Damped oscillations of the energy of a bosonic bath due to spectral gaps and special initial correlations

The energy of the bosonic bath and the flow of quantum information are analyzed over short and long times in local dephasing channels for special correlated or factorized initial conditions, respectively, which involve thermal states. The continuous distribution of frequency modes of the bosonic bath exhibits a spectral gap over low frequencies. The bath energy shows oscillatory behaviors around the asymptotic value and information is alternatively lost and gained by the open system. Due to the low-frequency gap, the damped oscillations become regular over long times and the frequency of the oscillations coincides with the upper cut-off frequency of the spectral gap. Sequences of long-time intervals are obtained over which the bath energy increases (decreases), for the correlated initial conditions, and information is lost (gained) by the open system, for the factorized initial configurations, even at different temperatures. Such long-time correspondence between the variations of the bath energy and of the information is reversed if compared to the one obtained without the low-frequency gap. The correspondence fails if the spectral density is tailored according to power laws with odd natural powers near the upper cut-off frequency of the spectral gap.