Role of dissipation in the stability of a parametrically driven quantum harmonic oscillator

We study the dissipative dynamics of a single quantum harmonic oscillator subjected to parametric driving with in an effective Hamiltonian approach. Using Liouville von Neumann approach, we show that time evolution parametrically driven has strong connection classical damped Mathieu equation. Based on numerical analysis Monodromy matrix, demonstrate dynamical instability generated by are reduced effect dissipation. Further, obtain closed relationship between localization Wigner function and stability