ~atoms! forming the BEC will becalled ‘‘constituent particles.’’

@14# W.G. Unruh and R. Schutzhold, Phys. Rev. D 68, 024008~2003!.

@15# C. Barcelo, S. Liberati, and M. Visser, Class. Quantum Grav.18, 1137 ~2001!; Phys. Rev. A 68, 053613 ~2003!.

@16# L.J. Garay, J.R. Anglin, J.I. Cirac, and P. Zoller, Phys. Rev.Lett. 85, 4643 ~2000!.

@17# We note that recently there has been established a dynamicalconnection between perfect fluid cosmologies in Einsteiniangravity and the dynamics of expanding Bose-Einstein conden-sates, not related to the kinematical analogy considered here;J.E. Lidsey, Class. Quantum Grav. 21, 777 ~2004!.

@18# U. Leonhardt, T. Kiss, and P. Ohberg, J. Opt. B: QuantumSemiclassical Opt. 5, S42 ~2003!.

@19# J.R. Anglin and W. Ketterle, Nature ~London! 416, 211~2002!.

@20# A.E. Leanhardt et al., Science 301, 1513 ~2003!.@21# Yu. Kagan, E.L. Surkov, and G.V. Shlyapnikov, Phys. Rev. A

54, R1753 ~1996!; Y. Castin and R. Dum, Phys. Rev. Lett. 77,5315 ~1996!.

@22# G. Baym and C.J. Pethick, Phys. Rev. Lett. 76, 6 ~1996!.@23# L.P. Pitaevski and A. Rosch, Phys. Rev. A 55, R853 ~1997!.@24# High quality factors for breathing mode ~monopole radial! os-

cillations in the strongly elongated limit v i /v'→0 have beenobtained experimentally by F. Chevy, V. Bretin, P. Rosenbusch,K.W. Madison, and J. Dalibard, Phys. Rev. Lett. 88, 250402~2002!.

TABLE I. Various time intervals effectively measuring laboratory, de Sitter, and adiabatic time in a 111DBEC, respectively, where b5b(t), and B is independent of laboratory time. The third entry specifies if theAQD detector, tuned to the given time interval, detects phonons. In the laboratory frame, the detector has asmall, nonthermal response for a finite amount of ~initial! laboratory time @cf. Eq. ~47!#.

Effective space Time interval Phonons detected

Laboratory dt Yes ~nonthermal!de Sitter dt5dt/Bb Yes ~thermal!Adiabatic dta5dt/Bb2 No

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@25# E. Zaremba, Phys. Rev. A 57, 518 ~1998!.@26# P.O. Fedichev and U.R. Fischer, Phys. Rev. A 69, 033602

~2004!.@27# Vl.S. Dotsenko and V.A. Fateev, Nucl. Phys. B240, 312

~1984!.@28# W. de Sitter, Mon. Not. R. Astron. Soc. 78, 3 ~1917!.@29# A. Recati et al., cond-mat/0212413.

@30# P.O. Fedichev and U.R. Fischer, Phys. Rev. Lett. 91, 240407~2003!.

@31# S. Stringari, Phys. Rev. Lett. 77, 2360 ~1996!.@32# L.I. Men’shikov and A.N. Pinzul, Usp. Fiz. Nauk 165, 1077

~1995! @Phys. Usp. 38, 1031 ~1995!#.@33# A. Gorlitz et al., Phys. Rev. Lett. 87, 130402 ~2001!.

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Appendix F

Petr O. Fedichev and Uwe R. Fischer

“Cosmological” quasiparticle production in harmonically

trapped superfluid gases”

Phys. Rev. A 69, 033602 (2004)

101

“Cosmological” quasiparticle production in harmonically trapped superfluid gases

Petr O. Fedichev1,2 and Uwe R. Fischer1

1Leopold-Franzens-Universität Innsbruck, Institut für Theoretische Physik, Technikerstrasse 25, A-6020 Innsbruck, Austria2Russian Research Center Kurchatov Institute, Kurchatov Square, 123182 Moscow, Russia

(Received 9 September 2003; published 8 March 2004)

We show that a variety of cosmologically motivated effective quasiparticle space-times can be produced inharmonically trapped superfluid Bose and Fermi gases. We study the analog of cosmological particle produc-tion in these effective space-times, induced by trapping potentials and coupling constants possessing an arbi-trary time dependence. The WKB probabilities for phonon creation from the superfluid vacuum are calculated,and an experimental procedure to detect quasiparticle production by measuring density-density correlationfunctions is proposed.

DOI: 10.1103/PhysRevA.69.033602 PACS number(s): 03.75.Kk, 98.80.Es

I. INTRODUCTION

In a gravitational field with explicit time dependence inthe metric, particles and antiparticles can be simultaneouslycreated by quantum fluctuations from the vacuum. By theuncertainty principle, the time scale of the system’s evolutiondictates the typical energy of the particles produced [1]. Theprocess of cosmological particle production, whosecondensed-matter analog we shall consider here, is poten-tially relevant in the expanding early universe, in whichphonons experience an acoustic geometry; as a consequence,the expansion of the universe could generate density wavesgrowing into galaxies [2].

Attention has been of late focused on condensed-matteranalogs of the curved space-times familiar from gravity, pri-marily due to their conceptual simplicity and realizability inthe laboratory [3–10]. Condensed-matter systems lend them-selves for an exploration of kinematical properties of curvedspace-times and, in particular, provide a testbed to study theeffects of a well-defined and controlled “trans-Planckian”physics, i.e., atomic many-body physics on a microscopicscale, on low-energy quantum effects such as Hawking ra-diation [11] and cosmological particle production. In thepresent paper, we investigate quantum fields propagating oneffective curved space-times backgrounds, for the case ofharmonically trapped, dilute superfluid gases with possiblytime varying particle interactions. For a perfect, irrotationalliquid, described by Euler and continuity equations, it wasrecognized by Unruh [12] that the action of fluctuations ofthe velocity potential F, around a spatially inhomogeneousand time-dependent background, can be identified with theaction of a minimally coupled scalar field according to

S =E dtd3x1

2kF− S ]F

]t− v · = FD2

+ c2s=Fd2G;

12E dtd3xÎ− ggmn

]mF]nF . s1d

Here, v is the background velocity, 1 /k the compressibility,and c the speed of sound of the liquid. We use the summa-tion convention over equal indices, unless indicated other-wise. The quantities gmn are the contravariant components of

the effective metric tensor related to its covariant compo-nents by gbngna=db

a, and g;det gmn is the determinant ofthe metric tensor. The action s1d leads to the “relativistic”scalar wave equation

hF ;1

Î− g]msÎ− g gmn

]nFd = 0. s2d

In general, the effects of quantum fluctuations described bythe quantum version of Eq. s1d are very small and can hardlybe observed because of finite-temperature and dissipation ef-fects. Therefore atomic superfluids, where both extremelysmall temperatures and dissipationless flows are possible, at-tract growing interest for an emerging research field of “ex-perimental cosmology.”

In the following, we study how various curved space-times can be implemented in harmonically trapped superfluidBose and Fermi gases. As a concrete example, we show howde Sitter and Friedmann-Robertson-Walker (FRW) universescan be “recreated” in superfluid gases. We analyze the qua-siparticle production probabilities, leading to a thermal spec-trum in the WKB approximation, and discuss an experimen-tal procedure to observe and characterize the excitationsproduced.

II. QUASIPARTICLE METRIC TENSORS INHARMONICALLY TRAPPED SUPERFLUIDS

A. Superfluid action

The hydrodynamic, i.e., long-wavelength action of atrapped superfluid is generally given by

S =E dtd3xFr]f

]t+

r

2s=fd2 + esrd + VtraprG , s3d

where the external harmonic potential Vtrapsx , td= 12 fvx

2stdx2

+vy2stdy2+vz

2stdz2g is characterized by the three frequen-cies vi, i=x ,y ,z. The trapping frequencies are assumed tobe time dependent in an arbitrary manner fwe can alsoconceive of making vi

2std effectively negative by “turningover” the potential, see Sec. IIIg. In the above action, thequantity r plays the role of a response or stiffness coeffi-

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cient to gradients of f, and equals the total fluid density atabsolute zero; the equation of state of the superfluid is givenby the energy density functional e=esrd. sNote that we leaveout an overall minus sign in the definition of S.d We gener-ally set "=m=1. The action entails the existence of a veloc-ity potential f, such that the vorticity is zero except on sin-gular lines, and ensures the validity of Euler and continuityequations for the superfluid velocity v= =f. The identifica-tion of f with the phase of a complex “order parameter”si.e., the direction of a unit vector in the plane of some ab-stract spaced leads to the quantization of circulation, becausef is then defined only modulo multiples of 2p. Finally, theabove action implies the conjugateness of phase and densityquantum variables f15g:

frsxd,fsx8dg = idsx − x8d . s4d

These properties, taken together, constitute the canonicaldefinition of a superfluid at T=0 f16,17g. Therefore, Eq. s3drepresents the universal action of a simple scalar superfluidat absolute zero made up of elementary bosonic or fermionicatoms, independent of a particular microscopic model.

The simplest example for the equation of state is that of aweakly interacting Bose gas, with eB= 1

2gr2, where g is thecoupling constant, g=4pas, with as the s-wave scatteringlength, characterizing pair collisions of atoms. The scatteringlength can be tuned using external magnetic fields [13]. An-other example is a two-component Fermi gas with attractiveinteractions between atoms of different hyperfine species[14]. The ground state of such a gas is superfluid (in thesimplest version, it is the BCS state of a scalar superfluidwith s-wave pairing), and since the interactions are weak, theBCS gap is small and the equation of state (to exponentialaccuracy) coincides with that of a free Fermi gas: eF= fs3p2d2/3 /10gr5/3. To consider all possible cases whichhave a power-law density dependence of the equation ofstate, in a generic way, we write

esrd = Agbrg, s5d

where A is a numerical constant. That is, b=1,g=2 for adilute Bose gas, and b=0,g=5/3 for noninteracting two-component fermions.

In our present context, an important quantity characteriz-ing a superfluid is the so-called “Planckian” energy scale,EPl, i.e., the frequency beyond which the spectrum of theexcitations above the superfluid ground state ceases to bephononic and (pseudo) Lorentz invariance is broken. For aweakly interacting Bose gas EPl,gr, of the order of themean interparticle interaction. In a BCS superfluid, EPl isdetermined by the BCS gap: EPl,r2/3 expf−1/ sugur1/3dg. Thecomplete analogy with a quantum field theory on a fixedcurved space-time background given by Eq. (1) only exists ifall time scales t0, describing the evolution of the superfluid,are much larger than the “Planck time”: t0@1/EPl.

B. Scaling transformation for Bose and Fermi superfluids

Our approach in the following is based on the so-calledscaling transformation [18–21] to describe the expansion andcontraction of the gas under time-dependent variations of the

trapping frequencies. It is by now a well-established fact thatthe hydrodynamic solution for density and velocity of mo-tion for such a system may be obtained from a given initialsolution by a scaling procedure both in the bosonic [18–20]

as well as in the fermionic case [21]. Defining the scaledcoordinate vector xb=eixi /bi, density and velocity are givenby the scaling transformations [18]:

rsx,td ⇒rsxbd

V, s6d

fsx,td ⇒bi

2bixi

2 + fsxb,td . s7d

The sdimensionlessd scaling volume V=pi bi in the densitys6d is dictated by particle conservation.

