We have seen that one of the main aims of research in analogue models of gravity is the possibility of simulating semiclassical gravity phenomena, such as the Hawking radiation effect or cosmological particle production. In this sense systems characterised by a high degree of quantum coherence, very cold temperatures, and low speeds of sound offer the best test field. Hence it is not surprising that in recent years Bose-Einstein condensates (BECs) have become the subject of extensive study as possible analogue models of general relativity [136, 137, 16, 19, 18, 115, 114].

Let us start by very briefly reviewing the derivation of the acoustic metric for a BEC system, and show that the equations for the phonons of the condensate closely mimic the dynamics of a scalar field in a curved spacetime. In the dilute gas approximation, one can describe a Bose gas through a quantum field satisfying

Here parameterises the strength of the interactions between the different bosons in the gas. It can be re-expressed in terms of the scattering length as As usual, the quantum field can be separated into a macroscopic (classical) condensate and a fluctuation: , with . Then, by adopting the self-consistent mean field approximation (see for example [153]) one can arrive at the set of coupled equations: Here The equation for the classical wave function of the condensate is closed only when the back-reaction effect due to the fluctuations are neglected. (This back-reaction is hiding in the parameters and .) This is the approximation contemplated by the Gross-Pitaevskii equation. In general one will have to solve both equations simultaneously. Adopting the Madelung representation for the wave function of the condensate and defining an irrotational “velocity field” by , the Gross-Pitaevskii equation can be rewritten as a continuity equation plus an Euler equation: These equations are completely equivalent to those of an irrotational and inviscid fluid apart from the existence of the so-called quantum potential which has the dimensions of an energy. Note that which justifies the introduction of the so-called quantum stress tensor This tensor has the dimensions of pressure, and may be viewed as an intrinsically quantum anisotropic pressure contributing to the Euler equation. If we write the mass density of the Madelung fluid as , and use the fact that the flow is irrotational then the Euler equation takes the form Note that the term has the dimensions of specific enthalpy, while represents a bulk pressure. When the gradients in the density of the condensate are small one can neglect the quantum stress term leading to the standard hydrodynamic approximation. Because the flow is irrotational, the Euler equation is often more conveniently written in Hamilton-Jacobi form: Apart from the wave function of the condensate itself, we also have to account for the (typically small) quantum perturbations of the system (187). These quantum perturbations can be described in several different ways, here we are interested in the “quantum acoustic representation” where are real quantum fields. By using this representation Equation (187) can be rewritten as Here represents a second-order differential operator obtained from linearizing the quantum potential. Explicitly: The equations we have just written can be obtained easily by linearizing the Gross-Pitaevskii equation around a classical solution: , . It is important to realise that in those equations the back-reaction of the quantum fluctuations on the background solution has been assumed negligible. We also see in Equations (200, 201), that time variations of and time variations of the scattering length appear to act in very different ways. Whereas the external potential only influences the background Equation (198) (and hence the acoustic metric in the analogue description), the scattering length directly influences both the perturbation and background equations. From the previous equations for the linearised perturbations it is possible to derive a wave equation for (or alternatively, for ). All we need is to substitute in Equation (200) the obtained from Equation (201). This leads to a PDE that is second-order in time derivatives but infinite order in space derivatives - to simplify things we can construct the symmetric matrix (Greek indices run from -, while Roman indices run from -.) Then, introducing (3+1)-dimensional space-time coordinates the wave equation for is easily rewritten as Where the are differential operators acting on space only: Now, if we make a spectral decomposition of the field we can see that for wavelengths larger than (this corresponds to the “healing length”, as we will explain below), the terms coming from the linearization of the quantum potential (the ) can be neglected in the previous expressions, in which case the can be approximated by numbers, instead of differential operators. (This is the heart of the acoustic approximation.) Then, by identifying the equation for the field becomes that of a (massless minimally coupled) quantum scalar field over a curved background with an effective metric of the form Here the magnitude represents the speed of the phonons in the medium: With this effective metric now in hand, the analogy is fully established, and one is now in a position to start asking more specific physics questions.

