Density correlations and dynamical Casimir emission of Bogoliubov phonons in modulated atomic Bose-Einstein condensates
Iacopo Carusotto, Roberto Balbinot, Alessandro Fabbri, Alessio Recati
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Density correlations and dynamical Casimir emission of Bogoliubov phonons inmodulated atomic Bose-Einstein condensates
Iacopo Carusotto, Roberto Balbinot, Alessandro Fabbri, and Alessio Recati
1, 4 CNR-INFM BEC Center and Dipartimento di Fisica,Universit`a di Trento, via Sommarive 14, I-38050 Povo, Trento, Italy Dipartimento di Fisica dell’Universit`a di Bologna and INFN sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy Departamento de Fisica Teorica and IFIC, Universidad de Valencia-CSIC, C. Dr.Moliner, 50, 46100 Burjassot, Spain Physik-Department, Technische Universit¨at M¨unchen, D-85748 Garching, Germany
We present a theory of the density correlations that appear in an atomic Bose-Einstein condensateas a consequence of the dynamical Casimir emission of pairs of Bogoliubov phonons when the atom-atom scattering length is modulated in time. Different regimes as a function of the temporal shapeof the modulation are identified and a simple physical picture of the phenomenon is discussed.Analytical expressions for the density correlation function are provided for the most significantlimiting cases. This theory is able to explain some unexpected features recently observed in numericalcalculations of Hawking radiation from analog black holes.
The dynamical Casimir effect [1] is a very general pre-diction of quantum field theory: whenever the boundaryconditions and/or the dispersion law and/or the back-ground of a quantum field are quickly varied in time,pairs of quanta are generated off the vacuum state byparametric amplification of zero-point noise. The sim-plest and most celebrated example of dynamical Casimireffect was predicted for an optical cavity whose plane-parallel mirrors are made to rapidly oscillate in timealong the cavity axis [2, 3, 4]. Despite the significant ef-fort devoted to these fascinating effects, no experimentalobservation of the dynamical Casimir effect has yet ap-peared, the main reason being the difficulty in moving themirror at a fast enough speed [3]. Alternative schemesto modulate the effective optical length of a cavity ona very fast time-scale by acting on the refractive indexof the cavity material have been proposed and shownto give a sizeable intensity of dynamical Casimir emis-sion [5, 6, 7, 8, 9, 10, 11]. Experiments in this directionare in progress in several groups [12, 13].Since the original proposal by Unruh [14], the ad-vances in the field of the so-called analog models [15] havepointed out the possibility of simulating the physics of aquantum field on a generic curved space-time in table-top condensed-matter experiments: the propagation ofelementary excitations in spatially and temporally inho-mogeneous systems can in fact be recast in terms of arelativistic wave equation on an effective curved space-time. In the simplest case of acoustic waves in a fluid,the space-time metric is fixed by the spatial and temporalprofiles of the sound speed and of the flow velocity. Uponquantization of the resulting field theory, an analog dy-namical Casimir effect is then expected to appear when-ever the sound speed in a spatially homogeneous systemis made to quickly vary in time, which in the languageof the analogy corresponds to the expansion or contrac-tion of the underlying universe [16, 17]. On the otherhand, in the presence of an acoustic horizon separatingan upstream region of sub-sonic flow from a downstreamone of super-sonic flow, the emission of analog Hawking radiation has been predicted [14, 18, 19].As proposed in [20, 21], a most promising way of exper-imentally investigating this physics involves the measure-ment of the correlation function of density fluctuations.A recent numerical experiment [22] has observed boththe dynamical Casimir effect and the Hawking emissionin a microscopic simulation of the dynamics of an atomicBose Einstein condensate during and after the formationof an acoustic black hole.After a series of papers [16, 17, 23, 24] focussed onthe deposited energy and the spectrum of the emittedphonons, a number of theoretical works have started in-vestigating the time-evolution of the correlation functionsof atomic many-body systems after a sudden quench ofsome parameter, in particular the atom-atom interactionconstant [20, 25, 26, 27, 28]. In the meanwhile, severalexperiments have recently addressed the response of Bosegases to a time-modulation of the atom-atom scatteringlength: the interest of most of these experiments washowever concentrated on the energy deposited in the sys-tem by the perturbation [29] and only provided qualita-tive information on the created density perturbation [30].As density correlations are becoming a standard ob-servable in the experimental study of ultracold atomicgases [31, 32, 33] we expect that the measurement of thedensity correlation pattern created by suitable perturba-tion sequences will soon provide a new powerful tool toinvestigate the microscopic properties of strongly corre-lated atomic gases and in particular of their elementaryexcitations. A few pioneering studies in this directionfor the simplest case of a sudden quench have recentlyappeared [27, 28].It is remarkable to note that a similar strategy ispresently under way in the completely different field ofcosmology, where one is trying to extract information onthe primordial inflation phase of the universe from thespectrum and the correlation function of the temperaturefluctuations of the cosmological microwave radiation [34].In the language of the analogy, the fast expansion of theuniverse in the inflation phase corresponds in fact to afast temporal variation of the sound speed, which is in-deed the result of a fast variation of the atom-atom scat-tering length [16, 17, 20].The present paper presents a comprehensive theoret-ical investigation of the different features that appearin the density correlation funcion of a spatially homo-geneous Bose Einstein condensate as a consequence of atime-dependent atomic scattering length. Calculationsbased the Bogoliubov theory of dilute condensates allowfor a quantitative understanding of the different regimesas a function of the temporal profile of the scatteringlength modulation and provide a clear physical pictureof the phenomenon in terms of the dynamical Casimiremission of entangled pairs of phonons. This theory fullyconfirms the numerical observations that was put forwardin the original paper [22].The paper is organized as follows. In sec.I we presentthe physical system under investigation and we briefly re-view the Bogoliubov approximation. The general theoryof the density correlations that result from the dynami-cal Casimir emission is presented in Sec.II. The followingsections discuss the phenomenology in the most remark-able cases of an adiabatic transition (Sec.III), of a sud-den jump (Sec.IV) and of a slow ramp (Sec.V) of theatom-atom scattering length. Analytical formulas validin the hydrodynamic limit are presented in Sec.VI. Acomparison with the numerical results of [22] is presentedin Sec.VII. The case of a quasi-periodic modulation ofthe scattering length is analyzed in Sec.VIII and a pos-sible application to the measurement of temperature dis-cussed. The possibility of reinforcing the density corre-lation signal by mapping phase fluctuations into densityones is quantitatively studied in Sec.IX. Conclusions arefinally drawn in Sec.X.
