4.2 Quantum models

4.2.1 Bose–Einstein condensates

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. One could reasonably hope to manipulate these systems to have Hawking temperatures on the order of the environment temperature (∼ 100 nK) [48Jump To The Next Citation Point]. 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 [231Jump To The Next Citation Point, 232Jump To The Next Citation Point, 45Jump To The Next Citation Point, 48Jump To The Next Citation Point, 47Jump To The Next Citation Point, 195Jump To The Next Citation Point, 194Jump To The Next Citation Point].

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 by a quantum field ^ Ψ satisfying

( 2 ) iℏ ∂-Ψ^ = − ℏ--∇2 + V (x ) + κ(a)Ψ^†^Ψ ^Ψ. (228 ) ∂t 2m ext
Here κ parameterises the strength of the interactions between the different bosons in the gas. It can be re-expressed in terms of the scattering length a as
4 πaℏ2 κ(a) = ------. (229 ) m
As usual, the quantum field can be separated into a macroscopic (classical) condensate and a fluctuation: ^Ψ = ψ + φ^, with ⟨Ψ^⟩ = ψ. Then, by adopting the self-consistent mean-field approximation (see, for example, [261])
φ^†^φ^φ ≃ 2⟨^φ†^φ⟩φ^+ ⟨φ^^φ⟩φ^†, (230 )
one can arrive at the set of coupled equations:
( ) ∂-- ℏ2-- 2 iℏ∂tψ (t,x) = − 2m ∇ + Vext(x) + κnc ψ (t,x) ∗ + κ {2&tidle;nψ (t,x) + &tidle;mψ (t,x)}; (231 ) ( ) ∂ ℏ2 2 iℏ--φ^(t,x) = − ---∇ + Vext(x) + κ2nT ^φ(t,x) ∂t 2m + κmT ^φ†(t,x). (232 )
2 2 nc ≡ |ψ (t,x )| ; mc ≡ ψ (t,x); (233 ) n&tidle;≡ ⟨^φ†^φ⟩; m&tidle; ≡ ⟨^φφ^⟩; (234 ) nT = nc + &tidle;n; mT = mc + &tidle;m. (235 )
The equation for the classical wave function of the condensate is closed only when the backreaction effect due to the fluctuations is neglected. (This backreaction is hiding in the parameters &tidle;m and &tidle;n.) 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
∘ -------- ψ (t,x ) = nc(t,x)exp [− i𝜃(t,x)∕ℏ], (236 )
and defining an irrotational “velocity field” by v ≡ ∇ 𝜃∕m, the Gross–Pitaevskii equation can be rewritten as a continuity equation plus an Euler equation:
∂-- ∂tnc + ∇ ⋅ (ncv ) = 0, (237 ) ( 2 2 2√ --) m ∂-v + ∇ mv---+ Vext(t,x ) + κnc − ℏ--∇-√--nc = 0. (238 ) ∂t 2 2m nc
These equations are completely equivalent to those of an irrotational and inviscid fluid apart from the existence of the quantum potential
2 2√ --- V = − ℏ--∇----nc, (239 ) quantum 2m √nc
which has the dimensions of an energy. Note that
[ 2 2√ --] [ 2 ] -ℏ--∇----nc -ℏ-- nc∇iVquantum ≡ nc∇i − 2m √nc-- = ∇j − 4m nc∇i∇j ln nc , (240 )
which justifies the introduction of the quantum stress tensor
ℏ2 σqijuantum = − ----nc∇i∇j ln nc. (241 ) 4m
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 ρ = mn c, and use the fact that the flow is irrotational, then the Euler equation takes the form
[ ] [ ] [ 2 ] ρ ∂-v + (v ⋅ ∇ )v + ρ∇ Vext(t,x-) + ∇ κρ-- + ∇ ⋅ σquantum = 0. (242 ) ∂t m 2m2
Note that the term V ∕m ext has the dimensions of specific enthalpy, while κρ2∕ (2m ) 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:
( √ --) ∂ [∇ 𝜃]2 ℏ2 ∇2 nc m ---𝜃 + ------+ Vext(t,x ) + κnc − -----√---- = 0. (243 ) ∂t 2m 2m nc
