Quantum Brownian motion in a general environment. II. Nonlinear coupling and perturbative approach

We develop a perturbation scheme to treat the dynamics of a quantum Brownian particle (coordinate x) coupled nonlinearly to a bath of oscillators (coordinates q₍) with an interaction action in the form (x) q₍^k. We derive the influence functionals for the k=2, 3, 4 cases to order ^2 and derive the master equations for the special cases of local dissipation and white noise, as well as the general cases of nonlocal dissipation and colored noise for f (x) =x and x^2. We show that a generalized fluctuation-dissipation relation exists between the lth-order noise kernels of the kth-order coupling and their corresponding dissipation kernels.