I. THE PHYSICAL SYSTEM AND THEBOGOLIUBOV DESCRIPTION
We consider a spatially homogeneous Bose-Einsteincondensate of atoms of mass m . The gas is assumedto be initially at rest in its thermal equilibrium state ata temperature T and to have a density n . Atom-atominteractions are modeled by a repulsive, local interac-tion potential of scattering length a >
0. The value ofthe scattering length is assumed to be constant in spacebut to have a non-trivial temporal dependence a ( t ). Inthe remote past and future t = ∓∞ , it tends to a con-stant values. Experimentally, the possibility of tuningof the atom-atom interactions on a wide range has beendemonstrated using magnetic and optical Feshbach reso-nances [36], as well as by modulating the lateral confine-ment of reduced dimensionality samples [31].At all times, the system is assumed to be well withinthe dilute regime n a ≪ φ is described by the Gross-Pitaevskii equation, i ~ ∂φ ∂t = − ~ ∇ m φ + V ( x ) φ + 4 π ~ a ( t ) m | φ | φ . (1)In the spatially homogeneous case V ( x ) = 0 that we areconsidering here, the condensate wave function remainsat all times constant in space and only acquires a time-dependent global phase, φ ( x , t ) = √ n exp iθ ( t ).Within the Bogoliubov approximation [37], the fluctu-ations around the purely condensed state are describedby a quadratic Hamiltonian in the ˆΛ k operators describ-ing the non-condensed k = 0 plane-wave modes H = E ++ 12 X k (ˆΛ † k , ˆΛ − k ) (cid:18) E k + µ ( t ) µ ( t ) − µ ( t ) − E k − µ ( t ) (cid:19) (cid:18) ˆΛ k ˆΛ †− k (cid:19) (2)The ˆΛ k operators have a simple expression in termsof the momentum-space atomic field operators, ˆΛ k = N − / ˆ b † k =0 ˆ b k . Here, N is the total number of particles inthe gas. In the following of the paper, we shall also makeuse of the real-space operators ˆΛ( x ) that are defined asthe Fourier transform of the ˆΛ k operators [37]. The (in-stantaneous) chemical potential is defined in terms ofthe nonlinear interaction constant g ( t ) = 4 π ~ a ( t ) /m as µ ( t ) = g ( t ) n . The kinetic energy of the k -mode is indi-cated as E k = ~ k / m .Neglecting the zero-point energy, the Hamiltonian (2)can be recast for each instantaneous value of a ( t ) into thecanonical form H ( t ) = X k ~ ω k ( t ) ˆ a † k ˆ a k . (3)where the (bosonic) Bogoliubov operators ˆ a k (ˆ a † k ) respec-tively destroy (create) an elementary Bogoliubov excita-tion of momentum ~ k and energy ~ ω k ( t ) = p E k ( E k + 2 µ ( t )) . (4)In terms of the atomic field operators, the (instanta-neous) Bogoliubov operators ˆ a k have the following ex-pression: ˆ a k = u k ( t ) ˆΛ k − v k ( t ) ˆΛ †− k . (5)in terms of the (instantaneous) Bogoliubov coefficients u k ( t ) and v k ( t ), u k ( t ) ± v k ( t ) = (cid:18) E k ~ ω k ( t ) (cid:19) ± / : (6)as a consequence of the time-dependence of the scatteringlength a ( t ), the Bogoliubov operators have an explicittime-dependence even in the Schr¨odinger picture of (5).At each time, the (instantaneous) ground state | g ( t ) i ofthe Bogoliubov theory is defined byˆ a k | g i = 0 ∀ k . (7)The time-dependence of | g ( t ) i is the key responsible forthe dynamical Casimir emission: when the scatteringlength is modulated at a fast rate, the system is not ableto adiabatically follow the istantaneous ground state.Non-adiabatic processes then result in the creation ofcorrelated pairs of excitations in the system out of thevacuum state [7]. This point of view will be discussed infull detail in Sec.IV.An alternative, yet equivalent picture of the dynamicalCasimir emission can be obtained in the limiting caseof a weak modulation a ( t ) = a + δa ( t ) with | δa ( t ) | ≪ a . In analogy to the discussion of [5], the dynamicalCasimir emission can in this case be recast in terms ofthe following Hamiltonian H = H + 2 π ~ nm δa ( t ) X k ( u k + v k ) × (ˆ a † k + ˆ a − k )(ˆ a k + ˆ a †− k ) . (8)Here, H is the Hamiltonian of the gas for a constantvalue of the scattering length a and the effect of theweak modulation δa ( t ) is to simultaneously excite pairsof entangled Bogoliubov particles with opposite momenta ± k . In the language of nonlinear optics, such a processgoes under the name of parametric down-conversion [44].The most remarkable case of a periodic modulation of a ( t ) will be discussed in Sec.VIII. II. THE THEORETICAL FRAMEWORKA. The time-evolution of Bogoliubov operators
