Aspects of cosmic inflation in expanding Bose-Einstein condensates

Michael Uhlmann, Yan Xu and Ralf Schützhold¹

Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany

¹ Author to whom any correspondence should be addressed.

Email: schuetz@theory.phy.tu-dresden.de

Received 13 September 2005
Published 30 November 2005

Contents

• 1. Introduction
• 2. Dimensional reduction
• 3. Effective geometry
• 4. Co-moving coordinates and scaling
• 5. Horizon analogues
• 6. Density-density correlations in Q2D
• 7. Density-density correlations in 3D
• 8. Summary
• Acknowledgments
• References

1. Introduction

A quarter of a century ago, Unruh [1] noticed an intriguing analogy among (at first glance) very different systems: phonons in irrotational fluids behave in the same way as (quantum) fields in curved space-times whose geometry is determined by the effective metric and depends on the velocity of the fluid flow and the speed of sound etc. This analogy facilitates a know-how transfer in both directions: on one hand, it allows us to apply all the concepts, tools and effects known from general relativity - such as horizons - to the propagation of phonons (or other quasi-particles) in fluids and hence leads to a better understanding of condensed-matter phenomena in terms of universal geometrical concepts. On the other hand, this analogy opens up the opportunity of a theoretical and possibly experimental investigation of exotic quantum effects known from cosmology such as Hawking radiation [2].

Unfortunately, it turns out that an experimental verification of the Hawking effect by means of such an analogue system (i.e., a black hole analogue) is very difficult [3]. In this paper, we shall focus on another exotic quantum phenomenon known from cosmology, which also involves the amplification of the initial quantum vacuum fluctuations due to the presence of a horizon: within our standard model of cosmology, the early universe was nearly homogeneous and basically all inhomogeneities - including the seeds for the formation of large-scale structures such as our galaxy - originate from the quantum vacuum fluctuations of a scalar field, which is called the inflation. During inflation [4], which is a (conjectured) period of accelerated expansion of the very early universe, these quantum fluctuations were amplified due to the presence of a cosmic horizon (cf section 5). These amplified fluctuations left an imprint in the cosmic microwave background radiation - small perturbations upon a homogeneous background as measured by the WMAP satellite [5].

Owing to the aforementioned analogy, the very same amplification mechanism should also occur in appropriate fluids. As an example, we shall consider the, in some sense, best understood superfluids - dilute atomic/molecular Bose-Einstein condensates (BECs), see e.g. [6] for a review. Apart from the good theoretical understanding, further advantages of BECs are the various possibilities to manipulate (e.g., changing the speed of sound) and control them experimentally [7]. As will become evident later, the free expansion of a BEC after switching off the trap - which is a standard procedure in time-of-flight experiments/measurements, for example - leads to the formation of an effective cosmic horizon and, consequently, to the amplification of the initial quantum fluctuations of the phonon modes. These amplified fluctuations manifest themselves in potentially measurable small-scale density (and phase) variations.

In order to study this effect for rather general conditions, we shall consider BECs with non-specified power-law self-interactions in an arbitrary number of (spatial) dimensions. After a discussion of the effectively lower-dimensional behaviour of strongly constrained BECs in section 2, the effective metric governing the propagation of phonons is derived in section 3 for arbitrary power-law self-interactions in any dimension. Section 4 is devoted to the introduction of co-moving coordinates (in complete analogy to cosmology) and to the scaling behaviour of the expanding condensate. The analogy to cosmology is further elaborated in section 5 by applying the concept of an effective horizon, whose existence generates the aforementioned amplification mechanism. The spectrum and magnitude of the resulting density fluctuations are calculated in sections 6 and 7 for the quasi-two-dimensional (Q2D) and the three-dimensional (3D) case, respectively, assuming the usual quartic coupling.

2. Dimensional reduction

Since we shall consider an arbitrary number of spatial dimensions later on, let us first discuss the behaviour of strongly confined and hence effectively lower-dimensional BECs. In three spatial dimensions, BECs are described by the Lagrange density (  =  1 throughout) [8]

with m being the mass of the bosons, V_ext the external and generally space-time-dependent one-particle trapping potential, and the two-particle coupling g in s-wave approximation, which is related to the s-wave scattering length a_s via g  =  4πa_s/m. If the external one-particle trapping potential can be split up into a parallel and a static transversal part via

it is useful to decompose the order parameter Ψ (and its quantum fluctuations ) into a complete set of real and time-independent functions _α(r_⊥) governing the transversal dependence (cf [9], where the density of an elongated BEC is decomposed in a similar manner)

The time evolution of the coefficient functions ψ_α(r_||, t) is determined by the reduced Lagrangian density

