Bogoliubov theory of acoustic Hawking radiation in Bose-Einstein condensates

We apply the microscopic Bogoliubov theory of dilute Bose-Einstein condensates to analyze quantum and thermal fluctuations in a flowing atomic condensate in the presence of a sonic horizon. For the simplest case of a step-like horizon, closed-form analytical expressions are found for the spectral distribution of the analog Hawking radiation and for the density correlation function. The peculiar long-distance density correlations that appear as a consequence of the Hawking emission features turns out to be reinforced by a finite initial temperature of the condensate. The analytical results are in good quantitative agreement with first principle numerical calculations.

Figure 1

(Color online) Sketch of the one dimensional dumb-hole configuration considered in this paper. A BEC is flowing in the positive $x$ direction. The horizon is located at $x=0$. The flow is uniform in the asymptotic regions far from the horizon, with a velocity, which is subsonic in the upstream region and supersonic in the downstream one. The wiggly arrows illustrate one among the different scattering processes described by Eq. (10): an incident $u$-in wavepacket is partially reflected onto the $u$-out branch and partially transmitted on the $d1$-out and $d2$-out branches. The chosen labeling of the modes is explained in the text and illustrated in Fig. 2.

Figure 2

(Color online) The Bogoliubov mode dispersion relations (4) in the asymptotic upstream (left panel) and in the down-stream region (right panel). Solid (dashed) lines correspond to modes with a positive (negative) Bogoliubov norm.

Figure 3

Transmission and reflection coefficients for a $d1$ in-going mode on a step-like dumb-hole configuration. ${\left|{\mathbf{S}}_{ud1}\right|}^2$ (dotted curve) corresponds to the transmission on the $u$-out mode; ${\left|{\mathbf{S}}_{d1d1}\right|}^2$ (solid curve) and ${\left|{\mathbf{S}}_{d2d1}\right|}^2$ (dashed curve) correspond to the reflection on the $d1$-out and $d2$-out modes, respectively. The parameters $v_0/c_u=0.75$, $v_0/c_d=1.5$ used here are the same as in Ref. [17]. Above $\Omega$ ($\Omega /{\mu }_u=0.316$ for the chosen set of parameters) one recovers [insets (a.1) and (a.2)] the usual case of a single transmitted ($u$-out) and a single reflected ($d1$-out) waves, respectively. Inset (b) displays a comparison of the exact ${\left|{\mathbf{S}}_{d1d1}\right|}^2$ coefficient (solid line) with its low energy approximation (14) (dot-dashed line).

Figure 4

(Color online) A $d1$ ingoing wave packet is incident on a step-like sonic horizon located at $x=0$ (upper panel). When the central frequency omega of the wave packet is smaller that Omega, a single transmitted $u$ wavepacket and two $d1,d2$ reflected wave packets are visible (middle panel). A single $u$ transmitted and a single $d1$ reflected wave packet are instead visible for $\omega >\Omega$ (lower panel). Same system parameters as in Fig. 3. Wave-packet carrier wave vector $q{\xi }_u=-1.2$ (upper and middle panels), $q{\xi }_u=-1.35$ (lower panel).

Figure 5

(Color online) Plot of the rescaled density correlation $n_0{\xi }_u\left[G_0^{\left(2\right)}\left(x,x^{\prime }\right)-1\right]$ for an initial zero temperature. The data are the same as the one presented in [17], but the color scale has been modified in order to make the positive $u\text{−}d1$ correlation clearly visible. This signal was overlooked in the original paper given its relative weakness. The dashed lines identify the different feature of the correlation function as discussed in the text. The white dotted line corresponds to the cut $x^{\prime }-x=39.5{\xi }_u$ used in Figs. 7, 6.

Figure 6

Cut of the zero-temperature density correlation function $G_0^{\left(2\right)}$ along the $x^{\prime }-x=39.5{\xi }_u$ line. Solid line: numerical result from [17]. Dashed lines: prediction of Eqs. (27, 33) for, respectively, the $u-d2$ and the $d1-d2$ contributions. On this scale the $u-d1$ contribution is hardly visible.

Figure 7

(Color online) Cut of the density correlation function $G_0^{\left(2\right)}$ along the $x^{\prime }-x=39.5{\xi }_u$ line. Black solid lines are the prediction of Eqs. (27, 33). Red dashed lines are the hydrodynamical approximations (29, 36). The main panel shows the $u-d2$ feature in the in-out region. Inset (a) shows the $u-d1$ feature in the in-out region. Inset (b) shows the $d1-d2$ feature in the in-in region.