Analogue cosmological particle creation: Quantum correlations in expanding Bose-Einstein condensates

We investigate the structure of quantum correlations in an expanding Bose-Einstein condensate (BEC) through the analogue gravity framework. We consider both a $3+1$ isotropically expanding BEC as well as the experimentally relevant case of an elongated, effectively $1+1$ dimensional, expanding condensate. In this case we include the effects of inhomogeneities in the condensate, a feature rarely included in the analogue gravity literature. In both cases we link the BEC expansion to a simple model for an expanding spacetime and then study the correlation structure numerically and analytically (in suitable approximations). We also discuss the expected strength of such correlation patterns and experimentally feasible BEC systems in which these effects might be detected in the near future.

Figure 1

The solution of the scaling problem for an anisotropic expansion of an initially cigar-shaped configuration for both the radial/orthogonal (blue-dashed line) and axial/parallel (red-solid line) directions to the axis of release. Note that the scaling function of the direction in which the trapping frequency remains fixed (blue dashed) in fact decreases during the expansion. The parameters used in this solution are ${\omega }_{\parallel }=100\mathrm{Hz}$, ${\omega }_{\perp }={10}^3\mathrm{Hz}$, and a sudden release of the $\parallel$ direction only.

Figure 2

The quantum pressure and interaction energy expressed in Joules as a function of axial distance in meters. Note that the quantum pressure is certainly non-negligible near the boundary of the TF approximate ground state density wave function (at $z\simeq 2.7\times {10}^{-5}\mathrm{m}$).

Figure 3

The ratio $R=V_{\mathrm{quantum}}/V_{\mathrm{interaction}}$ as a function of comoving axial position $\tilde{z}=z{\lambda }_{\parallel }$ at various times $t$ after release of the trapping potential. Note that the axial half length of the condensate at time $t=0$ (which of course is the comoving length for all times) is $L_z\simeq 2.7\times {10}^{-5}\mathrm{m}$.

Figure 4

A comparison of the relative magnitudes of ${\omega }_e$ and ${\omega }_s$ (red solid and blue dashed, respectively) and with the healing frequency (inset plot) in $s^{-1}$ as a function of time (in $s$). Note that, for the majority of the duration of the expansion, ${\omega }_s$ is much smaller than the expansion frequency ${\omega }_e$ which is much smaller than the healing frequency. Hence, the bulk of the particle production spectrum lies within the low frequency cutoff provided by the total size of the condensate and the high frequency cutoff provided by the healing length.

Figure 5

The characteristic shape of the spectrum of produced particles in an expanding spacetime described by (4.4), as a function of the modulus $k=\sqrt{\mathbf{k}·\mathbf{k}}$. Recall that the spectrum is proportional to the volume of space $V/2\pi$ as $N_k=V{k}^2{\beta }_k^2/2\pi$ so that what we plot here is in fact particle number density. The peak wave number is fixed by the typical time scale of the expansion $k_{\mathrm{peak}}\propto 1/{\eta }_0$.

Figure 6

The exact numerical integration of (minus) the entanglement contribution to $G$ as a function of point separation $x$ for four successive times $\eta =0$ (red solid), $\eta =1$ (green dashed), $\eta =2$ (blue dotted), and $\eta =4$ (black dot dashed) for the choice of parameters $a_i^2=1$, $a_f^2=\pi$, and ${\eta }_0=1$.

Figure 7

A comparison of the exact expression (in red solid) with both the adiabatic approximation (in blue dots) and the sudden approximation (in green dashes) near the turn over region ($k{\eta }_0\approx 0.3$) where neither approximation is very accurate. We use the representative values of $t=2$ and $x=3$. Note that the sudden approximation is most accurate for small $k$ and the adiabatic for large $k$. Also note that the adiabatic approximation remains relatively accurate (at least in mimicking the qualitative behavior of the exact function) all the way down to $k=0$ where the approximation in principle should be poor.

Figure 8

The integrated $I+I^{*}$ contribution to the Wightman function in the adiabatic approximation at three successive times $\eta =0$ (red solid), $\eta =2$ (blue dots,) and $\eta =4$ (green dashes). Note that since the expression (4.11) is only valid after the expansion has taken place (the midpoint of which occurs around $\eta =0$) the contribution for $t<0$ are not shown. To make these plots and for comparison with the exact result again we have used the values $a_i^2=1$, $a_f^2=\pi$, and ${\eta }_0=1$. Also plotted (in black dash dotted) is the standard flat spacetime correlations $(4{\pi }^2{x}^2{)}^{-1}$.

Figure 9

Comparison of the bump structure of correlations for three different values of the parameter ${\eta }_0=2$, 1, $1/2$, and $1/4$ (from top left to bottom right) at three different times $\eta =0$ (red solid), $\eta =2$ (blue dots), and $\eta =4$ (green dashes). Also plotted (in black dash dotted) is the standard flat spacetime correlations for comparison.

Figure 10

The field (phase) correlations $G^{\varphi }$ as a function of time $t$ for three different fixed laboratory spatial separations $0.2L_z$, $0.5L_z/2$, and $0.8L_z/2$.

Figure 11

The bump of correlations propagating outwards from the center of the BEC cloud. The curves are for $t=0\mathrm{s}$ (red solid), $t=0.3\mathrm{s}$ (blue dotted), $t=0.8\mathrm{s}$ (green dashed), and $t=1.2\mathrm{s}$ (black dash dotted) Here we plot the truncated sum up to the first 90 terms. The horizontal axis of this plot is the fraction of the half length $L_z/2$ of the condensate after expansion. The vertical axis is the integrated contribution Ent in units of the normalization factor $N$.

Figure 12

The contributions to the phase Wightman function from the “1” and “$|{\beta }_k{|}^2$” terms in (5.28) truncated to the first 200 term of the series. Such a truncation serves as an ultraviolet regulator on the correlation singularity at the origin of the “1” background contribution. No such UV divergence is present in the “$|{\beta }_k{|}^2$” contribution.

Figure 13

The density correlation function after expansion has taken place, for the finite expansion case, in units of $N\times A_{\perp }^2/g^2$. The curves from top to bottom correspond to $t=0.3\mathrm{s}$, $t=0.8\mathrm{s}$, and $t=1.2\mathrm{s}$ after the onset of expansion (which we take, as described above, to last for $0.15\mathrm{s}$).

Figure 14

The exact scaling function $\lambda$ (red solid) and its approximation by the tanh function ${\tilde{\Omega }}_0^2/\Omega$ (blue dashed) over the interval $t=0.15\mathrm{s}$ as described in the text.