Violation of Cauchy-Schwarz inequalities by spontaneous Hawking radiation in resonant boson structures

The violation of a classical Cauchy-Schwarz (CS) inequality is identified as an unequivocal signature of spontaneous Hawking radiation in sonic black holes. This violation can be particularly large near the peaks in the radiation spectrum emitted from a resonant boson structure forming a sonic horizon. As a function of the frequency-dependent Hawking radiation intensity, we analyze the degree of CS violation and the maximum violation temperature for a double barrier structure separating two regions of subsonic and supersonic condensate flow. We also consider the case where the resonant sonic horizon is produced by a space-dependent contact interaction. In some cases, CS violation can be observed by direct atom counting in a time-of-flight experiment. We show that near the conventional zero-frequency peak, the decisive CS violation cannot occur.

Figure 1

Quasiparticle dispersion relation on the subsonic (left, upstream) and supersonic (right, downstream) sides. The blue (red) branches correspond to positive (negative) normalization. As in Refs. [12, 13, 17], $d$ ($u$) denotes downstream (upstream). Here ${\xi }_{u\left(d\right)}$ denotes the asymptotic healing length, $c_{u\left(d\right)}$ and $v_{u\left(d\right)}$ the sound and flow velocities, and ${\omega }_{\max }$ the frequency above which no HR can be generated.

Figure 2

Scheme of the double-barrier configuration (left) and the configuration with two discontinuities in the otherwise flat density configuration (right).

Figure 3

Hawking radiation for a condensate leaking through a double barrier structure like that studied in Ref. [12] and shown in the left of Fig. 2. The strength of both delta barriers is $Z=2.2{\hslash }^2/m{\xi }_u$. They are separated by a distance $d=3.62{\xi }_u$. The flow is such that $q_u{\xi }_u=0.01$ and ${\omega }_{\max }=0.99\mu /\hslash$. Solid blue: Zero-temperature HR spectrum $|S_{ud2}{\left(\omega \right)|}^2$. Dashed red: ${\theta }_{ud2}\left(\omega \right)$ at temperature $T=\mu /k_B$, where $\mu$ is the comoving chemical potential on the subsonic side. Dotted green: Maximum violation temperature $T_v\left(\omega \right)$ in units of $\mu /k_B$. Not shown in the figure, $T_v\left(\omega \right)$ rises up to $T_v\left({\omega }_0\right)\simeq 21\mu /k_B$, where $|S_{ud2}\left({\omega }_0\right){|}^2\simeq 8$. Left inset: Zoom of the peak region, with $T_v\left(\omega \right)$ removed and $z_{ud2}\left(\omega \right)$ (relative atom CS violation) added (dotted purple). Right inset: Same as left inset; it shows ${\Theta }_{ud2}\left(\omega \right)$ (dashed-dotted brown) and $Z_{ud2}\left(\omega \right)$ (dashed purple), which measure the absolute amount of CS violation in the phonon and atom signals.

Figure 4

Same as Fig. 3 for a setup with a single delta barrier of strength $Z=0.62{\hslash }^2/m{\xi }_u$ and a uniform interaction, with flow $q_u{\xi }_u=0.3$ and ${\omega }_{\max }=0.6\mu /\hslash$. Atomic CS violation is not found for this setup.

Figure 5

Same as Fig. 3 but for $T=0.6\mu /k_B$ and for a structure without barriers but with two sharp variations in the local speed of sound, which takes the successive values $c_u,0.43c_u,0.6c_u$, as depicted in the right of Fig. 2. The intermediate region has a length $L=26{\xi }_u$. The flow is such that $q_u{\xi }_u=0.95$ and ${\omega }_{\max }=0.18\mu /\hslash$. This is a resonant generalization of the model first studied in Ref. [6], where only one sound-speed discontinuity was considered.

Figure 6

Scheme that shows graphically the values of ${\omega }_{d1d2}\left(\omega \right)$ and $-{\omega }_{uu}\left(\omega \right)$. See Fig. 1 for mode notation.