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

Input-output analysis of nonlinear dissipative quantum systems in the Fokker-Planck approximation

Franz X. Kärtner, Timo Langer, Christian Ginzel, Axel Schenzle

Published Mar 1, 1992• 7 citations
Publisher

Abstract

We develop a semiclassical approximation method for calculating correlation functions of the outgoing field of dissipative quantum systems. The method is strictly valid for linear quantum systems, and can be extended consistently to the nonlinear case. It can be used even at bifurcation points where linearization is no longer possible. It is applied to the dc-pumped degenerated Josephson-junction parametric amplifier. We demonstrate that the linearized fluctuation analysis can severely overestimate the maximum obtainable degree of squeezing.

DOI
10.1103/physreva.45.3228

Related papers

Quantum fluctuation of nonlinear degenerate optical parametric amplification

Chao Zhao, W. H. Tan

An analytical solution of the Fokker–Planck equation for the nonlinear degenerate optical parametric amplifier (DOPA) is presented, taking into account the influence of pump depletion on the generation of squeezed light. Results conform to those obtained using perturbation series expansion theory near threshold, and also apply to the whole region far away from threshold. When the nonlinear term(η → 0) is neglected, the solution transitions naturally to the linear approximation solution; when the nonlinear term is retained (∞ η), in the case μ → 0, the quantum fluctuations are close to vacuum fluctuations; in the case μ ≫ 1, squeezing increases, and tends to the result obtained using linear theory, 1/(1 + μ).

Journal of Modern Optics•Jun 27, 2006•5 citations

Squeezing of intracavity and traveling-wave light fields produced in parametric amplification

Matthew J. Collett, Crispin W. Gardiner

A general input-output theory for quantum dissipative systems is developed in which it is possible to relate output to input via the internal dynamics of a system. This is applied to the problem of computing the squeezing produced by a degenerate parametric amplifier located inside a cavity. The results for the internal modes are identical with those of Milburn and Walls [Opt. Commun. 39, 401 (1981)]. The output field is also found to have only 50% of maximal squeezing. However, by taking the output for a degenerate parametric amplifier inside a single-ended cavity and feeding this into an empty single-ended cavity, one can produce a maximally squeezed state inside this second cavity.

Physical Review A•Sep 1, 1984•1,071 citations

Quantum fluctuations in the time-dependent linearly driven degenerate parametric amplifier

Chao Zhao, Weihan Tan

The time-dependent, linearly driven Fokker-Planck equation for the degenerate parametric amplifier is derived when the pump depletion is present both above and below threshold. A model that the driving force εα1 is a product of the classical quantity ε and quantum operator α1 is proposed. It is found that below or near threshold, our results conform to the linear theory or perturbation series expansion, and above threshold, the short-time behavior of our solution is close to the linearization approximation; with the increase of interaction time τ, the long-time behavior of our solution shows that the squeezing is always quite different from the linear theory. It is shown that the maximal squeezing can be obtained at large pump parameter μ=ε/k and interaction time Δτ.

Journal of the Optical Society of America B•Sep 28, 2006•9 citations

Nonlinear quantum-stochastic model for the parameteric oscillator

Murray Wolinsky, H. J. Carmichael

The positive Prepresentation has proved a useful tool for analyzing optical systems which exhibit squeezing. The Glauber-Sudarshan phase-space representation gives a Fokker-Planck equation without positive semidefinite diffusion for systems which generate nonclassical fields. The positive Prepresentation provides a method for guaranteeing positive semidefinite diffusion and enables a quantum- dynamical model to be formulated in terms of classical stochastic differential equations. In previous work we had some success in applying this approach to describe the decay of the unstable state above threshold in a degenerate parametric oscillator.1 Below threshold a linearized analysis of fluctuations recovers the predictions of squeezing obtained by alternative methods.

Annual Meeting Optical Society of America•Jan 1, 1987•0 citations

Generation of squeezed microwave states by a dc-pumped degenerate parametric Josephson junction oscillator

Franz X. Kärtner, P. Russer

The master equation for a dc-pumped degenerate Josephson parametric amplifier is derived. It is shown that the Wigner distribution representation of this master equation can be approximated by a Fokker-Planck equation. By using this equation, the dynamical behavior of this degenerate Josephson amplifier with respect to squeezing of the radiation field is investigated. It is shown that below threshold of parametric oscillation, a squeezed vacuum state can be generated, and above threshold a second bifurcation point exists, where the device generates amplitude squeezed radiation. Basic relations between the achievable amplitude squeezing, the output power, and the operation frequency are derived.