Introducing a new “scaling time” variable by

dts

dt=

fg/gs0dgb

Vg−1 , s8d

we rewrite the action s3d in the form

S =E dtsd3xbFr

]

]tsf +

r

2Fistsds=bifd2 + esrd

+ Vtrapsxb,0drG , s9d

where the ts dependent scaling factors are

Fistsd =Vg−1

bi2fg/gs0dgb

=1

bi2

dtdts

, s10d

and esrd=esrdug=gs0d; =bi;] /]xbi. The rescaled density r hasno explicit ts dependence swhereas it has explicit depen-dence on the lab time td, and coincides with the equilibriumcondensate density profile in the scaling coordinate xb.Where any confusion might arise, we will generally desig-nate scaling variables with a tilde to clearly distinguish themfrom lab-frame variables.

For the relation (10) between the scaling factors and bi ,gto hold true, we must impose the following equations ofmotion for the scaling parameters bi:

bi + vi2stdbi =

fg/gs0dgbv02

Vg−1bi. s11d

They need to be solved with the initial conditions bi=1 andbi=0. Note that here no summation convention is used in thesecond term on the left-hand side. For a sufficiently largecloud, the stationary background solution can be found fromthe Thomas-Fermi density profile. It is given by using that−df /dts=m equals the initial chemical potential, and

de

dr= m − Vtrapsxb,0d . s12d

The part of the action quadratic in the fluctuations is ob-tained to be

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Ss2d =E dtsd3xbFdr

]

]tsdf +

r

2Fis=bidfd2 +

12

kdr2G ,

s13d

where dr= r− r0 and df= f+mts. The rescaled bulk com-pressional modulus sinverse compressibilityd k=d2e /d2rdoes not depend on the time ts, and is identical to gs0d in thebosonic case. After integrating out the density fluctuations,we obtain the effective action for the rescaled phase variable

Ss2d =E dtsd3xb

1

2kF− S ]

]tsdfD2

+ c2Fis=bidfd2G ,

s14d

where the squared scaling speed of sound c2= krsxbd. Usingthe identification with a minimally coupled scalar field,analogous to the one performed in the second line of Eq. s1d,the line element in the scaling variables reads

ds2 =c

kÎFxFyFzf− c2dts

2 + Fi−1dxbi

2 g . s15d

The line element takes a particularly simple form for an iso-tropic superfluid Fermi gas, where g=5/3 and thus all Fi=1, leading to dts /dt=b−2. We note that even for this simplecase, the metric defined by Eq. s15d is not trivial, since tsand t are different, and both c and k depend on the radialscaling coordinate rb. We will see below that in the case thatFi=1 the scaling transformation is exact, and that thereforeno quasiparticle creation occurs in the scaling variable basis,i.e., there is no mixing of negative and positive frequencyparts in the time ts squasiparticle creation can take place withreference to the lab-frame where the time coordinate is t,though, and a lab detector will still see that quasiparticles are“created”d.

Identifying df with F, and going back from scaling co-ordinates to laboratory-frame variables, we recover the ac-tion (1), with v= sbi /bidriei, and the line element, which is ofPainlevé-Gullstrand type, reads [24,25]

ds2 =ck

f− sc2 − v2ddt2 − 2vidxidt + dxi

2g , s16d

where c2=kr is the squared instantaneous speed of sound.We now assume that space is spherically symmetric, i.e., v

has a radial component vr only, that vr /c= fsrd holds, andfurthermore that c=cstd is a function of time only. We firstapply the transformation c0dt=cstddt, where c0 is some con-stant sinitiald sound speed, connecting the laboratory time tto the time variable t. This results in the line element ds2

= sc /kdf−c02s1− f2dd t 2−2fc0dtdz+dr2+r2dV2g. We then em-

ploy a second transformation c0dt=c0dt+ fdr / s1− f2d f26g,to bring the metric into the form

ds2 =ckF− f1 − f2srdgc0

2dt2 +1

1 − f2srddr2 + r2dV2G .

s17d

The metric in the above form facilitates comparison withmetric tensors in spherically symmetric space-times writtenin their standard form. E.g., if f2srd=2M /r is chosen, thisline element is conformally equivalent to the Schwarzschildmetric, the asymptotically flat vacuum solution of the Ein-stein equations around a spherically symmetric body withtotal mass M f27g.

III. CREATING de SITTER AND FRIEDMANN-ROBERTSON-WALKER UNIVERSES

The equation of state for Bose superfluids contains theinteratomic interaction. Therefore, by varying this interac-tion, possibly together with the trapping frequencies, expand-ing clouds of Bose atoms allow for the simulation of a largeset of cosmological space-times. We begin by discussing theso-called de Sitter universe, which is a solution of thevacuum Einstein equations characterized by the line element[27,28]

ds2 = − c2S1 −L

3r2Ddt2 + S1 −

L

3r2D−1

dr2 + r2dV2,

s18d

with L being the cosmological constant ; energy density ofthe vacuum. Up to the conformal factor c /g, the metric s17dcoincides with the de Sitter metric s18d, provided we requirethat vr

2 /c2=Lr2 /3 and that the speed of sound is a constantin space and time. The speed of sound in the center of thecloud is time independent if

c2 = c2 = const ⇔ gstd/gs0d = b3std . s19d

Close to the center of the condensate, c is, in addition, prac-tically spatially independent. Using Eq. s7d, we find that L

=const provided b~expfltg, with l=cÎL /3. This expo-nential expansion of the cloud can sasymptoticallyd beachieved if we turn over the potential, making it expel theparticles rather than trapping them: v2st→`d=−l2. Thede Sitter horizon, where vr=c, is stationary and situated atrh=cs0d /l, which is well inside the expanding cloud pro-vided l@v0, where v0 is the initial trap frequency.

The experimental sequence leading to “condensate infla-tion” is schematically depicted in Fig. 1. We assume that theexperiment can be done with one trapped (low-field seeking)

and one untrapped (high-field seeking) hyperfine componentof the same atomic species. We start from a sufficiently largeBose-Einstein-condensed cloud at small (effectively zero)

temperature with all atoms being in the trapped state. Then,we transfer all the atoms to the untrapped state, by flippingthe sign of the trapping potential. At the same time, we rampup the interaction strength, according to condition (19), usinga suitable Feshbach resonance [13]. As a result of the simul-taneous action of the inverted parabolic potential and theincreasing interaction energy, the gas cloud experiences a

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rapid exponential expansion, representing the analog of cos-mological inflation.

In fact, Eq. (19) defines a broad class of Bose superfluideffective space-times. In the present case of isotropic expan-sion with bi=b, we have t=ts and, in scaling variables, weobtain up to a conformal factor a Friedmann-Robertson-Walker metric:

ds2 =c

kb3 f− c2dt2 + b2drb2 + b2rb

2dV2g . s20d

According to the above form of the metric, the quantity bplays the role of the scaling parameter not only in ourcondensed-matter context, but can be interpreted equallywell as the scale factor in the expansion of the universe, withH; b /b the Hubble parameter. As demonstrated above, ex-ponential growth of b, with constant H, corresponds to ex-ponential inflation f22g. The present setup also allows for thesimulation of power-law inflation f22,23g, with bstd~ td. The“Hubble parameter” H changes for all exponents g inverselyproportional to time t, H~1/ t, and the exponent d=1/2 cor-responds to a “radiation dominated” universe, while the ex-ponent d=2/3 corresponds to a “matter dominated” uni-verse. An isotropically trapped expanding superfluid gas thusmodels an isotropic expanding universe. Generically, we canmodel anisotropic universes with Eq. s15d, with scaling fac-tors which are different in different spatial directions.

While this experiment is feasible in principle, increasingthe interaction dramatically increases three-body losses aswell, whose total rate scales like g4r2. This complication canbe avoided, by switching to effectively lower-dimensionalsystems; e.g., a 1+1D analog (where 1+1D stands for spa-tially and temporally one-dimensional) of a de Sitter uni-verse can be achieved for quasi-1D excitations in a linearlyexpanding elongated Bose condensate, without changing theinteraction [29]. Another possibility is to use superfluidFermi gases. We have, in the isotropic case, F=1, dts=dt /b2, and the metric may be written in the form

ds2 =c

b2kf− c2dt2 + b2drb

2 + b2rb2dV2g , s21d

where we defined a scale factor b;b2. Performing experi-ments in superfluid Fermi gases has the advantage that three-

body losses are strongly suppressed by Fermi statistics f30g.

IV. COSMOLOGICAL PARTICLEPRODUCTION ANALOG

Now we turn to describe the evolution of quantum fluc-tuations, on top of the classical (mean-field) hydrodynamicsolutions described above. The equation of motion for thephase fluctuations can be obtained after variation of the ac-tion (14),

]2

]ts2df − k=biS c2

kFistsd=bidfD = 0. s22d

Phrased in curved space-time language, the above equation isthe minimally coupled massless scalar wave equation for dr,analogous to Eq. s2d, with the metric s15d.

Consider for simplicity the isotropic case,

Fi ; F =b3g−5

fg/gs0dgb . s23d

The solution for the full quantum field reads

df = onÎ k

2Ven

fnsxbdfanxn + an†xn

*g , s24d

where V is the initial Thomas-Fermi volume of the cloud, theoperators an san

†d annihilate screated phonon excitations in theinitial vacuum state, and the mode functions xn satisfy

d2

dts2xn + Fstsden

2xn = 0. s25d

The initial conditions xnsts→−`d=expf−ientsg are selectedsuch that Eq. s24d, at t→−`, describes the phase fluctua-tions in a static trapped superfluid in its ground state. Inquantum-field-theory sQFTd language this ensures that alaboratory frame detector does not detect quasiparticles att→−`. Hereafter we define the “scaling vacuum” to bethe quantum state annihilated by the operators an, whereour choice of the initial conditions guarantees that theinitial superfluid vacuum and the scaling vacuum coincideat t→−`.

The case when all Fi;1 is remarkably special: In thiscase Eq. (25) does not depend on the superfluid evolution,and thus the quantum state of the excitations remains un-changed. As we have seen above, this indeed happens in thecase of an isotropic Fermi superfluid. Another example is a2D isotropic dilute Bose gas with constant particle interac-tion [19]. In these cases the scaling transformation is exact,both for the condensate and the excitations. In the languageof QFT this amounts to the fact that there is no particleproduction in the scaling basis, since the scaling solution isconstructed from eigenfunctions of the (exactly conserved)

scaling transformation operator Fi (in other words the scalingvacuum is protected by an exact scaling invariance whichforbids frequency mixing). The fact that no excitations areproduced in the scaling basis does not mean that a lab detec-tor does not detect quasiparticles. The transformation from

FIG. 1. Exponential expansion of a two-component Bose-Einstein condensate with effective spin F=1/2, to create a de Sitterquasiparticle universe close to its center. The trapping potential isinverted by applying a radio frequency srfd pulse, which transfersthe atoms to their untrapped hyperfine state.

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the laboratory time t to the scaling time ts time is nontrivialand thus the phase of the functions xn is a complicated func-tion of the laboratory time t. In other words, the phase fieldgiven by Eq. (24), if coupled to a detector of the type con-sidered in Ref. [29], gives a nonzero response. This indeter-minacy of the vacuum state finds its counterpart in theUnruh-Davies effect in flat space-time [1,31] and its curvedspace-time generalization, the Gibbons-Hawking effect [32].

The description of a quantum field state in terms of par-ticles and antiparticles is based upon the separation of posi-tive and negative frequency parts. As we confine ourselves toa measurement involving the laboratory time variable t, thisdistinction is only possible if the asymptotic phase of xnfunctions is large and sufficiently quickly increases as afunction of t. Using a WKB approximation to the solutionsof Eq. (25), we find

limt→`

EtÎF

dts

dtdt Þ 0. s26d

The latter condition can be also called a “Trans-Planckiansafety condition” sTP conditiond, since if fulfilled it impliesthat an experiment in a lab-frame probing an energy scaleE0!EPl does not require information about solutions ofEq. s25d with en*EPl. For isotropic expansion of a 3DBose gas, Eq. s26d is equivalent to divergence ofedtg1/2 /b5/2 and is quite restrictive: For the FRW analogydiscussed above avoiding the divergence implies, accord-ing to Eq. s19d, that b should not grow faster than linearly.The TP condition s26d is based upon the WKB approxi-mation condition for Eq. s25d, leading to the requirementFstsd* sentsd

−2 for large ts shere * means “grows fasterthan”d. Substituting the latter condition into Eq. s26d wefind that the marginal WKB case corresponds to a loga-rithmically divergent integral in Eq. s26d. Thus the mar-ginal TP case corresponds to the marginal WKB case andvice versa.

