Phase-space localization: Topological aspects of quantum chaos

1990 ◽  
Vol 65 (25) ◽  
pp. 3076-3079 ◽  
Author(s):  
P. Leboeuf ◽  
J. Kurchan ◽  
M. Feingold ◽  
D. P. Arovas
Keyword(s):  
Phase Space ◽  
Quantum Chaos ◽  
Space Localization

scholarly journals Phase space localization of chaotic eigenstates: Violating ergodicity

2000 ◽  
Vol 63 (1) ◽  
Author(s):  
Arul Lakshminarayan ◽  
Nicholas R. Cerruti ◽  
Steven Tomsovic
Keyword(s):  
Phase Space ◽  
Space Localization

scholarly journals A Brief Introduction to Stationary Quantum Chaos in Generic Systems

2020 ◽  
Vol 23 (2) ◽  
pp. 172-191 ◽  
Author(s):  
Marko Robnik
Keyword(s):  
Phase Space ◽  
Time Scale ◽  
Quantum Chaos ◽  
Mixed Type ◽  
Localized States ◽  
Quantum Evolution ◽  
Density Level ◽  
Level Spacing ◽  
Transport Time ◽  
Classical Phase Space

We review the basic aspects of quantum chaos (wave chaos) in mixed-type Hamiltonian systems with divided phase space, where regular regions containing the invariant tori coexist with the chaotic regions. The quantum evolution of classically chaotic bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behaviour occurs, as the motion is always almost periodic. However, the study of the stationary solutions of the Schrödinger equation in the quantum phase space (Wigner functions or Husimi functions) reveals precise analogy of the structure of the classical phase portrait. In classically integrable regions the spectral (energy) statistics is Poissonian, while in the ergodic chaotic regions the random matrix theory applies. If we have the mixed-type classical phase space, in the semiclassical limit (short wavelength approximation) the spectrum is composed of Poissonian level sequence supported by the regular part of the phase space, and chaotic sequences supported by classically chaotic regions, being statistically independent of each other, as described by the Berry-Robnik distribution. In quantum systems with discrete energy spectrum the Heisenberg time tH = πℏ/ΔE, where ΔE is the mean level spacing (inverse energy level density), is an important time scale. The classical transport time scale tT (transport time) in relation to the Heisenberg time scale tH (their ratio is the parameter α = tH / tT ) determines the degree of localization of the chaotic eigenstates, whose measure A is based on the information entropy. We show that A is linearly related to the normalized inverse participation ratio. We study the structure of quantum localized chaotic eigenstates (their Wigner and Husimi functions) and the distribution of localization measure A. The latter one is well described by the beta distribution, if there are no sticky regions in the classical phase space. Otherwise, they have a complex nonuniversal structure. We show that the localized chaotic states display the fractional power-law repulsion between the nearest energy levels in the sense that the probability density (level spacing distribution) to find successive levels on a distance S goes like ∝ S β for small S , where 0 ≤ β ≤ 1, and β = 1 corresponds to completely extended states, and β = 0 to the maximally localized states. β goes from 0 to 1 when α goes from 0 to ∞, β is a function of <A>, as demonstrated in the quantum kicked rotator, the stadium billiard, and a mixed-type billiard.

scholarly journals EFFECT OF CLASSICAL TRAPPING IN QUANTUM CHAOS: AN ANOMALOUS DIFFUSION APPROACH

2008 ◽  
Vol 22 (30) ◽  
pp. 5261-5277 ◽  
Author(s):  
JIAO WANG ◽  
ANTONIO M. GARCIA-GARCIA
Keyword(s):  
Phase Space ◽  
Anomalous Diffusion ◽  
Quantum Chaos ◽  
Quantum Dynamics ◽  
Classical Case ◽  
Dynamical Localization ◽  
Classical Motion ◽  
Classical Phase Space ◽  
Breaking Time ◽  
The Impact

We study generic effects on the quantum dynamics of classical trapping-leaking mechanism by investigating in detail the 2δ-kicked rotors whose classical phase space is partitioned into momentum cells separated by trapping regions which slow down the motion. We focus on a range of parameters where the dynamics is generic, namely, the phase space has no stable islands. As a consequence of the trapping-leaking mechanism, we show that the classical motion is described by a process of anomalous diffusion. We investigate in detail the impact of the underlying classical anomalous diffusion on the quantum dynamics with special emphasis on the phenomenon of dynamical localization. Based on the study of the quantum density of probability, its second moment and the return probability, we identify a region of weak dynamical localization where the quantum diffusion is still anomalous but the diffusion rate is slower than in the classical case. Moreover, we examine how other relevant time scales, such as the quantum-classical breaking time and the one related to the beginning of full dynamical localization, are modified by the classical anomalous diffusion. Finally, we discuss the relevance of our results for understanding the role of classical cantori in quantum mechanics.

scholarly journals Quantum Chaos in Mixed Phase Space and the Julia Set

2003 ◽  
Vol 150 ◽  
pp. 267-280 ◽  
Author(s):  
Akira Shudo ◽  
Kensuke S. Ikeda
Keyword(s):  
Phase Space ◽  
Quantum Chaos ◽  
Julia Set ◽  
Mixed Phase
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