We study the semiclassical mean-field theory of absorptive optical bistability in the limit of a fully developed hysteresis cycle, i.e., in the limit of large values of the bistability parameter
C. We show by multiple time-scale perturbation analysis that it is possible to describe the transition between the two stable branches by simple equations. Near the low-transmission branch, the behavior of the system is governed by an equation for the atomic population, whereas, near the high-transmission branch, it is an equation for the field amplitude which determines the long-time evolution of the system. We then present an analytic study, completed by numerical results whenever necessary, of the time-dependent response of the system when the input field is swept across the bistable region. We study the influence of the initial conditions, the sweeping velocity, and the ratio of the atomic relaxation times on the dynamical hysteresis as well as its relation with the stationary hysteresis.
We study the optical bistability and dynamics in a hybrid optomechanical system, where the cavity mode is coupled to the mechanical resonator via radiation pressure and to a two-level atom via the Jaynes–Cummings coupling. The nonlinearity due to optomechanical coupling and the addition of an atom leads to bistable behaviors of the steady-state intracavity photon number and atomic population inversion. We show that the optical bistability can be effectively controlled by the Rabi frequency of the atomic driving field, the cavity–atom coupling strength, and the amplitude and frequency of the cavity pump field. Moreover, the effects of different initial conditions of the cavity field and the amplitude of the cavity pump field on the dynamics of the system also have been discussed.
We discuss the quantum-statistical properties of a system of absorbing atoms in a ring cavity under the action of a resonant classical field. At a semiclassical level, this system exhibits a bistable behavior in the sense that the transmitted light
ET is a discontinuous function of the incident light
EI with a hysteresis cycle, which resembles a first-order phase transition. Here we specify this analogy stressing the nonthermodynamic properties of this coherently driven system. The quantized field
ET is described by a Glauber
P function, which obeys to a Fokker-Planck equation. This equation is derived from a very general master equation in the case of a good quality cavity by neglecting derivatives of order higher than second. A remarkable feature of this equation is that the diffusion coefficient is a nonlinear function of the field intensity. The steady-state solution of the Fokker-Planck equation has the following features. The
P function can be single or double peaked according to the value of
EI. In fact, our Fokker-Planck equation is like the one of a Brownian particle moving in a potential which presents one minimum or minima separated by a barrier, depending on the value of
EI. The peaks of the
P function correspond to the minima of the potential. In the double-peaked case, the peaks have different widths. The maxima of the
p function occur at values of
ET, which coincide with the classical stationary solutions that are stable according to the linear stability analysis. These semiclassical solutions give a discontinuous two-valued function of
ET vs
EI (hysteresis cycle). On the contrary, the mean value
〈ET〉 is a continuous single-valued function of
EI, which behaves similarly to the Maxwell construction of a first-order phase transition:
〈ET〉 jumps from low to high values in a very sharp transition region of
EI, which lies between the discontinuity points of the hysteresis cycle. However, the Maxwell rule that we obtain is quite different from that of equilibrium thermodynamics. The mean-square fluctuations of
ET are always very small except in the transition region where they are quite remarkable. We also present an exact expression of the stationary
P function, which takes into account all higher-order derivatives, and we show that the Fokker-Planck approximation is very good for calculating mean values and mean-square fluctuations. Finally, we discuss the connections which link bistable absorption to the laser with injected signal and to the laser with a saturable absorber.
We analyze the absorptive and hybrid absorptive–dispersive double-cavity optical bistability behavior in a four-level N-type atomic system driven by two optical fields circulating inside two independently unidirectional ring cavities and a trigger field outside the cavities. In this system, the two-photon absorption and cross Kerr nonlinearity play significant roles in the formation of bistability and can be controlled efficiently due to the quantum interference effect, and then the resulting nonlinear behavior exhibits different features from the two/three-level schemes. It is shown that the threshold values and hysteresis cycle width of the bistable curve can be manipulated via the adjustment of the system parameters. Meanwhile, we also obtain double-beam optical multistability. When adding positive and negative pulses to the cavity input, all-optical steady-state switching for the two cavity outputs can be achieved simultaneously. Therefore, the proposed scheme may have potential applications in all-optical communication and computing.
Absorptive optical bistability is shown to exist for a single two-level atom inside a resonant optical cavity. Solutions for the quantum-mechanical density operator are obtained numerically for a parameter regime at the interface between the quantum limit, in which quantum-mechanical noise invalidates the semiclassical prediction of bistability, and the classical limit, in which quantum noise is a negligible perturbation on semiclassical results. Bimodal photon number distributions and Q functions are obtained, and two-state transition rates are calculated.>
We discuss a mean-field quantum-mechanical model which describes the dynamics of a homogeneously broadened system of two-level atoms contained in a pencil-shaped resonant cavity and driven by a coherent resonant field
EI. The model is treated in the semiclassical approximation. The model is justified on the basis of Maxwell-Bloch equations with two coherently coupled directions of propagation, with boundary conditions for the fields taken into account. Above a suitable critical density of atoms the system exhibits a bistable behavior including both the stationary situation and the transient, both the light transmitted in the forward direction and the fluorescent light. Bistability is shown to be a consequence of atomic cooperation. We give a simple description of optical bistability leading to new predictions for the transient behavior of the transmitted light and for the spectrum of the fluorescent light. The damping constant which characterizes the rate of appraoch to the stationary situation exhibits a hysteresis cycle. In the low-transmission regime the approach is monotonic, whereas in the highly transmitting situation the approach is oscillatory. One finds a critical slowing down in correspondence with the values
EI(+), ${{E}_{I}}^{(\ensuremath{-})}$ of the incident field where the transmitted field (as well as the total fluorescence intensity, the rate of approach to the stationary state, etc.) changes discontinuously. On the basis of the regression hypothesis we give a qualitative description of the spectrum of the fluorescent light. This is shown to undergo hysteresis and discontinuous changes at the same values
EI(+), ${E}_{I}^{(\ensuremath{-})}$ of the incident field. Below the critical density of atoms one recovers the usual picture of resonance fluorescence, with a continuous transition from a single-line spectrum to a three-peaked structure when the Rabi frequency becomes equal to the natural linewidth. Above the critical density the triplet appears only when the Rabi frequency becomes equal to the cooperative linewidth of pure super-fluorescence (i.e., for
EI=EI(+)). Moreover, it appears discontinuously: when
EI crosses the value
EI(+) the spectrum changes from a single narrow line to a triplet with well-separated sidebands.