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arXiv:quant-ph/0407146 · Journal of Modern Optics

Squeezing spectra froms-ordered quasiprobability distributions: application to dispersive optical bistability

Ferran V. García-Ferrer, Isabel Pérez–Arjona, Germán J. de Valcárcel, Eugenio Roldán

Published Jan 26, 2005• 1 citation
arXivHTMLPDF

Abstract

It is well known that the squeezing spectrum of the field exiting a nonlinear cavity can be directly obtained from the fluctuation spectrum of normally ordered products of creation and annihilation operators of the cavity mode. In this article we show that the output field squeezing spectrum can be derived also by combining the fluctuation spectra of any pair of s -ordered products of creation and annihilation operators. The interesting result is that the spectrum obtained in this way from the linearized Langevin equations is exact, and this occurs in spite of the fact that no s -ordered quasiprobability distribution verifies a true Fokker–Planck equation; that is, the Langevin equations used for deriving the squeezing spectrum are not exact. The (linearized) intracavity squeezing obtained from any s -ordered distribution is also exact. These results are exemplified in the problem of dispersive optical bistability.

DOI
10.1080/0950034042000275388

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It is well known that the squeezing spectrum of the field exiting a nonlinear cavity can be directly obtained from the fluctuation spectrum of normally ordered products of creation and annihilation operators of the cavity mode. In this article we show that the output field squeezing spectrum can be derived also by combining the fluctuation spectra of any pair of s -ordered products of creation and annihilation operators. The interesting result is that the spectrum obtained in this way from the linearized Langevin equations is exact, and this occurs in spite of the fact that no s -ordered quasiprobability distribution verifies a true Fokker–Planck equation; that is, the Langevin equations used for deriving the squeezing spectrum are not exact. The (linearized) intracavity squeezing obtained from any s -ordered distribution is also exact. These results are exemplified in the problem of dispersive optical bistability.
We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-QQQ cavity. From a linearized Fokker-Planck equation derived in the positive PPP 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.
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 show that the Langevin equation for a nonlinear-optical system may be obtained directly from the Heisenberg equation of motion for the annihilation operators, provided a certain linearization procedure is valid. We apply the technique to the parametric oscillator used to generate squeezed light and compare our results to those obtained from Fokker-Planck-type equations. We argue that, only when the Wigner, as opposed to the P or Q, representation of quantum optics is used, do we get a correct description of the underlying stochastic process. We show how the linearization procedure may be carried out to describe the operation of the parametric oscillator both below threshold, where a squeezed vacuum state results, and above threshold, where we find a squeezed coherent state. In the region of the threshold a heuristic extension of the method leads to a possible description of the system by means of a nonlinear Langevin equation.
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.
We theoretically study quadrature and polarization squeezing in dispersive optical bistability through a vectorial Kerr cavity model describing a nonlinear cavity filled with an isotropic ${\ensuremath{\chi}}^{(3)}$ medium in which self-phase- and cross-phase-modulation, as well as four-wave mixing, occur. We derive expressions for the quantum fluctuations of the output field quadratures as a function of which we express the spectrum of fluctuations of the output-field Stokes parameters. We pay particular attention to study how the bifurcations affecting the non-null linearly polarized output mode squeezes the orthogonally polarized vacuum mode, and show how this produces polarization squeezing.
We consider the dynamics of a single-mode cavity field damped via its coupling to a broad-band squeezed heat bath. The density operator equation of motion is transformed to a Fokker-Planck equation (FPE) for the s-parametrized quasiprobability distributions. We show that the field amplitude, intensity, spectrum and quadrature variances can be evaluated easily using the differential equations derived directly from the FPE. The FPE is then solved subject to different initial conditions, and the solution is used to investigate squeezing and field state evolution. The time behaviour of the purity Tr ([pcirc] 2) is treated in detail both analytically and numerically. The general steady-state solution is found in the operator form. We also show that, when certain conditions are met, dissipation without change in fluctuation properties is possible. A comparison is made with the cases of single-atom damping and off-resonance field damping.