Dynamical Casimir effect in dissipative media: When is the final state nonseparable?

Xavier Busch* and Renaud Parentani†

Laboratoire de Physique Theorique, CNRS UMR 8627, Batiment 210, Universite Paris-Sud 11, 91405 Orsay CEDEX, France(Received 4 June 2013; published 27 August 2013)

We study the consequences of dissipation in homogeneous media when the system is subject to a

sudden change, thereby producing pairs of correlated quasiparticles with opposite momenta. We compute

both the modifications of the spectrum and those of the correlations. In particular, we compute the final

coherence level and identify the regimes where the state is nonseparable. To isolate the role of dissipation,

we first consider dispersive media and study the competition between the intensity of the jump which

induces some coherence, and the temperature which reduces it. The contributions of stimulated and

spontaneous emission are clearly identified. We then study how dissipation modifies this competition.

DOI: 10.1103/PhysRevD.88.045023 PACS numbers: 03.70.+k, 03.75.Gg, 05.70.Ln, 42.50.Lc

I. INTRODUCTION

The analogy [1] between sound propagation in nonuni-form fluids and light propagation in curved space-timesopens the possibility to experimentally test long standingpredictions of quantum field theory [2], such as the originof the large scale structures in our Universe [3,4] and theHawking radiation emitted by black holes. In the first case,the cosmic expansion engenders a parametric amplificationof homogeneous modes which is very similar to that at theorigin of the dynamical Casimir effect (DCE) [5–7], com-pare for instance [8] with [9]. Yet, in order to predict withaccuracy what should be observed, one must take intoaccount the short distance properties of the medium be-cause the predictions involve short wavelength modes. Asa result, the analogy breaks down and a case by caseanalysis is required.

In homogeneous isotropic media, quasiparticle excita-tions are governed by dispersion relations of the form

�2 þ 2i��ðk2Þ ¼ c2k2 þ fðk2Þ: (1)

In this equation,� is the frequency in the medium frame, kthe wave vector, and c2 the low frequency group velocity.The real function f characterizes the elastic (norm preserv-ing) high momentum dispersive effects. The absorptiveproperties are described by the imaginary contributiongoverned by �> 0. So far, following [10] most worksanalyzed the consequences of short distance dispersion;see [11–14] for reviews. Comparatively, very little atten-tion has been paid to dissipative effects. In this paper westudy the consequences of dissipation when a suddenchange is applied to a homogeneous system. Following[15–17], to efficiently compute the correlation functions(at equal time and for different times), we shall use aHamiltonian treatment rather than the truncated Wignermethod [18] as done in [19,20].

Because of the homogeneity and the Gaussianity of thesystems we shall study, the states are composed of two-mode sectors of opposite wave vectors fk;�kg which donot interact with each other. Hence, each sector can bestudied separately. Besides the expected temporal decayof the out of equilibrium distribution, we shall see that the

mean particle number nk ¼ Tr½�Taykak� is not significantly

affected by turning on dissipation in Eq. (1). On the con-trary, the issue of the correlations between quasiparticlesof opposite k is more tricky to handle. First, when latetime dissipation is sufficiently small that the state can bemeaningfully decomposed using a particle number basis,the complex number ck ¼ Tr½�Taka�k� accounts for thestrength of these correlations. Second, to determine if thefinal state is quantum mechanically entangled, one shouldconsider the relative value of the norm of ck with respect tonk [21–23]. Indeed, whenever �k, given by

�k ¼: nk � jckj; (2)

is negative, the bipartite state fk;�kg is nonseparable,i.e., so correlated that its statistical properties cannotbe reproduced by a stochastic ensemble. Other means canalso be envisaged, such as Cauchy-Schwarz inequalities[17,24,25] or sub-Poissonian statistics [26]. Third, as weshall see, even in the absence of dissipation, it is difficult toproduce nonseparable states, as an initial temperature in-creases the value of�k (because it increases the contributionof stimulated emission). The difficulties are reinforced in thepresence of dissipation. Indeed, the coupling to the environ-ment generally induces an increase of�k. Our principal aimis to study the outcome of the competition between thesudden change which produces the coherence, and the com-bined effect of temperature and dissipation which reduce it.The paper is organized as follows. In Sec. II we explain

how to couple the phonon field to an environment so as toengender some specific decay rate. In Sec. III we study theeffect of the temperature on the final value of�k in the casewhere there is no dissipation. In Sec. IV we study themodifications on the spectrum and �k when includingdissipation. We conclude in Sec. V.

*xavier.busch@th.u-psud.fr†renaud.parentani@th.u-psud.fr

PHYSICAL REVIEW D 88, 045023 (2013)

1550-7998=2013=88(4)=045023(14) 045023-1 � 2013 American Physical Society

II. ACTIONS FOR DISSIPATIVE PHONONS

To study the phenomenology of the DCE in dissipativemedia, we work with an action of the form ST ¼ S� þS� þ Sint, where � describes the (free) quasiparticles, �describes the (environment) degrees of freedom that shallcause the dissipation, and Sint describes the interactions

between � and �. As is usually done in atomic damping[27] or when describing quantum Brownian motion

[28,29], the action ST is taken quadratic in �, �, so thatthe field equations are linear.

In experiments, a finite range of k shall be accessible.The strength of the correlations can thus be studied as afunction of k. To cover various cases, we consider decayrates parametrized by

�ðkÞ ¼ g2ðc=�Þð��Þ2��1ðk�Þ2þ2n: (3)

The coupling constant g is dimensionless, � is the con-densed atom density, and � a short distance length whichcorresponds to the healing length in atomic Bose gases. Tofix the ideas and notations, the quasiparticles shall bephonons propagating in an elongated, effectively one-dimensional, atomic Bose condensate, as in the experimentof [5]. However, our treatment can be easily adapted toother systems displaying dissipation such as polariton ex-citations in microcavities [19].1 The powers n and � can bechosen to reproduce effects computed from first principles.For instance, in Bose gases, two types of dissipative effectsare found: The first one, called Beliaev decay [30], scaleswith� ¼ 0 and n ¼ 3=2, while the second one, the Landaudecay [31], depends on the temperature, has also � ¼ 0,

and is proportional to ckffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ k2�2

p.

In this paper, � describes relative density fluctuationspropagating in homogeneous time-dependent condensates;see Appendix A for details. Working with ℏ¼1, its action is

S� ¼ 1

2

Zdtdx�

��y

�i@t þ @2x

2m�mc2

��

�mc2�2 þ H:c:

�; (4)

where � gives the density of condensed atoms, m theatom mass, and c the (time-dependent) speed of sound.The latter is related to the time-dependent healing lengthby �ðtÞcðtÞ ¼ 1=2m.

