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Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences

Quantum statistical properties of a quantum theory of optical bistability

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

Published Dec 18, 1984• 0 citations
Publisher

Abstract

Abstract The strictly quantum driven Dicke model of N two-level atoms on the same site shows conventional optical bistability if and only if cavity feedback is included. In this model we find that for N->oo the photon statistics of the transmitted field are Poisson on the lower branch of the output-input curve, but on the upper branch approach Bose-Einstein in the hysteresis region, with reversion to Poisson only for very strong input fields.

DOI
10.1098/rsta.1984.0135

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Abstract The strictly quantum driven Dicke model of N two-level atoms on the same site shows conventional optical bistability if and only if cavity feedback is included. In this model we find that for N->oo the photon statistics of the transmitted field are Poisson on the lower branch of the output-input curve, but on the upper branch approach Bose-Einstein in the hysteresis region, with reversion to Poisson only for very strong input fields.
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.
The quantum statistical behavior of a small collection of N two-state atoms strongly coupled to the field of a high-finessse optical cavity is investigated. Input-output characteristics are recorded over the range 3\ensuremath{\lesssim}N\ensuremath{\lesssim}65, with bistability observed for N\ensuremath{\gtrsim}15 intracavity atoms and for a saturation photon number n0{\mathit{n}}_{0}n0​\ensuremath{\simeq}0.8. For weak excitation the transmitted field exhibits photon antibunching as a nonclassical manifestation of state reduction and quantum interference with the magnitude of the nonclassical effects largely independent of N.
A quantum treatment of a coherently driven dispersive cavity is given based on a cubic nonlinearity in the polarisability of the internal medium. This system displays bistability and hysteresis in the semiclassical solutions. Quantum fluctuations are included via a Fokker-Planck equation in a generalised P representation. The transmitted light shows a transition from a single-peaked spectrum to a double-peaked spectrum above the threshold of the lower branch. Fluctuations in the field are reduced on the upper branch and both photon bunching and photon antibunching are predicted, for different operating points. An exact solution obtained for the steady-state generalised P function shows decidedly non-equilibrium behaviour, e.g. the lack of a Maxwell construction.
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 general expression of the output-light Mandel parameter(xi) out is obtained in case of purely absorptive bistability, for arbitrary value of (xi) in of incident light. Calculations are based on stochastic rate equations. No adiabatic elimination of atomic variables is made; the coherent nature of 'incoming' of incident photon is into a cavity is correctly taken into account. On condition of bistability, although the strongly sub-Poisson input-photon statistics generally reduces the output noise as compared to the case of coherent input, the inequality (xi) out > 0 still near the switching points.
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 classify types of optical bistability according to the photon number density in the cavity. In contrast to previous studies of optical bistability in 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 study the nature of optical bistability in 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. Unraveling the nature of the optical bistability in the latter regime may be important for the manipulation of quantum systems. Concretely, 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.