Apart from the wave function of the condensate itself, we also have to account for the (typically small) quantum perturbations of the system (232View Equation). These quantum perturbations can be described in several different ways, here we are interested in the “quantum acoustic representation”
( 1 √n--- ) φ^(t,x ) = e−i𝜃∕ℏ -√----^n1 − i---c^𝜃1 , (244 ) 2 nc ℏ
where ^n ,^𝜃 1 1 are real quantum fields. By using this representation, Equation (232View Equation) can be rewritten as
1 ( ) ∂t^n1 + --∇ ⋅ n1∇ 𝜃 + nc ∇ ^𝜃1 = 0, (245 ) m ^ -1 ^ ℏ2-- ∂t𝜃1 + m ∇ 𝜃 ⋅ ∇ 𝜃1 + κ (a)n1 − 2m D2 ^n1 = 0. (246 )
Here D2 represents a second-order differential operator obtained by linearizing the quantum potential. Explicitly:
D ^n ≡ − 1n −3∕2[∇2 (n+1∕2)]^n + 1-n−1∕2∇2 (n−1∕2^n ). (247 ) 2 1 2 c c 1 2 c c 1
The equations we have just written can be obtained easily by linearizing the Gross–Pitaevskii equation around a classical solution: nc → nc + ^n1, ^ ϕ → ϕ + ϕ1. It is important to realise that in those equations the backreaction of the quantum fluctuations on the background solution has been assumed negligible. We also see in Equations (245View Equation) and (246View Equation) that time variations of V ext and time variations of the scattering length a appear to act in very different ways. Whereas the external potential only influences the background Equation (243View Equation) (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 ^ 𝜃1 (or alternatively, for ^n1). All we need is to substitute in Equation (245View Equation) the ^n1 obtained from Equation (246View Equation). 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 4 × 4 matrix
⌊ ⌋ f00 ... f0j ||⋅⋅⋅⋅⋅⋅ ⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅|| fμν(t,x) ≡ | . | . (248 ) ⌈ fi0 .. fij ⌉
(Greek indices run from 0 – 3, while Roman indices run from 1 – 3.) Then, introducing (3+1)-dimensional space-time coordinates,
xμ ≡ (t;xi) (249 )
the wave equation for 𝜃1 is easily rewritten as
μν ^ ∂μ(f ∂ ν𝜃1) = 0. (250 )
Where the f μν are differential operators acting on space only:
[ ℏ2 ]−1 f00 = − κ (a) − ---D2 (251 ) [ 2m ] 0j ℏ2 −1 ∇j𝜃0 f = − κ (a) − ---D2 ----- (252 ) [ 2m m] i0 ∇i 𝜃0 ℏ2 −1 f = − -m--- κ (a) − 2m-D2 (253 ) [ ]−1 ij ncδij ∇i-𝜃0 ℏ2-- ∇j𝜃0- f = m − m κ (a) − 2m D2 m . (254 )
Now, if we make a spectral decomposition of the field ^𝜃1 we can see that for wavelengths larger than ℏ∕mcs (this corresponds to the “healing length”, as we will explain below), the terms coming from the linearization of the quantum potential (the D 2) can be neglected in the previous expressions, in which case the μν f can be approximated by (momentum independent) numbers, instead of differential operators. (This is the heart of the acoustic approximation.) Then, by identifying
√ --- μν μν − gg = f , (255 )
the equation for the field ^𝜃1 becomes that of a (massless minimally coupled) quantum scalar field over a curved background
1 (√ --- μν ) Δ 𝜃1 ≡ √-−-g∂μ − gg ∂ν ^𝜃1 = 0, (256 )
with an effective metric of the form
⌊ ⌋ − {cs(a,nc)2 − v2} ... − vj n | | gμν(t,x ) ≡ -----c----|| ⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅ ⋅. ⋅⋅⋅⋅⋅⋅|| . (257 ) mcs (a, nc)⌈ − vi .. δij ⌉
Here, the magnitude cs(nc,a) represents the speed of the phonons in the medium:
κ (a)nc cs(a,nc)2 = -------. (258 ) m
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.

