Classical-quantum correspondence in the Redfield equation and its solutions

In a recent paper we showed the equivalence, under certain well-characterized assumptions, of Redfield's equations for the density operator in the energy representation with the Gaussian phase space ansatz for the Wigner function of Yan and Mukamel. The equivalence shows that the solutions of Redfield's equations respect a striking degree of classical-quantum correspondence, Here we use this equivalence to derive analytic expressions for the density matrix of the harmonic oscillator in the energy representation without making the almost ubiquitous secular approximation. From the elements of the density matrix in the energy representation we derive analytic expressions for Gamma(1)(n)(1/T-1(n)) and Gamma(2)(nm)(1/T-2(nm)), i.e., population and phase relaxation rates for individual matrix elements in the energy representation. Our results show that Gamma(1)(n)(t) = Gamma(1)(t) is independent of n; this is contrary to the widely held belief that Gamma(1)(n) is proportional to n. We also derive the simple result that Gamma(2)(nm)(t) = \n-m\Gamma(1)(t)/2, a generalization of the two-level system result Gamma(2) = Gamma(1)/2. We show that Gamma(1)(t) is the classical rate of energy relaxation, which has periodic modulations characteristic of the classical damped oscillator; averaged over a period Gamma(t) is directly proportional to the classical friction, gamma. An additional element of classical-quantum correspondence concerns the time rate of change of the phase of the off diagonal elements of the density matrix, omega(nm), a quantity which has received little attention previously. We find that omega(nm) is time-dependent, and equal to \n - m\Omega(t), where Omega(t) is the rate of change of phase space angle in the classical damped harmonic oscillator. Finally, expressions for a collective Gamma(1)(t) and Gamma(2)(t) are derived, and shown to satisfy the relationship Gamma(2) = Gamma(1)/2. This familiar result, when applied to these collective rate constants, is seen to have a simple geometrical interpretation in phase space. (C) 1997 American Institute of Physics.