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Annual Meeting Optical Society of America

Squeezed quantum noise effects in intrinsic optical bistability

Suneel Pratap Singh, Charles M. Bowden

Published Jan 1, 1992• 0 citations

Abstract

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.

DOI: 10.1364/oam.1992.fjj6

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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.

In the limit of large field-relaxation rates as compared to the rates of collective and atomic relaxation, we model the stochastic evolution of a bistable system with the aid of the Langevin equations for atomic variables. This semiclassical approach leads to a Fokker-Planck equation for the joint probability distribution of the atomic polarization and inversion. The Monte Carlo integration scheme, consisting in construction of approximate random trajectories, enables us to evaluate numerically the adiabatic values of the output field, intensity, and intensity fluctuations as functions of time. The final model used in numerical work is equivalent to ignoring microscopic fluctuations and considering delta-correlated external fluctuations impressed on the driving field.

In Chapter 1 we consider the question of squeezing in a two-photon laser. A two-photon laser is modelled as N two-level atoms interacting via a two-photon transition with a single resonant cavity field mode. A Fokker-Planck equation containing all leading noise terms is derived. A linearised fluctuation analysis reveals that there is no squeezing in a two-photon laser with injected signal. A small amount of squeezing is present in two-photon optical bistability. In Chapter 2 a microscopic model of degenerate four wave mixing including the quantisation of the medium is presented. Time-dependent solutions show ideal squeezing to be obtained in the limits of large detuning, low saturation and zero loss. The effect of nonideal atomic parameters is analysed. Equations describing the spatial propagation of the field through the medium are also derived. In Chapters 3 and 4 we are concerned with quantum effects, in particular violation of Bell's Inequalities, present in optical systems. An optical field produced by an intracavity four wave mixing is shown to exhibit the nonclassical features of photon antibunching, squeezing and violation of Cauchy-Schwarz and Bell's Inequalities. These intrinsic quantum mechanical effects are shown to be associated with the nonexistence of a positive normalizable Glauber-Sudarshan P-function.

A single two-level atom in a coherently driven cavity and damped by a broadband squeezed vacuum centered about the atomic transition frequency is studied. A second-order Fokker-Planck equation for this system is obtained without using system size expansion. Effects of detunings and cavity decays are also incorporated in the Fokker-Planck equation. This equation is used to study atomic inversion, fluorescent spectrum, and the intensity correlations of the transmitted and fluorescent photons in the bad-cavity limit. Several interesting effects in the atomic inversion, spectrum, and intensity correlations due to the squeezed vacuum are presented. These results are also compared with an atom that is damped by a thermal reservoir.

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.

In the general framework of the system size expansion of Van Kampen and Kubo, we consider the Fokker-Planck equation for a model of absorptive bistability in the bad-cavity limit. The physical system is described by the reduced atomic density operators after adiabatic elimination of the cavity field variables. Mapping of the master equation into

c

-number form according to the normal-ordering mapping scheme yields known results for the atomic fluctuations and correlation functions; however, it also leads to a Fokker-Planck equation with a non-positive-definite diffusion matrix. The symmetrical-order-mapping scheme eliminates this difficulty. The leading contribution to the system size expansion yields a Fokker-Planck equation for the symmetrical-ordered density function having a positive-definite diffusion matrix. The atomic expectation values and fluctuations previously derived from the quantum Langevin equations emerge naturally from this Fokker-Planck equation.