Exact Markovian evolution of quantum systems with several degrees of freedom : Phase space representations

TL;DR

This paper derives an exact solution for the Markovian evolution of multi-degree quantum systems using phase space representations, generalizing previous work to multiple degrees of freedom and providing explicit formulas for reduced states and moments.

Contribution

It extends the chord representation solution of the Lindblad equation to systems with several degrees of freedom, including explicit formulas for reduced density operators and moments.

Findings

Explicit expression for the reduced density operator in multi-degree systems

Positivity of the Wigner and P functions is maintained during evolution

Examples include Markovian evolution of a triatomic molecule and harmonic oscillator arrays

Abstract

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, also known as the characteristic function. It is here generalized for several degrees of freedom, so as to provide an explicit expression for the reduced density operator of any subsystem, 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 context of several degrees of freedom, a full dissipation matrix is defined, whose trace is equal to twice the previously derived dissipation coefficient. This governs the rate at which the phase space volume of the argument of the Wigner function contracts, while that of the chord function expands. Examples of Markovian evolution of a triatomic molecule and of an array of harmonic oscillators are discussed.