The action for the environment degrees of freedom istaken of the form [15,16]

S� ¼ 1

2

Zdtdx

Z 1

�1d�fj@t�� j2 � j���� j2g; (5)

where the extra variable � has dimension of a frequency,

and where �� obeys �y� ¼ ��� , as a Hermitian field in

momentum space. The variable � has been introduced in

order to have infinitely more degrees of freedom in the �

field than in �, a condition necessary to get dissipation[28]. (As shall be clear in the sequel, it is simpler to workwith a continuous set, rather than an infinite discrete one asin Ref. [32].) Notice that Sc contains no spatial gradients,

and that it does not depend on �. Hence, the kinematics ofthe environment degrees of freedom is independent of bothk and �. This is a simplifying approximation. In fact, ourphilosophy is to choose the simplest model that possessessome key properties. These are as follows: unitarity of thewhole system, standard action for the phonons, and abilityto reproduce the decay rates of Eq. (3) parametrized by g2,� and n. Given (5), to recover these rates, we shall show inthe next subsection that the third action should be

Sint¼�Zdtdx

gffiffiffi�

p�ð��Þ�ð�@xÞnð�þ �yÞ@t

�Zd���

��:

(6)

A. Field equations and effective dispersion relation

Because the condensate is homogeneous, it is appropri-ate to work with the Fourier components at fixed wavevector k ¼ �i@x, where k is real. Then the total actionsplits into sectors that do not interact: ST ¼ R

dkSk, with

Sk ¼ Sy�k. In the rest frame of the condensate, at fixed k,the field equations are, with k ¼ jkj,�i@t � k2

2m�mc2

��k ¼ mc2�y

�k þ �kffiffiffiffi�

p @tZ

d���;k;

(7a)

½@2t þ ð��Þ2���;k ¼ @tf��k

ffiffiffiffi�

p ð�k þ �y�kÞg ¼

:j�;k;

(7b)

where

�k ¼ gð��Þ��1=2ði�kÞn (8)

is the effective dimensionless coupling for the wavenumber k. The solution of Eq. (7b) is

��;k ¼ �0�;k þ

Zdt0R0

� ðt; t0Þj�;kðt0Þ; (9)

where �0�;k is a homogeneous solution we shall describe

below, and where R0� is the retarded Green function. It is

independent of k because the action S� contains no spatialgradient. When summing over � , it obeys [15,28]

@t

�Zd�R0

� ðt; t0Þ�¼ �ðt� t0Þ: (10)

1While finalizing this work, we became aware of [20] wheresimilar issues are considered in that context. Let us signal severaldifferences. First, in the present work, dissipation is handled in away which allows us to efficiently compute correlation func-tions. Second, we keep the contribution of ck ¼ Tr½�Taka�k�which accounts for the correlations, and use it to distinguish thestates that are nonseparable. Third, our dissipative effects onlyaffect the quasiparticles; see Appendix A.

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This guarantees that the kernel encoding dissipation islocal. Indeed, inserting Eq. (9) into Eq. (7a), and usingEq. (10) and (7a) gives�i@t � k2

2m�mc2

��k ¼ mc2�y

�k þ �k@t�0k=

ffiffiffiffi�

p

þ �k@tn��k

��k þ �y

�k

o; (11)

where �0k ¼ R

d��0�;k. As announced, the last term in the

right-hand side is local in time. Basically all other choicesof S� and Sint would give a nonlocal kernel.2 In thesecases, Eq. (11) would be an integro-differential equation.These more complicated models do not seem appropriateto efficiently calculate the consequences of dissipation onphonon correlation functions.

To get the effective dispersion relation, we considerEq. (11) when all background quantities are constant andwhen �0

k ¼ 0. We get, as in Eq. (1),

ð�k þ i�kÞ2 ¼ !2k; (12)

where

!2k ¼ c2k2ð1þ �2k2Þ � �2

k; �k ¼ j�kj2k2c�: (13)

Using Eq. (8), we verify that the second equation deliv-ers Eq. (3). Hence, by choosing g, � and n our model shallbe able to reproduce many ab initio computed decay rates.As expected, we also verify that, for j�kj2 ! 0, onerecovers the standard Bogoliubov dispersion for all k.Our models thus provide dissipative extensions of somedispersive model.

In what follows, the quantities g, c, � depend on time,while preserving the homogeneity. Hence the nontrivialdynamics will occur within two mode sectors fk;�kg.

B. Time-dependent settings

For homogeneous time-dependent systems, it is appro-priate to introduce the auxiliary field

k ¼: � �k þ �y�kffiffiffi

2p

ffiffiffiffiffiffiffiffiffiffi�

c�k2

s: (14)

Using Eq. (11), its time derivative is given by

@tk ¼ iffiffiffiffiffiffiffiffiffiffiffiffiffiffi��ck2

q �k � �y�kffiffiffi

2p : (15)

The field is both Hermitian (yk ¼ �k) and canonical: it

verifies the equal time commutators (ETC) ½k; @tyk� ¼ i,

½kyk� ¼ 0, which are the usual ones for a relativistic scalar

field ink space.Moreover, k is simply related to the relative

density fluctuation, and to the phase fluctuation; seeAppendix B. Finally, Sk, the action of the k sector, reads

Sk ¼ 1

2

Zdt

�j@tkj2 � ðckÞ2½1þ ðk�Þ2�jkj2

þZ

d�j@t��;kj2 � ð��Þ2j��;kj2

þ 2yk

ffiffiffiffiffiffiffiffi2�k

p@tZ

d���;k

�; (16)

where c, �, and � are arbitrary (positive) time-dependentfunctions, andwhere the phase of�k, ð�isgnðkÞÞn, has beenabsorbed in ��;k. In an atomic condensate, c and � are

related by c� ¼ 1=2m ¼ cst. We can alsomake the analogywith field propagation in a homogeneous cosmologicalbackground [9,33]. Indeed, cðtÞ acts as the inverse of thescale factor aðtÞ (and not as a varying speed of light). Hence,a decreasing speed of sound corresponds to an expandinguniverse.From the above action, or from Eq. (11), we get the

equation for k:

½ð@t þ �kÞ2 þ!2k�k ¼ ffiffiffiffiffiffiffiffi

2�k

p@t�

0k: (17)

The general solution can be written as

kðtÞ ¼ deck ðt; t0Þ þ dr

k ðt; t0Þ; (18)

where the driven part drk ðt; t0Þ and its temporal derivative

vanish at t ¼ t0. The decaying part deck ðt; t0Þ is thus the

solution of the homogeneous equation which obeysthe ETC at that time. Hence it possesses the followingdecomposition:

deck ðt; t0Þ ¼ e

�R

t

t0�kdt

0 ðak’kðtÞ þ ay�k’�kðtÞÞ; (19)

where the destruction and creation operators ak, ay�k obey

the standard canonical commutators ½ak; ayk� ¼ 1, andwhere ’k is a solution of

ð@2t þ!2kÞ’k ¼ 0; (20)

of unit Wronskian ið’�k@t’k � ’k@t’

�kÞ ¼ 1. The useful-

ness of this decomposition is twofold. On the one hand, t0can be conceived as the initial time when the state is fixed.

The operators ayk, ak can then be used to specify the

particle content of this state. On the other hand, Eq. (18)and (19) furnish an ‘‘instantaneous’’ particle representationaround any time t0. Indeed, in the limit �=! � 1 and�ðt� t0Þ � 1, the contribution of dr

k ðt; t0Þ can beneglected, and kðtÞ � dec

k ðt; t0Þ behaves as a standardcanonical field since the prefactor of Eq. (19) is approx-imatively equal to 1. We shall return to this in Sec. IVB 2.The driven part of Eq. (18) is given by

drk ðt; t0Þ ¼

Z 1

t0

dt0Gretðt; t0; kÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�kðt0Þ

q@t0�

0kðt0Þ; (21)

where Gret is the retarded Green function of Eq. (17).Using the unit Wronskian solution of Eq. (20), it can beexpressed as

2In this respect, we notice that one can generalize our model soas to deal with an ab initio computed nonlocal dissipative kernelDðt� t0Þ. To do so, one should compute its Fourier transformDð!Þ [which is equal to 1=ð!þ iÞ in our model] and replaceRd��� by

Rd�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!�Dð!� Þ

q�� in Eq. (6).

