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

Abstract

We develop a perturbation scheme to treat the dynamics of a quantum Brownian particle (coordinate $x$) coupled nonlinearly to a bath of oscillators (coordinates ${q}_{n}$) with an interaction action in the form $\ensuremath{\lambda}f(x){q}_{n}^{k}$. We derive the influence functionals for the $k=2, 3, 4$ cases to order ${\ensuremath{\lambda}}^{2}$ 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 $f(x)=x \mathrm{and} {x}^{2}$. We show that a generalized fluctuation-dissipation relation exists between the $l\mathrm{th}$-order noise kernels of the $k\mathrm{th}$-order coupling and their corresponding dissipation kernels.