It is interesting to consider the case in which the above “hydrodynamical” approximation for BECs does not hold. In order to explore a regime where the contribution of the quantum potential cannot be neglected we can use the so called eikonal approximation, a high-momentum approximation where the phase fluctuation is itself treated as a slowly-varying amplitude times a rapidly varying phase. This phase will be taken to be the same for both and fluctuations. In fact, if one discards the unphysical possibility that the respective phases differ by a time varying quantity, any time-constant difference can be safely reabsorbed in the definition of the (complex) amplitudes. Specifically, we shall write

As a consequence of our starting assumptions, gradients of the amplitude, and gradients of the background fields, are systematically ignored relative to gradients of . (Warning: What we are doing here is not quite a “standard” eikonal approximation, in the sense that it is not applied directly on the fluctuations of the field but separately on their amplitudes and phases and .) We adopt the notation Then the operator can be approximated as A similar result holds for acting on . That is, under the eikonal approximation we effectively replace the operator by the function For the matrix this effectively results in the replacement (As desired, this has the net effect of making a matrix of numbers, not operators.) The physical wave equation (205) now becomes a nonlinear dispersion relation After substituting the approximate into this dispersion relation and rearranging, we see (remember: ) That is: Introducing the speed of sound this takes the form:At this stage some observations are in order:

- It is interesting to recognise that the dispersion relation (228) is exactly in agreement with that found in 1947 by Bogoliubov [36] (reprinted in [310]; see also [223]) for the collective excitations of a homogeneous Bose gas in the limit (almost complete condensation). In his derivation Bogoliubov applied a diagonalization procedure for the Hamiltonian describing the system of bosons.
- It is easy to see that (228) actually interpolates between two different regimes depending
on the value of the wavelength with respect to the “acoustic Compton
wavelength” . (Remember that is the speed of sound; this is not a standard
particle physics Compton wavelength.) In particular, if we assume (no background
velocity), then for large wavelengths one gets a standard phonon dispersion relation
. For wavelengths the quasi-particle energy tends to the kinetic energy of
an individual gas particle and in fact .
We would also like to highlight that in relative terms, the approximation by which one neglects the quartic terms in the dispersion relation gets worse as one moves closer to a horizon where . The non-dimensional parameter that provides this information is defined by

As we will discuss in Section 5.1.3, this is the reason why sonic horizons in a BEC can exhibit different features from those in standard general relativity. - The dispersion relation (228) exhibits a contribution due to the background flow , plus a
quartic dispersion at high momenta. The group velocity is
Dispersion relations of this type (but in most cases with the sign of the quartic term
reversed) have been used by Corley and Jacobson in analysing the issue of trans-Planckian
modes in the Hawking radiation from general relativistic black holes [185, 186, 88]. The
existence of modified dispersion relations (MDR), that is, dispersion relations that break
Lorentz invariance, can be taken as a manifestation of new physics showing up at high
energies/short wavelengths. In their analysis, the group velocity reverses its sign for large
momenta. (Unruh’s analysis of this problem used a slightly different toy model in which the
dispersion relation saturated at high momentum [377].) In our case, however, the group
velocity grows without bound allowing high-momentum modes to escape from behind
the horizon. Thus the acoustic horizon is not absolute in these models, but is instead
frequency dependent, a phenomenon that is common once non-trivial dispersions are
included.
Indeed, with hindsight the fact that the group velocity goes to infinity for large was pre-ordained: After all, we started from the generalised nonlinear Schrödinger equation, and we know what its characteristic curves are. Like the diffusion equation the characteristic curves of the Schrödinger equation (linear or nonlinear) move at infinite speed. If we then approximate this generalised nonlinear Schrödinger equation in any manner, for instance by linearization, we cannot change the characteristic curves: For any well behaved approximation technique, at high frequency and momentum we should recover the characteristic curves of the system we started with. However, what we certainly do see in this analysis is a suitably large region of momentum space for which the concept of the effective metric both makes sense, and leads to finite propagation speed for medium-frequency oscillations.

