On the robustness of entanglement in analogue gravity systems

We investigate the possibility to generate quantum-correlated quasi-particles utilizing analogue gravity systems. The quantumness of these correlations is a key aspect of analogue gravity effects and their presence allows for a clear separation between classical and quantum analogue gravity effects. However, experiments in analogue systems, such as Bose-Einstein condensates, and shallow water waves, are always conducted at non-ideal conditions, in particular, one is dealing with dispersive media at nonzero temperatures. We analyze the influence of the initial temperature on the entanglement generation in analogue gravity phenomena. We lay out all the necessary steps to calculate the entanglement generated between quasi-particle modes and we analytically derive an upper bound on the maximal temperature at which given modes can still be entangled. We further investigate a mechanism to enhance the quantum correlations. As a particular example we analyze the robustness of the entanglement creation against thermal noise in a sudden quench of an ideally homogeneous Bose-Einstein condensate, taking into account the super-sonic dispersion relations.


I. INTRODUCTION
Can quantum effects in curved spacetimes be simulated in compact, laboratory-based experimental setups? Following the formal analogy between quantum field theory on curved spacetimes and classical fluid systems that was established by W. G. Unruh in [1], this question has captivated researchers for decades (see, e.g., Ref. [2] for a recent review). In particular, the prospect of accessible experimental setups to test the quantum effects of varying spacetime backgrounds has motivated scientists, who have subsequently directed their ingenuity and effort towards the study of analogue gravity systems. The range of physical systems in which such simulations can be performed is vast, reaching from actual shallow water waves [3][4][5][6], and Bose-Einstein condensates (BECs) [7,8], to laser pulse filaments [9,10], to name but a few.
A central aim in such studies is the observation of radiation that can be associated to quantum pair creation processes, for instance, to the Hawking-, Unruhand the dynamical Casimir effect. All of these effects rely on similar mechanisms in quantum field theory, i.e., particle creation due to time-dependent gravitational fields and boundary conditions, or the presence of horizons. It is then only natural to ask what are the criteria for associating the effects of quantum field theory with the analogue systems. How can these criteria distinguish quantum from classical scattering processes? One aspect that has already been studied theo-retically [11], and has been rigourously tested in experiments is the spectrum of the radiation and its relation to the effective gravitational field (see, e.g., Ref. [4]). However, it might be argued that the ingredient that is missing so far is the verification of the quantumness of the observed radiation. Efforts have been directed towards addressing this issue by studying non-classical behaviour as captured by sub-Poissónian statistics and the connected violation of Cauchy-Schwarz inequalities [12,13]. However, the most paradigmatic quantum mechanical feature-entanglement-has not yet been verified in analogue gravity systems, despite the fact that typical pair creation processes in curved spacetime scenarios [14] involve the creation of mode entanglement (see, e.g., Ref. [15]). However, the presence of quantum correlations depends not only on the presence of entangling processes, but also on the initial state of the system. The temperature in the system needs to be low enough to allow entanglement to be generated.
Here we study how robust the entanglement generation phenomena in analogue gravity systems are against such thermal noise. For this purpose we connect the techniques for quantum information processing with continuous variables (see, e.g., Ref. [16]) and relativistic quantum information (RQI). During the last decade the RQI community has developed techniques to quantify entanglement in quantum field theory (for a review see, e.g., Ref. [17]). In this article we apply the framework recently developed within RQI [18,19] to establish the first description of entanglement generation in analogue gravity systems, including such effects as initial temperatures and nonlinear dispersion relations. Connecting RQI and analogue gravity promises to be a fruitful endeavour. We address the central question: Is it in principle possible to observe quantum correla-arXiv:1305.3867v1 [quant-ph] 16 May 2013 tions in analogue gravity systems? Naively, the answer is: Yes. But, the effects are highly sensitive to the levels of thermal noise. We provide closed analytical expressions for the required maximal background temperatures. Our framework applies to all common homogeneous analogue gravity systems. In the case of inhomogeneities our results can be considered as a limiting case that supplies an upper bound for the allowed temperature. Our criteria thus represent critical benchmarks that need to be taken into account by the next generation of analogue gravity experiments. In addition, we consider a scheme-entanglement resonances-to enhance the generation of quantum correlations to overcome temperature restrictions. We illustrate our results for a particular example, an ideally homogeneous BEC undergoing a single sudden quench, i.e., a change of density, or series of such transformations, for which we explicitly include the effects of nonlinear dispersion.

