Quantum fluctuations around black hole horizons in Bose-Einstein condensates

We study several realistic configurations making it possible to realize an acoustic horizon in the flow of a one-dimensional Bose-Einstein condensate. In each case we give an analytical description of the flow pattern, the spectrum of Hawking radiation, and the associated quantum fluctuations. Our calculations confirm that the nonlocal correlations of the density fluctuations previously studied in a simplified model provide a clear signature of Hawking radiation also in realistic configurations. In addition, we explain by direct computation how this nonlocal signal relates to short-range modifications of the density correlations.

Figure 1

Density profile in the $\delta$ peak configuration. The flow is directed toward positive $x$. The $\delta$ potential is represented by a (red) vertical straight line. The density in the upstream region ($x<0$) is a portion of a dark soliton (see the text). The region $x>0$ is supersonic. It is shaded in the plot for recalling that it corresponds to the interior of the equivalent black hole. We keep this convention in Figs. 2, 3, and 4.

Figure 2

Same as Fig. 1 for the waterfall configuration.

Figure 3

Dispersion relation (22). In each plot the horizontal dashed line is fixed by the chosen value of $\omega$. The $q_{\ell }\left(\omega \right)$'s are the corresponding abscissas. Only the real eigenmodes are represented. Their denomination is explained in the text; their direction of propagation (left or right) is represented by an arrow. The part of the dispersion relation corresponding to negative norm states (see the text) is represented with a dashed line. The upper plot corresponds to a subsonic flow. The lower one corresponds to a supersonic flow; it is shaded in order to recall that it describes the situation inside the black hole.

Figure 4

The scattering modes. The color code is the same as in Fig. 3. The additional purple wiggly lines correspond to evanescent channels. For $\omega >\Omega$, the $D2$ mode disappears and the channel $d2|\mathrm{out}$ is replaced by $d|\mathrm{eva}$ in the two upper panels ($U$ and $D1$ modes). The region corresponding to the interior of the black hole is shaded as in the previous figures.

Figure 5

Normalized Hawking temperature $k_{\mathrm{B}}{T}_{\mathrm{H}}/\left(m{c}_u^2\right)$ as a function of the upstream Mach number $m_u=V_u/c_u$ for the $\delta$ peak and waterfall configurations. The inset displays the grayness factor $\Gamma$ as a function of $m_u$ for these two configurations.

Figure 6

Radiation spectrum in the $\delta$ peak configuration. Red curve, $|S_{u,d2}{|}^2$ as a function of the dimensionless quantity $\omega {\xi }_u/c_u$; black curve, $\Gamma \times n_{T_{\mathrm{H}}}\left(\omega \right)$. The plot is drawn for $m_u=0.5$; in this case $k_{\mathrm{B}}{T}_{\mathrm{H}}/\left(m{c}_u^2\right)\simeq 0.128$ and $\Gamma \simeq 0.977$. The difference between $|S_{u,d2}{|}^2$ and $\Gamma \times n_{T_{\mathrm{H}}}$ is maximum when $\omega =\Omega$ and is close to 0.06 in this case (for the chosen value of $m_u$ one has $\Omega {\xi }_u/c_u\simeq 0.369$).

Figure 7

${\xi }_u{n}_u{G}_0^{\left(2\right)}(x,x^{\prime })$ computed analytically from Eq. (73) (black solid line) compared with the numerics from the truncated Wigner method (red dashed line). The small discrepancy near $x-x^{\prime }=0$ is due to numerical uncertainty and to the plotting procedure which introduces a small amount of smoothing of the raw numerical data. The inset represents a color plot of the numerical results. The white dashed line is the line $x^{\prime }=50{\xi }_u-x$ along which we compare numerical and analytical results in the main plot.

Figure 8

2D plot of the numerical result for the quantity ${\xi }_u{n}_u{G}_0^{\left(2\right)}(x,x^{\prime })$ in the case of a $\delta$ peak configuration with $m_u=0.5$. The shaded area near the axis corresponds to the zone $\left|x\right|$ or $|x^{\prime }|<10{\xi }_u$. $G_0^{\left(2\right)}$ is only displayed for $\left|x\right|$ and $|x^{\prime }|>10{\xi }_u$, that is, in the asymptotic region where expressions (77, 78, 79, 80) are valid. The dashed straight lines correspond to the correlation lines where a heuristic interpretation of the Hawking signal leads to expect the largest signal (see the text).

Figure 9

Same as Fig. 8 for a waterfall configuration with $m_u=0.5$.