Thanks to the quadratic nature of the BogoliubovHamiltonian (3), the time-evolution of the Bogoliubovoperators in the Heisenberg picture can be written in aclosed form: d ˆ a k dt = − iω k ˆ a k + ( ˙ u k v k − u k ˙ v k ) ˆ a †− k . (9) d ˆ a †− k dt = iω − k ˆ a †− k + ( ˙ u k v k − u k ˙ v k ) ˆ a k . (10)At each time t , the u k ( t ) and v k ( t ) functions are to beevaluated using the instantaneous value of the scatteringlength a ( t ) according to (5). Dots indicate the derivativeover time t .The first terms on the RHS of (9-10) describe the triv-ial evolution of the ˆ a k and ˆ a †− k operators at frequencies ± ω k under the instantaneous Hamiltonian (3). The otherterms take into account the dependence (5) of the ˆ a k andˆ a †− k operators on the instantaneous scattering length a ( t ) via the time-dependence of the u k and v k Bogoliubov co-efficients, and are responsible for the mixing of the ˆ a k and ˆ a †− k operators. These mixing terms are proportionalto the rate at which a ( t ) is varied in time. Once a ( t ) hasapproached its late-time limiting value, one is left withthe trivial oscillation of the ˆ a k ( t ) and ˆ a †− k ( t ) operatorsat frequencies ± ω k .The relation between the ˆ a k ( t ) and ˆ a †− k ( t ) operators ata generic time t to their initial values ˆ a k (0) and ˆ a †− k (0)before the excitation sequence can be summarized as apair of linear equations.ˆ a k ( t ) = h C k , + ( t ) ˆ a k (0) + C k , − ( t ) ˆ a †− k (0) i (11)ˆ a † k ( t ) = h C ∗ k , + ( t ) ˆ a † k (0) + C ∗ k , − ( t ) ˆ a − k (0) i , (12)whose coefficients C k , ± ( t ) have to be computed by solv-ing the pair of differential equations (9-10).Thanks to the thermal equilibrium hypothesis, the Bo-goliubov modes are assumed to be initially uncorrelatedand thermally occupied, h ˆ a k (0)ˆ a − k (0) i = 0 (13) h ˆ a † k (0)ˆ a k (0) i = n th , k = 1exp( ~ ω k /k B T ) − . (14)At late times t > t fin after the end of the modulation,the Bogoliubov mode occupations n k ( t ) = h ˆ a † k ( t ) ˆ a k ( t ) i == (cid:0) | C k , + ( t fin ) | + | C k , − ( t fin ) | (cid:1) n th , k ++ | C k , − ( t fin ) | (15)remain constant in time, while the anomalous averages A k ( t ) = h ˆ a k ( t ) ˆ a − k ( t ) i = C k , + ( t fin ) C k , − ( t fin ) × (2 n th , k + 1) e − iω k ( t − t fin ) (16)keep on oscillating at the frequency 2 ω k . B. The density correlation function
In a homogeneous system the modulation of a ( t ) hasno effect on the average density that remains flat at alltimes, n ( x ) = h ˆΨ † ( x ) ˆΨ( x ) i = n. (17)On the other hand, the emission of entangled phononpairs is clearly visible in the density correlation function g (2) ( x , x ′ ) = 1 n D ˆΨ † ( x ) ˆΨ † ( x ′ ) ˆΨ( x ′ ) ˆΨ( x ) E == 1 + 1 nV X k h e i k ( x ′ − x ) (cid:0) ( u k + v k ) + 1 (cid:1) + c.c. i ++ 1 nV X k ( u k + v k ) h e i k ( x ′ − x ) (cid:16) h ˆ a k ˆ a − k i + h ˆ a † k ˆ a k i (cid:17) + c.c. i = 1 + 1 nV X k e i k ( x ′ − x ) (cid:2) ( u k + v k ) ×× D (ˆ a † k + ˆ a − k )(ˆ a k + ˆ a †− k ) E − i (18)which involves a combination of Bogoliubov mode occu-pation (15) and anomalous average (16). As we shall seein the following of the paper, the different time depen-dence of the two terms is responsible for qualitativelydifferent features in the density correlation function. C. Effect of external trapping
Before proceeding with the analysis of the density cor-relation function, it is important to briefly clarify theconsequences of the spatial inhomogeneity of the gas inthe presence of a trapping potential.As the density profile of a trapped gas strongly de-pends on interactions, the modulation of the scatteringlength a ( t ) may result into a macroscopic oscillation ofthe condensate shape [38] and even in its collapse whenthe scattering is tuned to a large and attractive value.This latter effect has been experimentally demonstratedin a remarkable way in the so-called Bose-nova experi-ments of [39].In the framework of the standard Bogoliubov the-ory [37], this physics is described by a term of the form δ H = 2 π ~ δa ( t ) m Z d x | φ ( x ) | φ ∗ ( x ) ˆΛ( x )+h.c. . (19)where φ ( x ) is the condensate wavefunction, normalizedin a way that R d x | φ ( x ) | = N . As usual in quantumoptics, an Hamiltonian term involving a single quantumfield operator leads to a coherent excitation of the field,in our case a collective excitation of the condensate. Asexpected, this term disappears in the case of a homo-geneous condensate considered in the rest of the paperthanks to the spatial orthogonality of the ˆΛ( x ) operatorto the condensate wavefunction φ ( x ), Z d x φ ∗ ( x ) Λ( x ) = 0 . (20)In the general case, the density fluctuation pattern canbe isolated even in the presence of a strong collectiveexcitation simply by subtracting out the deterministiccomponent of the density modulation. III. ADIABATIC LIMIT
The mixing of the ˆ a k and ˆ a †− k operators is negligible | C k , + | ≫ | C k , − | ≃ a ( t ) takes place on a very slow time-scale as compared to ω k [45]. As a consequence, theoccupation n k of the Bogoliubov modes is constant intime and equal to its initial value n th , k and the anomalousaverages A k remains zero.For a zero initial temperature T = 0, the adiabatic con-dition is equivalent to stating that the evolution is slowenough for the system to remain in its ground state: at alltimes, the density correlation function exactly coincideswith the static T = 0 one for the instantaneous value of a ( t ). On the other hand, for a non-zero initial tempera-ture T > n k of each Bogoliubov mode is conserved duringthe adiabatic evolution, the instantaneous energy ~ ω k ( t )of the mode has in fact a non-trivial time-dependence.As a result, the population n k of the different Bogoli-ubov modes at late times is no longer described by asimple thermal condition of the form (14). IV. SUDDEN JUMPA. General formulas
