On the observation of nonclassical excitations in Bose-Einstein condensates

In the recent experimental and theoretical literature well-established nonclassicality criteria from the field of quantum optics have been directly applied to the case of excitations in matter-waves. Among these are violations of Cauchy-Schwarz inequalities, Glauber-Sudarshan P-nonclassicality, sub-Poissonian number-difference squeezing (also known as the two-mode variance) and the criterion of nonseparability. We review the strong connection of these criteria and their meaning in quantum optics, and point out differences in the interpretation between light and matter waves. We then calculate observables for a homogenous Bose-Einstein condensate undergoing an arbitrary modulation in the interaction parameter at finite initial temperature, within both the quantum theory as well as a classical reference. We conclude that to date in experiments relevant for analogue gravity, nonclassical effects have not conclusively been observed and conjecture that additional, noncommuting, observables have to be measured to this end.


Nonclassicality criteria to falsify a specific classical theory
In quantum optics an important class of observables is provided by absorbing photodetection. As a case in point we consider monochromatic light emitted from a point source and photo detectors with no dead time that detect light of a specific wave vector. In the semiclassical theory of atom-light interactions [51,52] the joint click rates of a single or several photodetectors are directly proportional to the moments of mode intensities. In the theory of the quantized field the intensity I a of a mode a is represented by the photon number operatorˆˆ † = n a a, and the joint click rates are proportional to normally and time ordered moments of this operator [53]; e.g.for m coincident measurements at equal time and position they are proportional to (ˆ)ˆˆ(ˆ) (ˆ) † á ñ = á -¼ -+ ñ a a n n n m 1 1 m m . The destructive measurement of photons results in decreased autocorrelations, while the classical reference assumes a 'non-destructive' measurement of waves resulting in á ñ n m cl , where n=I is the (fluctuating) mode intensity. For example consider the observation of intensity correlations ( ) ( ( ) ( ))  t t = + C I t I t of continuously emitted light (i.e.in a stochastic steady-state) from a fluorescing atom at two subsequent times separated by τ as recorded by a single detector. (We use the notation ( )  ... for mean values of observables as measured in the laboratory.) If the density-density correlations are increasing with  t 0, we can rule out the classical model because the Cauchy-Schwarz theorem would always imply 0 [53]. However, the observation of ( ) t > C d d 0 0, known as photon antibunching, was reported in [32]. This effect is genuinely nonclassical by definition because it is not understood within the classical reference. Within quantum theory, the observed data can be explained by the fact that single photons are arriving with a tendency of being separated from each other and the normally ordered correlator is strongly decreased. The criterion for quantum states allowing for the observation of ( ) t > C d d 0 0can be formulated as the inequalitŷˆˆˆˆˆˆˆ( . It follows that this expression serves as a sufficient nonclassicality criterion for the quantum state in an experiment of the type outlined above.
The concept of nonclassical states and any corresponding criteria can be extended to the case of multiple detectors. In particular, let |a bñ , be a tensor product of coherent states for two modes a b , entering two detectors at different positions, and consider the measurement of coincident click rates, proportional to normally ordered correlators ofˆn n , a b in the quantum theory. Further, let us call states , P-classical if their Glauber-Sudarshan P-representation ( ) a b P , has the properties of a classical (i.e. positive) probability distribution for the two complex mode amplitudes. Using the defining property that the coherent states are eigenstates of their corresponding mode annihilation operators, it is evident that one can reproduce any coincident click rate within the classical reference by using ( ) a b P , as the probability distribution for the complex classical mode amplitudes [53][54][55]. On the other hand, for Pnonclassical states without such a P-representation there should exist in principle a coincidence counting experiment falsifying the classical theory. Thus, in the case of measuring such counting rates in quantum optics, a state is classical if and only if it is P-classical. A sufficient criterion to establish P-nonclassicality of a state is similarly given by inequality (1), where now a and b are two different modes at equal time, as is proven by contradiction from the Cauchy-Schwarz theorem. This inequality is equivalent to for the special case that ( ) are the normally ordered correlators. We therefore refer to this criterion as the violation of the intensity CSI.
We now discuss the connection of the intensity CSIs and another correlation measure, the TMV, also called the number squeezing parameter [40,56], which is experimentally accessible by repeatedly measuring the intensity of the two modes. The TMV is defined as the normalized number difference variance for two modes labeled a and b. Similar to referring to the case of a single variable with a variance smaller than its mean as sub-Poissonian, we refer to < V 1 as sub-Poissonian statistics in accordance with the literature. However V is sensitive to both the single-mode statistics and two-mode correlations. While V=1 for the state |a bñ , , in general neither uncorrelated nor Poissonian variables are necessary for V=1 5 . For the symmetric case, where (( ) ) (( ) )   = I I a n b n for n=1, 2, the TMV simplifies to In a quantum theory this can be written asˆ( so that it follows that < V 1 and a violation of the intensity CSI are equivalent, as was noted previously in [14]. Note that since  V 0, equation (4) also proves that cross-correlations never exceed the non-normally ordered auto-correlations; the normal ordering is crucial for a violation of the intensity CSI. In a quantum theory, due to the non-zero commutator, the normal ordering reduces the density-density auto-correlations, such that( ) = á ñ -á ñ G n n a a a a , 2, 2 2 , which allows for ( ) 2,2 . For the BEC, the implication that P-nonclassical states can demonstrate nonclassical behavior is not generally true, as it is fundamentally based on the ability to measure normally ordered moments of intensity in the quantum theory while at the same time these are represented without normal order in the classical reference. We will see below that this is not the case in the BEC. Thus the criteria for nonclassical states from quantum optics based on establishing P-nonclassicality, such as the violation of the intensity CSI(2), are neither a priori related to nonclassicality.
To further explore these issues we focus on the specific example of (parametrically) excited, strongly correlated two-mode number fluctuations in a BEC [8]. To determine the validity of the nonclassicality criteria for this system it is necessary to compare both the quantum and the corresponding semiclassical theories, as described in sections 2 and 3 respectively.

