Gaussian quantum Markov semigroups on finitely many modes admitting a normal invariant state

TL;DR

This paper characterizes Gaussian quantum Markov semigroups with normal invariant states, analyzing their long-term behavior, decoherence properties, and the structure of invariant states in quantum systems.

Contribution

It provides a complete characterization of GQMSs with normal invariant states and describes their long-term dynamics and invariant state structure.

Findings

Characterized GQMSs admitting a normal invariant state.

Described the set of all such invariant states.

Demonstrated environment-induced decoherence and convergence to Hamiltonian evolution.

Abstract

Gaussian quantum Markov semigroups (GQMSs) are of fundamental importance in modelling the evolution of several quantum systems. Moreover, they represent the noncommutative generalization of classical Orsntein-Uhlenbeck semigroups; analogously to the classical case, GQMSs are uniquely determined by a "drift" matrix $\mathbf{Z}$ and a "diffusion" matrix $\mathbf{C}$, together with a displacement vector $\mathbf{\zeta}$. In this work, we completely characterize those GQMSs that admit a normal invariant state and we provide a description of the set of normal invariant states; as a side result, we are able to characterize quadratic Hamiltonians admitting a ground state. Moreover, we study the behavior of such semigroups for long times: firstly, we clarify the relationship between the decoherence-free subalgebra and the spectrum of $\mathbf{Z}$. Then, we prove that environment-induced decoherence takes place and that the dynamics approaches an Hamiltonian closed evolution for long times; we are also able to determine the speed at which this happens. Finally, we study convergence of ergodic means and recurrence and transience of the semigroup.