Simple expressions for the expectation values appear-ing in (18) can be obtained in the limit of a sud-den variation of a ( t ) from a to a on a time scale σ t much faster than the frequency ω k of all the rele-vant modes, i.e. µ , σ t ≪
1. In this limit of a sud-den quench [25, 26, 27, 41], the evolution of the ˆΛ k ( t ),ˆΛ †− k ( t ) atomic operators during the sudden modulationof a is negligible and the simple picture of the dynam-ical Casimir effect introduced in [7] can be straightfor-wardly applied: the ˆ a k , ˆ a †− k operators at t = 0 ± rightbefore and right after the jump are expanded in termsof the ˆΛ k ( t = 0), ˆΛ †− k ( t = 0) atomic operators using (5)with the suitable u k , v k Bogoliubov coefficients. An ex-plicit relation between the ˆ a k , ˆ a †− k operators at t = 0 ± isstraightforwardly obtained by eliminating the ˆΛ k ( t = 0),ˆΛ †− k ( t = 0) operators. The evolution of the ˆ a k , ˆ a †− k op-erators at later t > ∓ iω + k t ).Once all these steps are combined together, one is fi-nally led to the following compact relations [41]:ˆ a k ( t ) = (cid:20) η + k + η − k a k (0 − ) + η + k − η − k a †− k (0 − ) (cid:21) e − iω + k t (21)ˆ a † k ( t ) = (cid:20) η + k + η − k a † k (0 − ) + η + k − η − k a − k (0 − ) (cid:21) e iω + k t (22)where the η ± k coefficients are defined as η ± k =( ω + k /ω − k ) ± / and the Bogoliubov frequencies ω ∓ k beforeand after the jump are evaluated using (4) with a = a , .As η ± k → E k /µ , ≫
1, the transformations (21-22) do not mix the ˆ a k , ˆ a †− k operators for large values of k ,which provides an a posteriori justification for the suddenjump condition σ t µ , ≫
1. For the hydrodynamic modeswith E k ≪ µ , , the mixing coefficient η ± k tends insteadto a finite limiting value ( c /c ) ± / .The Bogoliubov mode occupation n k after the suddenchange of scattering length is given by the formula n k = ( ω + k ) + ( ω − k ) ω + k ω − k n th , k + ( ω + k − ω − k ) ω + k ω − k , (23)while the anomalous average has the form A k ( t ) = 14 (cid:18) ω + k ω − k − ω − k ω + k (cid:19) (cid:16) n th , k + 1 (cid:17) e − iω + k t (24)Note that the Bose distribution n th , k is here to be eval-uated according to (14) using the initial value ω − k of theBogoliubov mode frequency. In agreement with (13), theinitial value of the anomalous average has been taken tobe zero. The terms in (23) and (24) proportional to theinitial population n th , k account for the amplification ofinitial thermal excitations by the sudden jump in a , whilethe other terms describe the contribution due to the dy-namical Casimir emission of Bogoliubov phonons out ofthe initial vacuum state.It is interesting to evaluate (23) and (24) in the limit ofa small change of a ( t ), i.e. a = a + ∆ a with ∆ a ≪ a .In this limit, one has: n k ≃ n th , k + ∆ µ E k + 2 µ ) (cid:16) n th , k + 1 (cid:17) (25) A k ( t ) ≃ ∆ µ E k + 2 µ ) (cid:16) n th , k + 1 (cid:17) e − iω + k t . (26)Here, the chemical potential variation is defined as ∆ µ =4 π ~ ( a − a ) n /m . While the effect on the anoma-lous average is linear in ∆ µ , the population change isquadratic and therefore much weaker. The differenceis even more dramatic at T >
0: while the populationchange is a (small) correction proportional to ∆ µ ontop of the (large) initial thermal distribution, the anoma-lous average fully originates from the dynamical Casimiremission and is amplified by the initial thermal popula-tion. B. Physical discussion
This physics is illustrated in the plots of the Bogoli-ubov occupation and the modulus of the anomalous av-erage shown in Fig.1(d,e) for the a /a = 0 .
25 case. Forthese plot, the exact formulas (23) and (24) have beenused.For a vanishing initial temperature T = 0 (black line),both n k and A k show a smooth peak centered at k = 0and a power-law tail that extends far on the high en-ergy modes. As predicted by the analytical approximateformulas (25) and (26), the anomalous average A k is inmodulus much larger than the occupation n k . Note thatthe smooth peak would be replaced by a 1 /k divergenceif interactions in either the initial or final states werevanishing.For a finite initial temperature T >
0, the effect of thesudden variation of a on n k is a very weak correction ontop of the initial thermal distribution n th , k . On the otherhand, the T = 0 contribution to the anomalous averagedue to the dynamical Casimir effect is strongly amplifiedby the initial thermal populaton. ( n ξ ) × G ( ) ( x , x ’) | x-x’ | / ξ -0.08-0.0400.040.08 -2 0 2 k ξ n k -2 0 2 k ξ A k µ t = 5 µ t = 20 µ t = 0 (a)(b)(c) (d) (e) FIG. 1: Panels (a-c): Time evolution of the density corre-lation function after a sudden jump of the scattering lengthfrom a to a = a /
4. Different (a-c) panels refer to dif-ferent evolution times after the jump, µ t = 0 , ,
20. Panel(d): Bogoliubov mode occupation n k after the jump. Panel(e): anomalous average |A k | after the jump. In all panels,thick black lines correspond to an initial T = 0; thin red linescorrespond to a finite initial temperature T /µ = 0 .
