  We consider a qubit coupled to a nonlinear quantum oscillator, the latter
coupled to an Ohmic bath, and investigate the qubit dynamics. This composed
system can be mapped onto that of a qubit coupled to an effective bath. An
approximate mapping procedure to determine the spectral density of the
effective bath is given. Specifically, within a linear response approximation
the effective spectral density is given by the knowledge of the linear
susceptibility of the nonlinear quantum oscillator. To determine the actual
form of the susceptibility, we consider its periodically driven counterpart,
the problem of the quantum Duffing oscillator within linear response theory in
the driving amplitude. Knowing the effective spectral density, the qubit
dynamics is investigated. In particular, an analytic formula for the qubit's
population difference is derived. Within the regime of validity of our theory,
a very good agreement is found with predictions obtained from a Bloch-Redfield
master equation approach applied to the composite qubit-nonlinear oscillator
system.
