Synthesis of Linear Quantum Systems to Generate a Steady Thermal State

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=τ I/2, τ > 1, as their steady covariance matrsices. This is physically important since the covariance matrix τ I/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=S diag(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=S diag(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.