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We derive a semiclassical approximation for the evolution generated by theLindblad equation as a generalization of complex WKB theory. Linear coupling tothe environment is assumed, but the Hamiltonian can be a general function ofpositions and momenta. The theory is carried out in the chord representationand describes the evolved quantum characteristic function, which gives directaccess to the Wigner function and the position representation of the densityoperator by Fourier transforms. The propagation is shown to be of Liouvilletype in a complex double phase space, the imaginary part of the action beingresponsible for decoherence. The theory is exact in the quadratic case, just asthe real WKB theory that we previously developed for the Markovian evolution ofextended states, but it also describes the decoherent and dissipative evolutionof localized states, such as the interference terms of a Schr\"odinger catstate. The present rederivation of the real WKB approximation leads to itsinterpretation as a first order classical perturbation of the complex theoryand to a discussion of its validity. The example of a simple cubic Hamiltonianillustrates the various levels of approximation derived from the complex WKBtheory.

Complex WKB Evolution of Markovian Open Systems