  We consider the model of a quantum harmonic oscillator governed by a Lindblad
master equation where the typical drive and loss channels are multi-photon
processes instead of single-photon ones; this implies a dissipation operator of
order 2k with integer k>1 for a k-photon process. We prove that the
corresponding PDE makes the state converge, for large time, to an invariant
subspace spanned by a set of k selected basis vectors; the latter physically
correspond to so-called coherent states with the same amplitude and uniformly
distributed phases. We also show that this convergence features a finite set of
bounded invariant functionals of the state (physical observables), such that
the final state in the invariant subspace can be directly predicted from the
initial state. The proof includes the full arguments towards the well-posedness
of the corresponding dynamics in proper Banach spaces of Hermitian trace-class
operators equipped with adapted nuclear norms. It relies on the Hille-Yosida
theorem and Lyapunov convergence analysis.