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Gretðt;t0;kÞ¼�ðt� t0Þe�R

t

t0 �kdt�2Imð’kðtÞ’�kðt0ÞÞ: (22)

Since is a canonical and linear field, the standardrelation between the commutator and the retarded Greenfunction holds, namely,

½kðtÞ; �kðt0Þ��ðt� t0Þ ¼ iGretðt; t0; kÞ: (23)

In consequence, when the state of the system is Gaussianand homogeneous, the reduced state of is completelyfixed by its anticommutator. Because of Eq. (18), it con-tains three terms,

Gacðt; t0; kÞ ¼: Trð�TfkðtÞ; �kðt0ÞgÞ¼ Gdec

ac þGdrac þGmix

ac : (24)

The first one decays and is governed by dec,

Gdecac ðt; t0; kÞ ¼ Trð�Tfdec

k ðtÞ; dec�kðt0ÞgÞ: (25)

The second one is driven and governed by �0,

Gdracðt; t0; kÞ ¼

Z 1

t0

d�d�0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�kð�Þ

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�kð�0Þ

qGretðt; �Þ

�Gretðt0; �0Þ@�@�0Trð�Tf�0kð�Þ; �0

�kð�0ÞgÞ:(26)

The third one describes the correlations between and �.It is nonzero either when the initial state is not factorized as� ¼ � � ��, or when the two fields have interacted. It is

given by twice the symmetrization in t; t0, of

~Gmixac ðt; t0; kÞ ¼:

Z 1

t0

d�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�kð�Þ

qGretðt0; �Þ

� @�Trð�Tfdeck ðtÞ; �0

�kð�ÞgÞ: (27)

When the state is prepared at an early time �ðt� t0Þ � 1,only the driven term significantly contributes to Eq. (24),which means that the system would have thermalized withthe bath.

At fixed k and � , �0�;k, the homogeneous solution

of Eq. (7) is a complex harmonic oscillator of pulsation!� ¼ �j�j. Hence it can be expressed as

�0�;kðtÞ ¼

e�i!� ta�;k þ ei!� tay��;�kffiffiffiffiffiffiffiffiffi2!�

p ; (28)

where a�;k and ay�;k are standard destruction and creation

operators. In the following sections we shall work withthermal baths at temperature T. In such states, the noisekernel entering Eq. (26) is, with Boltzmann constantkB ¼ 1,

Trð�Tf�0kð�Þ;�0

�kð�0ÞgÞ¼Z 1

0

d!�

�

coth�!�

2T

2!�

�cos½!� ð���0Þ�: (29)

We notice that it does not depend on k.

C. Sudden changes

We study the time evolution of the phonon state whenmaking two assumptions. First, we consider states whichare prepared a long time before the experiment, so thatEq. (24) is given by Eq. (26), with t0 ¼ �1. Second, wesuppose a sudden change of the condensed atoms occurs attime t ¼ 0. Hence, the speed of sound c and the effectivecoupling �k entering Eq. (17) will change on a similar timescale. Since approximating the change of the sound speedby a step function only modifies the response for very highk, as can be seen in Ref. [9], for simplicity, we shall workwith an instantaneous change for c. For �k instead, weshall use a continuous profile because an instantaneouschange would lead to divergences, as we shall see below.Hence we shall work with

cðtÞ ¼ cf þ ðcin � cfÞ�ð�tÞ;�kðtÞ ¼ �f þ ð�in � �fÞhð tÞ;

(30)

where hð tÞ is a smoothing out function which goes from 1to 0 around t ¼ 0 in a time lapse of the order of 1= . At thispoint, it should be noticed that any physical system, such asthe atomic Bose condensate described by Eq. (A2), wouldonly respond after a finite amount of time of the order of�=c. Hence the function h of Eq. (30) is physically mean-ingful, and should be of the order of c=�.3 Consideringthe asymptotic values of these profiles, we shall use thefollowing notations:

�in=f ¼: �2in=fðc�Þk2;

!2in=f ¼

:c2in=fk

2 þ ðc�Þ2k4 � �2in=f ;

(31)

see Eq. (13).Before considering dissipation, we study the dispersive

case, first, to analyze the reduction of the correlations dueto stimulated processes and, second, to know the outcomein the dissipation free case, so as to be able to isolate theconsequences of dissipation.

III. THE DISPERSIVE CASE

In the absence of dissipation, phonon excitations can beanalyzed before and after the jump using a standard parti-cle interpretation. Hence, the consequences of the jump areall encoded in the Bogoliubov coefficients �, � entering

’in ¼ �’out þ �’�out; (32)

which relate the in mode to the out mode. These modes

have a positive unit Wronskian and are equal to e�i!t=ffiffiffiffiffiffiffi2!

pfor t < 0 or t > 0, respectively. Using Eq. (20), one verifiesthat the modes are C1 across the jump. From the junctionconditions, one finds the Bogoliubov coefficients [9]

� ¼ !f þ!in

2ffiffiffiffiffiffiffiffiffiffiffiffiffi!f!in

p ; � ¼ !f �!in

2ffiffiffiffiffiffiffiffiffiffiffiffiffi!f!in

p : (33)

3We are grateful to Iacopo Carusotto for pointing this out to us.

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To prepare the comparison with the dissipative case, theinitial phonon state is taken to be a thermal bath at tem-perature T. This means that the initial mean occupation

number is nin ¼ 1=ðe!in=T � 1Þ and that cin, the initial cor-relation term between k and�k, vanishes. After the jump,the mean occupation number and the correlation term are

nout ¼ noutspont þ noutstim ¼ j�j2 þ j�j2 þ j�j2e!in=T � 1

; (34a)

cout ¼ coutspont þ coutstim ¼ ��þ 2��

e!in=T � 1: (34b)

These two quantities completely fix the late timebehavior of the anticommutator Gacðt; t0Þ ¼ ðnin þ 1=2Þ�Ref’inðtÞ’�

inðt0Þg. In fact, for t, t0 > 0, one has

Gacðt;t0Þ¼ðnoutþ1=2Þcosð!fðt�t0ÞÞ!f

þRe

�cout

e�i!f ðtþt0Þ

!f

�:

(35)

Using this expression, it is clear that the contribution of the

stimulated amplification [weighted by nin¼1=ðe!in=T�1Þ]and that of spontaneous processes are not easy to distin-guish. As recalled in the Introduction, to be able to do so, it isuseful to introduce the parameter� of Eq. (2). One can show

that it obeys �1=2<n� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinðnþ1Þp �n, where the

minimal and maximum values characterize, respectively,the pure (squeezed) state and the incoherent thermal state[22]. In addition, whenever �< 0, the two-mode statefk;�kg is nonseparable in the sense of Werner [34]. Thismeans that the correlations are so strong that they cannot bedescribed by a classical ensemble. Notice that� is linked tothe logarithmic negativity EN introduced in Ref. [35] andused in Refs. [36,37]. Indeed, for Gaussian two-mode stateswith nk¼n�k, one finds EN ¼ max ½�log 2ð1þ 2�Þ; 0�which is positive only if �< 0.

In the present case, the value of � associated withEqs. (34) is

�out ¼ �outspont þ �out

stim;

¼ �ðj�j � j�jÞj�j þ ðj�j � j�jÞ2e!in=T � 1

: (36)

At fixed j�=�j, the threshold value of nonseparability�¼0 defines a critical temperature TC. It is given by(see also Ref. [38])

j�=�j ¼ e�!in=TC : (37)

In Eq. (36), one clearly sees the competition between thesqueezing of the state due to the sudden jump governed by� which reduces the value of �, and the initial occupationnumber which increases its value. It remains to extract thisinformation from Eq. (35).

