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
$V=\tau I/2$
,
$\tau > 1$
, as their steady covariance matrsices. This is physically important since the covariance matrix
$\tau I/2$
,
$\tau > 1$
, 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
$V=S\operatorname{diag}(\Lambda,\Lambda ) S^{\top }/2$
with
$\Lambda > I$
being a diagonal matrix and
$S$
being a symplectic matrix will also be considered. Physically, a covariance matrix of the form
$V=S\operatorname{diag}(\Lambda,\Lambda ) S^{\top }/2$
,
$\Lambda > I$
, 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.