Equation (25) is formally equivalent to scattering of anonrelativistic particle with energy en by a potential en

2f1−Fstsdg. The initial conditions correspond to a single particleper unit time incident on the potential barrier. Time depen-dence of the scaling factors leads to scattering of the particlesfrom the incoming wave and at ts→` the WKB solutionreads

xn =1

F1/4San expF− ienE dtsÎFG + bn expFienE dts

ÎFGD ,

s27d

where an is the transmission and bn the forward-scatteringamplitude. The coefficients an and bn are related via theparticle flux conservation condition:

uanu2 − ubnu2 = 1. s28d

In QFT language the latter condition is the Bogoliubov trans-formation normalization condition for a bosonic field. Thenumber of particles detected by a scaling time detector at restis measured by the absolute square ubnu2 swhich is propor-tional to the probability that the detector absorbs a quantumd,

and therefore Nn= ubnu2 can be interpreted as the number ofscaling basis quasiparticles created.

In the WKB approximation the amplitudes are connectedin a simple way:

bn = expf− en/2T0gan, s29d

where the inverse temperature is given by the integral

1T0

= ImFEC

ÎF dtsG , s30d

and C is the contour in the complex ts plane enclosing theclosest to the real axis singular point of the function Fstsdf33g. Together with Eq. s28d, this gives

Nn = ubnu2 =1

expfen/T0g − 1, s31d

i.e., adiabatic evolution of trapped gases leads to “cosmo-logical” quasiparticle creation with thermal occupation num-bers in the scaling basis. The temperature T0 depends on thedetails of the scaling evolution fsee the specific example inEq. s37d belowg.

Interestingly, the evolution of the scaling parameters bi,and therefore the nontrivial line element (15), can be gener-ated already in a nonexpanding cloud with time-dependentinteraction gstd. A similar experiment has been suggested inRef. [34], where time-dependent interactions were used tosimulate FRW cosmologies and quantum quasiparticle pro-duction. The difference to our approach is due to the fact thatthe authors of Ref. [34] consider a trap with very steep walls(effectively a hard-walled container), so that the density ofthe cloud does not change, and the superfluid velocity van-ishes everywhere at all times. In our setup, we are able toinduce cosmological quasiparticle production in a harmoni-cally trapped gas, by changing simultaneously the harmonictrapping and the interaction. The simplest case we can con-sider is to leave all bi=1, such as in a stationary Bose con-densate. We then create cosmological quasiparticles just bychanging g (using Feshbach resonances, cf., e.g., Ref. [13]),and accordingly change the trap frequencies vi=v (in theisotropic case). Following Eqs. (8), (10), and (11), we havethe simple relations

v2std

v02 =

gstdgs0d

=dts

dt=

1Fstd

. s32d

The metric associated with such a thermal quasiparticle uni-verse created by “shaking the trap” and simultaneouslychanging the interaction appropriately reads, from Eq. s15d,

ds2 =cÎFgs0d

f− c2Fdts2 + dxi

2g . s33d

Now, defining the scale factor of the BEC quasiparticle uni-verse by the relation

ascal2 ;

cgstd/gs0d

, s34d

we have, up to the sirrelevantd factor 1 /gs0d,

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ds2 = − ascal6 dts

2 + ascal2 dxi

2. s35d

This is the form of the metric employed in Ref. f34g, whereit was used to calculate cosmological quasiparticle produc-tion, inspired by a model of Parker f35g. Note that here anontrivial scale factor is induced without expanding thecloud. We see from relations s32d and s34d that the scalefactor ascal is in our harmonically trapped case simply pro-portional to the ratio of initial and instantaneous trappingfrequencies, ascal=Îc v0 /vstd.

In Ref. [34], a specific choice of the scale functionascalstsd was taken for the calculation of the quasiparticlecreation process,

ascal4 stsd =

ascal,i4 + ascal,f

4

2+

ascal,f4 − ascal,i

4

2tanhF ts

ts0G , s36d

where ascal,i and ascal,f are initial and final scale factors,respectively. In the adiabatic approximation, one obtains athermal spectrum f34,35g, with a temperature governed bythe inverse laboratory time scale t0~ts0 on which trappingfrequencies, interaction, and thus the scale factors change:

T0 =1

4pt0

ascal,i4 + ascal,f

4

ascal,f2 ascal,i

2 . s37d

This temperature is, according to Eq. s30d, determined by thesingular points of the tanh function in Eq. s36d. The fact thatthe spectrum is thermal is obtained in Ref. f34g for a specificexample with a certain form of the time-dependent interac-tion. We emphasize here that the thermal spectrum is a ge-neric feature of adiabatic evolution in harmonically trappedsuperfluid gases with temporally varying trapping potentialand interactions.

V. DETECTION BY MEASURING DENSITY-DENSITYCORRELATIONS

Although the solutions of the hydrodynamic equations areunique, their interpretation in terms of the number ofphonons in a given mode is subject to all the conceptualdifficulties encountered by the definition of particle states incurved space-times [1]. In Ref. [29], we have shown thatsimply by choosing a specific realization of a quasiparticle(phonon) detector one can observe thermal quantum “radia-tion” from a de Sitter horizon (the Gibbons-Hawking effect[32]) as a purely choice-of-observer related phenomenon,without energy transport or dissipation taking place insidethe liquid. Below, we confine ourselves to the standard (con-ventional) laboratory means of particle detection (a change-coupled detector camera detecting individual atoms, ratherthan phonons), and concentrate on uniquely definedlaboratory-frame observables, such as the lab-frame density-density correlations discussed in what follows.

The density-fluctuation operator is given by

dr = onÎ 1

2Venk

]

]tshfnsxbdfanxn + an

†xn*gj , s38d

so that the lab-frame density-density correlator Gsx12d= kdrsx1ddrsx2dl /V2, averaged over the initial state is, inthe isotropic case,

Gsx12d = on

en

2Vk

ÎFV2 fnSx1

bDfnSx2

bDF1 + 2ubnu2

+ 2ReHanbn* expF− 2ienE dts

ÎFGJG . s39d

Here the normalization condition s28d is used, and r12= ux12u= ux1−x2u! ux1u , ux2u.

The TP condition (26) ensures that the cross term propor-tional to ab* averages to zero at large times t. The term with1 in the square brackets describes the evolution of thevacuum fluctuations and the summation over n is cut off atthe Planckian energy scale: maxfeng,EPl= rgs0d. Accord-ingly, the corresponding correlation function decays atPlanckian distances r12 /b, c /EPl and is very short range.Subtracting the vacuum contribution, we obtain the follow-ing expression for the regularized correlator:

Gregsx12d = on

ÎFen

VV2kfnSx1

bDfnSx2

bDNn. s40d

We note that in QFT the regularization procedure does notfollow in a unique manner from the field theory itself, andcan be applied using different assumptions about the high-energy behavior of the excitations created from the funda-mental “ether.” Here, the spectrum sand origind of the TPexcitations is well known, and hence the above regulariza-tion of two-point correlation functions can always be strictlyjustified. This regularization procedure of course is not lim-ited to density-density correlators only. A similar techniquecan be used, for example, to find a regularized energy-momentum tensor.

To be more specific, consider a large Bose-Einstein-condensed gas cloud in the Thomas-Fermi limit. Then, closeto the center of the gas we can use WKB (plane-wave) func-tions fn, with energies ek= ck, and the regularized Greenfunction is given by

Gregsr12d =r2

V2Îrg3s0d GST0r12

cbD , s41d

where the function

G =1

2p2EPl2 E

0

` sinskr/bd

r/bNkk

2dk . s42d

The function G reaches its maximum for r12=0, so that thesignal to noise ratio is maximally

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Gregs0d

r2 , Îras3s0dS T0

EPlD4

. s43d

The above discussion shows that even from a small “noise”signal, one can extract the relevant features of the quantumstate of the gas cloud sfor a more detailed discussion cf. Ref.f36gd. In order to be measurable, the quantity s43d has to beof the order of a few percent. This can in principle beachieved by using initially dense clouds with strong interpar-ticle interactions. Finally, we mention that a similar, i.e.,velocity-velocity instead of density-density noise correlationfunction has already been measured in the experiments ofRef. f37g.

VI. CONCLUSION

In the present investigation, we have derived the generalscaling equations for harmonically trapped superfluid Boseand Fermi gases, and related these, in particular, to quasipar-ticle metric tensors of the de Sitter and Friedmann-Robertson-Walker type, familiar from a cosmological con-text. The quasiparticle creation in a harmonically trappedsuperfluid gas, by changing interaction and trapping simulta-neously in an appropriate manner, can therefore be describedin a general framework, and be interpreted to be analogous tothe particle creation occurring during rapid expansion of thecosmos. In particular, it was found that for a readily experi-mentally available case, the harmonically trapped, dilute su-perfluid Bose gas, a FRW-type metric can be induced if trap-

ping frequency v and interaction coupling g are changedsuch that v2std~gstd, without expanding the gas. The cosmo-logical scale factor in this case is inversely proportional tothe trap frequency, ascalstd~1/vstd.

If the frequency mixing leading to quasiparticle creationcan be described in the WKB approximation, generally athermal distribution is found, where the temperature is deter-mined by the singular points of the scaling factors given byEq. (10), in the complex plane of scaling time ts.

We finally stress that, in contrast to a typical cosmologicalcalculation, hydrodynamic fluctuations in a laboratory ex-periment always have a well-defined initial state in the lab-frame, with time coordinate t. Therefore, ambiguities of thefinal quantum state as regards the dependence of its particlecontent on the initial conditions imposed on the “vacuum”can be ruled out: There exists the preferred lab-framevacuum, uniquely prescribing the initial particle content ofthe quantum field.

ACKNOWLEDGMENTS

P.O.F. was supported by the Austrian Science FoundationFWF and the Russian Foundation for Basic Research, andU.R.F. by the FWF. They both were supported by the ESFProgramme “Cosmology in the Laboratory,” and gratefullyacknowledge the hospitality extended to them during the Bil-bao workshop. We thank J. I. Cirac, U. Leonhardt, R. Paren-tani. R. Schützhold, M. Visser, G.E. Volovik, and P. Zollerfor helpful correspondence and discussions.

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[2] R. K. Sachs and A. M. Wolfe, Astrophys. J. 147, 73 (1967); Z.A. Golda and A. Woszczyna, Phys. Lett. A 310, 357 (2003).

[3] G. E. Volovik, Phys. Rep. 351, 195 (2001); The Universe in aHelium Droplet (Oxford University Press, Oxford, 2003).

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[5] M. Visser, Phys. Rev. Lett. 80, 3436 (1998); Int. J. Mod. Phys.D 12, 649 (2003).

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[7] C. Barceló, S. Liberati, and M. Visser, Class. Quantum Grav.18, 1137 (2001); Int. J. Mod. Phys. A 18, 3735 (2003).

[8] U. Leonhardt, T. Kiss, and P. Öhberg, Phys. Rev. A 67,033602 (2003); J. Opt. B: Quantum Semiclassical Opt. 5, S42(2003).