Lorentz breaking in BEC models – the eikonal approximation:
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 eikonal approximation, a high-momentum approximation where the phase fluctuation ^𝜃1 is itself treated as a slowly-varying amplitude times a rapidly varying phase. This phase will be taken to be the same for both ^n1 and ^ 𝜃1 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

𝜃^1(t,x) = Re {𝒜 𝜃 exp(− iϕ )}, (259 ) ^n1(t,x) = Re {𝒜 ρ exp(− iϕ )}. (260 )
As a consequence of our starting assumptions, gradients of the amplitude, and gradients of the background fields, are systematically ignored relative to gradients of ϕ. (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 ψ (t,x ) but separately on their amplitudes and phases ρ1 and ϕ1.) We adopt the notation
∂ϕ ω = --; ki = ∇iϕ. (261 ) ∂t
Then the operator D2 can be approximated as
D2 ^n1 ≡ − 1n −3∕2[Δ (n+1∕2)]^n1 + 1n− 1∕2Δ (n− 1∕2^n1) (262 ) 2 c c 2 c c 1- −1 ≈ + 2n c [Δ^n1 ] (263 ) 1 = − -n −c1k2^n1. (264 ) 2
A similar result holds for D2 acting on ^𝜃1. That is, under the eikonal approximation we effectively replace the operator D2 by the function
D → − 1-n−1k2. (265 ) 2 2 c
For the matrix f μν this effectively results in the (explicitly momentum dependent) replacement
[ 2 2 ]−1 f00 → − κ(a) + -ℏ-k-- (266 ) 4mnc [ 2 2 ]−1 j f0j → − κ(a) + -ℏ-k-- ∇-𝜃0- (267 ) 4mnc m i [ 2 2 ]−1 f i0 → − ∇-𝜃0- κ(a) + -ℏ-k-- (268 ) m 4mnc ij i [ 2 2 ]−1 j f ij → ncδ--− ∇-𝜃0- κ(a) + -ℏ-k-- ∇-𝜃0-. (269 ) m m 4mnc m
As desired, this has the net effect of making fμν a matrix of (explicitly momentum dependent) numbers, not operators. The physical wave equation (250View Equation) now becomes a nonlinear dispersion relation
00 2 0i i0 ij f ω + (f + f )ωki + f kikj = 0. (270 )
After substituting the approximate D2 into this dispersion relation and rearranging, we see (remember: k2 = ||k||2 = δijkikj)
[ ] 2 i nck2 ℏ2 2 i 2 − ω + 2v0ωki + ----- κ (a ) +------k − (v0ki) = 0. (271 ) m 4mnc
That is:
2 [ 2 ] (ω − vik)2 = nck-- κ(a) + -ℏ---k2 . (272 ) 0 i m 4mnc
Introducing the speed of sound cs, this takes the form:
∘ -------(------)2- i 2 2 -ℏ--2 ω = v0ki ± csk + 2m k . (273 )
At this stage some observations are in order:

  1. It is interesting to recognize that the dispersion relation (273View Equation) is exactly in agreement with that found in 1947 by Bogoliubov [79] (reprinted in [508]; see also [374]) for the collective excitations of a homogeneous Bose gas in the limit T → 0 (almost complete condensation). In his derivation Bogoliubov applied a diagonalization procedure for the Hamiltonian describing the system of bosons.
  2. Coincidentally this is the same dispersion relation that one encounters for shallow-water surface waves in the presence of surface tension. See Section 4.1.4.
  3. Because of the explicit momentum dependence of the co-moving phase velocity and co-moving group velocity, once one goes to high momentum the associated effective metric should be thought of as one of many possible “rainbow metrics” as in Section 4.1.4. See also [643]. (At low momentum one, of course, recovers the hydrodynamic limit with its uniquely specified standard metric.)
  4. It is easy to see that Equation (273View Equation) actually interpolates between two different regimes depending on the value of the wavelength λ = 2 π∕||k|| with respect to the “acoustic Compton wavelength” λc = h∕(mcs). (Remember that cs is the speed of sound; this is not a standard particle physics Compton wavelength.) In particular, if we assume v0 = 0 (no background velocity), then, for large wavelengths λ ≫ λc, one gets a standard phonon dispersion relation ω ≈ c||k||. For wavelengths λ ≪ λc the quasi-particle energy tends to the kinetic energy of an individual gas particle and, in fact, ω ≈ ℏ2k2∕(2m ).