This type of superluminal dispersion relation has also been analysed by Corley and Jacobson [90]. They found that this escape of modes from behind the horizon often leads to self-amplified instabilities in systems possessing both an inner horizon as well as an outer horizon, possibly causing them to disappear in an explosion of phonons. This is also in partial agreement with the stability analysis performed by Garay et al. [136, 137] using the whole Bogoliubov equations. Let us however leave further discussion regarding these developments to the Section 5.1.3 on horizon stability.

Helium is one of the most fascinating elements provided by nature. Its structural richness confers on helium a paradigmatic character regarding the emergence of many and varied macroscopic properties from the microscopic world (see [418] and references therein). Here, we are interested in the emergence of effective geometries in helium, and their potential use in testing aspects of semiclassical gravity.

Helium four, a bosonic system, becomes superfluid at low temperatures (2.17 K at vapour pressure). This superfluid behaviour is associated with the condensation in the vacuum state of a macroscopically large number of atoms. A superfluid is automatically an irrotational and inviscid fluid, so in particular one can apply to it the ideas worked out in Section 2. The propagation of classical acoustic waves (scalar waves) over a background fluid flow can be described in terms of an effective Lorentzian geometry: the acoustic geometry. However, in this system one can naturally go considerably further, into the quantum domain. For long wavelengths, the quasiparticles in this system are quantum phonons. One can separate the classical behaviour of a background flow (the effective geometry) from the behaviour of the quantum phonons over this background. In this way one can reproduce, in laboratory settings, different aspects of quantum field theory over curved backgrounds. The speed of sound in the superfluid phase is typically of the order of cm/sec. Therefore, at least in principle, it should not be too difficult to establish configurations with supersonic flows and their associated ergoregions.

Helium three, the fermionic isotope of helium, in contrast becomes superfluid at very much lower temperatures (below 2.5 milli-K). The reason behind this rather different behaviour is the pairing of fermions to form effective bosons (Cooper pairing), which are then able to condense. In the so-called phase, the structure of the fermionic vacuum is such that it possesses two Fermi points, instead of the more typical Fermi surface. In an equilibrium configuration one can choose the two Fermi points to be located at (in this way, the z-axis signals the direction of the angular momentum of the pairs). Close to either Fermi point the spectrum of quasiparticles becomes equivalent to that of Weyl fermions. From the point of view of the laboratory, the system is not isotropic, it is axisymmetric. There is a speed for the propagation of quasiparticles along the z-axis, , and a different speed, , for propagation perpendicular to the symmetry axis. However, from an internal observer’s point of view this anisotropy is not “real”, but can be made to disappear by an appropriate rescaling of the coordinates. Therefore, in the equilibrium case, we are reproducing the behaviour of Weyl fermions over Minkowski spacetime. Additionally, the vacuum can suffer collective excitations. These collective excitations will be experienced by the Weyl quasiparticles as the introduction of an effective electromagnetic field and a curved Lorentzian geometry. The control of the form of this geometry provides the sought for gravitational analogy.

Apart from the standard way to provide a curved geometry based on producing non-trivial flows, there is also the possibility of creating topologically non-trivial configurations with a built-in non-trivial geometry. For example, it is possible to create a domain-wall configuration [200, 199] (the wall contains the z-axis) such that the transverse velocity acquires a profile in the perpendicular direction (say along the x-axis) with passing through zero at the wall (see Figure 8). This particular arrangement could be used to reproduce a black hole-white hole configuration only if the soliton is set up to move with a certain velocity along the x-axis. This configuration has the advantage than it is dynamically stable, for topological reasons, even when some supersonic regions are created.