Quantum theory
We consider an interacting Bose gas in a box of volume = V L 3 with time-dependent interaction strength. The second quantized Hamiltonian is given bŷ whereâ k annihilates a single particle eigenstate of the (translationally invariant) momentum operator and  p Î L k 2 d . Here we assume the box to be large enough to approximate its single particle ground state bŷ | † ñ a 0 0 . For atoms of mass m and s-wave scattering valid in the ultracold regime, the effective interaction is In experiments, the time-dependence of the scattering length a s (t) can be achieved by use of an appropriate Feshbach resonance [7,57]. We assume all changes to be slow enough not to excite the bound state of the resonance [58].

Quasiparticles
Our formulation is based on standard Bogoliubov theory valid for a weakly interacting Bose gas [59]. The Hamiltonian can be diagonalized by retaining only interaction terms to quadratic order in the total number of particles N after the Bogoliubov approximationˆ( With the critical assumption of a small depletion of the condensate, i.e.ˆ †  D = å á ñ ¹ a a N k k k 0 , the Hamiltonian becomes a quadratic expression in the mode operators. Only modes with opposite momenta are coupled in a homogeneous condensate and we henceforth use the abbreviated notationˆô a a k andˆôb a k . A two-mode squeezing represented by the transformation , the wellknown Bogoliubov dispersion relation. Assuming the equilibrated isolated system can be treated in the canonical ensemble with respect to the particles (or grand canonical with respect to the quasiparticles with vanishing chemical potential as their number is not conserved), with the density operatorˆ(ˆ) and the anomalous quasiparticle averageˆá ñ = AB 0 th . (Note that although we are neglecting the interactions between the quasi-particles, we do assume their presence at longer time-scales, allowing the condensate to thermalize in the first place.) For continuous time dependence of U(t) the Heisenberg equations for the operators are, with ( ) ( ( ) ( ) ) † This implies that it is possible in general to describe the time evolution by a further linear transformation of the formˆ( The complex coefficients * a k , b k and a k , * b k are the entries of the first and of the second row of the fundamental matrix ( ) j = t t out of system(11) respectively, i.e. ( ) at all times and (12) is a Bogoliubov transformation. We choose with no loss of generality = t 0 out such that for > t 0 the interaction strength is kept constant, defining the out-region, in which the time evolution is a trivial phase oscillation. Note that the arbitrary time dependence of ( ) < U t 0 is now encoded in the complex Bogoliubov coefficients a k and b k subject to the normalization constraint. The dependence of these coefficients on ( ) < U t 0 is implicit in our notation. By a periodic modulation of the interaction strength, | | b k grows exponentially with the number of periods if k is in a region of instability. For example, for low amplitude sinusoidal modulations of the form More generally, some algebra shows the unstable regions are given for , where ( ) a k 1 is taken from the solution of system(11) integrated for a single period. Analytic results may be obtained in the case of a square wave modulation, for which we denote the amplitude pA 4. Furthermore, the position and growth rate of the first resonance 6 of a sinusoidal modulation with a large amplitude can be approximated by a series of sudden changes of this kind. Interestingly, this also predicts the positions of extremely fast growing parametric resonances occurring only for certain, large enough A. Fast growing, nonperturbative parametric resonances have been suggested to cause preheating in the reheating process of the inflationary Universe [61]. Note however that such resonances increase the condensate depletion dramatically and will eventually lead to violation of the assumed linear theory. Parametric resonance for the nonlinear, classical problem has been studied analytically for a variation of the trap in [62] and numerically for a variation of the scattering length in [63]. For the purpose of the present work we consider a resonance due to periodic modulations. For small modulation amplitudes the first resonance mode can be strongly excited, however the particle number can be insignificant with respect to the total depletion for a system with many modes.

