  We give a microscopic description of the optical bistability, where the
transmission coefficient has two different values as a function of input light
intensity, and the system exhibits a discontinuous jump with a hysteresis loop.
We developed an efficient numerical algorithm to treat the quantum master
equation for hybridized systems of many photons and a large number of two-level
atoms. By using this method, we characterize the bistability from the viewpoint
of eigenmodes and eigenvalues of the time evolution operator of the quantum
master equation. We investigate the optical bistability within the low
photon-density regime, where the hybridization of photon and atom degrees of
freedom occurs and the resonance spectrum has a double peak structure. We
compared it with the standard optical bistability between the low
photon-density regime and the high photon-density regime, where the photons can
be treated as a classical electromagnetic field and the resonance spectrum has
a single peak structure. We discuss the steady-state properties of the optical
bistability: dependencies of the photon number density on the intensity and the
double peak structure of the photon number distribution inside the bistable
region. As for the dynamical properties, we find that the relaxation timescale
shows an exponential growth with the system size, and reveal how the hysteresis
loop of the optical bistability depends on the size of the system and the
sweeping rate of the driving amplitude. Finally, by investigating the effects
of detuning frequency of the input field, we clarify the characteristic
properties of the present optical bistability within the low photon-density
regime, which are qualitatively different from the standard optical bistable
phenomena.
