Semiclassical Evolution of Dissipative Markovian Systems

TL;DR

This paper develops a semiclassical approximation for the evolution of density operators in dissipative Markovian systems using the chord function, extending classical phase space methods to include open quantum system effects.

Contribution

It introduces a semiclassical framework based on the chord function for dissipative systems, generalizing previous closed-system theories to include Lindblad operators.

Findings

Exact solutions for quadratic Hamiltonians with linear Lindblad operators.

Classical double Hamiltonian dynamics describe the evolution in double phase space.

Decoherence effects localize the Wigner function near caustics, simplifying analysis.

Abstract

A semiclassical approximation for an evolving density operator, driven by a "closed" hamiltonian operator and "open" markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra "open" term is added to the double Hamiltonian by the non-hermitian part of the Lindblad operators in the general case of dissipative markovian evolution. The particular case of generic hamiltonian operators, but linear dissipative Lindblad operators, is studied in more detail. A Liouville-type equivariance still holds for the corresponding classical evolution in double phase, but the centre subspace, which supports the Wigner function, is compressed, along with expansion of its conjugate subspace, which supports the chord function. Decoherence narrows the relevant region of double phase space to the neighborhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by a propagator in a mixed representation, so that a further "small-chord" approximation leads to a simple generalization of the quadratic theory for evolving Wigner functions.