II. ENTANGLEMENT IN ANALOGUE GRAVITY SYSTEMS
Let us start with a brief overview of the description and quantification of entanglement. For an introductory review see, e.g., Ref. [20]. A quantum state is called separable with respect to the bipartition into subsystems A and B if the corresponding density operator ρ AB can be written as i are pure states of the subsystems A or B respectively. Such states can contain classical correlations, i.e., correlations that depend on the local (in the sense of the subsystems A and B) choice of basis. Quantum states that cannot be decomposed in the form of ρ AB above are called entangled. A formal way to quantify the entanglement of a given state ρ AB is the entanglement of formation E oF , defined as [21] Here E(| ψ AB ) is the entropy of entanglement, given by the von Neumann entropy of the reduced state ρ A = Tr B | ψ AB ψ AB |. The minimum in Eq. (1) is taken over all pure state decompositions {p i , ψ AB i } that realize the state ρ AB , i.e., such that ρ AB = i p i | ψ AB i ψ AB i |. For mixed states of arbitrary dimension this measure is not operational but for some special cases, including those we discuss in this paper, the minimization in Eq. (1) can be carried out and E oF can be computed analytically.
In the analogue gravity context that we want to consider here, the subsystems A and B will be two (out of an ensemble of possibly infinitely many) bosonic modes, e.g., associated to phonons in a BEC [22]. The corresponding annihilation and creation operators, a i and a † i satisfy the canonical commutation relations [ a i , a † j ] = δ ij and [ a i , a j ] = 0. The operators a i and a † i may be combined into the quadrature operators q j := 1 √ 2 (a j + a † j ) and p j : , which, in turn, can be collected into the vector X := q 1 , p 1 , q 2 , p 2 , . . . T .
For the particularly important class of Gaussian states, which includes, e.g., the vacuum and thermal states, all the information about the state ρ is encoded in the first moments X i ρ and the second moments where X i ρ denotes the expectation value of the operator X i in the state ρ (see Ref. [16]). Furthermore, the covariance matrix Γ contains all the relevant information about the entanglement between the modes. Let us now consider a typical transformation occurring in analogue gravity systems, for example, the generation of phonons in a BEC that is undergoing a sudden change in density. Such transformations are described by Bogoliubov transformations, i.e., linear transformationsã m = n α * mn a n − β * mn a † n between two sets of annihilation and creation operators, , that leave the canonical commutation relations invariant. The Bogoliubov transformation induces a unitary transformation on the Hilbert space of states. In phase space, on the other hand, these unitaries are realized as symplectic transformations S that satisfy SΩS T = Ω, where the symplectic form Ω is defined by the relation iΩ mn = X m , X n . The transformation S can be expressed in terms of the coefficients α mn and β mn , which allows us to quantify the entanglement that is being generated between the modes for a wide variety of scenarios (see Ref. [18]).
At this point it is crucial to notice that the structure of the Bogoliubov transformations in the context of analogue gravity, e.g., in BEC simulations of an expanding universe [15,22,23] and the dynamical Casimir effect [8], or analogue Hawking emission in water surface waves [4], is particularly simple, at least as long as the system can be considered to be homogeneous within a reasonable approximation. In that case the absence of boundary conditions that couple counterpropagating modes along with momentum conservation implies that only coefficients α kk and β k(−k) (∀k) are non-zero. This is the case even if the dispersion relation allows for more than two modes to correspond to the same frequency. The entanglement is only ever produced between pairs of modes with opposite momenta. In other words, the transformation cannot shift the momenta of individual excitations but it allows for the creation of (quasi-)particle pairs with equal but opposite momenta. This simple structure permits us to consider the covariance matrix for any pairs of modes k and −k independently of any other modes of the continuum. In the following we consider this simple structure to be an approximation for inhomogeneous systems with minor density fluctuations. The effect of the inhomogeneity is to distribute the entanglement that is generated also across modes with different momenta [24], introducing additional noise in the reduced state of the modes k and −k. This will decrease the amount of entanglement produced between these modes, such that the results we obtain can be considered as upper bounds on the entanglement generation.
