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

Chaos and coherence in an optical system subject to photon nondemolition measurement

B. F. Wielinga, Gerard J Milburn

Published Jul 1, 1992• 13 citations
Publisher

Abstract

We consider the effect of quantum-nondemolition monitoring of the photon number on the quantum suppression of classical chaos in a recently proposed quantum optical model, the parametrically kicked nonlinear oscillator [G. J. Milburn, Phys. Rev. A 41, 6567 (1990)]. Classically the effect of the quantum-nondemolition measurement is equivalent to a phase diffusion in the phase plane of the oscillator. A similar result holds in the quantum description, but in addition the measurement rapidly diagonalizes the system density operator in the photon-number basis. This has the effect of causing the evolution of the quantum moments to approach the corresponding classical moments and thus restores the classical dynamics for practical purposes.

DOI
10.1103/physreva.46.762

Related papers

Quantum coherence and classical chaos in a pulsed parametric oscillator with a Kerr nonlinearity

Gerard J Milburn, Catherine A. Holmes

We consider a parametric amplifier driven by a periodically pulsed pump field inside a cavity containing a Kerr nonlinearity. The dynamics of the device is modeled as a kicked nonlinear system. The pulsed parametric amplifier constitutes the kick. In between kicks the dynamics is determined by the Kerr nonlinearity and damping. In the absence of damping, a classical description of the device exhibits a rich phase-space structure including fixed points of multiple period and chaos. We contrast the classical behavior of the mean intensity with that predicted by quantum dynamics. The mean photon number inside the cavity is shown to undergo regular collapse and revival in the regular region of the phase space and irregular revivals in the chaotic region. When damping is included, the quantum recurrences are rapidly suppressed, and the classical behavior is restored. In this case a stable steady state is possible. The damping represents the effect of photon-number measurements on the system. We also discuss the photon statistics in the steady state.

Physical Review A•Oct 1, 1991•83 citations

Coherence And Chaos In A Quantum Optical System

Gerard J Milburn

The quantum dynamics of periodically kicked hamiltonian systems can reflect the classical dynamics be it regular or chaotic. However in quantum systems the classical chaotic diffusion of one phase space variable is suppressed due to irregular recurrences in the quantum state. This aspect of quantum nonlinear dynamics should be accessible to experimental test and recent results on the microwave ionization of hydrogen would seem to confirm it.

Digest on Nonlinear Optics: Materials, Phenomena and Devices•Jan 1, 1990•1 citation

Measurement trajectories of chaotic quantum systems

Martin Schlautmann, R. Graham

We consider the behavior of a classically chaotic quantum system, the periodically driven pendulum, under the influence of a continuous measurement of its angular momentum. Without measurement the system shows dynamical localization, a quantum interference effect which suppresses the classical chaotic diffusion in momentum space. The coupling of the system to a measuring device destroys its coherence and thus leads to delocalization. This is studied on the level of individual systems including the recording of the measurement results. For that purpose we analyze the appropriate stochastic Schr\"odinger equation, from which a stochastic quantum map is derived as an effective tool for numerical simulation of measurement trajectories. We show that a continuous momentum measurement restores the diffusive behavior of the system in momentum space, but that for sufficiently low accuracy of measurement the corresponding diffusion constant is smaller than the classical one. This is reflected by an equal diffusive growth of the recorded measurement results. Thus we find signatures of the classical chaos and of the dynamical localization both in the behavior of the measured quantum system and in the corresponding signal of the measuring device.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jul 1, 1995•11 citations

Quantum nondemolition measurements in optical cavities

P. M. Alsing, Gerard J Milburn, Daniel F. Walls

We analyze schemes for performing quantum nondemolition (QND) measurements in optical cavities. We consider three schemes: (1) measurement of a quadrature phase amplitude using a parametric process, (2) measurement of a quadrature phase amplitude using the optical Kerr effect in a nonlinear fiber, and (3) measurement of the photon number also using the Kerr effect in a fiber. We show that in the second scheme an enhancement of the QND effect may be obtained by making the cavity finesse for the signal larger than that for the probe.

Physical Review A•Apr 1, 1988•61 citations

Quantum signatures of chaos in the dynamics of a trapped ion

J. K. Breslin, Catherine A. Holmes, Gerard J Milburn

We show how a nonlinear chaotic system, the parametrically kicked nonlinear oscillator, may be realized in the dynamics of a trapped, laser-cooled ion, interacting with a sequence of standing-wave pulses. Unlike the original optical scheme [G. J. Milburn and C. A. Holmes, Phys. Rev. A 44, 4704 (1991)], the trapped ion enables strongly quantum dynamics with minimal dissipation. This should permit an experimental test of one of the quantum signatures of chaos: irregular collapse and revival dynamics of the average vibrational energy.

arXiv:quant-ph/9611022•Physical Review A•Oct 1, 1997•27 citations

Quantum-nondemolition measurements and the ‘‘collapse of the wave function’’

