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Topical Meeting on Optical Bistability (OB)

Optical Bistability from the Dicke Model

Ravinder Rupchand Puri, G. P. Hildred, Saad S. M. Hassan, R. Bullough

Published Jan 1, 1983• 0 citations
Publisher

Abstract

We have reported a fairly complete analysis of the quantum Dicke model [1]. This consists of N 2-level atoms on the same site driven by arbitrary fields - coherent, chaotic, mixed coherent-chaotic. We also showed [2] in particular that it is easy to include, albeit in a rather formal way, the effects of a cavity on the Dicke model in abroad band chaotic field. The results agree well with those recently observed on high Rydberg Na atoms [3]. This suggests that a comparable analysis for a coherent driving field which also includes the effects of a cavity will provide a quantum theory of optical bistability. It also suggests that some of the consequences of the theory, primarily photon statistical properties, which, however, are calculated as interatomic correlations [2,3], might be observable in a suitable experiment.

DOI
10.1364/obi.1983.thb44

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We have reported a fairly complete analysis of the quantum Dicke model [1]. This consists of N 2-level atoms on the same site driven by arbitrary fields - coherent, chaotic, mixed coherent-chaotic. We also showed [2] in particular that it is easy to include, albeit in a rather formal way, the effects of a cavity on the Dicke model in abroad band chaotic field. The results agree well with those recently observed on high Rydberg Na atoms [3]. This suggests that a comparable analysis for a coherent driving field which also includes the effects of a cavity will provide a quantum theory of optical bistability. It also suggests that some of the consequences of the theory, primarily photon statistical properties, which, however, are calculated as interatomic correlations [2,3], might be observable in a suitable experiment.
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 PPP 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 authors consider a system comprised of a collection of identical two-level atoms interacting with the electromagnetic field in the dipole approximation and with an externally applied steady-state coherent driving field. The atoms are considered to interact with each other only via the electromagnetic field and are assumed to be contained within a volume much smaller than a resonance wavelength. The system is treated as a quantum-statistical ensemble in the rotating frame, rotating at the carrier frequency of the externally applied coherent field. The Hamiltonian in this frame is explicitly time independent and the exact analog of a spin temperature is defined. The thermodynamic-Green's-function method is used to determine the thermodynamic equilibrium properties of the system in the rotating frame. This results in a nonlinear relation between the applied field and the field in the volume containing the atoms. The expression is a nonlinear function of the effective or "spin" temperature as well, and has the form of a first-order phase transition with the macroscopic atomic polarization as order parameter. In the low-"spin"-temperature limit, and for perfect tuning, the expression reduces to the exact form of the nonlinear relation for optical bistability derived by Bonifacio and Lugiato and relating the incident and transmitted fields for atoms in a ring cavity in the mean-field approximation. The results contain absorptive as well as dispersive contributions. These results predict essentially different behavior as a function of off tuning compared with behavior predicted by others using statistical steady-state models far from thermodynamic equilibrium. In the limit of zero value for the applied field amplitude, the results reduce to the conditions for the well-known second-order superradiant" phase transition. The condition for the existence of the second-order phase transition is shown to depend on the cavity tuning and the photon escape rate. The authors also show why, in light of the experiments in optical bistability, this second-order phase is difficult to observe.
We systematically study the chaotic signatures in a quantum many-body system consisting of an ensemble of interacting two-level atoms coupled to a single-mode bosonic field, the so-called extended Dicke model. The presence of the atom-atom interaction also leads us to explore how the atomic interaction affects the chaotic characters of the model. By analyzing the energy spectral statistics and the structure of eigenstates, we reveal the quantum signatures of chaos in the model and discuss the effect of the atomic interaction. We also investigate the dependence of the boundary of chaos extracted from both eigenvalue-based and eigenstate-based indicators on the atomic interaction. We show that the impact of the atomic interaction on the spectral statistics is stronger than on the structure of eigenstates. Qualitatively, the integrablity-to-chaos transition found in the Dicke model is amplified when the interatomic interaction in the extended Dicke model is switched on.
We compare theoretical predictions and experimental data for the multimode instability of optical bistability, which arises from the coherent dynamics of a system of two-level molecules contained in an optical cavity and driven by an external stationary laser field. Considerable insight into this phenomenon is provided by two ``rules of thumb'' that govern the behavior of the self-pulsing frequency, and establish its relation with the Rabi frequency of the intracavity field. We describe in detail the experimental apparatus and the main features of the experimental findings. The theoretical results are based on the Maxwell-Bloch equations for a ring cavity in the plane-wave approximation. Despite the crudeness of this model, the numerical data display a satisfactory qualitative agreement with the experimental results. Additional period-doubling and chaotic phenomena, predicted by the theory and not observed in the experiment, are presumably washed out by the longitudinal and radial variation of the electric field in the cavity.