The Driven Dicke Model and its Macroscopic Extension: Bistability or Bifurcation?

Abstract

We are concerned with the relation between recent work on the “driven Dicke model” of N two-level atoms, on the same site, driven by a c. w. laser field Ω, and a corresponding theory for the more realistic macroscopically extended system. We review the results on the driven Dicke model: two different decorrelation schemes yield different results; in the steady state at resonance a semi classical approximation without damping best approximates the exact solution of the quantum model also described. The exact solution of the quantum model does not display normal optical bistability (OB):calculation of ${g^{(2)}}(O) = {G^{(2)}}(O)/\left\{ {{G^{(1)}}{{(O)}^2}} \right\}$ (where G⁽ⁿ⁾(0) = < (S₊)ⁿ(S_-)ⁿ > and S_± are collective spin operators) shows g⁽²⁾(0)→ 1.2 and there is a simple bifurcation point at θ {∞ limΩN⁻¹, N→∞} = 1. The inversion r₃ plays the role of the order parameter: r₃= ± 1/2 (l− θ²) 1/2, θ < 1; =0, θ > 1. There is a second-order type phase transition, and by moving off-resonance and relating to the decor- related model, we are able to identify one set of equivalent thermodynamic parameters for the model. We find “critical exponents” α = 1/2, β = 1/2, γ = 1.5 and α + 2β + γ > 2 in this manner. Results are compared with the operator theory for the extended system also presented (unlike the Dicke model this model does not have total spin as a constant of the motion). Decorrelation of operator products with self-correlation (radiation damping) leads of course to the c-number theory of cusp catastrophe OB. An operator theory involving a natural power dependent refractive index is sketched and we believe that it is this which should appear as the parameter in the usual treatment of the Fabry-Perot interferometer. But, alternatively, by extracting a single mode theory in the “mean-field” Approximation, we regain both the Bloch equations and the master equation of the driven Dicke model. The spectra and correlation functions shown in Figs. 1–4 are calculated from these in a decorrelation approximation which retains single-particle damping and which differs from the exact solution of the master equation. The hierarchy of different models relates to the realistic extended system model in ways very similar to those of a similar hierarchy in the theory of super fluorescence. It is concluded that mean field theory maltreats the analysis. However, it is expected that the decorrelation scheme adopted for the spectra we have calculated will be adequate to describe their essential features.

On sabbatical leave from: Ain Shams University, Faculty of Science, Applied Mathematics Department, Cairo, Egypt.