Differential formalism and the thermodynamic description of multimode Gaussian equilibrium states

A differential formalism for the covariance matrix of thermal equilibrium states is obtained when the Hamiltonian of the system is defined by quadratic forms. First, the cases of one- and two-modal Gaussian states are taken into account and the differential equations for the corresponding covariance matrices of these states in terms of the temperature are obtained. Later, the generalization of the differential equation to any number of modes is demonstrated by using the Wigner quasi-probability distribution. Some examples related to standard quadratic Hamiltonians are given and several thermodynamic properties as the free and internal energies, the entropy, and the heat capacity of different systems are listed. •A differential formalism for the covariance matrix of Gaussian thermal equilibrium states is obtained.•The formalism is generalized for any number of modes and quadratic Hamiltonians.•A general solution is proposed and its applicability is mentioned.•The frequency converter and parametric amplifier thermal states solutions are obtained explicitly.•Different thermodynamic quantities are calculated and discussed.