If we now choose the functions _α(r_⊥) to be the orthonormal eigenfunctions of the self-adjoint operator with eigenvalues Ω_α

the reduced Lagrangian density simplifies to

The last term induces a coupling of different modes, which complicates the solution. However, if we assume that the coupling term is sufficiently small and the lowest mode α  =  0 dominates |ψ₀|    |ψ_α > 0|, we may estimate the population of the higher modes in analogy to stationary perturbation theory: the mixing between the lowest mode α  =  0 and all higher modes α  >  0 is small if the energy differences ΔΩ_α  =  Ω_α  -  Ω₀ are large compared to the transition matrix elements of the interaction Hamiltonian, i.e.,

Since the modes _α are normalized , this condition can be re-expressed in terms of the size of the transversal dimension a_⊥ and the healing length . Therefore, imposing the conditions

where the latter requirement a_⊥    a_s is necessary for the Gross-Pitaevskii; Lagrangian density in equation (1) to be valid, we arrive at the reduced Lagrangian density for the lowest mode ψ  =  ψ₀

where Ω₀ has been absorbed into V^||_ext and

denotes the reduced (lower-dimensional) coupling constant. For example, the eigenfunctions _α(r_⊥)  =  _α(z) of a harmonic potential V_⊥(z)  =  mω_z²z²/2 with the transversal length scale are just the Legendre polynomials with an equidistant energy spectrum. In this case, the reduced coupling is given by . However, it should be emphasized that the method presented above is applicable to more general potentials satisfying the aforementioned conditions as well.

Note that if we relaxed condition (8) and the transverse size of the condensate, a_z, would be comparable to the scattering length, , we would obtain a lower-dimensional condensate with different interactions due to density-dependent corrections to the coupling coefficient [10]-[13].

3. Effective geometry

After having discussed the dimensional reduction, let us start with the Lagrange density in an arbitrary number of spatial dimensions D (where we omit the superscript || for the sake of conciseness). In addition, we shall assume a more general self-coupling term |ψ|^2N which will be motivated later

Inserting the Madelung representation

the Lagrange density reads

As usual, variation w.r.t. S yields the equation of continuity and w.r.t. the Bernoulli equation with the mean-field velocity and the specific pressure . Assuming that the density profile is sufficiently smooth, i.e., that the relevant length scales are much larger than the healing length (Thomas-Fermi approximation), we neglect the quantum pressure term . Linearization S  =  S₀  +  δS and  =  ₀  +  δ around a background solution S₀ and ₀ yields the second-order Lagrange density (cf [14]-[17])

where we have introduced the effective coupling

For the usual case N  =  2, we have g_N  =  g, but one should still bear in mind that we are considering an arbitrary number of spatial dimensions D. Note that there are no sound waves at all for N  =  1, since the theory is non-interacting in that case.

In the Thomas-Fermi approximation, the linearized Bernoulli equation can be solved for the density fluctuations

and inserting this result back into equation (14), we obtain the effective Lagrangian for the phase fluctuations  =  δS only

Even in an arbitrary number of spatial dimensions D (the difficulties for D  =  1 will be discussed below) and for a general self-coupling N  >  1, this Lagrangian is completely equivalent to that of a free (minimally coupled) scalar field in a curved space-time (e.g. [18])

provided that we insert the Painlevé-Gullstrand-Lemaître (PGL) metric [19]

with the speed of sound c_N²  =  g_N0/m and the conformal factor

This expression already indicates problems in one spatial dimension D  =  1 due to conformal invariance of the scalar field action (18) in 1 + 1 dimensions. Without the introduction of an additional dilaton field, the identification of an effective metric is only possible if c_N/g_N  =  const, for example if g  =  const and N  =  3; see also section 4.

The analogy between equations (17) and (18) represents the underlying idea of the analogue-gravity concept, which inspired numerous investigations and can be applied to many areas and phenomena, see also [20].

4. Co-moving coordinates and scaling

Since phase fluctuations in BECs behave (in the Thomas-Fermi approximation) exactly as a scalar field in a specific curved space-time described by the effective metric in equation (19), it will be useful to investigate this metric with the tools known from general relativity. The effective line element reads

In analogy to cosmology, we shall assume local isotropy and homogeneity - which will be a good approximation in the centre of the BEC cloud. Since the coupling g is supposed to be constant for simplicity, this assumption implies a spatially homogeneous but possibly time-dependent density ₀  =  ₀(t) and effective coupling g_N  =  g_N(t). Furthermore, after a suitable re-definition of the origin of our coordinate system, we may set v₀ r due to the presumed local isotropy and homogeneity. Insertion of this ansatz into the equation of continuity yields [15, 16]

which allows us to describe ₀(t) and v₀(t, r) as well as g_N(t) by means of a single time-dependent scaling parameter b(t) satisfying the initial condition b(t  =  0)  =  1 and . As we know from general relativity, an off-diagonal metric such as in equation (21) can be diagonalized by introducing co-moving spatial coordinates via (cf the scaling transformation for BECs in [21, 22])

