Complex WKB evolution of Markovian open systems

We derive a consistent complex WKB expression for the characteristic function, that is, the Fourier transform of the Wigner function, of a Markovian open system, defined by a generic Lindblad equation. The Hamiltonian can be a general function of positions and momenta, so the only restriction is that linear coupling to the environment is assumed. The propagation is shown to correspond to a classical evolution of the Liouville type in a complex double phase space, the imaginary part of the double Hamiltonian being responsible for decoherence, whereas dissipation is included in the real part. The theory, exact in the quadratic case, is designed to describe the decoherent and dissipative evolution of localized states, such as the interference terms of a Schrodinger cat state, as well as the evolution of extended states, which were the exclusive subjects of our previous real WKB approximation. The present rederivation allows us to interpret the real formalism as a first-order classical perturbation of the complex theory and to discuss its validity. This is also clarified by a simple cubic Hamiltonian that illustrates various levels of approximation derived from the complex WKB theory.