CLASSICAL-QUANTUM CORRESPONDENCE IN THE REDFIELD EQUATION AND ITS SOLUTIONS

In a recent paper we showed the equivalence, under certain well-charac terized 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 classi cal-quantum correspondence, Here we use this equivalence to derive ana lytic expressions for the density matrix of the harmonic oscillator in the energy representation without making the almost ubiquitous secula r 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; t his is contrary to the widely held belief that Gamma(1)(n) is proporti onal 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 en ergy relaxation, which has periodic modulations characteristic of the classical damped oscillator; averaged over a period Gamma(t) is direct ly proportional to the classical friction, gamma. An additional elemen t of classical-quantum correspondence concerns the time rate of change of the phase of the off diagonal elements of the density matrix, omeg a(nm), a quantity which has received little attention previously. We f ind that omega(nm) is time-dependent, and equal to \n - m\Omega(t), wh ere Omega(t) is the rate of change of phase space angle in the classic al damped harmonic oscillator. Finally, expressions for a collective G amma(1)(t) and Gamma(2)(t) are derived, and shown to satisfy the relat ionship Gamma(2) = Gamma(1)/2. This familiar result, when applied to t hese collective rate constants, is seen to have a simple geometrical i nterpretation in phase space. (C) 1997 American Institute of Physics.