In addition, we may transform from the laboratory time t to the effective proper (co-moving wrist-watch) time τ in order to eliminate the factor in front of dτ²

arriving at the standard Friedmann-Robertson-Walker representation, see e.g. [18]. Let us investigate the remaining factor A_D^(N)b² a bit further: since ₀(t) b ^- D(t) according to equation (22), we obtain g_N(t) b ^{- D(N - 2)}(t) from equation (15) as well as c_N(t) b ^{- D(N - 1)/2}(t), i.e., the factor in front of dρ² scales as

according to equation (20). Interestingly, the exponent vanishes for

e.g., for D  =  2 and N  =  2, or D  =  3 and N  =  5/3, or D  =  1 and N  =  3. As we have observed above, special care is required for the latter case D  =  1, but for N  =  3 we can indeed introduce an effective metric and the conformal factor can be chosen at will since it does not enter the calculation.

If the condition in equation (26) is satisfied and thus the exponent in equation (25) vanishes, the transformed metric in equation (24) is flat and hence the wave equation for the phonon modes becomes trivial in terms of the coordinates τ and ρ

i.e., a solution (τ,ρ) is independent of the external time dependence mediated via the scaling parameter b(t). It turns out that this perfect scaling is not only valid for the phonon modes, but can be extended to the full field operator: if we start from the equation of motion in the Heisenberg picture

again with general s-wave coupling (N) and in D spatial dimensions, we may account for an arbitrarily time-dependent external one-particle trapping potential V_ext(t, r) as long as it is purely harmonic at all times

by inserting the scaling ansatz derived above (cf [22])

The numerator exp[imv₀²(t, r/2] of the pre-factor reproduces v₀ and the denominator accounts for ₀(t)  =  ₀(t  =  0)/b^D(t) and ensures the correct commutation relations for , i.e.,

For a flat metric N  =  1  +  2/D, the proper time is (independently of D) determined as

and ρ is of course still given by ρ  =  r/b(t). Finally, if we arrange the scale parameter b(t) according to

the equation of motion for becomes independent of b(t)

i.e., we obtain a perfect scaling solution of the full field operator exactly in those cases where the effective metric is flat. Note that the implications of this property of the full field operator go far beyond the scaling of the hydrodynamic solution - we also obtain a perfect scaling of the quantum fluctuations to arbitrary order and for large wavenumbers (provided that the s-wave approximation is still valid), where hydrodynamic (Thomas-Fermi) solution breaks down. It is not even necessary to assume the mean-field expansion, i.e., the scaling also applies to non-condensed bosons.

Interestingly, all three examples (from D  =  1 to 3) are potentially relevant for real physical systems:

• D  =  1 and N  =  3: in Q1D BECs, one may obtain an effective |ψ|⁶ coupling if the perpendicular size a_z is comparable to the s-wave scattering length a_s, see e.g. [11]-[13].
• D  =  2 and N  =  2: this is the usual Q2D BEC with quartic coupling, see e.g. [6].
• D  =  3 and N  =  5/3: even though the exponent N  =  5/3 may seem somewhat unnatural, it appears in an effective description of a weakly interacting two-component Fermi gas in the BCS state, where the equation of state is determined by the Fermi energy E_F ^5/3, and which therefore also shows scaling behaviour, see e.g. [7, 15].

5. Horizon analogues

Apart from the question of whether it can be cast into a flat space-time form by means of an appropriate coordinate transformation or not, the emergence of a non-trivial effective metric in equation (19) suggests the application of concepts known from general relativity, such as horizons [4, 23, 24]. Generally speaking, horizons correspond to a loss of causal connectivity, i.e., events beyond a horizon have no influence or cannot be influenced. The most prominent example is the event horizon of a (classical) black hole, beyond which everything is trapped forever (i.e., nothing can ever escape to infinity).

On the other hand, in cosmology, two slightly different horizon concepts play a more important role due to the large-scale homogeneity and isotropy - the particle horizon and the apparent horizon. The particle horizon always refers to a chosen trajectory and indicates the border to the space-time region which cannot be reached by any signal starting from this trajectory or from where no signal can reach this trajectory. The apparent horizon depends on the chosen coordinates and is (roughly speaking) defined as the border beyond which all closed two-surfaces can either only expand or only contract w.r.t. the chosen coordinates.