5. Thedashed red line in panel (d) indicates the prediction of theadiabatic model, i.e. the initial population of the Bogoliubovmode n th , k . The density correlation function is immediately ob-tained inserting (23-24) into the general expression (18).A few snapshots after different evolution times are shownin Fig.1(a-c) for the simplest case of a one-dimensionalsystem. Exception made for some geometrical factors,the physics is however identical when two- or three-dimensional systems are considered.For t = 0 + , straightforward algebraic manipulationsconfirm that the sudden change of a does not have animmediate effect on correlation function, g (2) ( x − x ′ ; t =0 + ) = g (2) ( x − x ′ ; t = 0 − ): The jump is in fact toorapid for the microscopic state of the atoms to respond.However, as this state is no longer an eigenstate of thesystem Hamiltonian with a ( t >
0) = a , a non-trivialevolution is observed on g (2) at later times t > a , the density correlationfunction is characterized by a dip around x = x ′ due tothe effect of atom-atom repulsive interactions. At finitetemperature, this dip is less pronounced than at T = 0,and starts being accompanied by a Hanbury-Brown andTwiss bump due to the thermal fluctuations [42].After the sudden change, the static central structurearound x = x ′ is somehow amplified by the change inpopulation n k and, more importantly, by the increase inthe Bogoliubov u k + v k coefficient as a consequence ofthe reduced value of a . At the same time, a system ofmoving fringes originates from x = x ′ and propagatesin the outwards direction as a consequence of the time-dependent anomalous average A k ( t ). At each time t , thefringe pattern is concentrated in the | x − x ′ | & c t regionand shows a significant chirping in space, the externalpart of the pattern having a shorter wavelength than theinner part. As time goes on, the fringe pattern gets pro-gressively stretched in space.This peculiar fringe pattern has a very simple physicalinterpretation [25]. When the sudden change of a occurs,pairs of entangled phonons are created at all spatial posi-tions with opposite momenta ± k . As time goes on, thesepairs propagate in opposite direction at a k -dependentgroup velocity v ,k . As the group velocity v ,k of Bogoli-ubov modes is a growing function of k which starts from v ,k =0 = c , the correlated pairs will be separated at atime t by a distance equal or larger than 2 c t [47] Moreprecisely, the modes that most contribute to the fringepattern for a given | x − x ′ | are the ones with a wavevector ± ¯ k such that v , ¯ k ≃ | x − x ′ | / t , which are responsible forfringes of wavelength 1 / ¯ k . In this semiclassical picture,the observed chirping is then a simple conseguence of thefact that higher k modes propagate at a faster group ve-locity.While the strongly chirped external region of the fringepattern remains almost unaffected by a finite initial tem-perature, the long wavelength fringes at low | x − x ′ | aresubstantially reinforced. This confirms our intuitive un-derstanding of the fringe pattern: according to (24) ther-mal enhancement is in fact concentrated into the low- k modes which are responsible for the long wavelengthfringes. V. SLOW RAMP
The previous Section was devoted to the case of a sud-den change of a for which analytical expressions wereavailable for the Bogoliubov mode amplitudes in the finalstate. The more general case of an arbitrary dependence a ( t ) requires a solution of the pair of ordinary differential equations (9-10). In the present section we discuss theresult of a numerical solution of these equations for thecase of a smooth temporal dependence of the Erf form: a ( t ) = a + a a − a (cid:18) t − t σ t (cid:19) . (27)where the change of scattering length from a to a takesplace on a time scale σ t . ( n ξ ) × G ( ) ( x , x ’) | x-x’ | / ξ -0.04-0.0200.020.04 -2 0 2 k ξ n k -0.2 -0.1 0 0.1 0.2110100 -2 0 2 k ξ A k µ t = 50 µ t = 70 µ t = 0 (a)(b)(c) (d) (e) FIG. 2: Panels (a-c): Time evolution of the density cor-relation function after a slow ramp of the scattering lengthfrom a to a = a /
4. The ramp follows an Erf shape with σ t = 5 /µ centered at t = 5 σ t . Different (a-c) panels referto different evolution times after the jump, µ t = 0 , , n k after the jump.A magnified view of the peak is given in the inset. Panel (e):anomalous average |A k | after the jump. In all panels, thickblack lines correspond to an initial T = 0; thin red lines corre-spond to a finite initial temperature T /µ = 0 .
