Quadratic open quantum harmonic oscillator

We study the quantum open system evolution described by a Gorini–Kossakowski–Sudarshan–Lindblad generator with creation and annihilation operators arising in Fock representations of the sl2 Lie algebra. We show that any initial density matrix evolves to a fully supported density matrix and converges towards a unique equilibrium state. We show that the convergence is exponentially fast and we exactly compute the rate for a wide range of parameters. We also discuss the connection with the two-photon absorption and emission process.