Dissipation tends to wash out dynamical features observed at early evolution times. In this paper we analyze a resonant single--atom two--photon quantum optical amplifier both dynamically and thermodynamically. A detailed thermodynamic balance shows that the non--linear amplifier is thermodynamically equivalent to the linear amplifier discussed in (Phys. Rev. A, 74 (2006), 063822). However, by calculating the Wigner quasi--probability distribution for various initial field states, we show that unique quantum features in optical phase space, absent from the linear amplifier, are maintained for extended times. These features are related to the discrete nature of the two--photon matter--field interaction, and fingerprint the initial field state at thermodynamic times.
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