5. The dashedred line in (d) is the initial population n th , k . As one can see in Fig.2(d,e), the main effect of a finite σ t is to introduce a ultraviolet cut-off to the Bogoliubovmodes that are effectively excited during the modulationof a . All Bogoliubov modes with ω k σ t ≫ A k remains fullynegligible and the population n k remains very close toits value before the ramp [dashed line in Fig.2(d)].The density correlation function at different times af-ter the slow modulation is shown in Fig.2(a-c) for thesimplest case of a slow modulation rate as compared tothe chemical potential, µ , σ t ≫
1. In this case, only thelow- k , hydrodynamic modes result appreciably excitedand the chirped fringes in the large | x − x ′ | ≫ c t regiondisappear from the fringe patterns shown in Fig.2(a-c).These are then characterized by a single negative peakthat rigidly propagates at a speed 2 c with almost nodispersion. A different point of view on this same phe-nomenology was recently presented in [25]. As one cansee by comparing the thick black lines to the thin red onesin Fig.2(a-c), in this case the effect of a finite initial tem-perature reduces to an amplification of the propagatingpeak. VI. HYDRODYNAMIC LIMIT
In the limit of a slow µσ t / ~ ≫ | ∆ µ | ≪ µ jump, an analytical approximation can be obtained forthe height and shape of the moving peak. The idea isto use the sudden jump result (26) and then take intoaccount the slow variation of a ( t ) by means of a cut-off in the momentum integration of (18): while the low-frequency modes ω k ≪ /σ t experience the modulationas sudden, the high-frequency ones ω k ≫ /σ t experi-ence it as adiabatic. The assumed condition µσ t / ~ ≫ k t max ≃ /cσ t ≪ /ξ well within the hydrodynamicalregime for which ω k ≃ c | k | .Including this cut-off as an additional exponential fac-tor exp( − k/k t max ) in the integral of the zero-point con-tribution to (18), we immediately get to the followingexpression for the moving peaks: δg (2) T =0 ( x, x ′ ) ≈ ~ ∆ µ πmnµc ×× (cid:26) ℓ t − ( x − x ′ − ct ) [ ℓ t + ( x − x ′ − ct ) ] + ℓ t − ( x − x ′ + 2 ct ) [ ℓ t + ( x − x ′ + 2 ct ) ] (cid:27) . (28)The peak value is at δg (2) T =0 (cid:12)(cid:12)(cid:12) peak ≈ π ( nξ ) ( µσ t / ~ ) ∆ µµ (29)and the width is determined by the cut-off as ℓ t =1 /k t max = cσ t : Physically, this value of the peak widthcan be understood as a consequence of the uncertainty σ t in the emission time, which reflects into a broadening ℓ t = cσ t of the correlation signal.At a finite temperature T >
0, one has to includethe further contribution due to the amplified thermalfluctuations. Depending on whether the temperature k B T is higher or lower than the effective cut-off energy E t max = ~ ck t max = ~ /σ t imposed by the slow ramp, thecut-off on the thermal contribution to (18) has to be im-posed at k thmax = min( k t max , k B T / ~ c ). Including againthis cut-off as an exponential factor exp( − k/k thmax ), onegets the following expression for the moving peaks: δg (2)th ( x, x ′ ) ≈ ∆ µ k B T ℓ th µ n ×× (cid:20) x − x ′ − ct ) + ℓ + 1( x − x ′ + 2 ct ) + ℓ (cid:21) . (30)The width of the thermal peaks is set by the cut-off ℓ th =1 /k thmax . At low temperature k B T < E t max , the widthis enlarged to ℓ th = ~ c/k B T as a consequence of thefinite correlation length of thermal density fluctuationsin the initial state. At high temperature k B T > E t max ,the width is again dominated by the ramp time effect asin the T = 0 case. Depending on whether k B T ≷ E t max , the height of thethermal peaks is either: δg (2)th (cid:12)(cid:12)(cid:12) peak , high − T ≈
14 ( nξ ) ( µσ t / ~ ) ∆ µ k B Tµ (31)or δg (2)th (cid:12)(cid:12)(cid:12) peak , low − T ≈
14 ( nξ ) ∆ µ ( k B T ) µ . (32)Even though the parameters used in Fig.2 are on theedge of the validity domain of the hydrodynamic approx-imation, the analytical formulas discussed in the presentsection turns out to be in reasonable quantitative agree-ment with the numerical results. VII. COMPARISON WITH BLACK HOLECALCULATIONS ( n ξ ) × [ G ( ) ( x , x ’) - ] | x - x’ | / ξ -0.015-0.01-0.00500.0050.01 ( n ξ ) × [ G ( ) ( x , x ’) - ] (a)(b) FIG. 3: Comparison of the Bogoliubov prediction (18) for thedensity correlation function (thick black line) with the resultof the numerical simulations of [22] (thin red dashed line).The numerical lines are cuts of G (2) ( x, x ′ ) along a x + x ′ =2 x line with x /ξ , ≫ a to a = a / σ t µ =0 .
5. The correlation functions are evaluated a time µ t = 50after the change of a . The upper (a) panel is for T = 0. Thelower (b) panel is for a finite k B T /µ = 0 . In the previous sections we have investigated in detailthe density correlations that appear as a consequence ofa modulation of the atomic scattering length a ( t ). Oneof the motivations of the present work was to fully un-derstand some unexpected transient features that wereobserved in the numerical experiment of [22], namely asystem of moving fringes that appear inside the acous-tic black hole as soon as the horizon is created and thenrapidly leave the field of view. As the acoustic horizonwas created by suddenly ramping down the atomic scat-tering length in a full half space and a quantitativelyidentical system of fringes was observed in a spatiallyhomogeneous system, an interpretation was put forwardin terms of dynamic Casimir effect. In this Section, weconfirm this interpretation by performing a quantitativecomparison of the numerical results of [22] to the predic-tions of the Bogoliubov model that we have discussed inthe previous Sections. To this purpose, the same Arctan-shaped ramp of a ( t ) that was used in the numerical cal-culations has to be implemented in the Bogoliubov calcu-lation: the results of the comparison are shown in Fig.3.The agreement between the two calculations is remark-able, which firmly confirms our initial interpretation. VIII. QUASI-PERIODIC MODULATION ( n ξ ) × G ( ) ( x , x ’) | x-x’ | / ξ -0.02-0.0100.010.02 ( n ξ ) × G ( ) ( x , x ’) -2 0 2 k ξ n k -2 0 2 k ξ A k k B T / µ ( n ξ ) × δ G ( ) p ea k µ t = 150 µ t = 200 (a)(b) (c) (d)(e) FIG. 4: Panels (a,b): Time evolution of the density correla-tion function after an oscillating modulation of the scatteringlength of amplitude δa/a = 0 .
1, carrier frequency ~ ω /µ = 1,and Gaussian envelope of duration ω T = 10. The two panelsrefer to different evolution times after the peak of the modu-lation, µt = 150 (a) and 200 (b). Panel (c): Bogoliubov modeoccupation n k after the modulation. Panel (d): anomalousaverage |A k | after the modulation. In all (a-d) panels, blacklines are for a zero initial temperature T = 0; redlines are foran initial temperature k B T /µ = 0 .
