Quantum Signature of Analog Hawking Radiation in Momentum Space

We consider a sonic analog of a black hole realized in the one-dimensional flow of a Bose-Einstein condensate. Our theoretical analysis demonstrates that one- and two-body momentum distributions accessible by present-day experimental techniques provide clear direct evidence (i) of the occurrence of a sonic horizon, (ii) of the associated acoustic Hawking radiation, and (iii) of the quantum nature of the Hawking process. The signature of the quantum behavior persists even at temperatures larger than the chemical potential.

Figure 1

Schematic representation of an acoustic black hole in a BEC. The density profile $n(x)$ is the black solid line. $V_u$ and $c_u$ ($V_d$ and $c_d$) are the asymptotic upstream (downstream) flow and sound velocities. The upstream (downstream) asymptotic flow is subsonic (supersonic). The interior of the analog black hole is shaded in this figure and in Fig. 2.

Figure 2

Two upper plots: upstream and downstream dispersion relations $\omega (k)$ [from Eq. (1)]. In each plot the horizontal dashed line is fixed by the chosen value of $\omega$. The labeling of the modes is explained in the text; their direction of propagation is represented by an arrow. $u$ and $d1$ modes ($d2$ modes) have a positive (negative) norm. The $d2$ modes disappear for $\omega >\mathrm{\Omega }$. The lower plot schematically represents the black-hole configuration in real space and the different modes bearing propagation of elementary excitations.

Figure 3

Momentum distribution $⟨\widehat{n}(p)⟩$ within the condensate. The figure is drawn in the $T=0$ limit. The top panel is a schematic representation of a typical experimental result. The lower panel displays the analytic result in the waterfall configuration with $V_u/c_u=0.5$, $V_d/c_d=4=V_d/V_u$. The thick vertical dashed lines correspond to the upstream and downstream condensates located at $p=P_u$ and $p=P_d$ (where here $P_u{\xi }_u=0.5$ and $P_d{\xi }_u=2$).

Figure 4

Momentum-correlation pattern of $g_2(p,q)$ in the presence of a black-hole horizon at $T=0$. This plot is drawn for the same configuration and the same parameters as the lower panel of Fig. 3. The dotted lines are the momenta of the upstream and downstream condensates ($P_u$ and $P_d$). The momenta are expressed in units of ${\xi }_u^{-1}$. Except for the colored correlation lines, $g_2(p,q)$ is uniformly equal to 1. The colors are used for a nonambiguous identification of the correlation lines. As is obvious from the definition (2), the figure is symmetric with respect to the diagonal.

Figure 5

The value of $g_2(p,q)$ along the $u_{\mathrm{out}}-d{2}_{\mathrm{out}}$ correlation line (violet line in Fig. 4) is plotted as a function of $p$ (expressed in units of ${\xi }_u^{-1}$) for different temperatures. The Cauchy-Schwarz inequality is violated when $g_2(p,q)$ is larger than 2. The temperatures are expressed in units of the upstream chemical potential $m{c}_u^2$ ($0\le T\le 2$). For the set of parameters chosen here and in the previous figures, the Hawking temperature is $T_H=0.134$.