Franz X. Kärtner, H. A. Haus

This paper intends to clarify some issues in the theory of quantum measurement by taking advantage of the self-consistent quantum formulation of nonlinear optics. A quantum-nondemolition measurement of the photon number of an optical pulse can be performed with a nonlinear Mach-Zehnder interferometer followed by a balanced detector. The full quantum-mechanical treatment shows that the shortcut in the description of the quantum-mechanical measurement, the so-called ``collapse of the wave function,'' is not needed for a self-consistent interpretation of the measurement process. Coherence in the density matrix of the signal to be measured is progressively reduced with increasing accuracy of the photon-number determination. The quantum-nondemolition measurement is incorporated in the double-slit experiment and the contrast ratio of the fringes is found to decrease systematically with increasing information on the photon number in one of the two paths. The ``gain'' in the measurement can be made arbitrarily large so that postprocessing of the information can proceed classically.

Physical Review A•Jun 1, 1993•25 citations
We consider the effect of quantum-nondemolition monitoring of the photon number on the quantum suppression of classical chaos in a recently proposed quantum optical model, the parametrically kicked nonlinear oscillator [G. J. Milburn, Phys. Rev. A 41, 6567 (1990)]. Classically the effect of the quantum-nondemolition measurement is equivalent to a phase diffusion in the phase plane of the oscillator. A similar result holds in the quantum description, but in addition the measurement rapidly diagonalizes the system density operator in the photon-number basis. This has the effect of causing the evolution of the quantum moments to approach the corresponding classical moments and thus restores the classical dynamics for practical purposes.
We consider a parametric amplifier driven by a periodically pulsed pump field inside a cavity containing a Kerr nonlinearity. The dynamics of the device is modeled as a kicked nonlinear system. The pulsed parametric amplifier constitutes the kick. In between kicks the dynamics is determined by the Kerr nonlinearity and damping. In the absence of damping, a classical description of the device exhibits a rich phase-space structure including fixed points of multiple period and chaos. We contrast the classical behavior of the mean intensity with that predicted by quantum dynamics. The mean photon number inside the cavity is shown to undergo regular collapse and revival in the regular region of the phase space and irregular revivals in the chaotic region. When damping is included, the quantum recurrences are rapidly suppressed, and the classical behavior is restored. In this case a stable steady state is possible. The damping represents the effect of photon-number measurements on the system. We also discuss the photon statistics in the steady state.
We consider the behavior of a classically chaotic quantum system, the periodically driven pendulum, under the influence of a continuous measurement of its angular momentum. Without measurement the system shows dynamical localization, a quantum interference effect which suppresses the classical chaotic diffusion in momentum space. The coupling of the system to a measuring device destroys its coherence and thus leads to delocalization. This is studied on the level of individual systems including the recording of the measurement results. For that purpose we analyze the appropriate stochastic Schr\"odinger equation, from which a stochastic quantum map is derived as an effective tool for numerical simulation of measurement trajectories. We show that a continuous momentum measurement restores the diffusive behavior of the system in momentum space, but that for sufficiently low accuracy of measurement the corresponding diffusion constant is smaller than the classical one. This is reflected by an equal diffusive growth of the recorded measurement results. Thus we find signatures of the classical chaos and of the dynamical localization both in the behavior of the measured quantum system and in the corresponding signal of the measuring device.
We analyze schemes for performing quantum nondemolition (QND) measurements in optical cavities. We consider three schemes: (1) measurement of a quadrature phase amplitude using a parametric process, (2) measurement of a quadrature phase amplitude using the optical Kerr effect in a nonlinear fiber, and (3) measurement of the photon number also using the Kerr effect in a fiber. We show that in the second scheme an enhancement of the QND effect may be obtained by making the cavity finesse for the signal larger than that for the probe.
We show how a nonlinear chaotic system, the parametrically kicked nonlinear oscillator, may be realized in the dynamics of a trapped, laser-cooled ion, interacting with a sequence of standing-wave pulses. Unlike the original optical scheme [G. J. Milburn and C. A. Holmes, Phys. Rev. A 44, 4704 (1991)], the trapped ion enables strongly quantum dynamics with minimal dissipation. This should permit an experimental test of one of the quantum signatures of chaos: irregular collapse and revival dynamics of the average vibrational energy.
This paper intends to clarify some issues in the theory of quantum measurement by taking advantage of the self-consistent quantum formulation of nonlinear optics. A quantum-nondemolition measurement of the photon number of an optical pulse can be performed with a nonlinear Mach-Zehnder interferometer followed by a balanced detector. The full quantum-mechanical treatment shows that the shortcut in the description of the quantum-mechanical measurement, the so-called ``collapse of the wave function,'' is not needed for a self-consistent interpretation of the measurement process. Coherence in the density matrix of the signal to be measured is progressively reduced with increasing accuracy of the photon-number determination. The quantum-nondemolition measurement is incorporated in the double-slit experiment and the contrast ratio of the fringes is found to decrease systematically with increasing information on the photon number in one of the two paths. The ``gain'' in the measurement can be made arbitrarily large so that postprocessing of the information can proceed classically.