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[10] R. Schützhold and W. G. Unruh, Phys. Rev. D 66, 044019(2002); W. G. Unruh and R. Schützhold, ibid. 68, 024008(2003).

[11] S. W. Hawking, Nature (London) 248, 30 (1974); Commun.Math. Phys. 43, 199 (1975).

[12] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). An earlierderivation of Unruh’s form of the metric corresponding to non-

relativistic hydrodynamics can be found in A. Trautman, Com-parison of Newtonian and Relativistic Theories of Spacetime,Perspectives in Geometry and Relativity (Indiana UniversityPress, Bloomington, 1966).

[13] S. Inouye et al., Nature (London) 392, 151 (1998); S. L. Cor-nish et al., Phys. Rev. Lett. 85, 1795 (2000); A. Marte et al.,ibid. 89, 283202 (2002).

[14] K. M. O’Hara et al., Science 298, 2179 (2002); L. Pitaevskiand S. Stringari, ibid. 298, 2144 (2002).

[15] P. W. Anderson, Rev. Mod. Phys. 38, 298 (1966).[16] I. M. Khalatnikov, An Introduction to the Theory of Superflu-

idity (Addison-Wesley, Reading, MA, 1965).[17] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001); 75, 1083(E)

(2003).[18] Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. A

54, R1753 (1996); Y. Castin and R. Dum, Phys. Rev. Lett. 77,5315 (1996).

[19] L. P. Pitaevski and A. Rosch, Phys. Rev. A 55, R853 (1997).[20] Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev.

Lett. 79, 2604 (1997).[21] C. Menotti, P. Pedri, and S. Stringari, Phys. Rev. Lett. 89,

250402 (2002).[22] A. H. Guth, Phys. Rev. D 23, 347 (1981); a recent concise

review of inflationary cosmology is contained in I. I. Tkachev,e-print hep-ph/0112136.

[23] G. E. Volovik, e-print gr-qc/9809081.

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[24] M. Visser, Class. Quantum Grav. 15, 1767 (1998).[25] P. Painlevé, C. R. Hebd. Seances Acad. Sci. 173, 677 (1921);

A. Gullstrand, Ark. Mat. Astron. Fys. 16, 1 (1922).[26] Note that this transformation is singular at a quasiparticle ho-

rizon, where uf u =1 (i.e., uvr u =c). The consequences of this factare explored in detail in U. R. Fischer and G. E. Volovik, Int.J. Mod. Phys. D 10, 57 (2001).

[27] S. Weinberg, Gravitation and Cosmology (Wiley, New York,1972).

[28] W. de Sitter, Mon. Not. R. Astron. Soc. 78, 3 (1917).[29] P. O. Fedichev and U. R. Fischer, Phys. Rev. Lett. 91, 240407

(2003); Phys. Rev. D (to be published), e-print cond-mat/0307200.

[30] D. S. Petrov, Phys. Rev. A 67, 010703(R) (2003).[31] W. G. Unruh, Phys. Rev. D 14, 870 (1976).[32] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738

(1977).[33] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Quantum

Mechanics: Non-Relativistic Theory, 3rd ed. (Butterworth-Heinemann, Newton, MA, 1981).

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0306226.[37] D. Hellweg et al., Phys. Rev. Lett. 91, 010406 (2003).

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Appendix G

Uwe R. Fischer and Gordon Baym

“Vortex states of rapidly rotating dilute Bose-Einstein

condensates”

Phys. Rev. Lett. 90, 140402 (2003)

110

Vortex States of Rapidly Rotating Dilute Bose-Einstein Condensates

Uwe R. Fischer and Gordon Baym

Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street, Urbana, Illinois 61801-3080(Received 21 November 2001; published 9 April 2003)

We show that, in the Thomas-Fermi regime, the cores of vortices in rotating dilute Bose-Einsteincondensates adjust in radius as the rotation velocity, , grows, thus precluding a phase transitionassociated with core overlap at high vortex density. In both a harmonic trap and a rotating hard-walledbucket, the core size approaches a limiting fraction of the intervortex spacing. At large rotation speeds,a system confined in a bucket develops, within Thomas-Fermi, a hole along the rotation axis, andeventually makes a transition to a giant vortex state with all the vorticity contained in the hole.

DOI: 10.1103/PhysRevLett.90.140402 PACS numbers: 03.75.Lm

The energetically favored state of rotating superfluidssuch as 4He [1] and the alkali gas Bose-Einstein conden-sates [2–4] is a triangular lattice of singly quantizedvortices [5]. Here we pose the question of the structureof the lattice and ultimate fate of the superfluid at rotationspeeds so large that a sizable fraction of the fluid is filledby the vortex cores. A type II superconductor with avortex array simply becomes normal at a high magneticfield, Hc2, where the vortex cores begin to overlap [6].While in a rotating Bose superfluid the cores of quantizedvortex lines (if of fixed size) would completely fill thesystem at an upper critical rotation speed c2 analogousto Hc2, a low temperature bulk bosonic system does nothave a simple normal phase to which it can return, andthus the problem [7]. In He II the critical rotation rates atwhich the vortex cores approach each other are unob-

servably large, c2 1012 rad= sec. In atomic diluteBose-Einstein condensates, by contrast, inertial forcescan be comparable to interatomic forces, and the approachto tightly packed vortex lattices is within reach of experi-ment; indeed current experiments in harmonicallytrapped gases achieve rotational velocities that are asignificant fraction of c2 [2–4,8,9].

We consider a Bose gas at zero temperature in theThomas-Fermi regime, with weak repulsive interactionsdescribed by an s-wave scattering length as, and integrateout (in the sense of the renormalization group) the shortrange structure on scales of the vortex separation. Thestructure of the cores and the destiny of the vortex latticeat high rotational speeds depend fundamentally on thegeometry of the container confining the fluid. We studyhere the modification of the lattice induced by high rota-tional speeds in both harmonic traps and in cylindricalhard-walled (square-well) buckets [10]. In a harmonictrap, where considerable work has been done on under-standing the states of the fluid [7,11], the radial trappingpotential, V? 1

2m!2

?r2, where r is the perpendicular

distance from the axis of rotation, dominates the cen-trifugal potential. As the rotational frequency ap-proaches !?, the system becomes quasi-two-dimensionaland should eventually enter a quantum Hall-like state

[12–15]; we do not treat this limit here. We find that ina trap the cross-sectional area occupied by the vortexcores grows until they fill a limiting fraction, 1=2, ofthe space of the system, and they never touch, even forrotational velocities arbitrarily close to !?. This resultgoes beyond that of [7], in that we take into account(albeit approximately) the modification of the vortexcore in the finite sized cell of the lattice, for a givenparticle number in the cell.

In a cylindrical bucket, on the other hand, where thecentrifugal force tends to throw the fluid against thewalls, we find that at a critical rotation speed, h

2p h=m0R, the fluid begins to develop a cylindrical

hole in the center which grows with increasing ; hereR is the bucket radius, 0 h2mg nn1=2 is the coherenceor healing length, g 4 h2as=m, and nn is the uniformdensity in a nonrotating cloud. In the bucket, the ratio ofthe cross-sectional core area to the area of the unit cell ofthe vortex lattice is independent of for > h, andgiven by 90=

2p

R. A phase transition associated withcore overlap at high vortex density is thus precluded foreither trapping geometry. In addition, as we show, asystem in a bucket eventually undergoes a phase transi-tion in the limit of large to a giant vortex phase withthe vorticity concentrated along the cylinder axis.

The scale of rotation velocities in a dilute gas in acontainer is set by 0 2g nn= h h=m20 c2. For0 ’ 0:2 m [2,3], 0 ’ 1:8 104 rad= sec in 87Rb and’ 6:9 104 rad= sec in 23Na.

We describe the ground state of the system with arotating vortex lattice by an order parameter, r, deter-mined by minimizing the energy, E0 ELz, in therotating frame by means of a variational calculation; hereLz is the component of the angular momentum of thesystem along the rotation axis. With h 1,

E0 Z

d3r

1

2mjirm r j2

Vr 1

2m2r2

j j2 1

2gj j4

; (1)

where Vr is the trapping potential. For a single vortex

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line of unit winding number on the cylindrical axis, r fr; zei in polar coordinates. To calculate E0

we employ a Wigner-Seitz approximation [16], replacingthe triangular shaped cells of the (2D) lattice by cylin-drical cells of equal radius ‘; the assumption of equal cellarea is consistent with observed lattices in traps [3].We assume straight vortex lines for simplicity, and thatthe lattice rotates at a uniform angular velocity v,related to the (2D) density of vortices nv 1=‘2 byv nv=m 1=m‘2;v is in general smaller than thestirring velocity by terms of order 1=mR2.

In general, in a cell is a unique function of thenumber of particles therein and can be constructed, e.g.,by the method of matched asymptotic expansion [17]. Itis sufficient here to use the simple approximation thatwithin a cell, labeled by i,

fx; z (

x=

nizp

; 0 x ;

nizp

; x ‘;(2)

where x is the (polar) radial coordinate measured fromthe center ri of the cell. To account for the densityvariation in the system we allow the mean densitynri; z within a cell centered on ri to depend on the radialposition of the cell and height. Ansatz (2) is consistentwith the exact boundary conditions, fx ! 0; z x, anddfx; z=dxjx‘ 0. While the core radius in a cellshould depend on the local density, for simplicity wetake to depend only on the mean density in the system,an approximation adequate to bring out the essentialphysics of rotation at high speeds in the Thomas-Fermiregime. As one approaches the mean-field quantum Hallregime [14], it becomes necessary to include possibledependence of f on the trapping frequency as well.

The mean density nri; z in the cell is

nri; z 1

‘2

Z ‘

0

d2rj j2 niz1 =2; (3)

where =‘2 1 is the ratio of the core area to thetransverse area of the unit cell of the lattice.

Writing the local velocity as v vr !v toevaluate the kinetic energy in each cell, as in [18], we find

E X

i

Ei

!Z

d3rnr1

2m2

vr2 a 1v Vr

1

2gnrb

; (4)

where we replace the sum over cells by a continuousintegration, with ri ! r. In Eq. (4) the kinetic energyper cell is a nr=m and the interaction energy per cellis gn2rb ‘2=2, where

a 1

nrZ ‘

0

dxx

f

x

2

@f

@x

2

1 ln

p

1 =2;

b 2

‘2n2rZ ‘

0

dxxf4x; z 1 2 =3

1 =22 ;(5)

a has a minimum at ’ 0:56, while b increasesslowly with . The first term in (4) is the energy of solidbody rotation of the lattice, while the term nrv

arises from vr !v. The result (4) is effectively theThomas-Fermi approximation for the coarse grainedstructure, described by nr, with the functions a and baccounting for the fine grained vortex structure.

The total angular momentum in the Wigner-Seitzapproximation is [18]

Lz Z

d3rnr1 vmr2; (6)

the sum of the angular momentum per vortex cell and thesolid body rotation contribution; Lz is independent of aand b. In toto,

E0 Z

d3rnr2

v

2v

mr2

a 1v Vr 1

2gnrb

:

(7)

Minimization of Eq. (7) with respect to v implies thatv a 1N=I, where I

R

d3rmr2nr is themoment of inertia. For large systems, mR2 1, weneglect the small difference of v and ; then

E0 Z

d3rnr

2

2mr2 Vr a 1

2gnrb

:

(8)

Minimization of Eq. (8) with respect to nr at fixed totalparticle number yields the Thomas-Fermi result,

nr ~ m2r2=2 Vr=gb; (9)

where is the chemical potential and ~ a. Ina trap with transverse frequency !? and longitudinalfrequency !z, the density has the usual form [19]

ntrapr ~=gb1 r=Rt2 z=Zt2; (10)

with R2t 2 ~=m=!2? 2 and Z2t 2 ~=m=!2

z .Then N 8ZtR

2t ~=15gb and

~

!? 1

2

15Nbas

m!?

p !z

!?

