  We analyze the Jaynes-Cummings model of quantum optics, in the
strong-dispersive regime. In the bad cavity limit and on timescales short
compared to the atomic coherence time, the dynamics are those of a nonlinear
oscillator. A steady-state non-perturbative semiclassical analysis exhibits a
finite region of bistability delimited by a pair of critical points, unlike the
usual dispersive bistability from a Kerr nonlinearity. This analysis explains
our quantum trajectory simulations that show qualitative agreement with recent
experiments from the field of circuit quantum electrodynamics.