Dispersive effects on the other hand, e.g., in effective Friedmann-Robertson-Walker spacetimes, are easily taken into account in terms of modified Bogoliubov coefficients. These can be obtained by solving the corresponding equation of motions for the field modes, involving forth order derivatives in space for sub-and super-luminal dispersion relations. For the two chosen modes the symplectic transformation S can be written as where the 2 × 2 blocks are given by M n(−n) = −Re(β n(−n) ) Im(β n(−n) ) Im(β n(−n) ) Re(β n(−n) ) , with n = k, −k. The unitarity of the transformation implies |α kk | 2 − |β k(−k) | 2 = 1, and the Bogoliubov coefficients further satisfy α kk = e iΘ α (−k)(−k) , and β k(−k) = e iΘ β (−k)k , where the phase Θ ∈ R is left undetermined by the unitarity of the transformation. Let us now consider the effect of the Bogoliubov transformation on the entanglement between the modes k and −k, including dispersive and finite temperature effects. Ideally, these modes are initially in the ground state. However, in the analogue gravity situation we are considering, the background temperature is nonzero (see, e.g., Ref. [8]). We are therefore applying the transformation S to the covariance matrix Γ th (T ) of a thermal state at temperature T . Since the modes k and −k have the same initial frequency ω in = ω in (|k|) their thermal covariance matrix is proportional to the identity and given by Γ th (T ) = coth ωin 2 kB T 1 (see [25]), such that the average particle number is distributed according to Bose-Einstein statistics. The transformed state has the form where C is composed of the 2×2 matrices of Eq. (4) and the reduced state covariance matrices of the individual modes,Γ k andΓ −k , are identical thermal states with non-zero temperature even when the initial temperature T is vanishing. We shall use this fact to define a characteristic temperature T E of the individual modes via the relation where we have taken into account a possible change in frequency, ω in → ω out , for fixed k, due to nonlinear dispersion. This entanglement temperature T E , which corresponds to the Hawking temperature for a black hole evaporation process, can be attributed purely to the entanglement that is generated from the initial vacuum in a homogeneous system, in complete analogy to the mixedness that is quantified by the entropy of entanglement in Eq. (1). If the first moments X i ρ of the initial state vanish, the average particle number after the transformation can be computed fromÑ k = a † k a k = 1 where theΓ ij are the elements of the 4 × 4 covariance matrixΓ [see Eq. (2)], and we obtaiñ which reduces to the usual Bose-Einstein statistics for β k(−k) = 0, while it takes the familiar form [14] N k (T = 0) = |β k(−k) | 2 starting from the initial vacuum. From Eq. (6) we can further infer thatΓ is a symmetric state, i.e., det Γ k = det Γ −k , for which the Gaussian entanglement of formation E oF can be computed explicitly (see Ref. [16]). It is simply given as a function of the parameter ν − ≥ 0, the smallest eigenvalue of |iΩ P kΓ P k |, where P k = diag{1, −1, 1, 1} represents partial transposition of mode k. If 0 ≤ ν − < 1 the transformed stateΓ is entangled and E oF is a monotonously decreasing function of ν − , given by where For the stateΓ of Eq. (5) we find While Eqs. (9)-(11) completely quantify the entanglement that is generated by the Bogoliubov transformation in the initial thermal state of temperature T , it is Eq. (11) alone that is needed to determine whether or not any entanglement is created at all. In particular, we can identify the sudden death temperature T SD , i.e., the initial temperature at which a given Bogoliubov transformation no longer generates any entanglement between the fixed modes k and −k. It is determined by the condition ν − (T SD ) = 1. By combining the entanglement temperature and Eq. (11) we can express this condition as which, in turn, implies This simple relation provides us with a powerful tool to determine if a particular analogue gravity setup can in principle be expected to produce entanglement, for instance in a BEC simulating an expanding universe [15]. However, we can also formulate a simple procedure to identify transformations whose repetitions-should they be implementable-will resonantly enhance the entanglement produced between particular modes. Let us consider the symplectic representation S of such a repeatable transformation.

III. ENTANGLEMENT RESONANCES
Any symplectic transformation of two modes can be decomposed into a passive transformation S P , with S T P S P = 1, representing rotations and beam splitting, and an active transformation S A = S T A , consisting of single-and two-mode squeezing, i.e., S = S P S A (see Ref. [26]). From the reduced states in Eq. (6) we can easily see that the transformation described by Eq. (3) contains no single-mode squeezing. Consequently, S A is a pure two-mode squeezing operation, S A = S TMS (r), the paradigm Gaussian entangling operation, where r ∈ R is the squeezing parameter. For initial states of two modes that are proportional to the identity, as in our case, we can consider a resonance condition as discussed in Ref. [19], i.e., Since the typical Bogoliubov transformations in analogue gravity systems do not contain any single mode squeezing, i.e., S = S A S TMS , the condition of Eq. (14) has a very intuitive interpretation. It suggests that the state Γ TMS = S TMS S T TMS is invariant under the passive transformation S P , S P Γ TMS S T P = Γ TMS . Since two-mode squeezing operations form a one parameter subgroup of the symplectic transformations, S TMS (r 1 )S TMS (r 2 ) = S TMS (r 1 + r 2 ), one can easily see that a transformation which satisfies the resonance condition (14) will accumulate entanglement when repeated. In particular, the entanglement of formation of Γ TMS (r) is given by h(e −2|r| ), see Eqs. (9) and (10). The increase of entanglement for particular modes is then a matter of tuning the transformation at hand to fulfill the resonance condition (14).