1 2

!2?

2=5

: (11)

By contrast, the density in a cylinder grows quadrati-cally with r, until at a critical velocity h where ~ 0,the system develops a cylindrical hole in the center; in atrap the quadratic trapping potential dominates, prevent-ing formation of a hole. Integrating Eq. (9) over the

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volume of the cylinder, we find that the hole begins todevelop at h

2bp

00=R 2b0=mR21=2. Then

ncylr

nn m2r2 R2=2=2gb; <h;m2r2 R2h=2gb; h;

(12)

where the radius of the hole is Rh R

1h=p

. Thephase of the order parameter winds by 2mR2h at theinner edge of the hole, corresponding to an array ofphantom vortices of density nv inside the hole.

The energy per particle in the rotating frame is

E0trap

N 5 ~

7 a; (13)

E0cyl

N mR22

4 a 0b

4 b0

12

h

4

; <h

E0cyl

N mR22

2 a mR2h

3; h:

(14)

The core size is determined by minimizing E0 withrespect to at fixed N and ; differentiating Eq. (8)directly we have

@a

@ ghni

2

@b

@ 0; (15)

where hni R

n2r=R

nr is a mean of the density in thesystem. [Note that were we, more correctly, to solve forthe vortex structure within each cell, then the integrationin hni in Eq. (15) would be only over the given cell.] In abucket the relevant regime is small , where b ’ 1 =3, and @a=@ ’ 1=2 , the minimum is thus at cyl 3=ghni. For <h, hni nn1 1

3=h4, and

cyl ’6=0

1 13=h4

; <h: (16)

For h, hni 43 nn=h, and

cyl 9

2

h

0

; h: (17)

The total area occupied by the vortex cores thus scales asthe ratio of zero temperature coherence length and systemsize and is independent of ; see Fig. 1(a).

In a trap, hni 4 ~=7gb; for small , 1 and@a=@ ’ 1=2 , so that grows with increasing as!2

? 21=5, as in [7]. However, as ! !?, ~ andtherefore hni ! 0, and the solution of (15) requires@a=@ ! 0, which occurs at ’ 1=2; this is the limitingvalue of . Quite generally, the fact that hni ! 0 as !!? implies that the solution must minimize the kineticenergy in Eq. (1). The cores can never overlap, and thesystem can, within mean-field theory, maintain a latticeof quantized vortices for all <!?. This behavior isshown in Fig. 1(b); in strongly elongated traps, as in acylinder, the system reaches a self-similar regime inwhich the core size scales with the intervortex spacing.

The rotation speeds needed to observe the increase ofcore size with are in principle accessible within currentharmonic trap experiments, prior to the onset of a quan-tum Hall state. In [3], N 5 107, !?=!z 4:2,=!? & 0:7, and !?=0 0:008; an increase in aspectratio by a factor of 24, corresponding to the solid curve inFig. 1(b), should be sufficient to observe the strong in-crease of the core size for & !?.

At sufficiently large, a dilute gas in a cylindrical trapmakes a transition to a giant vortex state, with vorticityconcentrated in the center rather than spread through-out the system. Such behavior is seen in numerical simu-lations in traps that rise faster than harmonic [20,21](and also in the presence of vortex pinning [22]).Dynamical formation of metastable giant vortices inharmonic traps with 7 to 60 units of vorticity is reportedin Ref. [23]. At the level of Thomas-Fermi, which isadequate to describe the giant vortex regime in largesystems (mR2 1) a giant vortex, described by order

parameter r

nGrp

ei), has an energy in the rotat-ing frame,

E0G

Z

d3rnGr

)2

2mr2 g

2nGr )

: (18)

(a)

0.002 0.006 0.0080.004

0.01

0.012

0.008

0.006

0.004

0.002

0

ζ

0.2 0.4 0.6 0.8 1

0.2

0.3

0.4

0.5

0.1

(b)

0

cyl

Ω/Ω0

ζ trapζ max = 0.56

Ω/ω⊥

FIG. 1. Vortex core area as a fraction of the Wigner-Seitz cell area: (a) vs=0 in a hard-walled bucket with R=0 500 (solidline) and R=0 1000 (dashed line); (b) vs =!? in an anisotropic cylindrically symmetric harmonic trap with parametersN 5 107, as=d? 0:01, and !?=!z 102 (solid line), 104 (dashed line), and 108 (dash-dotted line).

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Minimizing with respect to nr we find the analog ofEq. (9), nGr ~ )2=2mr2=g, where ~ ).In Thomas-Fermi, the giant vortex has a hole in thecenter of radius RG *R, where * )=R

2m ~

p. In

the limit of large ( h), where * ! 1, we expandin powers of 0=mR221=2 and find

E0G

N 1

2mR22

2mR20

p

3 0

18 : (19)

Comparing with the lattice energy, Eq. (14), and using(17), we find the transition to the giant vortex state at

G 0

91 2a9=

2p

R’ 0

91 lnR=0: (20)

In this Letter we have studied only the zero tempera-ture structure of rotating Bose gases in the long wave-length Thomas-Fermi regime. The full structure at finitetemperatures reflected in the phase diagram in the -Tplane (studied for a harmonically trapped gas in [24])may be rich. For example, at sufficiently large rotationvelocities and low temperatures such that thermally ex-cited motion perpendicular to the boundary becomesfrozen out, the fluid forced against the boundary of thebucket becomes effectively two dimensional. Under thesecircumstances the fluid can possibly undergo a transitionto a two-dimensional Kosterlitz-Thouless phase.

G. B. is grateful to Chris Pethick for discussions thatformulated this approach to the present problem and toJason Ho for pointing out the possible development of aKosterlitz-Thouless state at high rotational velocities.U. R. F. acknowledges financial support by the Deutsche

Forschungsgemeinschaft (FI 690/2-1). This research wassupported in part by NSF Grants No. PHY98-00978,No. PHY00-98353, and No. DMR99-86199.

[1] E. J. Yarmchuk, M. J.V. Gordon, and R. E. Packard, Phys.Rev. Lett. 43, 214 (1979).

[2] K.W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard,Phys. Rev. Lett. 84, 806 (2000); F. Chevy, K.W. Madison,and J. Dalibard, Phys. Rev. Lett. 85, 2223 (2000); K.W.Madison, F. Chevy, V. Bretin, and J. Dalibard, Phys. Rev.Lett. 86, 4443 (2001).

[3] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W.Ketterle, Science 292, 476 (2001). This paper clearlydepicts vortex cores filling a fraction of the samplearea in a dilute alkali gas Bose-Einstein condensate.

[4] C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, andW. Ketterle, Phys. Rev. Lett. 87, 210402 (2001).

[5] V. K. Tkachenko, Zh. Eksp. Teor. Fiz. 49, 1875 (1965)[Sov. Phys. JETP 22, 1282 (1966)]; Zh. Eksp. Teor. Fiz.50, 1573 (1966) [Sov. Phys. JETP 23, 1049 (1966)].

[6] A. A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957)[Sov. Phys. JETP 5, 1174 (1957)].

[7] A. L. Fetter, Phys. Rev. A 64, 063608 (2001).[8] P. C. Haljan, I. Coddington, P. Engels, and E. A. Cornell,

Phys. Rev. Lett. 87, 210403 (2001) and P. Engels, I.Coddington, P. C. Haljan, and E. A. Cornell, Phys. Rev.Lett. 89, 100403 (2002) infer rotation rates of up to 95%of the centrifugal limit of the transverse trap frequency.The latter reports evidence for a new phase in highlydistorted lattices in which the vortex cores appear tomerge into sheets. Whether such a phase can exist is notclear. Explanations in terms of dynamics of discretevortices have been given by A. A. Penckwitt, R. J.Ballagh, and C.W. Gardiner, Phys. Rev. Lett. 89,260402 (2002) and E. J. Mueller and T.-L. Ho, cond-mat/0210276. Furthermore, a sheet singularity is notpermitted by the (single component) Gross-Pitaevskiıequation [L. P. Pitaevskiı (private communication)].

[9] P. Rosenbusch, D. S. Petrov, S. Sinha, F. Chevy,V. Bretin,Y. Castin, G. Shlyapnikov, and J. Dalibard, Phys. Rev.Lett. 88, 250403 (2002).

[10] We study the cylindical bucket as an extreme tractablecase of a radial potential steeper than harmonic. Suchpotentials may be created using Laguerre-Gaussian(doughnut) laser beams; T. Kuga et al., Phys. Rev. Lett.78, 4713 (1997).

[11] A. D. Jackson and G. M. Kavoulakis, Phys. Rev. Lett. 85,2854 (2000); G. M. Kavoulakis, B. Mottelson, and C. J.Pethick, Phys. Rev. A 62, 063605 (2000); A. D. Jackson,G. M. Kavoulakis, B. Mottelson, and S. M. Reimann,Phys. Rev. Lett. 86, 945 (2001).

[12] N. K. Wilkin, J. M. F. Gunn, and R. A. Smith, Phys. Rev.Lett. 80, 2265 (1998); N. K. Wilkin and J. M. F. Gunn,Phys. Rev. Lett. 84, 6 (2000); N. R. Cooper, N. K. Wilkin,and J. M. F. Gunn, Phys. Rev. Lett. 87, 120405 (2001).

[13] S. Viefers, T. H. Hansson, and S. M. Reimann, Phys. Rev.A 62, 053604 (2000).

[14] T.-L. Ho, Phys. Rev. Lett. 87, 060403 (2001).[15] J. Sinova, C. B. Hanna, and A. H. MacDonald, Phys. Rev.

Lett. 89, 030403 (2002).[16] E. Wigner and F. Seitz, Phys. Rev. 43, 804 (1933); 46, 509

(1934).[17] B.Y. Rubinstein and L. M. Pismen, Physica (Amsterdam)

78D, 1 (1994); J. R. Anglin, Phys. Rev. A 65, 063611(2002); A. L. Fetter and A. A. Svidzinsky, J. Phys.Condens. Matter 13, R135 (2001).

[18] G. Baym and E. Chandler, J. Low Temp. Phys. 50, 57(1983); 62, 119 (1986).

[19] G. Baym and C. J. Pethick, Phys. Rev. Lett. 76, 6 (1996).[20] E. Lundh, Phys. Rev. A 65, 043604 (2002).[21] K. Kasamatsu, M. Tsubota, and M. Ueda, Phys. Rev. A

66, 053606 (2002).[22] T. P. Simula, S. M. M.Virtanen, and M. M. Salomaa, Phys.

Rev. A 65, 033614 (2002).[23] P. Engels, I. Coddington, P. C. Haljan, V. Schweikhard,

and E. A. Cornell, cond-mat/0301532.[24] S. Stringari, Phys. Rev. Lett. 82, 4371 (1999).

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Appendix H

Petr O. Fedichev, Uwe R. Fischer, and Alessio Recati

“Zero temperature damping of Bose-Einstein condensate

oscillations by vortex-antivortex pair creation”

Phys. Rev. A 68, 011602(R) (2003)

115

Zero-temperature damping of Bose-Einstein condensate oscillationsby vortex-antivortex pair creation

Petr O. Fedichev,1,2 Uwe R. Fischer,1 and Alessio Recati1,3

1Institut fur Theoretische Physik, Leopold-Franzens-Universitat Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria2Russian Research Center Kurchatov Institute, Kurchatov Square, 123182 Moscow, Russia

3Dipartimento di Fisica, Universita di Trento and BEC-INFM, I-38050 Povo, Italy~Received 21 January 2003; published 22 July 2003!

