  The interplay of quantum fluctuations with nonlinear dynamics is a central
topic in the study of open quantum systems, connected to fundamental issues
(such as decoherence and the quantum-classical transition) and practical
applications (such as coherent information processing and the development of
mesoscopic sensors/amplifiers). With this context in mind, we here present a
computational study of some elementary bifurcations that occur in a driven and
damped cavity quantum electrodynamics (cavity QED) model at low intracavity
photon number. In particular, we utilize the single-atom cavity QED Master
Equation and associated Stochastic Schrodinger Equations to characterize the
equilibrium distribution and dynamical behavior of the quantized intracavity
optical field in parameter regimes near points in the semiclassical
(mean-field, Maxwell-Bloch) bifurcation set. Our numerical results show that
the semiclassical limit sets are qualitatively preserved in the quantum
stationary states, although quantum fluctuations apparently induce phase
diffusion within periodic orbits and stochastic transitions between attractors.
We restrict our attention to an experimentally realistic parameter regime.
