  We introduce a functional perturbative method for treating weakly nonlinear
systems coupled with a quantum field bath. We demonstrate using this method to
obtain the covariance matrix elements and the correlation functions of a
quantum anharmonic oscillator interacting with a heat bath. We identify a
fluctuation-dissipation relation based on the nonequilibrium dynamics of this
nonlinear open quantum system. To establish its connection with dynamical
equilibration, we further examine the energy flows between the anharmonic
oscillator and the bath field. The vanishing of the net flow {is an indication
of} the existence of an equilibrium state for such an open-system
configuration. The results presented here are useful for studying the
nonequilibrium physical processes of nonlinear quantum systems such as heat
transfer or electron transport.
