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Physica Scripta

Relaxation and Fluctuations in Quantum Chaos

Giulio Casati, Giulio Casati, Giulio Casati

Published Jan 1, 2001• 0 citations
Publisher

Abstract

We discuss quantum relaxation and fluctuation phenomena in the light of the corresponding classical behaviour. Quantum exponential localization and tunneling effects lead to the universal algebraic decay of the survival probability P ( t ) ∝ 1/ t . This behaviour is observed in particular in the time dependence of the ionization probability of Rydberg atoms driven by a monochromatic field. At a fixed time the quantum survival probability strongly fluctuates as some parameter is varied. These fluctuations have a fractal structure which is quantitatively related to the time dependence of the survival probability.

DOI
10.1238/physica.topical.090a00166

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•Jan 1, 2004•0 citations
We discuss quantum relaxation and fluctuation phenomena in the light of the corresponding classical behaviour. Quantum exponential localization and tunneling effects lead to the universal algebraic decay of the survival probability P ( t ) ∝ 1/ t . This behaviour is observed in particular in the time dependence of the ionization probability of Rydberg atoms driven by a monochromatic field. At a fixed time the quantum survival probability strongly fluctuates as some parameter is varied. These fluctuations have a fractal structure which is quantitatively related to the time dependence of the survival probability.
We numerically analyze quantum survival probability fluctuations in an open, classically chaotic system. In a quasiclassical regime and in the presence of classical mixed phase space, such fluctuations are believed to exhibit a fractal pattern, on the grounds of semiclassical arguments. In contrast, we work in a classical regime of complete chaoticity and in a deep quantum regime of strong localization. We provide evidence that fluctuations are still fractal, due to the slow, purely quantum algebraic decay in time produced by dynamical localization. Such findings considerably enlarge the scope of the existing theory.
We investigate the long-time limit of quantum localization of the kicked Rydberg atom. The kicked Rydberg atom is shown to possess in addition to the quantum localization time ${\ensuremath{\tau}}_{L}$ a second crossover time tD{t}_{D}tD​ where quantum dynamics diverges from classical dynamics towards increased instability. The quantum localization is shown to vanish as either the strength of the kicks at fixed principal quantum number or the quantum number at fixed kick strength increases. The survival probability as a function of frequency in the transient localization regime ${\ensuremath{\tau}}_{L}<t<{t}_{D}$ is characterized by highly irregular, fractal-like fluctuations.
Abstract We consider time-dependent relaxation of observables in quantum systems of chaotic and regular type. Using statistical arguments and exact numerical solutions we show that the spread of the initial wave function in the Hilbert space and the main characteristics of evolution of observables have certain generic features. The study compares examples of regular dynamics, a completely chaotic case of the Gaussian orthogonal ensemble, a bosonic system with random interactions, and a fully realistic case of the time evolution of various initial non-stationary states in the nuclear shell model. In the case of the Gaussian orthogonal ensemble we show that the survival probability obtained analytically also fully defines the relaxation timescale of observables. This is not the case in general. Using the realistic nuclear shell model and the quadrupole moment as an observable we demonstrate that the relaxation time is significantly longer than defined by the survival probability of the initial state. The full analysis does not show the presence of an analog of the Lyapunov exponent characteristic for examples of classical chaos.
The effect of parametric perturbations on the quantum dynamics of a system with nonintegrable classical limit and a mixed phase space is shown to depend crucially on the local invariant phase space structures sampled by the evolution. Using a probe coherent state four distinct scenarios are considered: (i) large regular regions surrounding primary fixed points; (ii) smaller regular regions corresponding to higher order orbits; (iii) scarred states on hyperbolic fixed points; (iv) chaotic regions. The result that scarring stabilizes quantum dynamics against fluctuations is discussed in the general context of instability in the quantum evolution as well as in relation to recent experiments on microwave ionization of Rydberg atoms.
This thesis investigates quantum transport in the energy space of two paradigm systems of quantum chaos theory. These are highly excited hydrogen atoms subject to a microwave field, and kicked atoms which mimic the delta-kicked rotor model. Both of these systems show a complex dynamical evolution arising from the interaction with an external time-periodic driving force. In particular two quantum phenomena, which have no counterpart on the classical level, are studied: the suppression of classical diffusion, known as dynamical localisation, and quantum resonances as a regime of enhanced transport for the delta-kicked rotor. The first part of the thesis provides new support for the quantitative analogy between energy transport in strongly driven highly excited atoms and particle transport in Anderson-localised solids. A comprehensive numerical analysis of the atomic ionisation rates shows that they obey a universal power-law distribution, in agreement with Anderson localisation theory. This is demonstrated for a one-dimensional model as well as for the real three-dimensional atom. We also discuss the implications of the universal decay-rate distributions for the asymptotic time-decay of the survival probability of the atoms. The second part of the thesis clarifies the effect of decoherence, induced by spontaneous emission, on the quantum resonances which have been observed in a recent experiment with delta-kicked atoms. Scaling laws are derived, based on a quasi-classical approximation of the quantum evolution. These laws describe the shape of the resonance peaks in the mean energy of an experimental ensemble of kicked atoms. Our analytical results match perfectly numerical computations and explain the initially surprising experimental observations. Furthermore, they open the door to the study of the competing effects of decoherence and chaos on the stability of the time evolution of kicked atoms. This stability may be characterised by the overlap of two identical initial states which are subject to different time evolutions. This overlap, called fidelity, is investigated in an experimentally accessible situation.