IV. SUDDEN QUENCH OF A BEC
Let us now discuss a specific application of the general principles we have mentioned so far. We shall consider the Bogoliubov transformation that describes a single, sudden change in density, a "quench", of a BEC at some initial temperature T , see Ref. [22]. As before we are going to make the approximation that the system is homogeneous throughout the process and that any effects of the inhomogeneity will enter as noise that reduces the entanglement. The non-adiabatic adjustment of the density causes a shift in the speed of sound, c in → c out , such that the frequencies of the phononic modes are altered, ω in → ω out , while the momenta k remain the same. The Bogoliubov coefficients thus have exactly the previously discussed structure for Θ = 0. More specifically we have [22] where t 0 is the time of the transition which becomes relevant for consecutive quenches. The frequencies are given by the nonlinear dispersion relation with = /(2m), where is Planck's constant, and m is the mass of the atoms of the BEC. For a BEC only the positive sign -super-sonic dispersion-occurs in Eq. (16), but it is straightforward to consider the sub-sonic case for other systems. The nonlinear dispersion relation now enters the problem of entanglement creation due to the Bogoliubov transformation in two places. First, the nonlinear effects influence the k-dependence of the initial temperature distribution, i.e., the average number of thermal phonons is strongly suppressed for higher mode numbers, as illustrated in Fig. 1. Additionally, the nonlinearity enters directly in the Bogoliubov coefficients (15), which decreases the number of quasi-particles that are produced by the quench. Together, the effects of the nonlinearity and the initial temperature compete to determine the entanglement generation in Eq. (11). It thus becomes evident that, given a specific transformation and dispersion relation, the entanglement generation is optimal for particular regions in k-space. We have illustrated this behaviour in Fig. 2 for convenient, but not necessarily experimentally accessible, values of the parameters T , , c in and c out . Finally, we can attempt to construct a resonant transformation from the single quench to enhance the entanglement production. For this transformation to be repeatable it has to take ω in → ω in . We can achieve a nontrivial transformation of this type by combining a first quench, with ω in → ω out , at time t 1 with a second quench that takes ω out → ω in at time t 2 . We then evaluate the resonance condition (14) for the total transformation for which we find the two necessary conditions cos(ω in t + ) sin(ω out t − ) f (ω in , ω out , t 1 , t 2 ) = 0 , (17a) where t ± = (t 1 ± t 2 ) , and with ω ± = (ω in ± ω out ). The equations (17) are trivially satisfied if |t − | = (nπ/ω out ) for any n ∈ N. In these cases the combined transformation reduces to local rotations that do not change the initial state. However, for |t − | = (nπ/ω out ) the remaining condition f (ω in , ω out , t 1 , t 2 ) = 0 can still suffice. Then the resonance condition reduces to the transcendent equation which can be solved numerically. However, for some special values an analytical solution lies close at hand. For instance, the transformation can be picked such that the ratio of ω + and ω − is rational, i.e., mω − = nω + , m, n ∈ Z. Inserting this into (19) the transcendent equation can be easily solved for t − = nπ/ω − . Since we are excluding the trivial transformations for which t − = (lπ/ω out ), l ∈ Z, we obtain the resonant solutions of Eq. (17) by additionally requiring that m and n have odd separation, i.e., (m − n) = 2l. While such a series of sudden transformation might be only a rough estimation of the sinusoidal modification of the speed of sound applied in Ref. [8], our framework allows to estimate whether a given system is above or below the sudden death temperature by measuring the average particle number for a given frequency and the initial temperature. With this information, Eq. (8) can be used to provide T SD and determine if entanglement is present even if the explicit Bogoliubov coefficients are not known. Since tests of entanglement in analogue systems can be rather involved, this check is vital to ensure the viability of such experiments. As mentioned above our results are in principle applicable to both super-and sub-luminal types of non-linear dispersion, for instance surface waves.

CONCLUSIONS
While our analysis assumes homogeneity throughout the system, the sudden death temperature we provide represents an upper bound on the allowed temperature for the inhomogeneous case as well. In particular in the case of resonant enhancement, e.g., by driving the transformation at a fixed frequency [8], the inhomogeneity leads to a smearing of the sharp peaks and the entanglement is distributed over several adjacent modes.
In conclusion, we have conducted an analysis of the entanglement generation in analogue gravity systems at finite initial temperature. We find that the entanglement generation is fully determined by the Bogoliubov transformations describing the simulated gravitational, or relativistic effects. For every pair of quasiparticle modes of system, the problem can be phrased in terms of an effective entanglement temperature T E . If the initial temperature is above the benchmark of 2(ω in /ωout)T E then no entanglement is produced at the particular modes corresponding to T E , regardless of the homogeneity of the system. The detection of entanglement in analogue gravity systems is a major ambition of future setups, for instance, to test Bell inequalities [27] in similar settings as in Ref. [8]. Our results provide clear-cut criteria for the feasibility of such endeavours that are applicable to a broad range of current analogue gravity experiments.