    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 v = − c 0 s. The non-dimensional parameter that provides this information is defined by

    ∘ ------- -λ2c --1-+-4λ2-−-1- ----1------λ2c- δ ≡ (1 − v0∕cs) ≃ (1 − v0∕cs)8 λ2. (274)
    As we will discuss in Section 5.2, this is the reason why sonic horizons in a BEC can exhibit different features from those in standard general relativity.
  5. The dispersion relation (273View Equation) exhibits a contribution due to the background flow viki 0, plus a quartic dispersion at high momenta. The group velocity is
    ( ) c2 + -ℏ2-k2 vi = ∂ω--= vi ± ∘-------2m2------ki. (275) g ∂ki 0 2 2 (-ℏ- 2)2 c k + 2m k

    Indeed, with hindsight, the fact that the group velocity goes to infinity for large k 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.

Relativistic BEC extension:
Bose–Einstein condensation can occur not only for non-relativistic bosons but for relativistic ones as well. The main differences between the thermodynamical properties of these condensates at finite temperature are due both to the different energy spectra and also to the presence, for relativistic bosons, of anti-bosons. These differences result in different conditions for the occurrence of Bose–Einstein condensation, which is possible, e.g., in two spatial dimensions for a homogeneous relativistic Bose gas, but not for its non-relativistic counterpart – and also, more importantly for our purposes, in the different structure of their excitation spectra.

In [191Jump To The Next Citation Point] an analogue model based on a relativistic BEC was studied. We summarise here the main results. The Lagrangian density for an interacting relativistic scalar Bose field ˆϕ(x, t) may be written as

1 ∂ˆϕ† ∂ˆϕ ( m2c2 ) ˆℒ = -- -------− ∇ ϕˆ† ⋅ ∇ ˆϕ − -----+ V(t,x ) ϕˆ†ˆϕ − U(ϕˆ†ϕˆ; λi) , (276 ) c2 ∂t ∂t ℏ2
where V(t,x ) is an external potential depending both on time t and position x, m is the mass of the bosons and c is the light velocity. U is an interaction term and the coupling constant λi(t,x) can depend on time and position too (this is possible, for example, by changing the scattering length via a Feshbach resonance [151, 175Jump To The Next Citation Point]). U can be expanded as
U(ˆϕ †ϕˆ; λi) = λ2ˆρ2 + λ3-ˆρ3 + ⋅⋅⋅ (277 ) 2 6
where ˆρ = ˆϕ†ˆϕ. The usual two-particle λ2ˆϕ4-interaction corresponds to the first term (λ2∕2)ˆρ2, while the second term represents the three-particle interaction and so on.

The field ˆ ϕ can be written as a classical field (the condensate) plus perturbation:

ϕˆ= ϕ(1 + ˆψ). (278 )
It is worth noticing now that the expansion in Equation (278View Equation) can be linked straightforwardly to the previously discussed expansion in phase and density perturbations ˆ 𝜃1, ˆρ1, by noting that
ˆρ1 ψˆ+ ψˆ† ψˆ− ˆψ† ---= -------, ˆ𝜃1 = -------. ρ 2 2i
Setting ψ ∝ exp[i(k ⋅ x − ωt)] one then gets from the equation of motion [191Jump To The Next Citation Point]
( ) ( ) ℏ- u0- --ℏ-- 2 -ℏ--2 ℏ- u0- -ℏ--- 2 -ℏ-- 2 − m q ⋅ k + c ω − 2mc2 ω + 2m k m q ⋅ k − c ω − 2mc2 ω + 2m k ( )2 − c0 ω2 + c20k2 = 0 , (279 ) c
where, for convenience, we have defined the following quantities as
μ ℏ- μν u ≡ m η ∂ ν𝜃 , (280 ) ℏ2 c20 ≡ ---2U ′′(ρ;λi)ρ , (281 ) 2m q ≡ mu ∕ ℏ. (282 )
Here q is the speed of the condensate flow and c is the speed of light. For a condensate at rest (q = 0) one then obtains the following dispersion relation
( ∘ ----------------------------) { ( 0)2 [ ( )2] ( 0) ( 0)2 [ ( )2]2} ω2± = c2 k2 + 2 mu--- 1 + -c0 ± 2 mu--- k2 + mu--- 1 + c0- . (283 ) ( ℏ u0 ℏ ℏ u0 )
The dispersion relation (283View Equation) is sufficiently complicated to prevent any obvious understanding of the regimes allowed for the excitation of the system. It is much richer than the non-relativistic case. For example, it allows for both a massless/gapless (phononic) and massive/gapped mode, respectively for the ω − and ω+ branches of (283View Equation). Nonetheless, it should be evident that different regimes are determined by the relative strength of the the first two terms on the right-hand side of Equation (283View Equation) (note that the same terms enter in the square root). This can be summarised, in low and high momentum limits respectively, for k much less or much greater than
0 [ ( )2] 0 mu--- 1 + c0- ≡ mu---(1 + b), (284 ) ℏ u0 ℏ
where b encodes the relativistic nature of the condensate (the larger b the more the condensate is relativistic).

