Well-posedness and convergence of the Lindblad master equation for a quantum harmonic oscillator with multi-photon drive and damping

TL;DR

This paper analyzes a quantum harmonic oscillator driven by multi-photon processes, proving convergence to a predictable invariant subspace and establishing well-posedness of the dynamics using advanced mathematical tools.

Contribution

It introduces a rigorous analysis of multi-photon driven Lindblad equations, demonstrating convergence to invariant states and proving well-posedness in appropriate functional spaces.

Findings

States converge to an invariant subspace of coherent states

Final states can be predicted from initial states using bounded invariants

The dynamics are well-posed in Banach spaces with nuclear norms

Abstract

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.