We investigate vortex-antivortex pair creation in a supersonically expanding and contracting quasi-two-dimensional Bose-Einstein condensate at zero temperature. For sufficiently large-amplitude condensate oscil-lations, pair production provides the leading dissipation mechanism. The condensate oscillations decay in anonexponential manner, and the dissipation rate depends strongly on the oscillation amplitude. These featuresallow one to distinguish the decay due to pair creation from other possible damping mechanisms. An experi-mental observation of the predicted oscillation behavior of the superfluid gas provides a direct confirmation ofthe hydrodynamical analogy of quantum electrodynamics and quantum vortex dynamics in two spatial dimen-sions.

DOI: 10.1103/PhysRevA.68.011602 PACS number~s!: 03.75.Kk, 03.75.Lm

The process of electron-positron pair creation is well es-tablished in quantum electrodynamics since the seminalwork of Schwinger @1#. Later on, it became apparent that thehydrodynamics of vortices in two-dimensional ~2D! super-fluids can be mapped onto 211D electrodynamics with vor-tices playing the role of charged particles, and phonons therole of photons @2#. In this analogy, the superfluid densityand the supercurrent act as the magnetic and electric fields onthe vortices whose circulation is the charge. The Schwingervacuum breakdown is a phenomenon occuring whenever theelectric field exceeds the magnetic field ~in cgs units!, whichcorresponds in the analogy to the instability of a supersonicflow with respect to the spontaneous creation of vortex-antivortex pairs from the superfluid vacuum.

Vortices in Bose-Einstein condensates have been observedand studied experimentally intensely in the past couple ofyears, e.g., in Refs. @3–7#. Here, we suggest an experiment ina quasi-2D Bose-condensed gas revealing the existence ofirreversible condensate dynamics at zero temperature as theresult of the Schwinger pair-creation instability. To argue thatvortex-antivortex pair creation is the dominant source of dis-sipation, we use the fact that a quasi-2D Bose-Einstein con-densate ~BEC! in a time-dependent harmonic trap has a pe-culiar feature: There is a time dependent transformation ~theso-called ‘‘scaling transformation’’!, using which the prob-lem can be exactly solved, because the time dependence caneffectively be removed from the Hamiltonian @8–10#. Thisscaling property also holds for linearized equations describ-ing the evolution of small density and phase perturbations~Bogoliubov quasiparticle excitations!, propagating on top ofthe moving superfluid. Therefore, initial perturbations cannotgrow and the instability mechanisms known from classicalhydrodynamics play no role @11#. This stability against per-turbations implies that at very low temperatures, condensateoscillations are practically undamped ~it has been measuredthat the quality factor Q*2000 @12#!.

On the other hand, the superfluid velocity in the scalingsolution grows linearly towards the condensate border, whilethe local density decreases. Therefore, the outer region of the

cloud is always supersonic and, according to the Landau cri-terion for superfluidity, can host instabilities. Vortices arenonlinear excitations above the superfluid ground state, sothat they are not protected by the scaling symmetry, whichholds for linear excitations. Spontaneous vortex-antivortexpair creation, analogous to the Schwinger process, is an in-trinsic instability mechanism and constitutes a source of dis-sipation already at zero temperature, without any need for asymmetry-breaking external perturbation.

In the following, we explicitly analyze the Schwinger in-stability of a supersonically expanding and contracting BECin a time-dependent quasi-2D harmonic trap. We show thatfor sufficiently large condensate oscillations, vortex-antivortex pair production provides the dominant dissipationmechanism. Furthermore, the condensate oscillations decayin a nonexponential manner and the dissipation rate dependsstrongly on the oscillation amplitude. These features allowone to distinguish experimentally the decay due to pair cre-ation from the previously studied damping mechanisms. Wenote that the suggested zero temperature damping mecha-nism is intrinsically different from that discussed in @10#,where the dissipation is due to the energy transfer from theradial condensate motion to the longitudinal modes in anelongated cylindrically symmetric condensate. This mecha-nism can only work if the condensate is sufficiently long,whereas we confine ourselves to the case of a quasi-2Dsample, for which any motion along the z axis is suppressed.

The analogy of 2D vortex dynamics with electrodynamicsis most easily established by noting that the expression forthe 2D Magnus force FM52prez3(X2vs) leads to theidentification of E5rvs3ez and B52rez with the ‘‘elec-tric’’ and ‘‘magnetic’’ fields, by comparing with the Lorentzforce FL5q(E1X3B). Here, X and vs are vortex and localsuperflow velocities, respectively, and r is the local density.The circulation (2p in our units with \5m51) is the‘‘charge’’ q ~cf., e.g., Refs. @13–15#!. The self-energy of a~widely separated! single vortex pair is 2E

v

052prL , with

L5ln(R/ac), where R is the size of the pair. We will use in

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this expression for the pair energy that the vortex core size ina dilute superfluid is given by ac51/cs , where cs is thespeed of sound. The inertial rest mass of a vortex, stemmingfrom compressibility, m

v5E

v

0/cs2 , is ~for large condensates!

of ‘‘electrodynamical’’ origin: It stems from the self-interaction of a moving vortex, with the long-range flow anddensity fields it induces inside the surrounding superfluidmedium. Since the ‘‘electromagnetic’’ fields ~the density andvelocity perturbations! represent ‘‘relativistic’’ particles~phonons!, the vortex mass diverges if the velocity of theaccelerated vortex approaches the speed of sound, in thesame manner in which the mass of a charged ultrarelativisticparticle diverges in conventional electrodynamics. We as-sume in what follows that other possible contributions to thevortex mass ~see, e.g., the backflow mass contribution dis-cussed in @16#! remain regular if the local superfluid velocityapproaches the speed of sound. These contributions aretherefore subdominant for relativistically moving vortices.

We consider a quasi-2D superfluid Bose gas in a time-dependent isotropic harmonic trapping potential V(x,t)5

12 v2(t)(x2

1y2), with x5(x ,y). It is a well-establishedfact that the hydrodynamic solution for density and velocityof motion in a harmonic potential with arbitrary time depen-dence may be obtained from a given initial solution by ascaling procedure @8,9#. Defining the scaled coordinate vec-tor rb5x/b , the rescaled density and velocity are given by

r~x,t !5

1

b2s~rb!5

r0

b2 S 12

rb2

R02D , ~1!

vs~x,t !5

bb

x. ~2!

Here, we assume the superfluid to be described initiallywithin the Thomas-Fermi ~TF! approximation ~that is, thecondensate is large enough to neglect the quantum pressure!,r0 is the initial central density, and R0 the initial TF radius,such that R5R(t)5b(t)R0 is the instantaneous TF radius ofthe cloud. The energy functional has in the TF approximationthe form

E~b , b !5

1

2b2E d2rbF S v21

b2

b2D b4rb2s1gs2G , ~3!

with g the interaction strength, which depends in the presentquasi-2D case on the tight confinement in the z direction andon the density of the condensate @17#. This leads to an effec-tive Hamiltonian for the dynamical variable b,

E~b , b !5S a

2b2

1

a

2v2~ t !b2

1

b

2b2D , ~4!

where a5pr0R04/6 and b5pr0

2gR02/3.

Consider a situation in which the external trap frequencyis changed from v in to v f!v in , on a time scale much lessthan the inverse initial trap frequency. As a consequence, thegas undergoes a large-amplitude monopole oscillation withfrequency 2v f @18#. At sufficiently low temperatures ~below

the Kosterlitz-Thouless temperature!, the initial state of thesuperfluid contains bound vortex-antivortex pairs, i.e., topo-logical excitations, which can be unbound by the action ofthe Magnus force in the ~time dependent! supersonic flowregion. Indeed, for an oscillating condensate, there exists aregion, the border of which is called horizon ~cf. Fig. 1!,where the superfluid velocity magnitude vs is larger than thelocal sound velocity cs5Agr . The speed of sound is ex-ceeded at the horizon radius

H~ t !5

b~ t !R0

Ag2~ t !11, ~5!

where g5A2 bb/v in .Beyond the horizon, the vortices and antivortices get ac-

celerated during condensate evolution and separate at localsuperflow velocities larger than that of sound. This is analo-gous to the Schwinger pair-creation process in quantum elec-trodynamics. It is important to recognize that the flow weconsider is inhomogeneous and time dependent by default.Consequently, the argument that there is no pair-creationpossible because one can always use the underlying Galileaninvariance to ‘‘transform away’’ the background flow, doesnot apply to our situation.

In a simple model of the 211D vacuum pair-creation in-stability, which exploits directly the analogy to Schwingerpair-creation in quantum electrodynamics, the pair produc-tion rate G per unit area can be written as @19#

G5

1

4p2cs2F

3/4(n51

`~21 !n11

n3/2expS 2

pn~Ev

0!2

AFD , ~6!

where we have defined F5E2cs22B2cs

4 and set, within loga-rithmic accuracy, the vortex pair size in E

v

0 equal to theThomas-Fermi radius of the condensate. The above relationholds for locally supersonic motion, i.e., if uEu/uBu.cs (F.0). The value of the prefactor in front of the exponential inthe above expression is subject to changes which are due tothe microscopic details of vortex motion. We display itsvalue, stemming from taking literally the analogy to quantumelectrodynamics also on the level of quantum fluctuations ~toone loop order!, for numerical concreteness. However, thebehavior of G for uEu/uBu*cs is dominated by the hydrody-

FIG. 1. An oscillating, cylindrically symmetric quasi-2D con-densate. The shaded region designates the region of space in whichthe speed of sound is exceeded by the oscillating condensate, andvortex pair creation takes place; H5H(t) is the horizon locationand R5R(t) the Thomas-Fermi radius of the condensate.

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namical exponent, whose value is independent of micro-scopic physics, and specifically by the n51 term in theabove sum.

Assuming that the vortex density is low, the energy dissi-pation rate e is obtained by multiplying Eq. ~6! by the restenergy of the widely separated vortices 2E

v

0 , and integratingover the area of the TF domain. This results in

e5

g1/2Lr03R0

2

2b4

g8

~g211 !13/4

FS L2

g1

Ag211

D , ~7!

where we introduced the function

F~l !5(n

~21 !n11

n3/2 E0

1dhh3/4~12h !9/4

3exp@2pnlA~12h !/h# . ~8!

Since g is proportional to bb , the Schwinger dissipation rate~7! can give rise to a measurable effect only if the condensateoscillation amplitude is sufficiently large, which implies v f!v in . In order to provide some analytical results, we con-sider a simple quasistationary perturbation-theory approach.Consequently, we assume that the dissipation rate is smalland therefore the energy of the system in Eq. ~3! is a slowlyvarying function within each oscillation period. Then, theequation of motion for the scaling parameter b can be foundfrom

ddt

E~b , b !52 e . ~9!

In the absence of dissipation ( e50), the range of b is be-tween bmin51 and bmax5vin /v f . Since bmin51!bmax , wecan approximately set bmin.0. One can then write g2

52(v f2/v in

2 )b2(bmax2

2b2). In a dilute gas, in the TF limit, theargument of F is large, L2

@gAg211, and the dynamical

equation ~9! for b takes the simpler form

b1v f2b2

v in2

b352

D

v in5 b7b4, ~10!

where the constant

D5

48

p8

g7/4AgN

~ ln@4AgN/p# !11/2 (n

~21 !n11

n8. ~11!

Using Eq. ~10!, bb can be expressed in terms of b only,b2b2

5v f2b2(bmax

22b2). The oscillation energy lost in a pe-

riod is then given by

IE5

7p

1024D

v f7

v in5 bmax

12 . ~12!

The energy decrease rate for bmax is obtained from the equa-tion

ddt

E~bmax!5

v f

pIE . ~13!

Thus, one obtains for the oscillation peak value the followingexpression:

bmax~ t !5

bmax~0 !

