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Complex WKB Evolution of Markovian Open Systems

We derive a semiclassical approximation for the evolution generated by the Lindblad equation as a generalization of complex WKB theory. Linear coupling to the environment is assumed, but the Hamiltonian can be a general function of positions and momenta. The theory is carried out in the chord representation and describes the evolved quantum characteristic function, which gives direct access to the Wigner function and the position representation of the density operator by Fourier transforms. The propagation is shown to be of Liouville type in a complex double phase space, the imaginary part of the action being responsible for decoherence. The theory is exact in the quadratic case, just as the real WKB theory that we previously developed for the Markovian evolution of extended states, but it also describes the decoherent and dissipative evolution of localized states, such as the interference terms of a Schrödinger cat state. The present rederivation of the real WKB approximation leads to its interpretation as a first order classical perturbation of the complex theory and to a discussion of its validity. The example of a simple cubic Hamiltonian illustrates the various levels of approximation derived from the complex WKB theory.