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In this paper, we explore the emergence of the Wigner entropy production ratein the stationary state of a two-mode Gaussian system. The interacting modesdissipate into different local thermal baths. Also, one of the bosonic modesevolves into the squeezed-thermal state because of the parametric amplificationprocess. Using the Heisenberg-Langevin approach, combined with quantum phasespace formulation, we get an analytical expression for the steady-state Wignerentropy production rate. It contains two key terms. The first one is anOnsager-like expression that describes heat flow within the system. The secondterm resulted from vacuum fluctuations of the baths. Analyses show thatself-correlation between the quadratures of the parametrically amplified modepushes the mode towards the thermal squeezed state. It increases vacuum entropyproduction of the total system and reduces the heat current between the modes.The results imply that, unlike other previous proposals, squeezing canconstrain the efficiency of actual non-equilibrium heat engines by irreversibleflows.

Irreversible Entropy Production rate in a parametrically driven-dissipative System: The Role of Self-Correlation between Noncommuting Observables