@11D8bmax10 ~0 !v f t#

1/10, ~14!

where D8535

512 (v f /v in)5D. Our perturbation-theory ap-

proach is valid as long as bmax10 (0)D8!1.

The damping of condensate oscillations due to vortex-antivortex pair creation is represented in Fig. 2, where weshow the numerical solution of the dynamical equation ~10!~gray solid line!, the approximate solution for the peak am-plitude ~black solid line!, and for comparison the free oscil-lation without pair creation ~dashed line!. The parametersused in the numerical integration for the plot are N5104,g51, and for the final trapping frequency v f50.1v in .These parameters are consistent with the argument of F be-ing large, L2

@gAg211, so that Eqs. ~10!–~14! hold. The

envelope bmax(t) is seen to decay very slowly and in a non-exponential manner, governed by the TF exponent 1

10 in Eq.~14!. For realistic parameters, we conclude from Fig. 2 thatan observable damping effect for the condensate oscillationsis obtained.

The scaling parameter evolution can be described by Eq.~9! only for sufficiently short times when the total density ofvortices produced is still low. At later times, the vortex-antivortex plasma can decrease the superfluid current in thesame way as the electron-positron plasma can screen theelectric field. This is an interesting collective effect, whichhowever requires a more elaborate treatment.

We described an intrinsic damping mechanism for large-amplitude condensate oscillations in a quasi-2D Bose gas at

FIG. 2. Damping of condensate oscillations due to vortex-antivortex pair creation, with N5104, g51, and v f50.1v in ,where the radius R is in units of the original Thomas-Fermi size R0.The black solid line is the envelope bmax from Eq. ~14!. The graysolid line is the damped breathing mode oscillation obtained fromnumerically solving Eq. ~10!. For comparison, the dashed line rep-resents the oscillation of the superfluid gas without pair creationtaking place.

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zero temperature. The dissipation originates from spontane-ous creation of vortex-antivortex pairs and depends on theshape and dynamics of the supersonic flow region. The re-sults we presented therefore depend strongly on the oscilla-tion amplitude. This feature can be used to distinguish theeffects of pair production from other possible dissipationmechanisms. The scaling solution not only exists for the dis-cussed monopole modes, but also for quadrupole oscilla-tions, so that, e.g., effects resulting from a rotating superfluidon the pair-creation process may be studied. Observation ofthe predicted oscillation behavior of the superfluid gas pro-vides a direct confirmation of the hydrodynamical analogy ofquantum electrodynamics and quantum vortex dynamics intwo spatial dimensions, and would put this analogy to its firstreal experimental test. Such confirmation would, then, givefurther motivation to the program of studying analogies be-tween high-energy physics, cosmology, and condensed-matter systems @20#.

The outlined mechanism for dissipation is not confined toquasi-2D samples. In strong elongated 3D condensates, thescaling solution also applies, and the vorticity is generated inthe form of vortex rings, with the total vorticity integratedover the sample volume still zero. However, as already men-tioned above, for a 3D condensate the effect of vortex ringcreation can be masked by possibly stronger dampingmechanisms, such as the parametric resonance discussed inRef. @10#.

We thank L. P. Pitaevskiı and G. E. Volovik for criticalremarks and helpful comments on the manuscript, and P.Zoller for discussions. P.O.F. has been supported by the Aus-trian Science Foundation FWF and the Russian Foundationfor Basic Research, U.R.F. by the FWF, and A.R. by theEuropean Union under Contract No. HPRN-CT-2000-00125.

@1# J. Schwinger, Phys. Rev. 82, 664 ~1951!.@2# V.N. Popov, Zh. Eksp. Teor. Fiz. 64, 672 ~1973! @Sov. Phys.

JETP 37, 341 ~1973!#.@3# K.W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys.

Rev. Lett. 84, 806 ~2000!.@4# C. Raman, J.R. Abo-Shaeer, J.M. Vogels, K. Xu, and W. Ket-

terle, Phys. Rev. Lett. 87, 210402 ~2001!.@5# J.R. Abo-Shaeer, C. Raman, J.M. Vogels, and W. Ketterle, Sci-

ence 292, 476 ~2001!.@6# P. Engels, I. Coddington, P.C. Haljan, and E.A. Cornell, Phys.

Rev. Lett. 89, 100403 ~2002!.@7# P. Rosenbusch et al., Phys. Rev. Lett. 88, 250403 ~2002!.@8# Yu. Kagan, E.L. Surkov, and G.V. Shlyapnikov, Phys. Rev. A

54, R1753 ~1996!.@9# Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 ~1996!.

@10# Yu. Kagan and L.A. Maksimov, Phys. Rev. A 64, 053610~2001!; e-print, cond-mat/0212377.

@11# L.D. Landau and E.M. Lifshitz, Fluid Mechanics ~PergamonPress, New York 1959!.

@12# F. Chevy, V. Bretin, P. Rosenbusch, K.W. Madison, and J. Dali-bard, Phys. Rev. Lett. 88, 250402 ~2002!.

@13# R.J. Donnelly, Quantized Vortices in Helium II ~CambridgeUniversity Press, Cambridge, 1991!.

@14# D.P. Arovas and J.A. Freire, Phys. Rev. B 55, 1068 ~1997!.@15# U.R. Fischer, Ann. Phys. ~N.Y.! 278, 62 ~1999!.@16# G. Baym and E. Chandler, J. Low Temp. Phys. 50, 57 ~1983!.@17# D.S. Petrov, M. Holzmann, and G.V. Shlyapnikov, Phys. Rev.

Lett. 84, 2551 ~2000!.@18# The existence of this mode in two spatial dimensions is related

to an underlying SO~2,1! symmetry of the system, see L.P.Pitaevskiı and A. Rosch, Phys. Rev. A 55, R853 ~1997!.

@19# R. Iengo and G. Jug, Phys. Rev. B 52, 7537 ~1995!.@20# G.E. Volovik, Phys. Rep. 351, 195 ~2001!; The Universe in a

Helium Droplet ~Oxford University Press, Oxford, 2003!.

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Appendix I

Uwe R. Fischer

“Existence of Long-Range Order for Trapped Interacting

Bosons”

Phys. Rev. Lett. 89, 280402 (2002)

120

Existence of Long-Range Order for Trapped Interacting Bosons

Uwe R. Fischer

Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080(Received 20 April 2002; published 31 December 2002)

We derive an inequality governing ‘‘long-range’’ order for a localized Bose-condensed state, relatingthe condensate fraction at a given temperature with effective curvature radius of the condensate andtotal particle number. For the specific example of a one-dimensional, harmonically trapped dilute Bosecondensate, it is shown that the inequality gives an explicit upper bound for the Thomas-Fermicondensate size which may be tested in current experiments.

DOI: 10.1103/PhysRevLett.89.280402 PACS numbers: 03.75.Fi

The classic Hohenberg theorem [1] employs the factthat, in the thermodynamic limit of an infinite system ofinteracting particles of bare mass m, the relation ( h kB 1)

Nk 12mT

k2n0n

(1)

for the occupation numbers Nk hbbykbbki of the plane

wave states enumerated by k ( 0) holds (theBogoliubov 1=k2 theorem [2]). The angled brackets hereand in what follows indicate a thermal ensemble (qua-si)average [2]; n0 and n are the condensate density andtotal density, respectively. The relation (1) entails that inone and two dimensions, a macroscopic occupation of asingle state, the condensate, is impossible: The inequalityleads to a contradiction in dimension D 2 due to the(infrared) divergence of the wave vector space integral of(1), which determines the number density of nonconden-sate atoms. Physically, long-range thermal fluctuationsdestroy the coherence expressed by the existence of thecondensate.

The question of the applicability of the Hohenbergtheorem to systems displaying long-range order has re-newed interest for trapped Bose-condensed vapors ofreduced dimensionality, which can now be realized usingvarious techniques [3–7]. In these systems, it is possibleto investigate in a controlled manner the influence offinite extension in different spatial directions on the ex-istence of a condensate, i.e., a macroscopically occupiedsingle state created by bby0 . For the noninteracting case, itis not difficult to show that there can exist a macroscopicoccupation of the ground state in a trapped gas of anydimension, as long as particle numbers remain finite,simply by evaluating the Bose-Einstein sums for theoccupation numbers (see, e.g., [8]). The difficulty comesin when interaction is turned on. An interacting gas has abehavior increasingly different from the ideal gas thelower the dimension of the system [9,10]. Furthermore,the classification of excited noncondensate states by planewaves is not the suitable one in a trapped gas: The con-densation, in the limit of large total particle number N,takes place primarily into a single particle state in coor-

dinate space [11], and condensate and total densities have(in principle arbitrary) spatial dependence, n0 ! n0r,n! nr. While a recent proof by Lieb and Seiringershows that ground state condensation is, in the thermody-namic limit, 100% into the state that minimizes theGross-Pitaevskiı energy functional in three as well as intwo dimensions [12], the question of the existence of acondensate for general energy functionals, at finitetemperatures and, in particular, in the generic one-dimensional (1D) case, remains open.

In what follows, we will take account of the phenome-non of spatially localized Bose-Einstein condensation, byderiving an inequality analogous to the integral of (1),with no explicit dependence on any Hamiltonian whichhas velocity independent interaction and trapping poten-tials. At a given finite temperature, the maximally al-lowed condensate fraction is related to an effectivecurvature radius of the condensate and the total particlenumber. The inequality thus allows for concrete state-ments on the limiting size of quantum coherent systemsof reduced dimensionality, and their realizability forgiven condensate and total density distributions. As anapplication to a specific example, we consider a 1D har-monically trapped, dilute Bose-Einstein condensate, andit is shown that the relation leads to bounds on the pa-rameters of a Thomas-Fermi cloud which may be verifiedin present experiments.

Our analysis is based upon the following decomposi-tion of field and density operators into condensate andnoncondensate parts:

r 0rbb0 r;r j0rj2 bby0 bb0 r;r

0r bby0 r H:c: yrr;(2)

where 0r is the single particle condensate wave func-tion (normalized to unity) and N0j0rj2 is the conden-sate density. The commutation relation,

r; yr0 r r0 0r 0r0; (3)

is obtained from the canonical commutation relations for

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the quantum fields r; yr. As a consequence of thiscommutator, the mixed response (the ‘‘anomalous’’ com-mutator) is taking the nonlocal form,

hr; r0i

N0p

0rj 0r0r0 r r0:

(4)

Here, after carrying out the commutator, hbb0i N0

p hbby0 i has been used, where this assignment is valid toO1=

N0p . The first term in the brackets on the right-

hand side (RHS) is due to the second term in the commu-tator (3). It vanishes if one takes the Bogoliubovprescription that bb0 and bby0 be replaced by c-numbers. Itis a well-established fact that this (standard) Bogoliubovapproach violates particle number conservation [13], andit becomes apparent below that neglecting the secondterm on the RHS of (3) would lead to numericallystrongly different predictions for the allowed condensatefraction.

We now make use of this form of the mixed response inthe Bogoliubov inequality [2], which reads

1

2hfAA; AAygi TjhCC; AAij2

hCC; HH; CCyi(5)

for any two operators AA and CC, where T is the tempera-ture. It is valid for any many-body quantum system, forwhich the indicated thermal (quasi)averages are welldefined. We choose the operators in relation (5) to be thesmeared excitation and total density operators,

AA Z

dDrfrr; (6)

CC Z

dDrgrr; (7)

where fr and gr are complex regularization kernels.Next, we derive a sum rule for the denominator on

the RHS of (5), analogous to the f-sum ruleR11 d!!Sk; ! Nk2=m for the dynamic structure

factor [14], but in coordinate space. Using

hr; HHr; r0i ihrr ||r; r0i 1

2mhr0 ryr0rr H:c: rr0 ryr0r H:c:i; (8)

we obtain that the (quasi)average of the double commutator equals

hCC; HH; CCyi 1

m

Z

dDrrgrg rnr; (9)

where nr hri hyr ri. Here, we used the continuity equation for the current density operator || yr ry=2im, together with the Heisenberg equation of motion for : ; HH i@t irr ||, thislast relation being valid for a Hamiltonian with no explicit velocity dependence in the interaction and externalpotentials [14].

