Analogous Hawking radiation from gapped excitations in a transonic flow of binary Bose-Einstein condensates

We have studied analytically the approximate solutions to the gapped mode equations in the hydrodynamic regime for a class of binary Bose-Einstein condensate acoustic black holes. The horizon from the transonic flow is formed by manipulating the phonon sound speed and the flow velocity with the experimentally accessible parameters. The asymptotic modes of various scattering processes are constructed from which to obtain scattering coefficients and then to further decompose the field operator in terms of the asymptotic states. Also, the Unruh state is introduced to be the appropriate state for the description of gravitational collapse of the black hole. The particle densities of the outgoing modes are computed. The effective energy gap term in the dispersion relation of the gapped excitations introduces the threshold frequency $\omega_r$ in the subsonic regime, below which the propagating modes do not exist. Thus, the particle spectrum of the analogous Hawking modes in the exterior of the horizon of the subsonic region significantly deviates from that of the gapless cases near the threshold frequency due to the modified graybody factor, which vanishes as the mode frequency is below $\omega_r$. However, in the interior region of the horizon of the supersonic region, the spectrum of the particle production of the Hawking partner has the nonthermal feature. The correlators between the analog Hawking mode and its partner of relevance to the experimental observations are also investigated and show some peaks near the threshold frequency $\omega_r$ resulting from the gap energy term to be seen in future experiments.