Black--hole lasing action in laboratory Bose--Einstein condensates

A recent experiment \footnote{J. Steinhauer, {\em Nature Physics} {\bf 11}, 864 (2014)} infers the the production of Hawking radiation in an analogue black-hole laser, which consists of a Bose-Einstein condensate of about 5,000 $^{87}$Rb atoms in a trap with a translating potential step. In the co-moving reference frame the flow velocity of the condensate exceeds the sound speed in a region confined between two sonic points, the analogue black and white hole horizons. We report simulations of that experiment based on the zero-temperature Gross-Pitaevskii (GP) equation that are consistent with the reported experimental results. The simulations show exponential growth of oscillatory modes trapped between the horizons, with a power spectrum consistent with expectations from the Bogoliubov dispersion relation, which saturates after an initial period. Quantum Hawking radiation occurs spontaneously in the vacuum, but in the presence of a coherent state of phonons it takes on a classical form captured by the zero-temperature GP equation. The growth of the trapped modes results from repeated super-radiant scattering from the black hole horizon, associated with emission of Hawking radiation to the exterior that is not well-resolved in the simulation.