We normalize the kernel fr to unity, i.e.,R

dDrjfrj2 1. The left-hand side of the Bogoliubov inequality (5) isthen bounded from above, using the Cauchy-Schwarz inequality, as follows:

1

2

Z

dDrZ

dDr0frf r0hfr; yr0gi Z

dDr0hyr0r0i a

2 N N0

a

2; (10)

where the quantity a 1 is given by a 1R

dDrR

dDr0frf r00r 0r0, and whereN N0

R

dDrhri R

dDrhyrri is the excited number of particles.Using the anomalous commutator (4), the Bogoliubov inequality (5) thus may be written in the following form:

N N0 a2mTN0

jR

dDr grfr0r R

dDrR

dDr0 grfr0j0rj20r0j2R

dDrrgrg rnr: (11)

It is important to recognize that the above relation is explicitly independent of the form of the excitation spectrum of thesystem, due to the relation (8). In particular, the strength of the interaction enters only implicitly in the form of thecondensate wave function and total density distribution. This is in contrast to previous considerations on ‘‘long-range’’order in Bose-Einstein condensates, employing correlation functions [10,15], where use was made of the excitationspectrum of the system, with Bogoliubov-type or WKB approximations.

Now choose the kernels to have the particular form

fkr

1=20 expik r r 2 D0

0 r =2 D0; gr

0r r 2 D0

0 r =2 D0; (12)

where0 is the volume of the domain D0 in which0 has finite support. Hence gr is also, automatically, normalizedto unity,

R

D0dDrjgj2

R

D0dDrj0j2 1. The choice of g is motivated by requiring that it contains only information

pertaining to (the spatial extension of) the condensate. The kernels fk are a set of functions in which we vary k such thatthe RHS of (11) is maximized and the inequality is assuming its strongest form.

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After introducing the kernels (12) into the Bogoliubov inequality (11), we have our primary result:

N N0 1 1

0j ~0kj22

TN020

j~nn0k ~0kR

D0dDr

0rj0rj2j2R

D0dDr0r 1

2mr

0rnr

: (13)

The Fourier transforms of single particle condensatedensity and wave function are defined to be ~nn0k R

dDr j0rj2 expik r and ~0k R

dDr0r expik r.

We define the effective radius of curvature of the con-densate to be

Rc

N

0

s Z

D0

dDr0rr 0rnr

!1=2: (14)

According to the above formula, Rc is obtained by weigh-ing a quantity proportional to the kinetic energy densityof the condensate with nr=N=0, i.e., the local densityrelative to the average density, and finally integrating overthe domain D0. Using this definition, relation (13) reads

1 F

F 1

N

2 R2c!2dB

Ck 1

2N01 j ~0kj2=0; (15)

where the functional Ck is given by

Ck ~nn0k ~0k

Z

D0

dDr 0rj0rj2

2

;

(16)

and the condensate fraction F N0=N; the de Broglie

thermal wavelength !dB

2 =mTp

. We stress that thevalue of Ck is strongly reduced as a consequence of thecommutation relation (3), which causes the second termunder the square in (16).

The relation (15) implies that, for temperatures close tozero, such that FR2c=!

2dB remains large [in which case the

second term in (15) is negligible], the approach ofNC11 F C1N N0 / T" to zero with a powerlaw has to fulfill " 1 for complete Bose-Einstein con-densation into a localized single state 0r to be pos-sible. This statement holds for arbitrary strength and formof the interaction, in any spatial dimension.

The above relations (13) and (15) are general. Thestrongest result, i.e., constraint on the system parameterswe may expect for a 1D system, in analogy to the originalHohenberg theorem [the 1=k infrared divergence of theintegral of (1) in one dimension]. To demonstrate themeaning of the relation (13) explicitly, we thus nowproceed by considering the example of axially symmet-ric, harmonically trapped gases in one dimension, in thecurrently experimentally accessible Thomas-Fermi limit.

Consider the Thomas-Fermi wave function,

0z

n0TFN

s

1 z2

Z2TF

1=2

: (17)

This mean-field form of 0 is valid if the 1D scatteringlength fulfills the strong coupling condition nja1Dj 1,

and the Thomas-Fermi parameters are ZTF 3Nd4z=ja1Dj1=3, n0TF 9=64N2ja1Dj=d4z1=3 [16]; the quanti-ties d?;z m!?;z1=2 are the harmonic oscillatorlengths. In the quasi-3D scattering limit, which has trans-verse length scale d? as, the 1D scattering length isgiven by a1D d2?=2as1 Cas=d?, with C 1:4603 [16]. We neglect the difference between nr andNj0rj2, i.e., take the limit of both sides of (13) tolinear order in 1 F. Evaluating the elementary inte-grals involved, we find that (x kZTF)

Cx

3

x2

sinx

x cosx

27 2

128

J1xx

2

; (18)

where J1x is a Bessel function of the first kind. We seefrom Fig. 1 that the function C is strongly peaked at itsglobal maximum km ’ 3:7Z1

TF (!m ’ 1:7ZTF, whereCkm ’ 1:54 102. Using Rc 4ZTF=3

2 p

, and ne-glecting the second term on the RHS of (15), we obtain, atk km,

ZTF!dB

6:0

N N0p

: (19)

We compare the above relation with the experiment ona 1D 23Na condensate in [3], where the (relatively mod-erate) parameters were as 2:8 103 (m, dz 11:2 (m (!z=2 3:5 Hz), d? 1:15 (m, and N 1:5 104, which result in ZTF ’ 150 (m. Inequality(19) becomes ZTF=!dB & 7:3 102

1 Fp

. Tempera-tures in this experiment have been of the order T 100 nK [17], which gives !dB ’ 1 (m and ZTF=!dB ’1:5 102. The parameters of [3] are thus consistentwith (19), provided F is not too close to unity [18].Note that (19) would be inconsistent with the parameters

1.5

1.25

1

0.75

0.5

0.25

2.50 5 7.5 10 12.5 15x = kZ TF

100× C(x )

FIG. 1. The function Ck on the RHS of (15) in the one-dimensional Thomas-Fermi case, from Eq. (18).

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in that experiment, were it not for the commutationrelation (3), due to imposing the canonical commutationrelations for the total quantum field. Neglecting the sec-ond term on the RHS of (3) leads to a decrease of the RHSof (19) by about 1 order of magnitude. In this sense,particle number conservation, which is violated by thestandard Bogoliubov prescription, is necessary for thecondensate to exist [19].

A further decrease of the aspect ratio!z=!? d2?=d2z

down to values of order 103 and lower is achievable, e.g.,in optical lattices [7], so that the condition (19) on the trapparameters should be experimentally verifiable withinpresent technology. When (19) ceases to be fulfilled, thesystem has to enter into a new (non-Thomas-Fermi) state.For very low densities, a ‘‘Tonks gas’’ (1D gas of impene-trable particles) may be formed, which has no condensate[10,16,20]. Another possibility is a condensate consistingof (overlapping) phase coherent droplets [21], resulting ina reduced value of Rc.

The primary result of the present investigation, in-equality (13), holds under the generic conditions thatthe Bogoliubov inequality (5) is valid and that the poten-tials in the Hamiltonian are independent of particle ve-locities. It is, furthermore, to be emphasized that theapplication of (13) is by no means limited to (real) groundstate forms of 0. It is also possible to employ thisrelation to examine existence conditions for excited statecondensates with complex 0, such as single vortices orvortex lattices. While (13) certainly cannot guarantee theexistence of a given fN0;0r; nrg state, it can rule outmodels for trapped Bose-condensed gases in various spa-tial dimensions which are inconsistent with theBogoliubov inequality (5).

I am indebted to Tony Leggett for many helpful dis-cussions, which provided a major inspiration for thepresent work. The author acknowledges support by theDeutsche Forschungsgemeinschaft (FI 690/2-1). This re-search was also supported in part by NSF GrantNo. DMR99-86199.

[1] P. C. Hohenberg, Phys. Rev. 158, 383 (1967).[2] N. N. Bogoliubov, Selected Works, Pt. II: Quantum and

Statistical Mechanics (Gordon and Breach, New York,1991).

[3] A. Gorlitz et al., Phys. Rev. Lett. 87, 130402 (2001).[4] S. Dettmer et al., Phys. Rev. Lett. 87, 160406 (2001).[5] W. Hansel, P. Hommelhoff, T.W. Hansch, and J. Reichel,

Nature (London) 413, 498 (2001).

[6] H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, andC. Zimmermann, Phys. Rev. Lett. 87, 230401 (2001).

[7] M. Greiner, I. Bloch, O. Mandel, T.W. Hansch, andT. Esslinger, Phys. Rev. Lett. 87, 160405 (2001).

[8] W. Ketterle and N. J. van Druten, Phys. Rev. A 54, 656(1996); N. J. van Druten and W. Ketterle, Phys. Rev. Lett.79, 549 (1997).

[9] E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605(1963).

[10] D. S. Petrov, M. Holzmann, and G.V. Shlyapnikov, Phys.Rev. Lett. 84, 2551 (2000); D. S. Petrov, G.V.Shlyapnikov, and J. T. M. Walraven, ibid. 85, 3745(2000); 87, 050404 (2001).

[11] W. J. Mullin, J. Low Temp. Phys. 106, 615 (1997).[12] E. H. Lieb and R. Seiringer, Phys. Rev. Lett. 88, 170409

(2002).[13] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001).[14] D. Pines and P. Nozieres, The Theory of Quantum

Liquids: Volume I (Benjamin, New York, 1966).[15] T.-L. Ho and M. Ma, J. Low Temp. Phys. 115, 61

(1999).[16] V. Dunjko, V. Lorent, and M. Olshaniı, Phys. Rev. Lett.

86, 5413 (2001); M. Olshaniı, ibid. 81, 938 (1998).[17] J. M. Vogels and W. Ketterle (private communication).[18] Observe that, if N N0 is close to the ideal gas predic-

tion N N0 2 d2z=!2dB ln2N [8], relation (19) be-comes independent of temperature, and leads to a lowerbound for the aspect ratio, !z=!? N~aas=5:8102dz ln

3=22N , where ~aas as=j1 Cas=d?j. In thethermodynamic limit, dz ! 1, N ! 1, with N=d2zfixed, condensation is then ruled out. This statement,however, should be taken cum grano salis: In the homo-geneous case, the crossover from interacting to noninter-acting has been shown to be nonanalytic for a contactinteraction potential of finite strength [9], and N N0may grow stronger than only logarithmically with N.

[19] The point that the standard non-number-conservingBogoliubov approach, when applied to the Bogoliubovinequality, is inconsistent with existing Bose-Einsteincondensates has also been made by A. C. Olinto, Phys.Rev. A 64, 033606 (2001). However, it was claimed that,as a consequence, there is a need to modify the canonicalcommutation relations, leading to explicit dependence ofthe Bogoliubov inequality on particle interaction. It wasshown here that this modification is not necessary, andthat explicit dependence on interaction does not need tooccur.

[20] M. D. Girardeau and E. M. Wright, Phys. Rev. Lett. 87,210401 (2001).

[21] S. Giovanazzi, D. O’Dell, and G. Kurizki, Phys. Rev.Lett. 88, 130402 (2002) discussed a ‘‘supersolid’’ phasefor a self-bound condensate, in the presence of an addi-tional laser-induced interaction potential.

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