Quantum transport in the black-hole configuration of an atom condensate outcoupled through an optical lattice

The outcoupling of a Bose-Einstein condensate through an optical lattice provides an interesting scenario to study quantum transport phenomena or the analog Hawking effect as the system can reach a quasi-stationary black-hole configuration. We devote this work to characterize the quantum transport properties of quasi-particles on top of this black-hole configuration by computing the corresponding scattering matrix. We find that most of the features can be understood in terms of the usual Schr\"odinger scattering. In particular, a transmission band appears in the spectrum, with the normal-normal transmission dominating over the anomalous-normal one. We show that this picture still holds in a realistic experimental situation where the actual Gaussian envelope of the optical lattice is considered. A peaked resonant structure is displayed near the upper end of the transmission band, which suggests that the proposed setup is a good candidate to provide a clear signal of spontaneous Hawking radiation.