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arXiv:nlin/0106018 · Journal of Physics A Mathematical and General

Autocorrelation function of eigenstates in chaotic and mixed systems

Arnd Bäcker, Roman Schubert

Published Jan 14, 2002• 39 citations
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Abstract

We study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits. We obtain an expansion of the autocorrelation function in terms of the correlation distance. For localized states in billiards, like bouncing ball modes or states living on tori, a simple model using only classical input gives good agreement with the exact result. In particular, a prediction for irregular eigenfunctions in mixed systems is derived and tested. For chaotic systems, the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales.

DOI
10.1088/0305-4470/35/3/307

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Under suitable conditions quantum eigenfunctions for mixed systems localize neither on the regular nor on chaotic components in phase space. Such exotic quantum states are called amphibious states, and the flooding of wave functions from regular to chaotic regions, or in the opposite way, are predicted to occur when the eigenstates become amphibious. The emergence of such new states has so far been verified in rather artificial model systems. Here we explore how generically the flooding of wave functions takes place and whether it is an observable phenomenon even in laboratory experiments such as atomic matter-wave systems. Since the flooding occurs as a result of the strong correlation between different types of quantum localization, dynamical tunneling and dynamical localization, there are several competing factors, some of them accelerate but other do not. We study a quasi-periodic kicked rotor with generic phase space to search optimal situations to realize the flooding phenomenon and discuss the compatibility of competing factors between different types of quantum localization.
Abstract Quantum classical correspondence, QCC, is a wide subject of interest from the historical approaches to current studies in simple and complex systems, including time dependent Hamiltonians. Quantum eigenfunctions, or wave-functions, are part of the quantities analysed in recent years, including the implementation due to M V Berry and A Voros in 1977 of a suitable quantity to be considered as the classical eigenfunction counterpart. Here we present the calculation of the statistical moments of eigenfunctions in energy representation for anharmonic oscillators. The classical counter part of the system is fully chaotic. The comparison of the quantum version in order to test for quantum classical correspondence is analysed using a slightly modified formula of the proposed by Berry and Voros. The agreement is good with the corresponding classical counterpart being at the bottom of the quantum version for all the energy range considered.
The autocorrelation function of the force acting on a slow classical system, resulting from interaction with a fast quantum system is calculated following Berry-Robbins, Wilkinson and Jarzynski within the leading order correction to the adiabatic approximation. The time integral of the autocorrelation function is often proportional to the rate of energy transfer between the systems. The fast quantum system is assumed to be chaotic in the classical limit for each configuration of the slow system. An analytic formula is obtained for the finite-time integral of the correlation function, in the framework of random matrix theory (RMT), for a specific dependence on the adiabatically varying parameter. Extension to a wider class of RMT models is discussed. For the Gaussian unitary and symplectic ensembles for long times the time integral of the correlation function vanishes or falls off as a Gaussian with a characteristic time that is proportional to the Heisenberg time, depending on the details of the model. The fall-off is inversely proportional to time for the Gaussian orthogonal ensemble. The correlation function is found to be dominated by the nearest-neighbour level spacings. It was calculated for a variety of nearest-neighbour level spacing distributions, including ones that do not originate from RMT ensembles. The various approximate formulae obtained are tested numerically in RMT. The results shed light on the quantum to classical crossover for chaotic systems. The implications on the possibility to experimentally observe deterministic friction are discussed.