Exact Markovian evolution of multicomponent quantum systems: phase space representations
Résumé
The exact solution of the Lindblad equation with a quadratic Hamiltonian and linear coupling operators was derived within the chord representation, that is, for the Fourier transform of the Wigner function. It is here generalized for multiple components, so as to provide an explicit expression for the reduced density operator of any component, as well as moments expressed as derivatives of this evolving chord function. The Wigner function is then the convolution of its straightforward classical evolution with a widening multidimensional gaussian window, eventually ensuring its positivity. Futher on, positivity also holds for the Glauber-Sundarshan P-function, which guarantees separability of the components. In the multicomponent context, a full dissipation matrix is defined, whereas its trace, equal to twice the previously derived dissipation coefficient, governs the rate at which the phase space volume of the argument of the Wigner function contracts, while those of the chord function expands. Examples of markovian evolution of a triatomic molecule and of an array of harmonic oscillators are discussed.