To this end, we plot in Fig. 1 the product!fGacðt; t0 ¼ tÞof Eq. (35) as a function of time, for two different values ofT, respectively, half and twice the temperature,

T�in¼: mc2in ¼

1

4m�2in

; (38)

fixed by the initial value of the healing length. We obtaintwo perfect sinusoidal curves since the mode k freelypropagates after the sudden change. Interestingly, we canread the values of n and jcj from the envelope of the curve.Indeed, the maxima reach nþ 1=2þ jcj and the minimanþ 1=2� jcj ¼ �þ 1=2. Therefore, if in an experiment,the minimal value of !fGac is measured with enoughprecision to be less than 1=2, one can assert that the stateis nonseparable (in the absence of dissipation).This identification can also be obtained using the so-

called g1 correlation function, or the g2; see Appendix B.One could also work at fixed t and vary t0. In this case, onewould get a periodic behavior of frequency !f=2�.However, the curve is now centered on 0, and the maximavary from nþ 1=2� jcj to nþ 1=2þ jcj according to thevalue of the fixed time t. Hence, the nonseparability of thestate reveals itself in the fact that there exists some values oft such that !fGacðt; t0Þ remains smaller than 1=2 for all t0.To complete the analysis, we study how these results

depend on the wave number k. We work with the standardBogoliubov dispersion relation [see Eq. (12)] with � ¼ 0,c� ¼ 1=2m constant, and with cf=cin ¼ 1=10. In Fig. 2 weplot !f �Gacðt; t0 ¼ tÞ as a function of k for two tempera-tures, namely, T�in

=2 (left panel) and T�in(right panel). We

first see that the modes with lower k are more amplifiedthan those with higher k. As expected from Eq. (37), whenlooking at the lower envelope, we also see that the coher-ence level is higher (the minima of Gac lower) when work-ing with a smaller temperature, and/or with higher k, i.e.,with rarer events governed by a smaller initial occupationnumber nin. To quantify this effect, and possibly also toguide future experiments, we characterize the domain

0 5 10 15 200

1

2

3

4

t m cin2

f.G

ac

FIG. 1. We represent the product !f �Gacðt; t0 ¼ tÞ, whereGac is given in Eq. (35), as a function of the adimensionalizedtime tmc2in, for k ¼ mcin, and for two values of the temperature,

namely, T�in=2 (dashed line) and 2T�in

(solid line); see Eq. (38).

The value of the jump is cf=cin ¼ 0:1. As explained in the text,the dotted line gives the threshold value 1=2 which distinguishesnonseparable states. When increasing the temperature, the con-tribution of the stimulated amplification with respect to thespontaneous one is larger. As a result, the coherence is reduced;i.e. the minima of !f � Gac are increased.

DYNAMICAL CASIMIR EFFECT IN DISSIPATIVE . . . PHYSICAL REVIEW D 88, 045023 (2013)

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where the resulting state is nonseparable, i.e., where�out < 0. To this end, in Fig. 3, we plot �out as a functionof T=T�in

and the ratio cf=cin. We consider two values of k,

namely, one in the hydrodynamical regime, and one of theorder of the inverse healing length. This clearly confirmsthat at higher momenta, states are more likely to be non-separable. Moreover, one sees that for a wave numbersmaller than the healing length, at a temperature �T�in

of

Eq. (38), in order to obtain a nonseparable state, cf=cinshould be either larger than 3 or smaller than 1=3.

To illustrate these aspects with a concrete example, weconsider the experiment of Ref. [5]. The relevant values areT ¼ 6:05T�in

and k� 2:15mcin, so that the initial number

of particles is of the order of 3. On the other hand, one has

cf=cin � 21=4. (To get these numbers, we used T¼200 nK,!=2� ¼ 2 kHz,m ¼ 7:10�27 kg and cin ¼ 8 mm=s.) Thecorresponding value of the coherence level is �� 1:4.Hence the state is separable. In order to reach � ¼ 0,one should either increase the ratio cf=cin � 6 or work

with a lower temperature, of the order of 0:6T�in.

IV. THE DISSIPATIVE CASE

In the presence of dissipation, the mode interpretation

involving the Bogoliubov coefficients of Eq. (32) is no

longer valid. In fact, the state of is now characterized

0.00 0.05 0.10 0.15 0.20 0.25 0.301.0

0.5

0.0

0.5

1.0

T Tin

log 10

cf

c in

0.50

0.40

0.30

0.20

0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.0 0.2 0.4 0.6 0.8 1.01.0

0.5

0.0

0.5

1.0

T Tin

log 10

cf

c in

FIG. 3 (color online). Contour plot of �out induced by a sudden variation of the sound speed, as a function of temperature T=T�inand

the logarithm of the ratio cf=cin. In the left panel, k ¼ mcin=10 is in the hydrodynamical regime, and in the right panel, k ¼ mcin. Thethreshold value �out ¼ 0 is indicated by a thick line. One clearly sees the competition between the height of the jump governed bycf=cin which increases the coherence, i.e. reduces the value of �, and the initial occupation number which increases �. One also seesthat the state of a higher momentum mode stays nonseparable for higher temperatures.

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

k m cin

f.G

ac

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

k m cin

f.G

ac

FIG. 2. The anticommutator of Eq. (35) multiplied by !f as a function of k=mcin, when evaluated at equal time t ¼ t0 ¼ 5=mc2in, forcf ¼ cin=10, on the left panel for T ¼ T�in=2 and on the right for T ¼ T�in (solid oscillating curves). The dashed line in the middle

gives the value of !fGac ¼ nin þ 1=2 before the sudden change. The envelopes of the minima and maxima are also indicated by solidlines. One clearly sees that the domain of k, where the lower line is below the threshold value 1=2, is reduced when increasing thetemperature. Notice that all curves asymptote to 1=2 because, in the limit k ! 1, one has nk ¼ ck ¼ 0.

XAVIER BUSCH AND RENAUD PARENTANI PHYSICAL REVIEW D 88, 045023 (2013)

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by Gdrac of Eq. (26), which is governed by the retarded

Green function and the noise kernel. The separability ofthe state should thus be deduced from its properties.

When the state of the environment is thermal, usingEq. (29), Eq. (26) can be expressed as

Gdracðt;t0Þ¼

Z d!�

2�!� coth

�!�

2T

�~Grðt;!� Þ ~Grðt0;�!� Þ; (39)

where we introduced the Fourier transform

~Grðt; !� Þ ¼:Z 1

�1d�ei!��

ffiffiffiffiffiffiffiffiffiffi�ð�Þ

pGretðt; �Þ (40)

of the retarded Green function of Eq. (22). In the following,we compute Eq. (39), which is easier to handle thanEq. (26), in two different cases. In the first one, there isno dissipation after the sudden change, i.e., �f ¼ 0 inEq. (30). In the second one, � is constant.

A. Turning off dissipation after the jump

When �f ¼ 0 for t � 1, we have the possibility ofusing the standard particle interpretation to read theasymptotic state. In fact, inserting Eq. (22) into Eq. (40),using Eq. (30) and t � 1, one gets

~Grðt; !� Þ ¼ffiffiffiffiffiffiffi�in

p2!f

½ei!f tRð!� Þ � e�i!f tR�ð�!� Þ�; (41)

where

Rð!� Þ ¼:ffiffiffiffiffiffiffiffiffi2!f

p Z 1

�1d�hð �Þei!��e�

R1��’out: (42)

The function ’outð�Þ is the standard out mode: it is thepositive unit Wronskian mode of Eq. (20) which is positivefrequency at asymptotically late time. The time depen-dence of Eq. (41) guarantees that Eq. (39) has exactly theform of Eq. (35). The final occupation number nout andcorrelations cout are found to be

noutþ1

2¼�in

!f

Z 1

0

d!�

�!� coth

�!�

2T

�ðjRð!� Þj2þjRð�!� Þj2Þ;

cout�¼2�in

!f

Z 1

0

d!�

�!� coth

�!�

2T

�Rð!� ÞRð�!� Þ: (43)

We notice that these expressions are similar to those ofEq. (34), and that R�ð!� Þ plays the role of a density

(in !� ) of � and �, respectively. In fact, when taking the

limit �in ! 0 in Eq. (43), one recovers the dispersiveexpressions of Eq. (34).

