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Physical Review A

Adiabatic mechanism of wave-function localization in the globally chaotic system

Jakub Zakrzewski, Roman Marcinek

Published Dec 1, 1990• 17 citations
Publisher

Abstract

The wave-function localization in a classically, strongly chaotic system---quadric oscillator model---is discussed in the adiabatic approach. A comparison is made between the coordinate-based traditional method and the periodic-orbit-based scheme of Eckhardt, Hose, and Pollak [Phys. Rev. A 39, 3776 (1989)] for channel localized wave functions. A second type of localization of the ring type, observed in the model, is accounted for by an adiabatic approach in polar coordinates.

DOI
10.1103/physreva.42.7172

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The wave-function localization in a classically, strongly chaotic system---quadric oscillator model---is discussed in the adiabatic approach. A comparison is made between the coordinate-based traditional method and the periodic-orbit-based scheme of Eckhardt, Hose, and Pollak [Phys. Rev. A 39, 3776 (1989)] for channel localized wave functions. A second type of localization of the ring type, observed in the model, is accounted for by an adiabatic approach in polar coordinates.
We investigate the classical and quantum dynamics of atoms moving in a phase-modulated standing light field. In both cases the width of the momentum distribution exhibits characteristic oscillations as a function of the modulation amplitude. We argue that at the maxima of these oscillations the system is chaotic, whereas in the valleys it is almost regular. Quantum localization appears only in the chaotic regime. We connect our analysis with a recent experiment [F. Moore et al., Phys. Rev. Lett. 73, 2974 (1994)].
Excited states of a model Hamiltonian with strongly coupled modes are investigated. Comparisons are made between the classical Poincaré surfaces of section and the Husimi function, which is a quantum mechanical phase-space density function. It is found that for the majority of states the Husimi function exhibits local maxima which resemble in shape and location the invariant tori of Poincaré surfaces of section. For certain anomalous states the maxima in the Husimi function bears no such resemblance, and these states are considered as quantum chaotic.
We study the open system dynamics of a circuit QED model operating in the ultrastrong coupling regime. If the resonator is pumped periodically in time the underlying classical system is chaotic. Indeed, the periodically driven Jaynes-Cummings model in the Born-Oppenheimer approximation resembles a Duffing oscillator which in the classical limit is a well-known example of a chaotic system. Detection of the field quadrature of the output field acts as an effective position measurement of the oscillator. We address how such detection affects the quantum chaotic evolution in this bipartite system. We differentiate between single measurement realizations and ensembles of repeated measurements. In the former case a measurement/decoherence induced localization effect is encountered, while in the latter this localization is almost completely absent. This is in marked contrast to numerous earlier works discussing the quantum-classical correspondence in measured chaotic systems. This lack of a classical correspondence under relatively strong measurement induced decoherence is attributed to the inherent quantum nature of the qubit subsystem and in particular to the quantum correlations between the qubit and the field which persist despite the decoherence.
We study chaotic quantum billiards using both microwave cavities and numerical simulations. For the same geometry, viz., a Sinai billiard, agreement to remarkable precision is found for both the eigenvalue magnitudes and the spatial detail of the eigenfunctions. The association of the eigenfunctions with classical periodic orbits is demonstrated, and scarred states are identified. Desymmetrizing the Sinai billiard by slightly moving the central disk is shown to lead to strong localization of the eigenfunction. The calculated eigenstates of the symmetric billiard show an even- and odd-parity pair whose linear combination gives the localized state.
We study quantum localization phenomena in chaotic systems with a parameter. The parametric motion of energy levels proceeds without crossing any other and the defined avoided crossings quantify the interaction between states. We propose the elimination of avoided crossings as the natural mechanism to uncover localized structures. We describe an efficient method for the elimination of avoided crossings in chaotic billiards and apply it to the stadium billiard. We find many scars of short periodic orbits revealing the skeleton on which quantum mechanics is built. Moreover, we have observed strong interaction between similar localized structures.