As it turns out, the concepts of the particle and the apparent horizon can be applied to expanding BECs. Let us start with the particle horizon and choose the trajectory r  =  0 → ρ  =  0 for which the particle horizon can conveniently be determined using the metric in equation (23). The question is: which values of the co-moving coordinate ρ can a sound wave reach by starting at the origin ρ  =  0 at time t and being described by a null line ds²_eff  =  0, i.e., how far can it travel? According to equation (23), this length can be calculated via the following integral (cf also [4])

If the above integral converges to a finite result Δρ_horizon(t), this corresponds to a particle horizon since no point beyond this co-moving coordinate can be reached by sound waves any longer.

For a free expansion of the condensate V_ext(t ↑ ∞)  =  0, we obtain b(t ↑ ∞) t at late times according to equation (33) and thus a horizon exists for all N  >  1. The occurrence of a horizon might be puzzling if the metric in terms of the proper time τ in equation (24) is flat - but this puzzle can be resolved by the observation that the proper time τ reaches a finite value τ(t ↑ ∞)  <  ∞ in the limit of arbitrarily late laboratory times in that case, cf equation (32).

In contrast to the particle horizon, which necessitates the knowledge of the full future (or past), the apparent horizon can be identified by means of the configuration at a certain instant of time only. Instead of a trajectory as for the particle horizon, we have to choose a specific coordinate system (time slices) in order to define the apparent horizon. The coordinates (τ, ρ) leading to a flat metric, for example, do not allow the introduction of an apparent horizon. On the other hand, since we observe the expanding condensate using the laboratory coordinate system (t, r), we shall choose these coordinates instead. For a spherically symmetric metric as in equation (19), the apparent horizon is determined by g₀₀^eff  =  0, i.e., where the velocity of the condensate v₀ exceeds the speed of sound c_N

The very intuitive interpretation is that no sound wave can enter the region r  <  r_horizon from the outside.

For perfect scaling N  =  1  +  2/D and a freely expanding condensate, the apparent horizon settles down at late times (when becomes constant) to a finite value r_horizon(t ↑ ∞)  =  r_horizon^∞  >  0, otherwise it may increase or decrease forever (depending on D and N). Note that both, the apparent and the particle horizon are never at rest w.r.t. the co-moving coordinate ρ but always decreasing. As a result, the wavelength of all sound modes exp(iκ · ρ) exceeds the horizon size at some point of time (horizon crossing) in view of the permanent stretching of modes due to the expansion of the condensate. After the wavelength exceeds the horizon size, the regions of higher and lower pressure cannot interact anymore and hence the modes stop oscillating (freezing of modes). As a result, comparing each mode with a harmonic oscillator, the momentum and its variance decrease drastically - and according to the Heisenberg uncertainty relation, the complementary variance must increase in order to compensate this. Ergo, the initial ground/vacuum state gets squeezed, which amplifies the quantum fluctuation in a certain direction. This admittedly rather intuitive picture (horizon crossing, freezing and squeezing) applies to both, the expanding universe (provided a horizon exists, such as during cosmic inflation) as well as expanding BECs in a very similar way and must of course be further supported by explicit calculations, see section 6.

Note that analogue horizons (and perhaps horizons in real gravity as well) are only low-energy effective concepts since, for very small wavelengths (below the healing length), the quantum pressure term becomes important leading to group and phase velocities exceeding the usual speed of sound. Furthermore, as the healing length increases during the expansion, the late-time limit in the derivation of the particle horizon is not strictly valid. Nevertheless, the main effects such as the amplification of quantum fluctuations are not modified drastically, see section 6.

6. Density-density correlations in Q2D

According to the results of section 4, the time-evolution of a Q2D condensate with the usual quartic coupling (i.e., N  =  D  =  2) is particularly simple to solve due to its perfect scaling behaviour - which enables us to derive its dynamics exactly (e.g., including the quantum-pressure corrections). If we assume that the initial trapping potential is harmonic, the free expansion after switching off the longitudinal trap V_||^ext(t  >  0)  =  0 is fully described by the time-evolution of the scaling parameter according to equation (33)

The associated proper time τ can be calculated from equation (32)

Since τ quickly approaches a finite value at late times t ↑ ∞, the fluctuations freeze in and assume the value at proper time τ  =  π/(2ω₀). In view of the perfect scaling discussed in section 4, both the background and the phonon modes - expressed in terms of ρ and τ - behave in the same way before and during the expansion, leaving the relative correlation function

unchanged. Consequently, the spectrum of the relative density contrast in co-moving coordinates after horizon crossing is given by the initial correlation spectrum. Besides the Q2D condensate with usual quartic coupling (N  =  D  =  2), this also holds for all other cases where perfect scaling occurs.