We can now explain why we introduced the function h inEq. (30). For ! 1, hð tÞ becomes the step function�ð�tÞ. In this limit, Eq. (42) gives

Rð!� Þ ¼!f þ ð!� � i�inÞ!2

in � ð!� � i�inÞ2þO

�1

�; (44)

which indicates that R behaves as 1=!� for !� ! 1.

Hence both nout and cout of Eq. (43) would logarithmicallydiverge. The divergences arise from the fact that the

environment field � contains arbitrary high frequencies!� . To regulate the divergences, several avenues can be

envisaged. One could either introduce a �-dependent cou-pling in Eq. (6) or cut off the high frequency!� spectrum in

Eq. (5). However, these would spoil the locality of Eq. (10).For this (mathematical) reason, we prefer to use hð tÞ ofEq. (30). Moreover, taking an instantaneous change in �ðtÞwould remove the C1 character of Gacðt; t0Þ found in thedispersive case; see the discussion after Eq. (32).In Appendix C we derive approximate expressions for

Rð!� Þ, both for a general profile h and when applied to the

particular case

hðzÞ ¼

8>>><>>>:1 if z < 0;

1� z if 0< z < 1;

0 if 1< z;

(45)

that we shall use to obtain the following figures.

1. Spectral deviations due to dissipation

We study how nout of Eq. (43) depends on the dissipationrate �in. To this end, we study its difference with thedispersive occupation number noutdisp of Eq. (34) evaluated

with the same values for the temperature T, and the initialand the final frequency [see Eq. (31)]. In Fig. 4 we repre-sent the relative change

�nr ¼:noutð�inÞ � noutdisp

noutdisp þ 1=2(46)

as a function of the initial occupation number nin and theratio�in=!in. Wework with a jump!f=!in ¼ 0:1, and with

0.001 0.01 0.1 1.0.01

0.1

1.

10.

100.0.001 0.01 0.1 1.

0.01

0.1

1.

10.

100.

in in

nin

100.5

100.0

10 0.5

10 1.0

10 1.5

10 2.0

10 2.5

10 3.0

10 3.5

10 4.0

FIG. 4 (color online). The ratio �nr of Eq. (46) for a jump!f=!in ¼ 0:1 and ¼ 10!in is represented in the plane nin,�in=!in. For low initial occupation numbers, one sees that �nr isproportional to �in=!in, whereas it evolves from linear to qua-dratic for high numbers. As explained in the text, these obser-vations are confirmed by analytical expressions.

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¼ 10!in. [We use this parametrization because, at fixed k,Eq. (17) only depends on !ðtÞ and �ðtÞ. Hence the healinglength and k need not be specified.] For these values, we findtwo regimes. First, for a low occupation number, we observethat the deviation�nr linearly depends on �in. An analyticaltreatment based on Eq. (C3) reveals that, for small �in=!in

and large =!in, �nr behaves as

�nr ¼��in

!in

�1

nin þ 1=2� gð =!inÞ; (47)

where gð =!inÞ is a rather complicated function.4 We nu-merically checked that Eqs. (47) and (48) apply for nin & 5and * 10!in. Hence, under these conditions, �nr de-pends on eff of Eq. (49), but not on the exact shape of hof Eq. (30).

Second, for high occupation numbers and high �=!2in,

we observe in Fig. 4 a quadratic dependence in �.Lowering �=!2

in, we observe a transition from this qua-

dratic behavior to a linear one. These numerical observa-tions are in agreement with the analytical result

�nr ���in

!in

�2 !2

in �!2f

!2in þ!2

f

þO��in

�; (50)

which applies in the limit �in � ! � � T.In brief, Eqs. (47) and (50) establish how nout of Eq. (43)

converges towards the dispersive occupation number ofEq. (34) when �in=!in � 1 and � !in. A similar analy-sis can be done for the coefficient cout of Eq. (43), and itgives similar results.

2. Final coherence level

In Fig. 5, we represent the coherence level � of Eq. (2)as a function of the temperature and cf=cin, for two differ-ent values of k, namely, k=mcin ¼ 0:3 and 1, and threevalues of �2, namely, 0, 0.25 and 0.5. The value of is10mc2in. As one might have expected, we observe a con-

tinuous deviation of� for increasing values of the coupling�2k. More surprisingly, we observe that increases of cðtÞ,

cf > cin, and decreases, cf < cin, behave very differently.In the first case, there is a large increase of �, whichimplies a loss of coherence. On the contrary, in thesecond case, the value of � is robust, and some marginal gain of coherence can even be found. To validate these

observations, we studied the behavior of � for differentprofiles of the smoothing function hð tÞ. Whenever cf=cinis not too close to 1, we obtained similar results, therebyshowing that the choice of the profile of h does not sig-nificantly matter. Instead, when cf=cin � 1, the behavior of� is less universal.In order to understand the different behaviors of cf > cin

and cf < cin, we represent in Fig. 6 the anticommutatornormalized to its value at the jump Gacðt ¼ t0Þ=Gt¼t0¼0

as a function of !ft, and for different values of the steep-ness parameter . We first notice that this function is C1

across the jump, as were the modes in dispersive theories.

FIG. 5 (color online). We represent the lines of � ¼ �0:2(dashed), 0 (solid) and 0.2 (dotted) in the planefT=T�in

; log 10ðcf=cinÞg, on the top panel for k ¼ 0:3mcin and

on the bottom one for k ¼ mcin. �2k takes three values: from 0

i.e. the dispersive case shown by the thick red line, to 0.5, and ¼ 10mc2in. For temperatures that are not too low, we observe

that the value of � is robust when cf < cin (which corresponds toan expanding universe), while it increases when cf > cin.

4It is given by

gð =!inÞ ¼ 1

�

2 log

� eff

!in

1þ ð!f=!inÞ2

� 1

!; (48)

where eff is the effective slope of the profile hð tÞ. It is given by

eff ¼: exp

����

Zdtdt0ð@thÞð@t0hÞ log ð jt� t0jÞ

�; (49)

where � is the Euler constant. The interesting part of theseequations is that they apply for any hð tÞ when =!in � 1.Moreover, eff also governs the logarithmic growth of jcj.

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This implies that one extremum of the anticommutatorcoincides with the value before the jump. In the absenceof dissipation, one easily verifies that it is a minimumfor cf < cin and a maximum otherwise. For weak dissipa-tion, by continuity in �, this must still be the case.Hence, when cf < cin, the minima of the anticommutatorare fixed by the initial state. Instead, for cf > cin, they arefixed by the intensity of the jump and the injectionof energy from the environment. As clearly seen in thefigure, this injection increases with and explains whycoherence is more robust when cf < cin (i.e., in expandinguniverses).

B. Constant dissipation rate

In this section, we study our model when the dissipativerate � is constant, as it is found for instance in polaritonsystems [19,20] and in Josephson metamaterial [7]. In thiscase, there is no unambiguous notion of (out) quanta, eventhough the anticommutator of Eq. (39) is well defined forall t, t0. Nevertheless, provided �=! is low enough, weshall see that an approximate reading of the final state canbe reached in term of the instantaneous particle represen-tation based on Eq. (19).