A third way in which superfluid Helium can be used to create analogues of gravitational configurations is the study of surface waves (or ripplons) on the interface between two different phases of [415, 417]. In particular, if we have a thin layer of in contact with another thin layer of , the oscillations of the contact surface “see” an effective metric of the form [415, 417] where and (All of this provided that we are looking at wavelengths larger than the layer thickness, and .) The advantage of using surface waves instead of bulk waves in superfluids is that one could create horizons without reaching supersonic speeds in the bulk fluid. This could alleviate the appearance of dynamical instabilities in the system, that in this case are controlled by the strength of the interaction of the ripplons with bulk degrees of freedom [415, 417].The geometrical interpretation of the motion of light in dielectric media leads naturally to conjecture that the use of flowing dielectrics might be useful for simulating general relativity metrics with ergoregions and black holes. Unfortunately, these types of geometry require flow speeds comparable to the group velocity of the light. Since typical refractive indexes in non-dispersive media are quite close to unity, it is then clear that it is practically impossible to use them to simulate such general relativistic phenomena. However recent technological advances have radically changed this state of affairs. In particular the achievement of controlled slowdown of light, down to velocities of a few meters per second (or even down to complete rest) [383, 204, 52, 211, 309, 374, 348], has opened a whole new set of possibilities regarding the simulation of curved-space metrics via flowing dielectrics.

But how can light be slowed down to these “snail-like” velocities? The key effect used to achieve this takes the name of Electromagnetically Induced Transparency (EIT). A laser beam is coupled to the excited levels of some atom and used to strongly modify its optical properties. In particular one generally chooses an atom with two long-lived metastable (or stable) states, plus a higher energy state that has some decay channels into these two lower states. The coupling of the excited states induced by the laser light can affect the transition from a lower energy state to the higher one, and hence the capability of the atom to absorb light with the required transition energy. The system can then be driven into a state where the transitions between each of the lower energy states and the higher energy state exactly cancel out, due to quantum interference, at some specific resonant frequency. In this way the higher-energy level has null averaged occupation number. This state is hence called a “dark state”. EIT is characterised by a transparency window, centred around the resonance frequency, where the medium is both almost transparent and extremely dispersive (strong dependence on frequency of the refractive index). This in turn implies that the group velocity of any light probe would be characterised by very low real group velocities (with almost vanishing imaginary part) in proximity to the resonant frequency.

Let us review the most common setup envisaged for this kind of analogue model. A more detailed analysis can be found in [232]. One can start by considering a medium in which an EIT window is opened via some control laser beam which is oriented perpendicular to the direction of the flow. One then illuminates this medium, now along the flow direction, with some probe light (which is hence perpendicular to the control beam). This probe beam is usually chosen to be weak with respect to the control beam, so that it does not modify the optical properties of the medium. In the case in which the optical properties of the medium do not vary significantly over several wavelengths of the probe light, one can neglect the polarization and can hence describe the propagation of the latter with a simple scalar dispersion relation [235, 124]

where is the susceptibility of the medium, related to the refractive index via the simple relation .It is easy to see that in this case the group and phase velocities differ

So even for small refractive indexes one can get very low group velocities, due to the large dispersion in the transparency window, and in spite of the fact that the phase velocity remains very near to . (The phase velocity is exactly at the resonance frequency ). In an ideal EIT regime the probe light experiences a vanishing susceptibility near the the critical frequency , this allows us to express the susceptibility near the critical frequency via the expansion where is sometimes called the “group refractive index”. The parameter depends on the dipole moments for the transition from the metastable states to the high energy one, and most importantly depends on the ratio between the probe-light energy per photon, , and the control-light energy per atom [232]. This might appear paradoxical because it seems to suggest that for a dimmer control light the probe light would be further slowed down. However this is just an artificial feature due to the extension of the EIT regime beyond its range of applicability. In particular in order to be effective the EIT requires the control beam energy to dominate all processes and hence it cannot be dimmed at will.At resonance we have