As discussed in section 5, the apparent horizon depends on the chosen time slicing, i.e., the set of coordinates. (The particle horizon is independent of the set of coordinates, but requires the knowledge of the whole future evolution.) Co-moving coordinates, albeit advantageous to describe the evolution of the modes, are not suitable for describing the appearance of an apparent horizon. On the other hand, all observations are performed in laboratory time t and coordinates r with the metric (19) showing the formation of an apparent horizon according to equation (36). For N  =  1  +  2/D, it follows

in the homogeneous case. While the trapping potential is still turned on (t  <  0), the (apparent) horizon size r_horizon is infinite. Clearly, without any expansion, all points are causally connected via sound waves. After releasing the condensate, however, the horizon settles down very quickly (ω₀t    1) at a finite position (in laboratory coordinates) c_N(0)/ω₀, given by the initial speed of sound c_N(0) and trapping frequency ω₀.

Since the (apparent) horizon approaches a finite position in laboratory coordinates (t,r) and every phonon mode with a given wavelength λ expands with the condensate cloud, each phonon mode will cross the horizon eventually. Equivalently, in terms of the co-moving coordinates, the horizon size decreases for all times (as long as the condensate is expanding) and the (co-moving) wavenumber κ remains constant. As mentioned before, the initial correlation function of a Q2D condensate translates directly into the frozen density contrast after horizon crossing.

In order to derive the density-density correlation function quantitatively, we quantize the phonon modes within the initial condensate v₀  =  0 using the approximation (centre of the BEC cloud) of a constant background density ₀  =  const. After a normal mode expansion, the Lagrange function (13) reads

where V_Q denotes the quantization volume. Note that, in contrast to equation (14), we did not neglect the quantum pressure term. In a straightforward manner, the phase and density fluctuations can be quantized via the introduction of creation ( ) and annihilation operators ( ) which diagonalize the Hamiltonian. The quantized phase and density fluctuations read

with the well-known Bogoliubov dispersion relation

where μ  =  g_2D0^2D is the (initial) chemical potential.

For a perfectly scaling condensate, the two-point function in equation (39) yields the density-density correlations at all times, and in particular also the frozen correlations (the density contrast) after horizon crossing. We can thus calculate the Fourier components

in order to obtain the spectrum.

For large wavelengths, the spectrum is linear in the (co-moving) momentum κ in complete agreement to the curved space-time analogy. For large (co-moving) momenta κ, where the dispersion relation (44) changes from linear to quadratic and the curved space-time analogy breaks down, the spectrum of the two-point correlation function approaches a constant value corresponding to a Dirac δ-contribution C(ρ, ρ ') δ(ρ  -  ρ ')  +  ···. Interestingly, after subtracting this local term, we obtain a scale-invariant spectrum C(κ) 1/κ² for large κ - in analogy to cosmic inflation, where the spectrum is scale-invariant too (in 3  +  1 dimensions it is 1/k³). However, one should bear in mind that this scale-invariance occurs in a region where the curved space-time analogy breaks down.

For addressing the question of whether this effect (amplification of the initial quantum fluctuations of the phonon modes) can be measured, we estimate the order of magnitude of the relative density-density correlations C(κ) for ρ ≠ ρ '. Since the 2D coupling constant, g_2D, is proportional to the ratio of the scattering length, a_s, and the perpendicular extension of the condensate, a_z, we have typically

Note, however, that this can only be seen as a rough estimate of the order of magnitude, since the two-point function is logarithmically divergent at small distances (where eventually the employed approximations break down). Remembering the conditions (8) in section 2, we see that the relative density contrast must be significantly smaller than one - which was to be expected. However, the fluctuations could be on the per cent level (see section 8) and thus might well be measurable.

7. Density-density correlations in 3D

For an expanding 3D condensate with quartic coupling the situation is more complicated, since it does not show the perfect scaling behaviour as in two dimensions. Therefore, it is not possible to infer the final (frozen) density (or phase) fluctuations from the initial state exactly as in the previous case. Fortunately, for long wavelengths, we may exploit the effective space-time analogy and derive the evolution using the tools known from cosmology. To this end, we transform onto co-moving coordinates and thus diagonalize the effective PGL metric (19). Note that, in three (spatial) dimensions, the background density scales as (t)  =  (0)/b³(t). Assuming the background density to be smooth (Thomas-Fermi limit), the evolution equation for the scale factor - after switching off the trap - reads, cf  [22]

Similar to the previous case, the scale factor accelerates (the interaction energy is transformed into kinetic energy) after switching off the trap, but quickly assumes a linear behaviour, b t (for t    1/ω₀).

