Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Quantum tunneling rates in dispersive optical bistability for low cavity damping

H. Risken, Karl Vogel

Published Aug 1, 1988• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.38.1349

Related papers

Quantum tunneling in dispersive optical bistability

H. Risken, Craig M. Savage, Fritz Haake +1 more

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.

Physical Review A•Feb 1, 1987•58 citations

Quantum-tunneling rates and stationary solutions in dispersive optical bistability

Karl Vogel, H. Risken

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.

Physical Review A•Sep 1, 1988•55 citations

Dispersive optical bistability for large photon numbers and low cavity damping

Karl Vogel, H. Risken

For a quantum-mechanical model of dispersive optical bistability we derive in the classical limit, i.e., for large photon numbers, a two-variable Fokker-Planck equation. For low cavity damping this Fokker-Planck equation can be approximated by a one-variable Fokker-Planck equation which is solved numerically. If, in addition, the number of thermal photons is small compared with the number of photons inside the cavity, an analytical result for the transition rate is obtained.

Physical Review A•Jul 1, 1990•17 citations

Quantum theory of damping – Heisenberg–Langevin approach

Marlan O. Scully, Muhammad Suhail Zubairy

In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.

Cambridge University Press eBooks•Sep 4, 1997•2 citations

Quantum optical tunneling: A representation-free theory valid near the state-equation turning points

P. D. Drummond

The problem of representation dependence in the treatment of quantum tunneling in absorptive optical bistability is considered. Disregarding this problem can lead to order-of-magnitude errors in calculated tunneling rates. A new description is presented, valid near the state-equation turning points, in which regime the discrepancies between different representations are shown to disappear. This description is also useful for treating critical fluctuations and dispersive cases.

Physical Review A•Jun 1, 1986•31 citations

Quantum theory of optical bistability. III. Atomic fluorescence in a low- Q cavity

H. J. Carmichael, Daniel F. Walls, P. D. Drummond +1 more

We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-$Q$ cavity. From a linearized Fokker-Planck equation derived in the positive $P$ 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.

Physical Review A•Jun 1, 1983•35 citations
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.
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 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.
In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.
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.