We can now generalise the above discussion to the case in which our highly dispersive medium flows with a characteristic velocity profile . In order to find the dispersion relation of the probe light in this case we just need to transform the dispersion relation (234) from the comoving frame of the medium to the laboratory frame. Let us consider for simplicity a monochromatic probe light (more realistically a pulse with a very narrow range of frequencies near ). The motion of the dielectric medium creates a local Doppler shift of the frequency where is the usual relativistic factor. Given that is a Lorentz invariant, it is then easy to see that this Doppler detuning affects the dispersion relation (234) only via the susceptibility dependent term. Given further that in any realistic case one would deal with non-relativistic fluid velocities we can then perform an expansion of the dispersion relation up to second order in . Expressing the susceptibility via (236) we can then rewrite the dispersion relation in the form [235] where and (most of the relevant articles adopt the signature , as we also do for this particular section) The inverse of this tensor will be the covariant effective metric experienced by the probe light, whose rays would then be null geodesics of the line element . In this sense the probe light will propagate as in a curved background. Explicitly one finds the covariant metric to be whereSeveral comments are in order concerning the metric (242). First of all it is clear that although more complicated than an acoustic metric it will be still possible to cast it into the Arnowitt-Deser-Misner-like form [392]

where the effective speed is proportional to the fluid flow speed and the three space effective metric is (rather differently from the acoustic case) non-trivial.In any case, the existence of this ADM form already tells us that an ergoregion will always appear once the norm of the effective velocity exceeds the effective speed of light (which for slow light is approximately where can be extremely large due to the huge dispersion in the transparency window around the resonance frequency ). However a trapped surface (and hence an optical black hole) will form only if the inward normal component of the effective flow velocity exceeds the group velocity of light. In the slow light setup so far considered such a velocity turns out to be .

The realization that ergoregions and event horizons can be simulated via slow light may lead one to the (erroneous) conclusion that this is an optimal system for simulating particle creation by gravitational fields. However, as pointed out by Unruh in [284, 379], such a conclusion would turn out to be over-enthusiastic. In order to obtain particle creation an inescapable requirement is to have so-called “mode mixing”, that is, mixing between the positive and negative frequency modes of the incoming and outgoing states. This is tantamount to saying that there must be regions where the frequency of the quanta as seen by a stationary observer at infinity (laboratory frame) becomes negative beyond the ergosphere at .

In a flowing medium this can in principle occur thanks to the tilting of the dispersion relation due to the Doppler effect caused by the velocity of the flow Equation (238), but this also tells us that the condition can be satisfied only if the velocity of the medium exceeds which is the phase velocity of the probe light, not its group velocity. Since the phase velocity in the slow light setup we are considering is very close to , the physical speed of light in vacuum, not very much hope is left for realizing analogue particle creation in this particular laboratory setting.

However it was also noticed by Unruh and Schützhold [379] that a different setup for slow light might deal with this and other issues (see [379] for a detailed summary). In the setup suggested by these authors there are two strong background counter-propagating control beams illuminating the atoms. The field describing the beat fluctuations of this electromagnetic background can be shown to satisfy, once the dielectric medium is in motion, the same wave equation as that on a curved background. In this particular situation the phase velocity and the group velocity are approximately the same, and both can be made small, so that the previously discussed obstruction to mode mixing is removed. So in this new setup it is concretely possible to simulate classical particle creation such as, e.g., super-radiance in the presence of ergoregions.

Nonetheless the same authors showed that this does not open the possibility for a simulation of quantum particle production (e.g., Hawking radiation). This is because that effect also requires the commutation relations of the field to generate the appropriate zero-point energy fluctuations (the vacuum structure) according to the Heisenberg uncertainty principle. This is not the case for the effective field describing the beat fluctuations of the system we have just described, which is equivalent to saying that it does not have a proper vacuum state (i.e., analogue to any physical field). Hence one has to conclude that any simulation of quantum particle production is precluded.

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