A quantum treatment of a coherently driven cavity with a medium of two-level atoms is given. The system displays both absorptive and dispersive bistability. Quantum fluctuations are included via a Fokker-Planck equation in a generalized
P representation, after adiabatic elimination of the atomic variables. The transmitted spectrum and correlation function is calculated in a linearized (Gaussian) approximation. In the limit of large detuning and radiative damping, the quantum fluctuations are identical to those of a simple nonlinear polarizability theory. An analysis is given of the nonclassical statistical properties of the intensity-correlation spectrum.
The topic of this thesis is a theoretical study of nonequilibrium transitions and quantum statistical properties of nonlinear quantum optical systems driven by an external radiation source. In certain limiting cases, a comparison is made between these transitions and the phase transitions found in equilibrium physical systems. In chapters one and two, the mathematical tools are introduced. In operator terms, the time development is described by a Markovian master equation in the interaction picture. This is equivalent to a corresponding time development equation or Fokker-Planck equation in a vector space of c-numbers. In order to deal with the type of Fokker-Planck equation that results, we introduce a quantum classical correspondence resulting in a distribution function over a complex phase-space, which is a generalisation of the Glauber-Sudarshan P-function. In chapter three this is applied to a model of a coherently driven mode with nonlinear dispersion and absorption. We find in the limit of zero temperature, that the spectrum is symmetric relative to the input frequency, and an exact solution is obtained for the distribution function. For a detuned driving field and nonlinear dispersion, optical bistability can occur. In chapter four a model of sub/second harmonic generation is introduced. This has several non-equilibrium transitions, including dispersive optical bistability, and bistable behaviour with coherent phase-locked input to both modes. Exact solutions occur in the limit of zero temperature and adiabatic elimination of one mode. In both chapters three and four, steady-state photon anti-bunching occurs with an absorptive nonlinearity. In chapter five we include interactions between the radiation mode and a fluorescent atomic system. In this case different behaviour occurs depending on the relative decay rates of the individual atoms and of the radiation mode. In the case of a high-Q interferometer, the atomic variables can be adiabatically eliminated. Both dispersive and absorptive bistability can occur. We show by analytic and numerical calculations (in the case of inhomogeneous broadening) that dispersive operation has advantages in requiring a lower atomic density and input field to observe bistability. When the input field has Gaussian (rather than coherent) photon statistics, there is no bistability, but enhanced photon bunching occurs. Finally, there is a different type of behaviour when the field mode decays rapidly and can be adiabatically eliminated. Within the cooperation lifetime the system can be described by a J²-invariant Hamiltonian, giving the special case of only collective damping. The result is a new type of critical point transition in the thermodynamic limit, with the appearance of a family of solutions like Lotka-Volterra cycles for a coherent driving field above threshold.
We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-
Q cavity. From a linearized Fokker-Planck equation derived in the positive
P representation, the steady-state correlation matrix and spectrum of fluctuations are calculated. Expressions which normally assume a positive-definite diffusion matrix are shown to be valid for nonpositive-definite diffusion. For absorptive bistability we find agreement with results derived previously by quite different methods. More generally we extend previously published results by including atomic and cavity detunings. We compare the transmitted spectrum for dispersive bistability with that for absorptive bistability, and photon antibunching in dispersive bistability is discussed.
The purpose of this work is to study the behavior of atomic fluctuations and the spectrum of transmitted light by a nonlinear medium with absorption and dispersion. The authors make use of the system-size expansion and apply it to the quantum-mechanical Langevin's equations for the atomic operators and obtain the characteristic curves for bistable behavior, macroscopic atomic expectation values, and the spectrum of the atomic fluctuations. The calculated spectrum of transmitted light exhibits line narrowing near bistable thresholds, discontinuous formation of bands along the high-transmission bistable branch, and hysteresis effects, etc., for detuning $|\ensuremath{\delta}|$ less than a critical value ${\ensuremath{\delta}}_{c}$. For $|\ensuremath{\delta}|>{\ensuremath{\delta}}_{c}$, the discontinuous character of all the physical variables of the system vanishes; the system no longer exhibits bistable behavior. The system, however, still shows cooperative and single-atom behavior separated by a critical value of the intensity of the incident coherent pump at which a critical slowing down of the medium occurs, exhibiting a phase transition.
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.
A generalized Fokker-Planck equation is used to study the problem of single-atom optical bistability in the good-cavity limit. The effects of quantum fluctuations are investigated by linearizing fluctuations around the steady-state value. It is shown that in the good-cavity limit, quantum fluctuations are relatively small, and hence there exist both absorptive and dispersive single-atom bistability. By comparing our single-atom Fokker-Planck equation with the N-atom Fokker-Planck equation, we find that in the good-cavity limit, these two different Fokker-Planck equations reduce to identical equations. We therefore come to an important conclusion that the results obtained from the many-atom Fokker-Planck equation in the good-cavity limit are also valid for a single atom. \textcopyright{} 1996 The American Physical Society.