5. Panel (e): Peak value ofthe fringe amplitude as a function of the initial temperaturefor a given quasi-periodic excitation sequence. Black circles:numerical integration of (9-10). Dashed red line: fit of thepoints with a thermal law (1 + 2 n th , k ). A narrow window of k modes can be specifically ad-dressed by using a periodic modulation of the scatteringlength in time: according to the form (8) of the systemHamiltonian, a weak perturbation at frequency ω is infact able to effectively excite those pairs of Bogoliubovmodes that satisfy the resonance condition ω = ω k + ω − k . (33)This physics is illustrated in Fig.4, where we show theresult of a numerical integration of (9-10) under a sinu-soidal modulation of a ( t ) with a Gaussian temporal en- velope: the mixing of the ˆ a k and ˆ a †− k operators is limitedto a small range of k vectors and results in very peakedshapes of the Bogoliubov mode occupation n k and of theanomalous average A k [panels (c,d)].The pair of weaker peaks that appears in the anoma-lous average A k at larger values of k is due to second-order processes in the modulation amplitude. This inter-pretation is confirmed by the scaling of the peak ampli-tude as δa and by the position of the peak that satisfiesthe second-order resonance condition ω k + ω − k = 2 ω .For larger modulation amplitudes, higher order peakswould appear at k values satisfying higher-order reso-nance conditions of the form ω k + ω − k = N ω , N beinga generic positive integer number.Correspondingly to the dominant pair of peaks, thedensity correlation function [panels (a,b)] shows a peri-odic fringe pattern of wavelength 2 π/k that travels awayfrom | x − x ′ | = 0 at the group velocity 2 v ,k . The en-velope of the fringe pattern follows the envelope of theoscillating a ( t ) modulation.At a finite temperature (thin red lines), the resonancepeaks in the Bogoliubov mode occupation n k are almostcompletely hidden by the thermal component, but re-main perfectly visible and even reinforced in the anoma-lous average A k . Correspondingly, the moving fringepattern experiences an overall amplification without anysignificant distortion of the oscillating shape nor of itsenvelope.The simple relations (18) and (16) that relate the den-sity correlation at the end of the modulation sequenceto the initial thermal occupation of the mode can be ex-ploited as a simple way to precisely measure the tem-perature of the system in an almost non-destructive way.This proposal extends an original suggestion of [24] tomeasure the temperature of a Bose-Einstein condensateusing a parametric modulation of some parameter: as il-lustrated in Fig.4(c), looking at the density correlationrather than at the Bogoliubov mode occupation has thesignificant advantage that the interesting signal is nothidden by a broad thermal pedestal. In contrast to thecase of a single jump discussed [28], a periodic modula-tion is able to concentrate the interesting signal into asingle Bogoliubov mode and, more remarkably, to makeit significantly stronger without distorting it.As long as the applied modulation is weak enough fornonlinear and saturation effect beyond Bogoliubov the-ory to be negligible, the observed signal is in fact propor-tional to 2 n th , k + 1. To clarify this statement, the peakvalue of the fringe amplitude is plotted in Fig.4(e) as afunction of the initial temperature. The points are theresult of a numerical integration of (9-10), the dashed lineis a fit using the known thermal dependence: provided asuitably low-energy mode is used, the peak value of thefringe amplitude is proportional to the system tempera-ture. IX. REINFORCING THE DENSITYCORRELATION
A critical issue in view of an experimental verifica-tion of the conclusions of the present paper as well asof the predictions of [22] is the actual value of the den-sity correlation signal that one is to detect: given itsrelatively small value, methods to reinforce it can be ofcrucial importance. In the present section we apply to thedynamical Casimir effect a diagnostic trick that was re-cently used to characterize phase fluctuations of a quasi-condensate in a strongly one-dimensional geometry e.g.in [43] and that was recently put forward in the contextof observing the analog Hawking radiation [35]. The effi-ciency of this tool to measure the microscopic propertiesof low-dimensional many-body systems was recently dis-cussed in [28]. A related idea was presented in [17] withthe purpose of amplifying the signal of analog cosmolog-ical particle production. ( n ξ ) × G ( ) ( x , x ’) | x-x’ | / ξ -0.02-0.0100.010.02 (a)(b)(c) FIG. 5: Density correlation function after a slow ramp of thescattering length from a to a = a / a f = 0 within σ ′ t ≪ σ t , and a final time interval t free of ballistic, non-interactingevolution. Different panels (a-c) correspond to different freeevolution times µ t free = 0 (a) 5 (b), and 10 (c). The switch-off of the scattering length to a f = 0 is performed at µ t = 70within a time µ σ ′ t = 1 (black lines). For comparison, thecase of a sudden switch-off ( σ ′ t = 0) is shown as a red dottedline in (b). In agreement to the Goldstone theorem, long wave-length phonons have a mostly phase-like character and avery weak component of density fluctuation [37]: a quickswitch-off of the the interactions shortly before measur-ing the density correlations can then be used in order toreinforce the signal by converting phase fluctuations intodensity fluctuations. The efficiency of this trick is illus-trated in Fig.5: the density correlation signal is plottedfor different values of the ballistic expansion time t fin be-tween the switch-off of a and the actual measurement.During this time, the original signal gets amplified by asignificative factor. Note that in order to avoid a substantial emission ofhigh- k particles and the consequent appearance of fastoscillations in the density correlation pattern, the switch-off time σ ′ t can not be chosen too short. This point isillustrated in Fig.5(b) where the signal obtained with asudden final jump is shown for comparison as a red dottedline: the importance of a careful choice of σ ′ t is apparent.An analytical understanding of the physical origin ofthe different features that appear in the density correla-tion function after the second jump is the subject of thenext subsection. A. Hydrodynamic model
