  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 $sl_2$ 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.