A detailed discussion of the different regimes would be inappropriately long for this review; it can be found in [191Jump To The Next Citation Point]. The results are summarised in Table 1. Note that μ ≡ mcu0 plays the role of the chemical potential for the relativistic BEC. One of the most remarkable features of this model is that it is a condensed matter system that interpolates between two different Lorentz symmetries, one at low energy and a different Lorentz symmetry at high energy.

Table 1: Dispersion relation of gapless and gapped modes in different regimes. Note that we have c2 = c2b∕(1 + b) s, and c2 = c2(2 + b)∕(1 + b) s,gap, while m = 2(μ∕c2)(1 + b)3∕2∕(2 + b) eff.

  b ≪ 1 b ≫ 1
mu0(1+b)- |k | ≪ ℏ
2mc0 |k| ≪ ℏ 2 2 2 ω = csk
2 2 2 ω = csk
m2 c4s,gap ω2 = --effℏ2---+ c2s,gapk2
2mc0-≪ |k| ≪ mu0- ℏ ℏ ℏω = (ℏck)2- 2μ  
0 |k| ≫ mu-(ℏ1+b)-  
ω2 = c2k2

Finally, it is also possible to recover an acoustic metric for the massless (phononic) perturbations of the condensate in the low momentum limit (k ≪ mu0 (1 + b)∕ℏ):

[ ( ) ] ------ρ------- uσu-σ uμuν- gμν = ∘ --------σ--2 ημν 1 − c2 + c2 . (285 ) 1 − uσu ∕co 0 0
As should be expected, it is just a version of the acoustic geometry for a relativistic fluid previously discussed, and in fact can be cast in the form of Equation (157View Equation) by suitable variable redefinitions [191].

4.2.2 The heliocentric universe

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 [660Jump To The Next Citation Point] 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 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 on 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 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 3He-A 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 {px = 0,py = 0,pz = ±pF } (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, c∥ ≃ cm ∕sec, and a different speed, −5 c⊥ ≃ 10 c∥, 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 nontrivial flows, there is also the possibility of creating topologically nontrivial configurations with a built-in nontrivial geometry. For example, it is possible to create a domain-wall configuration [327Jump To The Next Citation Point, 326Jump To The Next Citation Point] (the wall contains the z-axis) such that the transverse velocity c⊥ acquires a profile in the perpendicular direction (say along the x-axis) with c⊥ passing through zero at the wall (see Figure 11View Image). 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 that it is dynamically stable, for topological reasons, even when some supersonic regions are created.

View Image

Figure 11: Domain wall configuration in 3He.

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 3He [657Jump To The Next Citation Point, 659Jump To The Next Citation Point]. In particular, if we have a thin layer of 3He-A in contact with another thin layer of 3He-B, the oscillations of the contact surface “see” an effective metric of the form [657Jump To The Next Citation Point, 659Jump To The Next Citation Point]

[ ] 1 ( ) ds2 = -------------- − 1 − W 2 − αA αBU 2 dt2 − 2W ⋅ dxdt + dx ⋅ dx , (286 ) (1 − αA αBU 2)
W ≡ αAvA + αBvB, U ≡ vA − vB, (287 )
----hBρA------ ----hA-ρB----- αA ≡ h ρ + h ρ ; αB ≡ h ρ + h ρ . (288 ) A B B A A B B A
(All of this provided that we are looking at wavelengths larger than the layer thickness, khA ≪ 1 and khB ≪ 1.)
View Image

Figure 12: Ripplons in the interface between two sliding superfluids.

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 [657Jump To The Next Citation Point, 659].