Because � is constant in Eq. (30), there is a simplifica-tion with respect to the previous subsection: no regulari-zation is now needed since Eq. (39) is finite. Moreover, theretarded Green function of Eq. (22) is exactly known. It isgiven by

e��ðt�t0Þ

�8<:�ðt� t0Þ sin!ðt�t0Þ

! for t0; t< 0 or t0> 0;

sin ð!f tÞcos ð!int0Þ

!f� cos ð!f tÞ sin ð!int

0Þ!in

for t0< 0 and t> 0:

(51)

Hence, after the jump of c, for positive times, the Fouriertransform of Eq. (40) gives

~Grðt; !� Þ ¼ffiffiffiffi�

pei!� t

!2f � ð!� � i�Þ2 þ

ffiffiffiffi�

p e��tþi!f t

2!f

��!f þ ð!� � i�Þ!2

in � ð!� � i�Þ2 �1

!f � ð!� � i�Þ�

þ ð!f ! �!fÞ: (52)

This means that we (exactly) know the integrand ofEq. (39). The integral can be performed by analyticalmethods (by evaluating the residues of poles), and thenrecognizing the infinite sum as a finite sum of hypergeo-metric functions. The main results are presented below.

1. Two-point correlation function

To discover the effects of dissipation, in Fig. 7 we plot!f �Gacðk; t ¼ t0Þ both as a function of time, as in Fig. 1,and as a function of the wave number k, as in Fig. 2. Whenconsidered as a function of t, we observe that the oscilla-tions take place in a narrowing envelope. As expected, thelatter follows an exponential decay in e�2�t towards theequilibrium value !fG

eqac ¼ !fGacðt ¼ t0 ! 1Þ. This sim-

ple behavior implies that the nonseparability of the state isquickly lost at high temperatures. Indeed, a rough estimateof the lapse of time for the decoherence to happen is of theorder ð2�neqÞ�1, where neq is the mean occupation numberat equilibrium. Hence, when neq � 1, the time for the lossof coherence is smaller than the dissipative time 1=� by afactor 1=2neq. When considered as a function of k, on theright panel, we observe damped oscillations. For large k,they are more damped than those of the dispersive case(represented by a dashed line) since the decay rate � / k2.To further study the effects of increasing �, in Fig. 8 we

represent the equal time density-density two-point function(see Appendix B),

Gdd ¼: Trð�T��ðt; xÞ��ðt; x0ÞÞ�

¼Z 1

�1dk

�eikðx�x0Þc�k2Gk

acðt; t0 ¼ tÞ; (53)

for four values of �2, namely, log 10ð�2Þ ¼ �2, �1:5, �1and �0:5. We take a rather large t ¼ 7:5=mc2f to see the

propagation of the phonon waves. As expected, we observea peak centered on x ¼ x0 plus a series of peaks for jx�x0j> 2cft. The first one is present even in vacuum and isamplified by the mean occupation numbers nk > 0. We seethat it is broadened by dissipation. The series of peaks forjx� x0j> 2cft is due to the fact that the phonons are

1 0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

2.5

3.0

f t

Gac

Gac

t0

FIG. 6 (color online). We represent the anticommutator Gac

normalized to its value before the jump, as a function of !ft, andfor T ¼ T�in

and k ¼ 2mcin. The three upper curves (in blue)

represent an expanding universe (cf ¼0:5cin), while the threelower (in purple) ones represent a contracting universe (cf ¼2cin). The solid lines show the two dispersive cases, the dashedlines represent the dissipative cases with �2 ¼ 0:15 and ¼100mc2in, and the dotted lines show the results when is

increased to 104mc2in in order to see the logarithmic growth of

n and c of Eq. (34). In all cases, Gac is C1 across the jump. Sincethe coherence is based on the minima of Gac, the value of � isrobust when cf=cin<1, whereas it necessarily increases whencf=cin > 1.

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produced in pairs. As in inflationary cosmology [8], theiramplitudes are fixed by the ck coefficients. These peakspropagate at different speeds because of dispersion. Thefastest are more damped because dispersion is anomalous,and because dissipation goes in k2. We also notice that thefirst propagating peak is negative when working with cf <cin. (For cf > cin, instead, it would be positive.) We con-clude by noticing that this plot gives no indication ofwhether the state is separable or not, mainly because Gdd

mixes different two-mode sectors labeled by k, some ofwhich being nonseparable, but not all.

2. Approximate particle interpretation and separability

To interpret the properties of Gac, we now use theinstantaneous particle representation based on dec of

Eq. (19). Even though the anticommutator Gac is welldefined, for dissipative systems, there is no unique (canoni-cal) way of defining the concept of particle. Hence themean occupation number n and the correlation term c aresomehow ambiguous. The issue is twofold, as it requiresone to treat separately the equilibrium and the out-of-equilibrium parts of Gac.First, a close examination of the out-of-equilibrium part

�Gac ¼: Gac �Geqac reveals that it contains nonoscillating

terms which cannot be expressed as the anticommutator ofdec of Eq. (19). In fact the equal-time anticommutator ofdec decays as e�2�t, whereas the nonoscillating termsdecay as e�2�tT . Hence, when �<�T, these extra termscan be neglected for t; t0 � 1=ð�T � �Þ. When these con-ditions are fulfilled, one can define �nðt0Þ and �cðt0Þ, theout-of-equilibrium value of the occupation number and thecoherence at t0, by

�Gacðt; t0Þ � �nðt0Þ’decðt; t0Þ½’decðt0; t0Þ��þ �cðt0Þ’decðt; t0Þ’decðt0; t0Þ þ cc; (54)

where’decðt;t0Þ¼e��ðt�t0Þe�i!f ðt�t0Þ=ffiffiffiffiffiffiffiffiffi2!f

pis the decaying

solution of Eq. (19) which contains only positive frequencyand which is unit Wronskian at t ¼ t0. Since �Gacðt; t0Þ isindependent of t0, one immediately deduces that

�nðt0Þ ¼ �nð0Þe�2�t0 ; j�cðt0Þj ¼ j�cð0Þje�2�t0 : (55)

This matches the behavior of the envelope observed inFig. 7. In the limit of small dissipation, one finds that theinitial values obey

�nð0Þ ¼ �ndisp þO��

!in

þ �

T

�;

�cð0Þ ¼ cdisp þO��

!in

þ �

T

�;

(56)

where �ndisp ¼ nout � neq and cdisp ¼ cout are the corre-sponding quantities evaluated with the dispersive case

0 100 200 300 4001.5

1.0

0.5

0.0

0.5

1.0

m cin x x'

Gdd

mc i

n

FIG. 8. We represent the equal time density-density correlationof Eq. (53) as a function of mcinjx� x0j, for cf=cin ¼ 0:1, andT ¼ T�in

. We take t ¼ 7:5=mc2f and four values of �2k, namely,

10�2 (solid line), 10�1:5 (dashed line), 10�1 (dot-dashed line)and 10�0:5 (dotted line). The parameter n of Eq. (8) is n ¼ 0. Weobserve a peak at x ¼ x0 that is broadened by dissipation, and aseries of peaks propagating away from the center with a groupvelocity higher than cf . The faster they propagate, the moredamped they are since we have �� k2.

0 10 20 30 40 500.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

t m cin2

f.G

ac

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

k m cin

f.G

ac

FIG. 7. We represent the product !f �Gacðt; t0 ¼ tÞ, where Gac is given in Eq. (35), on the left panel, as a function of theadimensionalized time tmc2in for k ¼ 1:5mcin, �

2k ¼ 0:03 and T ¼ 0:8T�in

, and on the right panel, as a function of k=mcin for

t ¼ 5=mc2in, �2k ¼ 0:05 and T ¼ 0:5T�in

. The dashed line on the right panel is the dissipativeless case �2k ¼ 0. In all cases, the height of

the jump is cf=cin ¼ 0:1.

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� ¼ 0. The values of nout and cout are given in Sec. III, andneq is the mean occupation number in a thermal bath.

