Synthesis of linear quantum systems to generate a steady thermal state

Abstract

The purpose of this article is to synthesize a linear quantum system, which is strictly stable and has a steady thermal state. Specifically, we give a parameterization of a class of stable linear quantum systems that have V=τI/2 , τ>1 , as their steady covariance matrsices. This is physically important since the covariance matrix τI/2 , τ>1 , corresponds to a quantum thermal state. Hence, we can say that these systems will asymptotically evolve into a quantum thermal state. An extension to the case where V=Sdiag(Λ,Λ)S⊤/2 with Λ>I being a diagonal matrix and S being a symplectic matrix will also be considered. Physically, a covariance matrix of the form V=Sdiag(Λ,Λ)S⊤/2 , Λ>I , corresponds to a mixed Gaussian quantum state. So, we can alternatively say that the corresponding linear quantum systems will asymptotically evolve into a mixed Gaussian quantum state.