4.2.3 Slow light in fluids

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) [617, 338, 96, 353, 506, 603, 565], 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, centered 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 [383Jump To The Next Citation Point]. 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 [390Jump To The Next Citation Point, 211]

ω2 k2 = -2-[1 + χ (ω )], (289 ) c
where χ is the susceptibility of the medium, related to the refractive index n via the simple relation ------ n = √ 1 + χ.

It is easy to see that in this case the group and phase velocities differ

∂ω- --------c-------- ω- ---c---- vg = ∂k = √ ------ -ω-∂-χ ; vph = k = √1--+-χ-. (290 ) 1 + χ + 2n ∂ ω
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 c. (The phase velocity is exactly c at the resonance frequency ω0). In an ideal EIT regime the probe light experiences a vanishing susceptibility χ near the the critical frequency ω0, this allows us to express the susceptibility near the critical frequency via the expansion
2α- [ 3] χ(ω) = ω0 (ω − ω0) + O (ω − ω0) , (291 )
where α is sometimes called the “group refractive index”. The parameter α 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, ℏω0, and the control-light energy per atom [383]. 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

vg = ∂ω-→ --c---≈ -c; vph = ω-→ c. (292 ) ∂k 1 + α α k
We can now generalise the above discussion to the case in which our highly dispersive medium flows with a characteristic velocity profile u(x, t). In order to find the dispersion relation of the probe light in this case we just need to transform the dispersion relation (289View Equation) 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 ω near ω0). The motion of the dielectric medium creates a local Doppler shift of the frequency
ω → γ (ω0 − u ⋅ k) , (293 )
where γ is the usual relativistic factor. Given that 2 2 2 k − ω ∕c is a Lorentz invariant, it is then easy to see that this Doppler detuning affects the dispersion relation (289View Equation) only via the susceptibility dependent term. Given further that in any realistic case one would deal with non-relativistic fluid velocities u ≪ c we can then perform an expansion of the dispersion relation up to second order in u∕c. Expressing the susceptibility via Equation (291View Equation) we can then rewrite the dispersion relation in the form [390]
μν g kμkν = 0, (294 )
( ) ω0- kν = c ,− k , (295 )
[ 2 2 | T 2 ] μν -− (1-+-αu-∕c-)|----−-αu--∕c------- g = − αu ∕c2 I3×3 − 4αu ⊗ uT ∕c2 . (296 )
(Note that most of the original articles on this topic adopt the opposite signature (+ − − − ).) 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 2 μ ν ds = gμνdx dx. In this sense the probe light will propagate as in a curved background. Explicitly one finds the covariant metric to be
[ | T ] --− A--|----− Bu------- gμν = − Bu I3×3 − Cu ⊗ uT , (297 )
1 − 4αu2∕c2 A = ------2-------2--2-----2--4--4; (298 ) 1 + (α − 3 α)u ∕c − 4α u ∕c B = --------------1---------------; (299 ) 1 + (α2 − 3 α)u2∕c2 − 4α2u4 ∕c4 1 − (4∕α + 4u2∕c2) C = ------------------------------. (300 ) 1 + (α2 − 3 α)u2∕c2 − 4α2u4 ∕c4

Several comments are in order concerning the metric (297View Equation). First of all, it is clear that, although more complicated than an acoustic metric, it will still be possible to cast it into the Arnowitt–Deser–Misner-like form [627]

[ | ] −-[c2eff-−-gabuaeffubeff]|[ueff]i- gμν = [u ] |[g ] , (301 ) eff j eff ij
where the effective speed u eff is proportional to the fluid flow speed u and the three-space effective metric geff is (rather differently from the acoustic case) nontrivial.

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 c∕α, where α can be extremely large due to the huge dispersion in the transparency window around the resonance frequency ω 0). 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 √ -- u = c∕ (2 α).

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 [470Jump To The Next Citation Point, 612Jump To The Next Citation Point], such a conclusion would turn out to be over-enthusiastic. In order to obtain particle creation through “mode mixing”, (mixing between the positive and negative norm modes of the incoming and outgoing states), an inescapable requirement is that there must be regions where the frequency of the quanta as seen by a local comoving observer becomes negative.

