Phase space approach to theories of quantum dissipation

Abstract

Six major theories of quantum dissipative dynamics are compared: Redfield theory, the Gaussian phase space ansatz of Yan and Mukamel, the master equations of Agarwal, Caldeira-Leggett/Oppenheim-Romero-Rochin, and Louisell/Lax, and the semigroup theory of Lindblad. The time evolving density operator from each theory is transformed into a Wigner phase space distribution, and classical-quantum correspondence is investigated via comparison with the phase space distribution of the classical Fokker-Planck (FP) equation. Although the comparison is for the specific case of Markovian dynamics of the damped harmonic oscillator with no pure dephasing, certain inferences can be drawn about general systems. The following are our major conclusions: (1) The harmonic oscillator master equation derived from Redfield theory, in the limit of a classical bath, is identical to the Agarwal master equation. (2) Following Agarwal, the Agarwal master equation can be transformed to phase space, and differs from the classical FP equation only by a zero point energy in the diffusion coefficient. This analytic solution supports Gaussian solutions with the following properties: the differential equations for the first moments in p and q and all but one of the second moments (q2 and pq but not p2) are identical to the classical equations. Moreover, the distribution evolves to the thermal state of the bare quantum system at long times. (3) The Gaussian phase space ansatz of Yan and Mukamel (YM), applied to single surface oscillator dynamics, reduces to the analytical Gaussian solutions of the Agarwal phase space master equation. It follows that the YM ansatz is also a solution to the Redfield master equation. (4) The Agarwal/Redfield master equation has a structure identical to that of the master equation of Caldeira-Leggett/Oppenheim-Romero-Rochin, but the two are equivalent only in the high temperature limit. (5) The Louisell/Lax HO master equation differs from the Agarwal/Redfield form by making a rotating wave approximation (RWA), i.e., keeping terms of the form ââ†,â†â and neglecting terms of the form â†â†,ââ. When transformed into phase space, the neglect of these terms eliminates the modulation in time of the energy dissipation, modulation which is present in the classical solution. This neglect leads to a position-dependent frictional force which violates the principle of translational invariance. (6) The Agarwal/Redfield (AR) equations of motion are shown to violate the semigroup form of Lindblad required for complete positivity. Considering the triad of properties: complete positivity, translational invariance and asymptotic approach to thermal equilibrium, AR sacrifices the first while Lindblad’s form must sacrifice either the second or the third. This implies that for certain initial states Redfield theory can violate simple positivity; however, for a wide range of initial Gaussians, the solution of the AR equations does maintain simple positivity, and thus for these states appears to be distinctly more physical than the solution of the semigroup equations.