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
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
At this stage some observations are in order:
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
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.
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]
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 toAt 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 Several 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]
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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