  In this paper, we explore the emergence of the Wigner entropy production rate
in the stationary state of a two-mode Gaussian system. The interacting modes
dissipate into different local thermal baths. Also, one of the bosonic modes
evolves into the squeezed-thermal state because of the parametric amplification
process. Using the Heisenberg-Langevin approach, combined with quantum phase
space formulation, we get an analytical expression for the steady-state Wigner
entropy production rate. It contains two key terms. The first one is an
Onsager-like expression that describes heat flow within the system. The second
term resulted from vacuum fluctuations of the baths. Analyses show that
self-correlation between the quadratures of the parametrically amplified mode
pushes the mode towards the thermal squeezed state. It increases vacuum entropy
production of the total system and reduces the heat current between the modes.
The results imply that, unlike other previous proposals, squeezing can
constrain the efficiency of actual non-equilibrium heat engines by irreversible
flows.
