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Digest on Nonlinear Optics: Materials, Phenomena and Devices

Coherence And Chaos In A Quantum Optical System

Gerard J Milburn

Published Jan 1, 1990• 1 citation
Publisher

Abstract

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.

DOI
10.1109/nlo.1990.695897

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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.
New experimental results are presented for short-pulse microwave ionization of highly excited hydrogen atoms. A comparison of these results with quantum numerical computations and analytical predictions provides for the first time experimentally grounded evidence of the localization phenomenon that leads to the suppression of the quantum version of the chaotic diffusion in action space occurring in the classical limit. The onset of chaotic motion in externally driven classical systems is a key problem in nonlinear dynamics that is now well understood in its essential aspects. In many physically interesting cases, as the perturbation strength increases beyond some critical value, the system starts absorbing energy in a diffusive way. The question whether “diffusive” excitation processes can take place also in externally driven quantum systems is then a very interesting one for the physics of atoms and molecules in external electromagnetic fields. This is a deep question involving the nature and the validity of the quasiclassical approximation when the underlying classical dynamics is chaotic. The question of “quantum diffusion” was first addressed in a simple model system, the kicked rotator. It was chosen because the features of classical chaos in it were relatively clean and well understood. Besides that, the numerical simulation of its quantum dynamics could be easily accomplished. The major indication of this model was that quantum mechanics suppresses the classical chaotic diffusion in action via a destructive interference effect similar to that responsible for the Anderson localization well known in condensed matter physics.
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well known that quantum coherence can obliterate chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact classical chaotic behavior. How effectively decoherence works to sustain chaos, and how the resultant behavior qualitatively differs from the quantum picture, depending on the coupling of the system with the environment and the spectral density and temperature of the environment. We show how recurrence in the quantum cat map is lost and classical ergodicity is recovered due to the effect of the environment. Quantum coherence and diffusion suppression are instrumental to dynamical localization for the kicked rotor. We show how environment-induced effects can destroy this localization. Such effects can also be understood as resulting from external noises driving the system. Peculiar to decohering chaotic systems is the apparent transition from reversible to irreversible dynamics. We show such transitions in the quantum cat map and the kicked rotor and distinguish it from apparent irreversibility originating from dynamical instability and imprecise measurements. By performing a time reversal on and following the quantum kicked rotor dynamics numerically, we show how the otherwise reversible quantum dynamics acquires an arrow of time upon the introduction of noise or interaction with an environment.
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.
It is well established that classical chaotic motion is suppressed by quantum mechanics. The paradigm theoretical system has been the ?-kicked rotor, where the suppression manifests itself as dynamical localization. This effect is saturation of momentum growth after the quantum break time, with a resulting exponentially localized distribution. Classical analyses of this system have also identified regimes of anomalous diffusion, characterized by “Levy-type” flights in phase space [1]. This behavior is considerably more complex than simple diffusion and has attracted much interest in recent years. The quantum system was predicted to exhibit an increase in localization length near regions of anomalous diffusion [2], and recent theoretical work predicts very large growth of energy in this case, even for systems that are far from the semiclassical limit [3].
Analytical and numerical results are presented on the mechanism of excitation and ionization of hydrogen atoms under microwave fields. In particular the existence is predicted of a critical value of the microwave field, the quantum delocalization border, above which the quantum packet delocalizes and strong excitation and ionization takes place. Below the quantum border, the packet is localized even though the correseponding classical system is chaotic and obeys a diffusion equation. These studies reveal some other unexpected new features of quantum dynamics which aslo could be observed in laboratory experiments and provides a quantum theory for subthreshold ionization. Keywords: Quantum Chaos; Energy Level Statistics; Quantum Kicked Rotator; Anderson Localization; Microwave Excitation of Hydrogen Atoms; Keldish Parameter; Quantum Diffusion; Italy.