Second, a similar analysis of the equilibrium part of Gac

gives

Geqacðt; t0Þ ¼ e��jt�t0jGth;disp

ac ðt; t0Þ þO��

!f

�; (57)

where Gth;dispac is the corresponding dispersive anticommu-

tator in a thermal state. It is worth noticing that in thepresence of dissipation, the rescaled anticommutator canbe smaller than 1=2, as can be seen in Fig. 7 for high k. Thisis because the � field is still interacting with the environ-ment. Yet, in the limit �jt� t0j � 1 and �=!f � 1, G

eqac is

indistinguishable from Gth;dispac . We can then work with

2neq þ 1 ¼ coth ½!f=2T�. Doing so, we get an imprecisionof the order of �=!f .

Having identified n ¼ �nþ neq and c, we can computethe coherence level � of Eq. (2), which of course inheritsthe imprecision coming from neq. In Fig. 9, we represent�k and its imprecision as a function of time for fourdifferent cases. As already discussed, we notice that thenonseparability of the state is lost in a time much smallerthan 1=�. We also notice that when �=! is low enough, theimprecision in � does not significantly affect our ability topredict when nonseparability will be lost. In brief, for lowvalues of �=! and �=T, the anticommutator Gac can bereliably interpreted at any time t0 using the instantaneousparticle representation based on decðt; t0Þ of Eq. (19).

V. CONCLUSIONS

In this paper, we computed the spectral properties nk andthe coherence coefficient ck of quasiparticles producedwhen a sudden change is applied to a one-dimensional

homogeneous system. We took into account both the effectsof an initial temperature and the fact that the quasiparticlesare coupled to a reservoir of modes, something whichinduces dissipative effects. For definiteness, the quasipar-ticles are taken to be Bogoliubov phonons propagating in anelongated atomic condensate. Yet our results should apply,mutatis mutandis, to all weakly dissipative systems. We arecurrently extending our treatment to polariton fluids.For simplicity, we worked with a quadratic action.

Importantly, this allows us to compute the anticommutatorfor nonequal times [see Eqs. (24)–(27)], something whichis not generally done when using the truncated Wignermethod [18], but which could be very useful for futureexperiments. Because our system is coupled to a bath, nkand ck are a posteriori extracted from the anticommutatorof the phonon field [see Eq. (43) and (54)]. Then, todistinguish classical correlations from quantum entangle-ment, we used the fact that negative values of the parameter�k of Eq. (2) correspond to nonseparable states (for theGaussian states we consider).When neglecting dissipative effects, we studied the

competition between the squeezing of the quasiparticlesstate, which is induced by the sudden change, and theinitial temperature, which increases the contribution ofstimulated effects [see Eq. (36)]. In Fig. 1, one clearlysees that the value of the minima of the equal time anti-commutator allows one to know if the state is separable ornot. The outcome of the competition is summarized inFig. 3, where the coherence parameter � is plotted as afunction of the sudden change of the sound speed and thetemperature. We applied our analysis to the recent experi-ment of Ref. [5] and concluded that one should eitherincrease the change of c or work with a lower temperaturein order to obtain a nonseparable state.We then included dissipative effects. When there is no

(significant) dissipation after the sudden change, weshowed in Fig. 4 how the final number of particles isprogressively affected by increasing dissipation. Whenthe initial occupation number is low, the deviations arelinear in the decay rate �, whereas they are quadratic foroccupation numbers nin * 5. Interestingly, we observed inFig. 5 that dissipation, on the one hand, hardly affects thecoherence parameter � when the sudden change is due to adecrease of the sound speed (something which correspondsto an expanding universe when using the analogy withgravity) and, on the other hand, severely reduces the co-herence when the sound speed increases. This discrepancyis further studied in Fig. 6 which illustrates the key roleplayed by the C1 character of the anticommutator across thesudden change.We also studied the case when the dissipative rate is

constant. In this case, the main effect of dissipation on theanticommutator is the expected damping towards the equi-librium value (see Fig. 7). As a result, for high occupationnumbers, the nonseparability of the state is lost in a time

0.0 0.2 0.4 0.6 0.8 1.00.2

0.1

0.0

0.1

0.2

0.3

0.4

0.5

t

FIG. 9 (color online). We represent the coherence level � as afunction of �t, for T ¼ T�in

=2 and �2k ¼ 0:01. We consider two

values of cf=cin ¼ 0:1 (solid line) or 0.5 (dashed line), and twovalues of k=mcin ¼ 1 (black line) or 1.5 (thick red line). Theimprecision in the value of � is indicated by vertical bars. In thepresent weakly dissipative cases, the spread of � is of the order0.02. Therefore, the moment where the nonseparability of thestate is lost is known with some precision.

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much shorter than the inverse decay rate (see Fig. 9). In spiteof the fact that the quasiparticles are still coupled to theenvironment, we showed that a reliable study of this loss canbe performed for weakly dissipative systems, i.e. with�=! � 1. On the contrary, for strongly dissipative systems,i.e. rapidly decaying quasiparticles, we believe the notion ofnonseparability cannot be meaningfully implemented.

ACKNOWLEDGMENTS

We thank Scott Robertson for a careful reading of ourmanuscript, and Iacopo Carusotto for interesting discus-sions. This work, which is part of the project QEAGE(Quantum Effects in Analogue Gravity Experiments),was supported by the French National Research Agencyunder the Program Investing in the Future Grant No. ANR-11-IDEX-0003-02. It has also been supported by the FQXigrant ‘‘Hawking radiation in dissipative field theories’’Grant No. FQXi-MGB-1129.

APPENDIX A: ACTION FOR RELATIVEDENSITY FLUCTUATIONS

We briefly review how Eq. (4) is obtained from the actionof the atomic field; formore details see [39].We then explainhowEq. (6) can also be derived from an action defined at theatomic level (rather than that of the quasiparticles).

The action for the second quantized field describing adilute ultracold atomic gas is [40]

S ¼Z

dtdx

�i�y@t�� 1

2m@x�

y@x�� V�y�

� gat2

�y�y� �

�: (A1)

One writes � ¼ �condð1þ �Þ, where �condðt; xÞ is themean field describing the condensed atoms. It is a solutionof the Gross Pitaevskii equation

i@t�cond ¼�� 1

2m@2x þ V þ gat�

�cond�cond

��cond: (A2)

On the other hand, � describes relative density fluctua-

tions. The action is then expanded in powers of �. UsingEq. (A2), the quadratic part is

S� ¼Z

dtdx�

�i�yð@t þ v@xÞ�� 1

2m@xð�yÞ@x�

�mc2

2ð�2 þ ð�yÞ2 þ 2�y�Þ

�; (A3)

where � ¼: j�condj2, 2imv ¼: @xðln�cond � ln��condÞ and

c2 ¼: gat�=m are arbitrary functions of t and x obeying@t�þ @xð�vÞ ¼ 0. When considering homogeneous con-densates at rest (v ¼ 0), one obtains Eq. (4) with � ¼ cst.

