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Journal of Modern Optics

Intracavity Optical Bistability Driven by Squeezed Light from a Parametric Amplifier

F. A. M. de Oliveira, S. Dutta Gupta

Published Sep 1, 1989• 1 citation
Publisher

Abstract

We describe the theory of optical bistability when atoms are collectively excited within the cavity of a parametric oscillator. Both optical bistability and parametric amplification can squeeze significantly the cavity-field quantum noise. When they are coupled together we find significant changes both on the mean value bistability and on the spectrum of squeezing as the parametric coupling increases. These are calculated directly from the appropriate master equation for the density matrix using a quantum distribution function (positive P) to develop Fokker—Planck and Ito equations.

DOI
10.1080/09500348914551251

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Annual Meeting Optical Society of America•Jan 1, 1986•1 citation
We describe the theory of optical bistability when atoms are collectively excited within the cavity of a parametric oscillator. Both optical bistability and parametric amplification can squeeze significantly the cavity-field quantum noise. When they are coupled together we find significant changes both on the mean value bistability and on the spectrum of squeezing as the parametric coupling increases. These are calculated directly from the appropriate master equation for the density matrix using a quantum distribution function (positive P) to develop Fokker—Planck and Ito equations.
The dynamics of a coherently driven two-level atom with parametric amplifier and coupled to a vacuum reservoir is analyzed. The combination of the master equation and the quantum Langevin equation is presented to study the quantum properties of light. By using these equations, we have determined the time evolution of the expectation values of the cavity mode and atomic operators. Moreover, with the aid of these results, the correlation properties of noise operators, and the large-time approximation scheme, we calculate the mean photon number, power spectrum, second-order correlation function, and quadrature variances for the cavity-mode light and fluorescence. It is found that the half-width of the power spectrum for the fluorescent light in the presence of a parametric amplifier increases, while it decreases for the cavity-mode light. Moreover, we have found the probability for the atom to be in the upper level in the presence of a parametric amplifier.
We analyze fluctuation effects of a squeezed electromagnetic reservoir (also termed squeezed vacuum, or SV) on the phenomenon of intrinsic optical bistability (IOB) in a dense collection of two-level atoms. A quantum statistical treatment is developed in the limit of a bad cavity and large dephasing in the medium. Under these conditions we obtain a single operator Langevin equation for the atomic inversion by adiabatic elimination of the atomic polarization variables. By using Ito calculus, the corresponding Fokker-Planck equation is derived, the stationary solution of which yields the probability distribution function for the bistable system as a function of the inversion. In a narrow region of transition, the probability distribution is double peaked with the two peaks corresponding to two metastable states in IOB. Quantum fluctuations can induce the system to jump randomly from one metastable state to the other with certain characteristic (passage) times. Squeezed quantum noise (as in the SV) is found to result in drastic alteration of the intrinsic bistable behavior of the system. Interestingly, the probability distribution and the passage times now exhibit critical dependence on the relative phase between the external driving field and the squeezed vacuum.
Optical bistability is reexamined as a candidate system for the generation of squeezed states of light. Previous analyses have concluded that significant squeezing can be achieved only in a rather extreme range of parameter space relative to current experimental capabilities and only over a restricted range of frequencies near the carrier.1 By contrast we have identified a new regime in which appreciable squeezing should be obtainable for rather modest operating conditions and without excessive demands on the stability of external control parameters. Following the treatment by Carmichael,2 we have conducted a theoretical search for optimal squeezing in optical bistability in the general case without adiabatic elimination of atomic or cavity variables. Our analysis is based on a linearization of the generalized Fokker-Planck equation for a system of N atoms coupled to a single mode of a high finesse cavity. For comparable atomic and cavity decay rates, there is a mode splitting in the eigenvalue structure of the system which is responsible for the vacuum field Rabi splitting. It is near the spectral features associated with these vacuum field Rabi sidebands that we find significant squeezing. The concepts of optimum squeezing as indicated by Mandel's Q parameter and by the spectrum of squeezing are discussed with reference to the various operating regimes in optical bistability.
We determine the incoherent spectrum, squeezing properties, and second-order correlation function of the collective atomic degrees of freedom in absorptive optical bistability. This is accomplished via a linearized Fokker-Planck equation in the positive PPP representation, guided by the analysis of H. J. Carmichael [Phys. Rev. A 33, 3262 (1986)] which does not resort to adiabatic elimination. We focus on the regimes of weak and strong intracavity excitation, addressing the good-cavity and bad-cavity limits as well as the limit of collective strong coupling. Adiabatic elimination of the intracavity field sustained by an auxiliary resonator coupled to the ensemble is used to probe the atomic correlations via the formation of a collective emission channel. We compare to the corresponding expressions for the forward-scattered light with reference to experimental results, discussing key differences between the lower and upper branch of the steady-state semiclassical bistability curve. Our analysis is carried out around the stable states situated far away from the turning points, where analytical expressions can be obtained self-consistently, demonstrating a clear departure from classical behavior.
We study the spectrum of squeezing in the light transmitted by a system of two-level atoms in an optical cavity driven by a coherent field, as recently observed by Kimble and collaborators. Our analysis does not introduce any restriction on the detuning parameters or on the relative order of magnitude of the atomic and cavity relaxation rates. Using the dressed-mode formalism, we derive explicit analytic expressions for the spectrum of squeezing and for the second-order intensity correlation function. We describe how the spectrum depends on the various parameters of the system; this study provides general indications on how to optimize the conditions to observe squeezing in optical bistability. A special attention is devoted to the changes of the spectrum under variation of the detuning parameters or on the ratio of cavity to atomic damping rate. In all cases, the changes are continuous and smooth, with quantitative variations in the level of squeezing. This indicates the existence of a single avenue to squeezing in this system. The bad-cavity direction is identified as especially convenient to obtain a significant amount of squeezing.
Squeezed states of light represent a manifestation of the nonclassical character of the electromagnetic field and hence are of great intrinsic interest in quantum optics. There are as well a number of applications in measurement science and optical communication associated with the possibility of sensitivity beyond the standard quan­tum limit. The experiments that we have undertaken attempt to explore the degree of achievable and detectable squeezing in three distinct physical systems. From a conceptual point of view each is in some sense ideal and should provide a large degree of squeezing. However, in practice each offers a potential set of advantages and disadvantages associated with technological constraints and with a lack of precise correspondence between experiment and existing theoretical models. The three experiments that we describe are as follows: (1) degenerate parametric oscillation in a doubly resonant cavity pumped by the frequency-doubled light from a Nd:YAG laser at 1.06 μm, (2) intracavity second harmonic generation in a doubly resonant cavity pumped at a fundamental wave­length of 515 nm with an Ar+ laser, and (3) optical bistability for a collection of two-level atoms within a high finesse interferometer of decay rate comparable to the inverse atomic lifetime.