Nonlinear semiclassical dynamics of open systems

• Published in: Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences Vol. 369; no. 1935; pp. 260 - 277
• Main Authors: Ozorio de Almeida, A. M., Brodier, O.
• Format: Journal Article
• Language: English
• Published: England The Royal Society Publishing 28.01.2011; The Royal Society
• Subjects: Approximation; Caustic networks; Decoherence; Evolutionary theories; Fourier transformations; Lagrangian function; Mathematical functions; Open systems; Phase-Space Representations; Quantum decoherence; Quantum Markovian Systems; Quantum mechanics; Semiclassical Mechanics; Trajectories
• ISSN: 1364-503X; 1471-2962
• DOI: 10.1098/rsta.2010.0261

A semiclassical approximation for an evolving density operator, driven by a 'closed' Hamiltonian and 'open' Markovian Lindblad operators, is reviewed. 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 is a quadratic function and the Lindblad operators are linear functions of positions and momenta. The semiclassical formulae are interpreted within a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra 'open' term in the double Hamiltonian is generated by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by the definition of a propagator, here developed in both representations. Generalized asymptotic equilibrium solutions are thus presented for the first time.