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Physical Review A

Quantum-tunneling rates and stationary solutions in dispersive optical bistability

Karl Vogel, H. Risken

Published Sep 1, 1988• 55 citations
Publisher

Abstract

For the model of Drummond and Walls describing dispersive optical bistability, stationary solutions as well as quantum-tunneling rates are calculated by solving a pseudo-Fokker-Planck equation (PFPE), i.e., a Fokker-Planck-like equation with a non-positive-definite diffusion matrix, in terms of matrix continued fractions. Due to the quantum nature of the model, an interesting oscillating variation of the decay rates of the PFPE as a function of the scaling factor for the photon number is found for small cavity damping.

DOI
10.1103/physreva.38.2409

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For the model of Drummond and Walls describing dispersive optical bistability, stationary solutions as well as quantum-tunneling rates are calculated by solving a pseudo-Fokker-Planck equation (PFPE), i.e., a Fokker-Planck-like equation with a non-positive-definite diffusion matrix, in terms of matrix continued fractions. Due to the quantum nature of the model, an interesting oscillating variation of the decay rates of the PFPE as a function of the scaling factor for the photon number is found for small cavity damping.
Quantum tunneling times for the model of Drummond and Walls describing dispersive optical bistability are investigated for small cavity damping. Without damping the system can be described by an appropriate Hamilton operator. By expanding the density operator in eigenstates of this Hamilton operator the stationary solution as well as the lowest eigenvalues are obtained from a Pauli master equation for the diagonal elements of the density matrix for small cavity damping. The tunneling time follows from the lowest nonzero eigenvalue of this master equation. Expectation values as well as the Q function for the stationary case are also presented.
For the model of Drummond and Walls [J. Phys. A 13, 725 (1980)] describing dispersive optical bistability, the s-parametrized quasidistributions W(\ensuremath{\alpha},s) introduced by Cahill and Glauber [Phys. Rev. A 117, 1882 (1969)] are investigated. The equation of motion for W(\ensuremath{\alpha},s) generally has derivative terms up to third order. For the Q function (s=-1) and the P function (s=1) the third-order terms vanish. The equation of motion is then a pseudo-Fokker-Planck equation, i.e., a Fokker-Planck equation where the diffusion matrix is not positive definite or semidefinite. It is shown that the equation of motion with the inclusion of third-order-derivative terms can be solved by the matrix continued-fraction method. In particular, results for the Wigner function (s=0) are presented and compared with the results of the Q function (s=-1). Furthermore, the results are compared also with those obtained from the equation of motion for the Wigner function where the third-order-derivative terms have been neglected.
For the model of Drummond and Walls describing dispersive optical bistability, quantum tunneling rates are calculated in the low-cavity-damping limit. Without damping the system can be described by an appropriate Hamilton operator. By expanding the density operator into eigenstates of this Hamilton operator the equation of motion of the density operator is cast into an equation of motion for the density matrix. In this equation we take into account only diagonal elements and those nondiagonal elements for which the corresponding eigenstates of the Hamilton operator are nearly degenerate. The tunneling rates are then determined by the eigenvalues of the truncated density-matrix equation.
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.