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 (293View Equation); but this also tells us that the negative norm mode must satisfy the condition ω0 − u ⋅ k < 0, but this can be satisfied only if the velocity of the medium exceeds |ω0∕k |, which is the phase velocity of the probe light, not its group velocity. This observation suggests that the existence of a “phase velocity horizon” is an essential ingredient (but not the only essential ingredient) in obtaining Hawking radiation. A similar argument indicates the necessity for a specific form of “group velocity horizon”, one that lies on the negative norm branch. Since the phase velocity in the slow light setup we are considering is very close to c, 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 [612Jump To The Next Citation Point] that a different setup for slow light might deal with this and other issues (see [612Jump To The Next Citation Point] 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.

4.2.4 Slow light in fibre optics

In addition to the studies of slow light in fluids, there has now been a lot of work done on slow light in a fibre-optics setting [505Jump To The Next Citation Point, 504Jump To The Next Citation Point, 64Jump To The Next Citation Point, 63Jump To The Next Citation Point], culminating in recent experimental detection of photons apparently associated with a phase-velocity horizon [66Jump To The Next Citation Point]. The key issue here is that the Kerr effect of nonlinear optics can be used to change the refractive index of an optical fibre, so that a “carrier” pulse of light traveling down the fibre carries with it a region of high refractive index, which acts as a barrier to “probe” photons (typically at a different frequency). If the relative velocities of the “carrier” pulse and “probe” are suitably arranged then the arrangement can be made to mimic a black-hole–white-hole pair. This system is described more fully in Section 6.4.

4.2.5 Lattice models

The quantum analogue models described above all have an underlying discrete structure: namely the atoms they are made of. In abstract terms one can also build an analogue model by considering a quantum field on specific lattice structures representing different spacetimes. In [310Jump To The Next Citation Point, 149Jump To The Next Citation Point, 322] a falling-lattice black-hole analogue was put forward, with a view to analyzing the origin of Hawking particles in black-hole evaporation. The positions of the lattice points in this model change with time as they follow freely falling trajectories. This causes the lattice spacing at the horizon to grow approximately linearly with time. By definition, if there were no horizons, then for long wavelengths compared with the lattice spacing one would recover a relativistic quantum field theory over a classical background. However, the presence of horizons makes it impossible to analyze the field theory only in the continuum limit, it becomes necessary to recall the fundamental lattice nature of the model.

4.2.6 Graphene

A very interesting addition to the catalogue of analogue systems is the graphene (see, for example, these reviews [127, 339Jump To The Next Citation Point]). Although graphene and some of its peculiar electronic properties have been known since the 1940s [672], only recently has it been specifically proposed as a system with which to probe gravitational physics [153Jump To The Next Citation Point, 152Jump To The Next Citation Point]. Graphene (or mono-layer graphite) is a two-dimensional lattice of carbon atoms forming a hexagonal structure (see Figure 13View Image). From the perspective of this review, its most important property is that its Fermi surface has two independent Fermi points (see Section 4.2.2 on helium). The low-energy excitations around these points can be described as massless Dirac fields in which the light speed is substituted by a “sound” speed cs about 300 times smaller:

∂ ψ = c ∑ σk∂ ψ . (302 ) t j s xk j k=1,2
Here ψj with j = 1,2 represent two types of massless spinors (one for each Fermi point), the σk are the Pauli sigma matrices, and c = 3ta∕2 s, with a = 1.4 ˚A being the interatomic distance, and t = 2.8 eV the hopping energy for an electron between two nearest-neighboring atoms.
View Image

Figure 13: The graphene hexagonal lattice is made of two inter-penetrating triangular lattices. Each is associated with one Fermi point.

From this perspective graphene can be used to investigate ultra-relativistic phenomena such as the Klein paradox [339]. On the other hand, graphene sheets can also acquire curvature. A nonzero curvature can be produced by adding strain fields to the sheet, imposing a curved substrate, or by introducing topological defects (e.g., some pentagons within the hexagonal structure) [669]. It has been suggested that, regarding the electronic properties of graphene, the sheet curvature promotes the Dirac equation to its curved space counterpart, at least on the average [153, 152]. If this proves to be experimentally correct, it will make graphene a good analogue model for a diverse set of spacetimes. This set, however, does not include black-hole spacetimes, as the curvatures mentioned above are purely spatial and do not affect the temporal components of the metric.

  Go to previous page Go up Go to next page