Physical Review A•Nov 1, 1990•22 citations

Path-integral approach to the quantum theory of the degenerate parametric amplifier

Mark S. Hillery, Muhammad Suhail Zubairy

The quantum theory of the degenerate parametric amplifier is usually treated in the parametric approximation where the pump field is treated classically. In this paper we present a fully quantized theory of this nonlinear optical device using a path-integral approach. A perturbation series, the first term of which corresponds to the parametric approximation, is employed to evaluate explicitly the coherent-state propagator. The question of the validity of the parametric approximation is considered and the conditions under which this approximation is justified are elucidated. Finally, certain correlation functions for the signal-mode operators are calculated that are needed to study squeezed states. It is shown that the quantum nature of the pump field tends to decrease the squeezing.

Physical Review A•Mar 1, 1984•55 citations
We develop a semiclassical approximation method for calculating correlation functions of the outgoing field of dissipative quantum systems. The method is strictly valid for linear quantum systems, and can be extended consistently to the nonlinear case. It can be used even at bifurcation points where linearization is no longer possible. It is applied to the dc-pumped degenerated Josephson-junction parametric amplifier. We demonstrate that the linearized fluctuation analysis can severely overestimate the maximum obtainable degree of squeezing.
An analytical solution of the Fokker–Planck equation for the nonlinear degenerate optical parametric amplifier (DOPA) is presented, taking into account the influence of pump depletion on the generation of squeezed light. Results conform to those obtained using perturbation series expansion theory near threshold, and also apply to the whole region far away from threshold. When the nonlinear term(η → 0) is neglected, the solution transitions naturally to the linear approximation solution; when the nonlinear term is retained (∞ η), in the case μ → 0, the quantum fluctuations are close to vacuum fluctuations; in the case μ ≫ 1, squeezing increases, and tends to the result obtained using linear theory, 1/(1 + μ).
A general input-output theory for quantum dissipative systems is developed in which it is possible to relate output to input via the internal dynamics of a system. This is applied to the problem of computing the squeezing produced by a degenerate parametric amplifier located inside a cavity. The results for the internal modes are identical with those of Milburn and Walls [Opt. Commun. 39, 401 (1981)]. The output field is also found to have only 50% of maximal squeezing. However, by taking the output for a degenerate parametric amplifier inside a single-ended cavity and feeding this into an empty single-ended cavity, one can produce a maximally squeezed state inside this second cavity.
The time-dependent, linearly driven Fokker-Planck equation for the degenerate parametric amplifier is derived when the pump depletion is present both above and below threshold. A model that the driving force εα1 is a product of the classical quantity ε and quantum operator α1 is proposed. It is found that below or near threshold, our results conform to the linear theory or perturbation series expansion, and above threshold, the short-time behavior of our solution is close to the linearization approximation; with the increase of interaction time τ, the long-time behavior of our solution shows that the squeezing is always quite different from the linear theory. It is shown that the maximal squeezing can be obtained at large pump parameter μ=ε/k and interaction time Δτ.
The positive Prepresentation has proved a useful tool for analyzing optical systems which exhibit squeezing. The Glauber-Sudarshan phase-space representation gives a Fokker-Planck equation without positive semidefinite diffusion for systems which generate nonclassical fields. The positive Prepresentation provides a method for guaranteeing positive semidefinite diffusion and enables a quantum- dynamical model to be formulated in terms of classical stochastic differential equations. In previous work we had some success in applying this approach to describe the decay of the unstable state above threshold in a degenerate parametric oscillator.1 Below threshold a linearized analysis of fluctuations recovers the predictions of squeezing obtained by alternative methods.
The master equation for a dc-pumped degenerate Josephson parametric amplifier is derived. It is shown that the Wigner distribution representation of this master equation can be approximated by a Fokker-Planck equation. By using this equation, the dynamical behavior of this degenerate Josephson amplifier with respect to squeezing of the radiation field is investigated. It is shown that below threshold of parametric oscillation, a squeezed vacuum state can be generated, and above threshold a second bifurcation point exists, where the device generates amplitude squeezed radiation. Basic relations between the achievable amplitude squeezing, the output power, and the operation frequency are derived.
The quantum theory of the degenerate parametric amplifier is usually treated in the parametric approximation where the pump field is treated classically. In this paper we present a fully quantized theory of this nonlinear optical device using a path-integral approach. A perturbation series, the first term of which corresponds to the parametric approximation, is employed to evaluate explicitly the coherent-state propagator. The question of the validity of the parametric approximation is considered and the conditions under which this approximation is justified are elucidated. Finally, certain correlation functions for the signal-mode operators are calculated that are needed to study squeezed states. It is shown that the quantum nature of the pump field tends to decrease the squeezing.