Real particles
We are primarily concerned with the momentum-space observables, which are the expectation values of products of time-dependent real particle operators of the form ( ) > a t 0 and/or its conjugate, as these observables can be measured in current experiments, e.g.using the standard TOF method [1,64]. In a TOF experiment one measures real particles rather than quasi-particles, and in the following we will be focussing on observables in the real particle basis [65]. Note, this is in contrast to most of the analogue gravity studies on similar subjects, which focus on the quasi-particles modes, e.g. [17,26,46,47,49]. Combining the interaction squeezing(8) with the parametric excitations for < t 0 (12) and the subsequent phase evolution, we havê Because the Bogoliubov transformations form a group, the coefficients of the total transformation satisfy | ( )| | ( )| l g , and are given by and a straightforward calculation shows that Starting from a thermal initial state, the number of particles in mode k for > t 0 is given by Here n k th refers to the initial population when , which can be solved analytically.
is when time averaged th out 2 th 2 out 2 2 th out 2 2 in particular containing the following three terms: the thermal depletion n k th which vanishes only for = T 0 in , the depletion due to interactions v k out 2 which vanishes only for = U 0 out , and the quasiparticle production b k 2 which vanishes only for U=const. Furthermore all three mutual dual products of these terms appear as well as the triple product, signifying mutual amplification of these processes. For large but finite volume it is possible to stay in the validity regime of the Bogoliubov theory  D N by reducing the scattering length sufficiently. Significant interactions are still possible by increasing the particle density. For the rest of this paper, we assume that we are in this regime.
The only other non-vanishing correlation containing two operators is The modulus squared is referred to as the anomalous density. The anomalous density is of quartic order, and hence cannot be neglected in our truncation of the original Hamiltonian (6). Although the anomalous density is not directly observable in section 4.3 we discuss a proposal [49] to observe this quantity indirectly. As we shall see the anomalous density plays an important role in terms of establishing the nonseparability for the quantum fluctuations in a BEC.

