Solutions of generic bilinear master equations for a quantum oscillator -- positive and factorized conditions on stationary states

Abstract

We obtain the solutions of the generic bilinear master equation for a quantum oscillator with constant coefficients in the Gaussian form. The well-behavedness and positive semidefiniteness of the stationary states could be characterized by a three-dimensional Minkowski vector. By requiring the stationary states to satisfy a factorized condition, we obtain a generic class of master equations that includes the well-known ones and their generalizations, some of which are completely positive. A further subset of the master equations with the Gibbs states as stationary states is also obtained. For master equations with not completely positive generators, an analysis on the stationary states suggests conditions on the coefficients of the master equations that generate positive evolution for a given initial state.

arXiv commentAccepted by Physica A

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• Eigenvalues of the Liouvillians of Quantum Master Equation for a Harmonic Oscillator
• Damping modes of harmonic oscillator in open quantum systems
• Liouvillian exceptional points in continuous variable system
• Lindblad dynamics of open multi-mode bosonic systems: Algebra of bilinear superoperators, exceptional points and speed of evolution
• Evolution of Gaussian mixed states under the Markovian master equation for a driven quantum oscillator