Black-hole radiation in Bose-Einstein condensates

We study the phonon fluxes emitted when the condensate velocity crosses the speed of sound, i.e., in backgrounds which are analogous to that of a black hole. We focus on elongated one-dimensional condensates and on stationary flows. Our theoretical analysis and numerical results are based on the Bogoliubov–de Gennes equation without further approximation. The spectral properties of the fluxes and of the long distance density-density correlations are obtained, both with and without an initial temperature. In realistic conditions, we show that the condensate temperature dominates the fluxes and thus hides the presence of the spontaneous emission (the Hawking effect). We also explain why the temperature amplifies the long distance correlations which are intrinsic to this effect. This confirms that the correlation pattern offers a neat signature of the Hawking effect. Optimal conditions for observing the pattern are discussed, as well as correlation patterns associated with scattering of classical waves. Flows associated with white holes are also considered.

Figure 1

Upper plot: shape of the profile $(c+v)∕c_0$ as a function of $\kappa x∕c_0$ for $D=0.5$ (solid line) and $D=1$ (dashed line). Lower plot: separate profiles $c\left(x\right)∕c_0$ and $v\left(x\right)∕c_0$ for $D=0.5$ and $n=2$. The solid lines correspond to $q=0.3$ and the dashed lines to $q=0.7$. Both pairs of profiles give rise to the same function $c+v$, represented by the solid line in the upper plot.

Figure 2

Graphical resolution of the dispersion relation Eq. (23) for $q=1$ (left plot) and for $q=0$ (right plot). The straight lines represent $\omega -v_{\pm }k$ and the curves represent ${\Omega }_{\pm }\left(k\right)$. The solutions $k\left(\omega \right)$ are given by the abscissa of their intersections.

Figure 3

Graphical resolution of the dispersion relation in the supersonic region for three values of ω: $0<\omega <{\omega }_{\max }$, $\omega ={\omega }_{\max }$, and $\omega >{\omega }_{\max }$.

Figure 4

Contours of constant ${\omega }_{\max }∕\kappa$ in the $(D,\lambda )$ plane, for $q=0$ (solid lines) and $q=1$ (dashed lines).

Figure 5

Schematic space-time representation of a wave-packet made out of modes ${\varphi }_{\omega }^{u,\mathit{\text{in}}}$ for $\omega >{\omega }_{\max }$. At early times, the packet is purely right moving. It is then scattered into a transmitted right-moving part and a reflected left-moving one. The dashed line represents the initially unexcited “ancestor” of the late-time left-moving packet.

Figure 6

Same as Fig. 5 for $\omega <{\omega }_{\max }$ when pair production occurs.

Figure 7

Energy flux density $f_{\omega }$ (left plot) and effective temperature $T_{\omega }∕T_H$ (right plot) versus $\omega ∕\kappa$, for $(D,\lambda )=(0.1,70)$, (0.4, 18), and (0.7, 10). $q$ is fixed to 0.3.

Figure 8

${\Delta }_H$ versus ${\omega }_{\max }∕\kappa$ for $D=0.1$ and $D=0.7$. $q$ is fixed to 0.3.

Figure 9

Effective temperature as a function of $\omega ∕\kappa$ for $D=1$, $q=0.7$, and λ =6. This situation describes approximately the experimental realization of Ref. 25.

Figure 10

Effective temperature as a function of $\omega ∕\kappa$ for $D=0.4$ and $(q,\lambda )=(0,19.54)$, (0.5, 17.15), and (1, 15.47). ${\omega }_{\max }∕\kappa$ is the same for all curves, equal to 2.2.

Figure 11

${\left|A_{\omega }\right|}^2$ (upper plot) and $n_{\omega }^v$ versus $\omega ∕\kappa$. The parameters are identical to those of Fig. 10.

Figure 12

${\left|A_{\omega }\right|}^2$ (upper plot) and ${\overline{n}}_{\omega }^v$ versus $\omega ∕\kappa$ for various values of $D$. The legend applies to both plots. The values of λ are identical to those of Fig. 7 and $q$ is fixed to 0.3.

Figure 13

${\left|A_{\omega }\right|}^2$ evaluated for $\hslash \omega =k_B{T}_H$ versus ${\omega }_{\max }∕\kappa$ for $D=0.1$, $D=0.4$, and $D=0.7$. $q$ is fixed to 0.3 in the upper plot and 0.5 in the lower one.

Figure 14

Effective temperature $T_{\omega }$ divided by Hawking temperature as a function of $\omega ∕\kappa$ for $q=0.3$ and $(D,\lambda )=(0.2,3.5)$ (solid line) and (0.4,1.8) (dashed line) when the variation in $c+v$ occurs on one healing length.

Figure 15

Effective temperature $T_{\omega }$ divided by the healing temperature of Eq. (76), as a function of $\omega ∕{\omega }_{\max }$ for $q=0.7$ and $D=1$ in extremely dispersive cases, when $\kappa ⪢c_0∕{\xi }_0$.

Figure 16

Energy flux emitted to the right of the horizon. $q$ is fixed to 0.3. The dotted spectra are those of Fig. 7. The values τ =1 (solid lines) and τ =0.3 (dashed lines) correspond, respectively, to an initial temperature $T_{\mathit{\text{in}}}=30\mathrm{nK}$ and $T_{\mathit{\text{in}}}=10\mathrm{nK}$ [see Eq. (77)].

Figure 17

Different contributions to the energy flux to the right of the horizon. Left plot: $D=0.1$, τ =1. Right plot: $D=0.7$, τ =0.3.

Figure 18

Initial proper frequency ${\Omega }^{\mathit{\text{in}}}∕\kappa$ as a function of $\omega ∕\kappa$ for the $u,\omega$ (left plot), $u,-\omega$ (middle plot) and $v,\omega$ modes (right plot), for $D=0.4$, $q=0.3$ and τ =0.3, and different values of λ. The legend applies to all plots.

Figure 19

Initial occupation numbers $n_{\omega }^{\mathit{\text{in}}}$ (left plot), $n_{-\omega }^{\mathit{\text{in}}}$ (middle plot), and $n_{\omega }^{\mathit{\text{in}},v}$ (right plot) versus $\omega ∕\kappa$ for the same values of the parameters as in Fig. 18.

Figure 20

Upper plot: adimensionalized (by multiplication by ${\rho }_{0}v$) power spectrum of density fluctuations as a function of $\omega ∕\kappa$. Lower plot: adimensionalized comoving atom flux as a function of $\omega ∕\kappa$. In both plots, $D=0.7$, $q=0.3$, and τ =0.3, and $G_{\omega }^{\text{in}}$ and ${\mathcal{F}}_{\omega }^{\mathrm{com}}$ are evaluated in the right asymptotic region, in the coincidence point limit. “HR related” refers to the contribution weighted by $n_{\omega }^{\mathit{fin}}$ and “initial excitations” to the one weighted by $n_{\omega }^{v,\mathit{\text{in}}}$.

Figure 21

Occupation number of the $v$ quanta for a BH (${\overline{n}}_{\omega }^v$, solid line) and of the $u$ quanta for the corresponding WH under the transformation (D1) (${\overline{n}}_{\omega }^{\mathrm{WH},\mathrm{u}}$, dashed line), as functions of $\omega ∕\kappa$ for $q=0.3$ and $D=0.4$ (upper plot) and $D=0.1$ (lower plot).

Figure 22

Relative difference $|{\left|{\beta }_{\omega }\right|}^2-{\left|{\beta }_{-\omega }\right|}^2|∕{\left|{\beta }_{\omega }\right|}^2$ as a function of ω. Same parameters as in Fig. 21.