Correlations
The two-mode squeezed thermal state we are considering here is the exponential of a quadratic expression in the mode operators. For such Gaussian states, a finite temperature Wick theorem exists for mode operator moments in anti-normal, normal, and symmetrized order, respectively. Higher moments can be computed by simply summing over all possible pairings of second moments of the mode operators. This immediately gives the normal ordered correlation functions, where for clarity we reintroduce the k-dependence, as 2 . This is an entanglement criterion which was first derived in the context of cosmological inflation [45] and subsequently applied to the phonons of quenched and parametrically excited BECs in [26,46]. The connection of entanglement to the violation of the CSI has been noticed previously [50]. We can rewrite expression (22)  and in particular simplify the condition < V 1 to obtain Taking the limit of vanishing interaction  U 0 out quasiparticles reduce to real particles and the term | ( )| g t k 2 may be replaced by | | b k 2 in (24). Then this expression agrees with the phonon entanglement criterion found in [66].
We now give an independent (and short) derivation that (24) is a nonseparability criterion for the real particles,i.e. atoms, of opposite momenta.
For a given partition of the Hilbert space into each of the two modes and the rest of the system r we may ask if a product state r r r = Ä ab r is separable with respect to the first two spaces. This means by definition that the bipartite state can be written as a mixture of products (i.e. a convex sum), , and the subsystems' states r m i might be taken to be pure without loss of generality. If this is not possible one has a nonseparable or entangled state with respect to   Ä a b [67]. A requirement for a separable state is that the partial transpose of the density operator is again positive semidefinite [68]. If this is not the case the state must be nonseparable. For continuous variable two-mode systems, like for the two modes a b , considered here, the implication of this sufficient nonseparability criterion for the covariance matrix is also necessary for nonseparability if the state is a Gaussian Wigner function [69]. A straightforward procedure to evaluate this entanglement criterion conveniently written in terms of moments of the annihilation and creation operators directly is to consider the determinant Here rows and columns have been reordered such that D 5 trivially splits into the product of two subdeterminants for our case The state is entangled iff < D 0 5 [70]. Since in general | | á ñ < á ñá + ñ ab n n 1 is thus sufficient for nonseparability in general, and equivalent to nonseparability for the two-mode squeezed Gaussian states considered here. The intensity Cauchy-Schwarz violation(2) itself on the other hand does not imply nonseparability in general when the moments do not factorize as for Gaussian states. For example, V=0 for (mixtures of) Fock states | | ñ Ä ñ n n with the same number of particles in mode a and b. Without the Gaussian factorization the breakdown of the intensity CSI as an entanglement criterion is expected. Furthermore for Gaussian states with coherent properties the intensity CSI violation is independent of entanglement [43,47].
We see from(23) that for the thermal state of the non-interacting system with g = 0 k the fluctuations are always super-Poissonian, = + V n 1 k th , and thus the state is separable. Increasing | | g k due to quantum depletion or parametric excitations however leads to a decrease of V and sub-Poissonian statistics and nonseparability are possible.
For the interacting system in equilibrium, i.e.without parametric excitation, it can be shown that V as a function of k increases with decreasing wave number k. For  k 0, V=1 is reached for = k T Un B , i.e.the temperature is of the order of the chemical potential μ of the Bose gas. Here the atoms with opposite momenta are nonseparable already for all k due to interactions. This effect is absent for the quasiparticles.
The term | | g k 2 on the left hand side of formula(24) appears in (16). Within (16) it may be interpreted as a quantum, 'spontaneous emission' term, comprising both quasiparticle production and quantum depletion. Note that for high temperatures ( )  n T 2 1 k th Bogoliubov theory implies a spontaneous emission term | ( )|  g t 1 k 2 and entanglement when sufficiently strong correlations < V 1 are measured, but it also predicts that the spontaneous emission is insignificant compared to the amplification of thermal noise. In the next section we show that the same measurable strong correlations < V 1 which imply entanglement in Bogoliubov theory are possible in a classical theory without a notion of entanglement. In this theory the spontaneous emission term is absent and only amplification of the thermal occupation occurs.

Classical theory
When quantum and thermal fluctuations are neglected, the mean-field dynamics of a Bose-Einstein condensate are well-described by the so-called (GPE), given by , is the complex-valued macroscopic wave function and V(r) is the trapping potential taken to be zero for the homogeneous case considered here. Formally the GPE can be obtained from the in the quantum theory and by a replacementˆf Y  in the operator equation of motion, as is well known. For a more detailed derivation see for example [2].
A suitable classical Hamiltonian yielding the GPE from the equation of motion is given by the mean field energy functional [73] ( ) with the understanding that the complex field ( ) ( ( ) , and the right hand side of this expression amounts to the same formal manipulations as in the right hand side of the . We consider the system to be in a classical thermal state of the canonical ensemble with probabilities for a field configuration f given by [ ] . As in the quantum case, we may linearize and diagonalize the theory for sufficiently low classical depletion . The linear transformations to the classical quasiparticles (normal modes) are completely analogous, preserving the correspondence between the equations of motion of the quantum and classical case. This similarity between the quantum and classical linear theory is well known for the common-place approach of linearizing after deriving the equations of motion, i.e.writing eitherˆ( ) , , 0 in the GPE case (for the homogeneous time dependent case considered here, ( ) f t 0 is constant in space). Thus, we obtain the equations of motion for the classical theory without further calculation by making a replacementˆ( in the corresponding equations from the quantum theory (11), where the classical quasiparticles are defined by a similar linear Bogoliubov transformation of Fourier modes of the field as in the quantum case where again we have set f = a k , with the coefficients g k and l k similarly given by (13) and (14). We identify the mode intensity ( )  I a with * á ñ a a cl , and similarly for the intensity correlations. Within this classical treatment of the BEC, we find again by evaluating a Gaussian integral that * á ñ = A A n k cl ,cl th and á ñ = AB 0 cl for the initial thermal state. However now the energy is equally partitioned over all the field modes in accordance with the high temperature limit of the Bose-Einstein statistics with vanishing chemical potential. The equipartition of energy is indeed observed in equilibrating micro-canonical simulations of the projected GPE [74,75]. Note that furthermore it has been argued there that the projected GPE at finite temperature provides indeed a (classical) theory of the highly occupied modes of an actual Bose-Einstein condensate. We do however not rely on a physical interpretation of the classical theory of this sort; instead, we just use it as a tool providing a definite classical reference, to define and isolate the quantum effects.