The following calculations are most conveniently performed using the laboratory time but co-moving spatial coordinates. One advantage of the analogy to curved space-times is the inherited freedom of choosing suitable coordinates, where the independence of the final result follows automatically from covariance. The equation of motion of the field modes _κ with co-moving wavenumber κ assumes the form

Initially, for the trapped condensate, the damping term vanishes and the modes oscillate freely. During the expansion, the third term decreases whilst the second one increases and eventually dominates the latter one - in complete analogy to an over-damped oscillator, the modes freeze in. Because most phonon modes have frequencies much larger than the trapping frequency (e.g. [25]), this horizon crossing and freezing process happens during the period of linear expansion b(t)  =  αt (with α ≈ 0.82 ω₀). In view of the adiabatic theorem, the initial vacuum state for these (initially) rapidly oscillating modes is preserved (adiabatic vacuum) during the initial nonlinear expansion. Thus the equation of motion describing the freezing-in of modes can be simplified considerably by inserting b(t)  =  αt

and solved analytically in terms of Bessel functions [26]

where H_2/3^(1,2) denote the Hankel functions with the index ν  =  2/3 and C_κ^(1,2) are the corresponding integration constants. As one would expect, for early times, the modes _κ oscillate as . It turns out that the Hankel functions H_2/3^(1,2) have the proper asymptotic behaviour for early times such that the integration constants C_κ^(1,2) can be replaced by (time-independent) creation and annihilation operators (in analogy to the previous section) associated to the initial adiabatic vacuum state for the quantized phonon modes in the Heisenberg picture

where the pre-factor can be derived by imposing the canonical commutation relations for the conjugate variables and obtained from the Lagrangian (18) for co-moving coordinates. Note that the argument of the Hankel functions coincides (up to a constant) with the proper time, , during the phase of linear expansion.

For late times t ↑ ∞, the modes _κ approach a constant value (due to horizon crossing and freezing) which can be derived by inserting the asymptotic behaviour of the Hankel functions H_2/3^(1,2) [26]. The frozen late-time t ↑ ∞ expectation value for phase-phase correlations reads

Employing equation (16), we can calculate the density fluctuations, (in co-moving coordinates, where v₀ vanishes), and obtain the spectrum of the relative density-density correlation function at late times

where the right-hand side terms (healing length etc) refer to the initial state. Interestingly, one obtains the same spectra (k ^- 4/3 for the phase and k ^+ 4/3 for the density fluctuations) for a condensate at rest after sweeping through the phase transition at g  =  0 by means of a time-dependent coupling g(t) [27].

In contrast to the Q2D case with perfect scaling, the above results rely on the curved space-time analogy and thus are only valid for wavelengths far above the healing length. Furthermore, in three dimensions, the healing length scales like ξ(t)  =  ξ(0)b^3/2(t) and grows faster (again in contrast to the Q2D case) than the wavelengths λ(t)  =  λ(0)b(t) in laboratory coordinates. In order to obtain a rough estimate of the maximum amplification effect, we insert the maximum co-moving wavenumber κ  =  1/ξ(α/ω_ξ)^1/4 (with ξ the initial healing length and ω_ξ  =  c₀/ξ) of the phonon modes whose wavelength exceeds the healing length during the relevant period of their evolutions (i.e., until horizon crossing and freezing)

Here ω_ξ  =  c₀/ξ(0) denotes the (initial) frequency of the phonons with the (co-moving) wavenumber 1/ξ(0) and where is a dimensionless constant. In contrast to the Q2D case, the size of the fluctuations now depends on the diluteness parameter (instead of the ratio a_s/a_z), which must be small for the employed approximations to apply . Furthermore, the ratio ω_ξ/ω₀  =  c₀ ξ/ω₀ should be large, ω_ξ/ω₀    1, if we want a sufficiently long period of linear expansion b(t)  =  αt. However, the smallness of these parameters can partly be compensated by the numerical pre-factor 30.3 such that the final effect can be on the per cent level (see section 8).

The spectrum and size of the frozen-in correlations C_3D could be measured by obtaining a density map of a slice of the condensate (tomographic imaging). By making a projective (e.g., absorption) image instead, one would average over the fluctuations in the (spatial) direction of projection. While the spectrum remains unchanged, this averaging yields an additional suppression by a factor of order ξ/l_z with l_z denoting the transversal (i.e., in direction of projection) extension of the condensate.

8. Summary

The objective was to investigate and to exploit the analogy between phonons in expanding BECs on one hand and quantum fields in an expanding universe on the other hand. The advantages of this curved space-time analogy are twofold: firstly, it facilitates the application of all the concepts, tools and effects known from general relativity (such as horizons) and thereby fosters a better understanding of condensed-matter phenomena in terms of universal geometrical concepts. Secondly, this analogy enables a theoretical and possibly experimental investigation of exotic quantum effects known from cosmology.