One can also derive Eq. (6) when coupling the environ-ment field �� to the atomic field. More precisely, to get

Eq. (6), one should work with

Sint ¼ �Z

dtdx

�½ð�yÞ����~gð@xÞ@t

�Zd� ���

��: (A4)

Then, using Eq. (5), the equations of motion are

i@t� ¼�� 1

2m@2x þ V þ gat�

y��� (A5a)

þ �ð�yÞ��1��~gð@xÞ@t�Z

d� ���

�;

ð@2t þ!2� Þ ��� ¼ @t~gð�@xÞ½ð�yÞ����: (A5b)

Hence, ��� will also condense. We can write it as

��� ¼ �cond� þ �� : (A6)

From Eq. (A5a), we get a modified Gross-Pitaevskiiequation

i@t�cond ¼ H cond�cond; (A7)

with

H cond ¼ �@2x2m

þ V þ gatj�condj2

þ �j�condj��1~gð@xÞ@t�Z

d� ��cond�

�: (A8)

Since this last term does not contain any operator acting on�cond, the conservation equation @t�þ @xð�vÞ ¼ 0 is stillvalid. Moreover, it gives rise, in Eq. (A3), to a term

�S� ¼Z

dtdx��y���~gð@xÞ@t�Z

d��cond�

�: (A9)

On the other hand, Eq. (A4) gives two contributionsgiven by

Sð1Þint ¼ �Z

dtdx

���ð�� 1Þ

2ð�yÞ2 þ �2�y�

þ �ð�� 1Þ2

�2���~gð@xÞ@t

�Zd��cond

�

��;

Sð2Þint ¼ �Z

dtdx

����½�y þ ��~gð@xÞ@t

�Zd���

��:

(A10)

Since the first one is second order in � and zero order in

�� , it combines with Eq. (A9) to give

�S� þ Sð1Þint ¼Z

dtdx�ð�� 1Þ

2ð�y þ�Þ2��~gð@xÞ

� @t

�Zd��cond

�

�: (A11)

Its role is to modify the speed of sound which is nowgiven by

mc2 ¼: gat�þ �ð�� 1Þ2

���1~gð@xÞ@t�Z

d��cond�

�:

(A12)

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The second contribution Sð2Þint gives rise to Eq. (6), with

�~gð�@xÞ ¼ g��þ1=2ð�@xÞn. Hence, we have demonstratedthat our model based on Eqs. (4) and (6) can be derived

from actions involving only the atomic field �.Notice that in a polariton system [19,20], the dissipative

processes occur at the level of the number of photons, andnot only at the level of the quasiparticles (phonons of thephoton fluid). This means that the coupling between� and�pola should be of the form

Sint; pola ¼ �Z

dtdx

���pola@t

�Zd���

�þ H:c:

�: (A13)

We are currently studying this case.

APPENDIX B: ON OTHER OBSERVABLES

We establish the dictionary between the field ofEq. (14) and its anticommutator Gac of Eq. (24), and twoother languages often used in the literature, namely, on theone hand, the phase and density fluctuations � and ��, and,on the other hand, the so-called g1 and g2 functions[40,41]. These functions are expressed in terms of the

atomic field � as

�ðx; tÞ ¼ �cond

�1þ ��

2�

�ei�;

g1ðx; t; x0; t0Þ ¼: Re½Trð�T�yðx; tÞ�ðx0; t0ÞÞ�;

g2ðx; t; x0; t0Þ ¼: Trð�T�yðx; tÞ�yðx0; t0Þ�ðx0; t0Þ�ðx; tÞÞg1ðx; t; x; tÞg1ðx0; t0; x0; t0Þ :

(B1)

In homogeneous condensates, in momentum space, and tolinear order, we have

k ¼ � ��kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�c�k2

p ; @tk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2��ck2

q�k: (B2)

Hence, the three anticommutators are related to our Gac by

Trð�T��ðtÞ��ðt0ÞÞ ¼ 2�c�k2Gkacðt; t0Þ;

Trð�T�ðtÞ��ðt0ÞÞ ¼ �@tGkacðt; t0Þ;

Trð�T�ðtÞ�ðt0ÞÞ ¼ @t@t0Gkacðt; t0Þ

2�c�k2:

(B3)

The knowledge of Gac of Eq. (24) thus fixes the three ofthem.

Similarly, for k � 0, we have

gk1ðt; t0Þ ¼�c�k2

2þ 1

2c�k2@t@t0

�Gk

acðt; t0Þ þ i@tGkcðt; t0Þ;

(B4)

where Gkcðt; t0Þ is the commutator of (which is

imaginary), and

gk2ðt; t0Þ ¼2

�½c�k2Gk

acðt; t0Þ � i@tGkcðt; t0Þ�: (B5)

Using Eq. (35), we see that a measurement of g1ðt; t0 ¼ tÞ,or g2ðt; t0 ¼ tÞ, for various t is sufficient to extract n and c,and therefore to distinguish nonseparable states from sepa-rable ones. For instance, the minimum value of g2ðt; t0 ¼ tÞis proportional to n� jcj þ ð1�!f=c�k

2Þ=2. Hence,when knowing !f=c�k

2 and �, measuring the minima ofg2ðt; t0 ¼ tÞ is sufficient to distinguish nonseparable states.

APPENDIX C: APPROXIMATING EQ. (42)

The goal of this appendix is to approximatively evaluateEq. (42). To do so, we first consider that h is constant fornegative times.5 Hence, the part of the integral that runs onnegative times is easy to handle and givesZ 0

�1d�hð �Þei!��e�

�in

R1 �

h2ðzÞdz ffiffiffiffiffiffiffiffiffi2!f

p’outð�Þ

� e��in

R10h2 !f þ ð!� � i�inÞ!2

in � ð!� � i�inÞ2: (C1)

Here, we neglected the effect of the change of ! forpositive times on’out. The deviation is generically of order�2in=!f .

6 To get this bound, we can write, for positive

times, ’out ¼ e�i!f�=ffiffiffiffiffiffiffiffiffi2!f

p þ ’1, and perturbatively in’1, get to j’1j

ffiffiffiffiffiffiffiffiffi2!f

p< �2

in=!f R10 f2. This bound can

be relaxed order by order by solving Eq. (20) with a source.This is not the goal of this appendix. For positive times, wenow make the same approximation and getZ 1

0d�hð �Þei!��e�

�in

R1 �

h2ðzÞdz ffiffiffiffiffiffiffiffiffi2!f

p’outð�Þ

�Z 1

0d�hð�Þe��in

R1�h2eið!��!f Þ�: (C2)

We can now compute this last integral perturbatively in�= . To get coherent results and to get rid of the 1=!�

term, the same expansion is necessary in Eq. (C1).Since for the cases we consider (i.e., sudden change,

� � ! � ), �in= > �2in=!f > �2

in= 2, the expansion

in �in= should be done to first order maximum. To thisorder, R becomes

R ¼��

1� �in

Z 1

0h2�

!f þ ð!� � i�inÞ!2

in � ð!� � i�inÞ2

þZ 1

0d�hð �Þ

�1� �in

Z 1

�h2�eið!��!f Þ�

�

��1þO

��2in

!f

��: (C3)

5This simplifies the study, but is not necessary. When it is notthe case, we should apply the same treatment to positiveand negative times.

6Note that for low values of , a WKB approximation givesthe same first term, with an upper bound on the approximationgiven by �2=!2ð1þ =!Þ.

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When working with h of Eq. (45), one obtains

R� !f þ ð!� � i�inÞ!2

in � ð!� � i�inÞ2þ

h1�ið!��!f Þ

!� �!f

� �in

3

24 !f þ ð!� � i�inÞ!2

in � ð!� � i�inÞ2þ

h4�ið!��!f Þ

!� �!f

35; (C4)

where ex ¼ Pnk¼0 x

k=k!� hnðxÞxn=n! defines the

remainder term of order n, hnðxÞ, of the Taylor expansion

of the exponential function. We notice that hnðxÞ / xfor x ! 0 so that R is regular at !� ¼ !f . Moreover,

at large x, hnðxÞ � exn!=xn þ ð1þOð1=xÞÞ so that

R� 1=!2� at large !� .

To complete the study, we checked the validity of

Eq. (C4) by numerically evaluating R. When considering

only the first line of Eq. (C4), we observed that the relative

error is smaller than �= , as predicted. When including the

second line, the relative error remains smaller than �2= !f .

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