Correlations
Using the Bogoliubov transformation(31) we obtain Notice the absence of terms leading to the spontaneous generation of field fluctuations within this classical treatment of the excitations. It is also worth stressing that even in the absence of a modulation, b = 0 k , the thermal noise is correlated by the interaction term in the real particle basis, which contributes to the (thermal) depletion. However, for  T 0 this depletion goes to zero in contrast to the quantum formulation. From equations (33) and (34) it is immediate that | | < m n k k ,cl ,cl . Hence the mode CSI cannot be violated in this classical theory. This is in agreement with the mathematical Cauchy-Schwarz theorem.
The Wick theorem for the evaluation of higher correlation functions now follows from Isserlis' theorem [76] directly. The fourth moments needed in expression (4)

Comparison between classical and quantum theory for specific observables
Finally we apply our findings to recent experimental procedures to detect sub-Poissonian statistics, intensity CSI and mode CSI. We demonstrate that none of these proposed experimental observables can establish the nonclassicality of the excitations in a BEC undergoing an arbitrary modulation of the interaction parameter. Below we discuss in depth two proposals that can be implemented using a TOF and / in-situ imaging technique of the condensate. In a TOF measurement the trap is switched off and the atoms fall due to gravity while expanding rapidly due to the non-zero velocity distribution of the excitations in the BEC. Hence in a TOF experiment real space density fluctuations are mapped onto the atomic momentum space distribution. The motional states of the real particles carry information about the motional states of the original quantum gas, see for example [65]. Conversely, in-situ measurements involve a direct imaging of the trapped condensate [9,10,20,77].

Sub-Poissonian statistics
Independent of the underlying theory the TMV, see equation (4), can be extracted experimentally via TOF and / or in-situ measurements. From a single experiment one can extract I I a a , I I a b and I a . By repeating the experiment and then averaging over the extracted quantities one can then get the corresponding expectation values. As pointed out above if the TMV is smaller than 1 the statistics of the process is referred to as sub-Poissonian.
Our classical reference predicts Note that V and V cl are mathematically inequivalent expressions, but physically refer to the same observable.
Thus we obtain sub-Poissonian statistics for th within a completely classical treatment of small fluctuations in a BEC.
Comparing the quantum result (24) with the classical result (36), it appears that it is possible in both cases to produce sub-Poissonian statistics for a sufficiently low temperature and suitable modulation. We compare the time dependent value of V for a resonant mode undergoing periodic modulation for both the quantum and classical cases in figure 1. In the left column of the figure we plot the dependence of the Vʼs in the resonance mode for a square-wave modulation of the scattering length of a factor of 1.2 over 30 cycles for different initial temperatures. There are three important observations that can be made from these plots, which are true for both theories. First, V is in general time-dependent due to the form of g 2 given in equation (15). Second, for sufficiently long driving the oscillations become suppressed and V approaches zero. The oscillations in V for the quantum case also suggest that the entanglement criterion (24) is time-dependent. It should be noted that this refers to mode entanglement in contrast to particle entanglement. Third, the classical value of V is always below the quantum value. This follows from comparing the criteria(36) and (24) and given that < n n k k ,cl th th at finite temperature. The physical explanation is that our classical reference is a wave theory and as such is missing the shot noise due to the discrete excitations (in this case atom numbers), i.e.the first term on the right hand side of (ˆ)ˆˆˆ= á ñ + á ñ -á ñ = + n n n n n n n Var : : The classical reference only contains the final term n k 2 that has been associated by Einstein to the wave character of the atoms [78]. The shot noise term is necessary for a (super-) Poissonian two-mode variance for low occupation numbers. To see this, we make use of an alternative form of expression(4), a for positive or vanishing correlations. Since classically the variance scales with the square of the occupation number, V will always be sub-Poissonian for low enough occupation numbers. Indeed, for equilibrium and vanishing interactions, (35) becomes = V n k cl ,cl th showing the threshold is an occupation of unity. We stress that the absence of shot noise in the classical theory is not always the cause of the obtained sub-Poissonian statistics. At high initial temperature  T 1 the shot noise becomes insignificant and the classical and quantum results approach each other. For strong enough driving, sub-Poissonian statistics is possible in both theories, even if the initial thermal state exhibits super-Poissonian fluctuations.
We conclude that a TMV smaller than 1, and / or sub-Poissonian statistics is not a sufficient nonclassicality criterion to rule out our specific classical reference. appears in both the quantum and classical case. A distinction between the quantum theory and the classical reference as in coincidence counting experiments is then not available.