After a discussion of the effectively lower-dimensional behaviour of strongly constrained BECs in section 2, the emergence of a metric governing the propagation of low-energy phonons is discussed in section 3 for arbitrary power-law self-coupling in any dimension. In complete analogy to cosmology, the introduction of co-moving coordinates is advantageous for the description of many phenomena. It turns out that the full field operator (in a harmonic trapping potential) possesses perfectly scaling solutions exactly if the effective metric is flat in terms of the co-moving coordinates. In a freely expanding BEC, an effective sonic horizon is formed, i.e., two points at fixed (co-moving) spatial positions whose distance exceeds the horizon size cannot be connected anymore by (low-energy) phonons, cf section 5. The formation of this effective horizon implies the amplification of the quantum fluctuations (horizon crossing and freezing) already known from cosmology, which has been calculated explicitly for Q2D and 3D condensates with quartic coupling in sections 6 and 7.

In order to obtain an explicit estimate for the size of the derived effect, let us consider a condensate of 10⁵ sodium atoms inside a highly anisotropic trap with the trapping frequencies ω_⊥/2π  =  790 Hz and ω_||/2π  =  10 Hz, cf [28]. The thickness of the disc, a_z  =  0.746 μm, is smaller than the healing length, ξ  =  1.34 μm, but still much larger than the scattering length, a_s  =  2.8 nm, complying with the hierarchy of scales derived in section 2. To estimate the order of magnitude of the density-density correlations, we consider a mode with (co-moving) wavenumber κ  =  2π/ξ. For the relative density contrast inside a volume of size ξ², we obtain C(κ)/ξ²  =  1.79%. For the modes in the linear regime of the spectrum, horizon crossing and thus freezing-in occur shortly after the trap is switched off. Initially the horizon is at infinity, but very quickly settles at r_horizon(t ↑ ∞)  =  3.28 μm. Hence already after one e-fold, when the radius of the atom cloud has increased by a factor e ≈ 2.7, all modes with wavelength larger than ξ are frozen-in and the fluctuations are transformed into a density contrast. With the correlation function depending on the ratio a_s/a_z, the effect can be enhanced by confining the condensate more tightly in the perpendicular direction. However, if a_s/a_z is not small, the condensate is no longer described by the usual Gross-Pitaevskii; Lagrangian and our analysis no longer applies in this form (e.g., the perfect scaling breaks down). For such a case, the (initial) correlation spectrum (with appropriate non-quartic coupling, cf [13, 29]) was calculated in [29] and differs from our result in equation (45).

As an example for the 3D case, let us consider⁸⁷Rb atoms inside a spherically symmetric trap. For a condensate consisting of 10⁷ atoms and a trapping frequency of ω₀/2π  =  200 Hz, which are potentially realizable parameters, cf [30], the Thomas-Fermi radius is R  =  12.2 μm. The minimal frequency of phonons ω_min  =  2πc_s/R  =  5582 Hz is well above the trapping frequency ω₀. Hence the freezing of modes takes place during the period of linear expansion and the approximation b(t)  =  αt in the evolution equation of the phonon modes is justified. A rough estimate of the maximum effect according to equation (54) yields

i.e., a potentially measurable effect.

So far, we have assumed zero temperature. Of course, initial thermal fluctuations would - provided that there is no thermalization during the expansion of the condensate - also be amplified by basically the same mechanism as the quantum fluctuations. In order to ensure that the quantum fluctuations are larger than the (initial) thermal fluctuations, the temperature must be small enough such that the thermal occupation of the phonon modes under consideration is negligible. For the parameters used above, the relevant temperature scales are T(1/ξ)  =  0.1 nK for the Q2D and T(κ)  =  3.9 nK for the 3D condensate, respectively. By increasing the particle number and/or the trapping frequency, the requirements regarding the experimental temperature can be relieved. (Another way to discriminate between thermal and quantum fluctuations is the spectrum.)

In summary, expanding BECs facilitate the experimental simulation of exotic effects of quantum fields in curved space-times. On the other hand, the amplification mechanism under consideration sets the ultimate quantum limit of accuracy for time-of-flight experiments: no matter how smooth the initial cloud can be prepared, the frozen and amplified quantum ground-state fluctuations generate noticeable density perturbations.

Acknowledgments

The authors acknowledge valuable conversations with U R Fischer and W G Unruh. This work was supported by the Emmy-Noether Programme of the German Research Foundation (DFG) under grant no SCHU 1557/1-1. Further support by the COSLAB programme of the ESF, the Pacific Institute of Theoretical Physics, the Humboldt foundation and the programme EU-IHP ULTI is also gratefully acknowledged.

