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Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition

This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Caldeira and Leggett, Physica A 121, 587 (1983)] is extended for a nonequilibrium influence functional, which has been obtained for general cases where the ground and the excited electronic state baths can be different. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional with respect to time up to the second order. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation, but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived through short time expansion of the effective action followed by Gaussian integrations in analytically continued complex domain of space. This leads to a compact form of GQFPE with time dependent kernels and additional terms, which make the resulting equation the Dekker form [H. Dekker, Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit.