  We studied the process of decoherence induced by the presence of an
environment in acoustic black holes, using the open quantum system approach,
thus extending previous work. We focused on the ion trap model but the
formalism is general to any experimental implementation. We computed the
decoherence time for that setup. We found that a quantum to classical
transition occurs during the measurement and we proposed improved parameters to
avoid such a feature. We provide analytic estimations for both zero and finite
temperature. We also studied the entanglement between the Hawking-pair phonons
for an acoustic black hole while in contact with a reservoir, through the
quantum correlations, showing that they remain strongly correlated for small
enough times and temperatures. We used the stochastic formalism and the method
of characteristic to solve the field wave equation.