References

[1]

Unruh W G 1981 Phys. Rev. Lett. 46 1351 
CrossRef

[2]

Hawking S W 1974 Nature 248 30 
CrossRef
Hawking S W 1975 Commun. Math. Phys. 43 199 
CrossRef

[3]

Unruh W G 2002 Measurability of dumb hole radiation? Artificial Black Holes ed M Novello, M Visser and G Volovik (Singapore: World Scientific) 
CrossRef

[4]

Liddle A R and Lyth D H 2000 Cosmological Inflation and Large-Scale Structure  (Cambridge: Cambridge University Press) 

[5]

Bennett C L et al 2003 Astrophys. J. Suppl. 148 1 http://map.gsfc.nasa.gov/ 
IOPscience

[6]

Dalfovo F, Giorgini S, Pitaevskii L P and Stringari S 1999 Rev. Mod. Phys. 71 463 
CrossRef

[7]

Volovik G E 1998 J. Low Temp. Phys. 113 667 
CrossRef

[8]

Kamtchanov A M 2004 JETP 98 908 
CrossRef

[9]

Zaremba E 1998 Phys. Rev. A 57 518 
CrossRef

[10]

Petrov D S, Holzmann M and Shlyapnikov G V 2000 Phys. Rev. Lett. 84 2551 
CrossRefPubMed
Petrov D S and Shlyapnikov G V 2001 Phys. Rev. A 64 012706 
CrossRef

[11]

Petrov D S, Shlyapnikov G V and Walraven J T M 2000 Phys. Rev. Lett. 85 3745 
CrossRefPubMed

[12]

Kolomeisky E B, Newman T J, Straley J P and Xiaoya Qi 2000 Phys. Rev. Lett. 85 1146 
CrossRefPubMed
Kolomeisky E B and Straley J P 1992 Phys. Rev. B 46 11749 
CrossRef

[13]

Lee M D, Morgan S A, Davis M J and Burnett K 2002 Phys. Rev. A 65 043617 
CrossRef

[14]

Fischer U R 2004 Mod. Phys. Lett. A 19 1789 
CrossRef
Barcelo C, Liberati S and Visser M 2003 Phys. Rev. A 68 053613 
CrossRef

[15]

Fedichev P O and Fischer U R 2004 Phys. Rev. A 69 033602 
CrossRef

[16]

Fischer U R and Schützhold R 2004 Phys. Rev. A 70 063615 
CrossRef

[17]

Stone M 2000 Phys. Rev. E 62 1341 
CrossRef

[18]

Birrell N D and Davies P C W 1982 Quantum Fields in Curved Space  (Cambridge: Cambridge University Press) 

[19]

Painlevé P 1921 C. R. Hebd. Seances Acad. Sci. 173 677 

Gullstrand A 1922 Ark. Matter. Astron. Fys. 16 1 

Lemaître G 1933 Ann. Bull. Soc. R. Sci. Med. Nat. Brux. 53 51 

[20]

Barcelo C, Liberati S and Visser M 2005 Living Rev. Rel. in press (Preprint gr-qc/0505065)
Preprint
Slayer T R and Savage C M 2005 Class. Quantum Grav. 22 3833 
IOPscience
Calzetta E and Hu B L 2003 Phys. Rev. A 68 043625 
CrossRef
Campbell G et al 2005 Preprint cond-mat/0509302
Preprint

[21]

Castin Y and Dum R 1996 Phys. Rev. Lett. 77 5315 
CrossRefPubMed

[22]

Kagan Yu, Surkov E L and Shlyapnikov G V 1996 Phys. Rev. A 54 R1753 
CrossRef

[23]

Wald R M 1984 General Relativity  (Chicago, IL: University of Chicago Press) 

[24]

Hawking S W and Ellis G F R 1973 The Large Scale Structure of Space-Time  (Cambridge: Cambridge University Press) 
CrossRef

[25]

Ozeri R, Steinhauer J, Katz N and Davidson N 2002 Phys. Rev. Lett. 88 220401 
CrossRefPubMed
Steinhauer J, Ozeri R, Katz N and Davidson N 2002 Phys. Rev. Lett. 88 120407 
CrossRefPubMed

[26]

Abramowitz M and Stegun I E 1972 Handbook of Mathematical Functions  (New York: Dover) 

[27]

Schützhold R 2005 Phys. Rev. Lett. 95 135703 
CrossRefPubMed

[28]

Görlitz A et al 2001 Phys. Rev. Lett. 87 130402 
CrossRefPubMed

[29]

Gies C and Hutchinson D A W 2004 Phys. Rev. A 70 043606 
CrossRef

[30]

Streed E W et al 2005 Preprint cond-mat/0507348
Preprint