Analytical expressions can be obtained in the case inwhich the second jump brings the scattering length to afinite final value a f and both jumps are performed on atime-scale σ t , σ ′ t long as compared to the chemical poten-tial, σ t , σ ′ t ≫ /µ . Under these assumptions, the ana-lytical technique introduced in Sec.VI can be generalizedto the case of a two-jump modulation sequence. ( n ξ ) × G ( ) ( x , x ’) | x-x’ | / ξ -0.00200.0020.004 ( n ξ ) × G ( ) ( x , x ’) A B A ±1 B B -1 A B B -1 A ±1 B (a)(b) FIG. 6: Density correlation function after a two-jump se-quence. Jump times σ t , σ ′ t = 8 /µ . Delay time t del = 100 /µ .Upper panel (a): a = a / a f = a , observation time t = t free + t del = 270 /µ . Lower panel (b): a = a / a f = a /
8, observation time t = t free + t del = 320 /µ . Theblue labels indicate the terms in the two-sudden-jumps ana-lytical model that correspond to each feature. Red lines indi-cate the density correlation function in the absence of secondjump, a f = a . An explicit forms for the Bogoliubov coefficients a time t fin after the end of the two-jump modulation sequencecan be obtained by repeatedly applying two transforma-0 x-x’ t del c t del + c f t free c t del c t del + c f t free -c t free c t free c t del + c f t free c t del + c f t free t del t del + t free A B B B A ±1 B -1 B -1 A ±1 B t a f a a FIG. 7: Scheme of the spatial position of the different featurespredicted by (38) for a configuration a > a > a f inspiredto Fig.6(b). tions of the form (21-22) : C k , + = h C (2) k , + C (1) k , + e − iω (2) k t del ++ C (2) k , − C (1) k , − e iω (2) k t del i e − iω ( f ) k t fin (34) C k , − = h C (2) k , + C (1) k , − e − iω (2) k t del ++ C (2) k , − C (1) k , + e iω (2) k t del i e − iω ( f ) k t fin (35)Here, t del is the time interval between the jumps and t fin is the time interval between the second jump and theactual measurement. ω (1 , ,f ) k are the Bogoliubov disper-sions before the first jump, after the first jump, and aftersecond jump, respectively. The single jump Bogoliubovcoefficients C (1 , k , ± are defined according to (21) and (22)as: C (1) k , ± = 12 vuut ω (2) k ω (1) k ± vuut ω (1) k ω (2) k (36) C (2) k , ± = 12 vuut ω ( f ) k ω (2) k ± vuut ω (2) k ω ( f ) k (37)The corresponding density correlation function is thenobtained by inserting these formulas into the general for-mula (18) and imposing a suitable cut-off to the integralsat k t max = 1 /ℓ t . Limiting ourselves to the simplest T = 0case, some straightforward algebra leads to the final re- sult: δg (2) ( X = x − x ′ ) = ~ πmnc f { A F ℓ t [ X − c t del ]++ A F ℓ t [ X ] + A − F ℓ t [ X + 2 c t del ]++ B F ℓ t [ X − c f t free − c t del ]++ B F ℓ t [ X − c f t free ]+ B − F ℓ t [ X − c f t free + 2 c t del ]++( X ↔ − X ) } . (38)The amplitudes have the following expressions in termsof the Bogoliubov operators of the two jumps: A ± = C (2) k , + C (1) k , − C (2) k , − C (1) k , + (39) A = | C (2) k , + C (1) k , − | + | C (2) k , − C (1) k , + | (40) B = C (2) k , + C (1) k , + C (2) k , + C (1) k , − (41) B = C (2) k , + C (1) k , + C (2) k , − C (1) k , + ++ C (2) k , − C (1) k , − C (2) k , + C (1) k , − (42) B − = C (2) k , − C (1) k , − C (2) k , − C (1) k , + , (43)and the function F ℓ ( x ) is defined as F ℓ ( x ) = ℓ − x [ ℓ + x ] . (44)As a consequence of the interference between the dif-ferent terms of (34-35), a number of peak/dips appear inthe final result (38) and have a peculiar evolution as afunction of t free . An illustration of this physics is shownin Fig.6: even though significantly distorted by effectsbeyond hydrodynamics, all the features are clearly recog-nizable. Labels refer to the corresponding amplitudes de-fined in eqs.(39-43) and schematically illustrated in Fig.7.The A feature provides a slight modification of themany-body dip. The standard dynamical Casimir effectby the second jump is responsible for the feature B thatemerges from the many-body dip at x = x ′ at travels ata speed 2 c f . In agreement with (29), its sign depends onthe sign of the second jump ∆ a f = a f − a .The dynamical Casimir feature that was visible at2 c t del before the second jump splits into three features B − , A ± and B that travel away at speeds respectivelyequal to − c f , 0, 2 c f . In the absence of second jump (i.e.for a f = a , dashed red line in Fig.6), only the B sur-vives with a finite amplitude, though at a slightly shiftedposition as a consequence of the unchanged sound veloc-ity. All other B − , A ± features instead vanish as a con-sequence of the C (2) k , − = 0 condition. Before the jump, theheight of the B feature is proportional to C (1) k , + C (1) k , − /c .At the jump, it gets multiplied by a factor approximatelyequal to: η = (cid:20) (cid:18) c c f (cid:19)(cid:21) . (45)1As expected, this factor is larger than 1 as soon as thesecond jump corresponds to a decrease in the scatteringlength a f < a . In particular, it becomes very large when a f is brought to a very small value a f ≪ a .Even though these analytical considerations are lim-ited to the hydrodynamic regimes, they provide an use-ful qualitative guidelines to interpret the full numericalresults shown in Fig.5 and 6. X. CONCLUSIONS
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