Violation of intensity CSI for time-resolved TOF measurements
A time-resolved TOF method as proposed in [14] is not a suitable detection scheme for this purpose. In the TOF method the atoms are released from the trap, hence the BEC is destroyed, and the wave description for the excitations breaks down. Collective excitations have been transferred to a finite number of atoms and hence independent of the nature of the collective excitations the measurement of the auto-correlations is destructive: the probability to subsequently measure two excitations with momentum k is proportional to ( ) n n 1 k k . In this sense the resulting violation of the CSI within a time-resolved TOF measurement of the density-density is largest at this time. Thus, when measured at this particular time t m , we can replace the real part by a minus signˆ( We can then see that-assuming the quantum theory-at t m the mode CSI violation (27) is equivalent tôˆ( Keeping track of higher order terms shows that one condition for this analysis to hold is  D N n k , which has to be compared to  D N for the validity of standard Bogoliubov theory, where Δ is the depletion of the condensate (17).
Note thatˆr r á ñ = -N k k in the limit of zero temperature, no driving and vanishing interactions due to the nonvanishing commutator. (The quantum noise causes the measured in-situ density to fluctuate.) Thus, the observable consequence of entanglement in this experiment is a measured suppression of these fluctuations. The absence of quantum noise in any classical reference already indicates that such a threshold does not exist in our classical reference.
Indeed, at = t t m we obtain hence it is also possible to obtain ( )  r r < -N k k . We compare the time dependent value ofˆr r á ñ k k for a resonant mode undergoing periodic modulation for both the quantum and classical cases in figure 1. In the right column of the figure we plot the dependence of theˆr r á ñ k k in the first resonance mode for a square-wave modulation of the scattering length of a factor of 1.2 over 30 cycles for different initial temperatures. In both case (quantum as well as classical) the specific observable under consideration exhibit nonclassical correlations. Very much like in the TMV case there are commutator terms only appearing within the quantum theory that make the observation imply a violation of the mode CSI.
In summary, an indirect measurement of the mode CSI of the type suggested in [9,49] is not a sufficient nonclassicality measure when applied to the case of excitations in a homogeneous parametrically excited BEC.

Conclusions
In this paper we have taken the viewpoint that nonclassical effects in the BEC are those that are incompatible with the results obtained using an ensemble of classical trajectories given by solutions of the GPE. For excitations in a parametrically excited BEC we showed that observable strong number correlations as indicated by < V 1 (related to P-nonclassicality and even nonseparability of atomic modes in the quantum theory) and an indirect measurement of the intensity CSI as suggested in [9,49] are compatible with this classical theory. Similarly, we argued that a violation of the intensity CSI in a time-resolved TOF measurement is compatible with the classical picture of an atomic cloud after destruction of the BEC. Although we considered the definite model of parametric creation of two-mode squeezing, we also argued above that in different set-ups one should also not automatically expect these number correlation-based observables to rule out a corresponding classical model, and possibly a quantum model with decoherence. Nevertheless, in the context that the Bogoliubov theory is a valid approximation, a measurement of < V 1 (or equivalently a violation of the intensity CSI) is sufficient for P-nonclassicality and nonseparability, both of which are criteria that assess a given quantum state. However, these criteria are not suitable to theoretically single out the spontaneous process of amplified vacuum noise for analogue gravity studies.
We conjecture that the direct experimental verification of the quantumness of the fluctuations remains an open challenge for future BEC experiments in general. For this purpose we propose that additional observables, including noncommuting ones, e.g.density and phase fluctuations, have to be measured. For example it is possible to involve interference techniques where one measures the phase quadrature of quasi particles [11,15,79]. Returning to the particular setup discussed above, a parametrically excited condensate, we would like to point out that such a system can be used to mimic models of cosmological particle production in table-top experiments [24]. As argued in this paper, even within highly controllable and repeatable BEC experiments one is for now facing a similar dilemma to the one of establishing the quantum origin of the fluctuations seeding our Universe [80]. So far in both cases a suitable Bell-type experiment, that would for once and all resolve this issue, is absent.