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

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.

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Dynamical localization and signatures of classical phase space

Physics Letters A ◽

10.1016/s0375-9601(00)00538-7 ◽

2000 ◽

Vol 274 (3-4) ◽

pp. 98-103 ◽

Cited By ~ 19

Author(s):

Farhan Saif

Keyword(s):

Phase Space ◽

Dynamical Localization ◽

Classical Phase Space

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The Projective Hilbert Space as a Classical Phase Space for Nonrelativistic Quantum Dynamics

International Journal of Theoretical Physics ◽

10.1007/s10773-005-8982-2 ◽

2005 ◽

Vol 44 (11) ◽

pp. 2041-2049 ◽

Cited By ~ 4

Author(s):

Igor Bjelaković ◽

Werner Stulpe

Keyword(s):

Hilbert Space ◽

Phase Space ◽

Quantum Dynamics ◽

Nonrelativistic Quantum ◽

Projective Hilbert Space ◽

Classical Phase Space

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A Brief Introduction to Stationary Quantum Chaos in Generic Systems

Nonlinear Phenomena in Complex Systems ◽

10.33581/1561-4085-2020-23-2-172-191 ◽

2020 ◽

Vol 23 (2) ◽

pp. 172-191 ◽

Cited By ~ 1

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.

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Quantum Chaos and Quantum Randomness—Paradigms of Entropy Production on the Smallest Scales

Entropy ◽

10.3390/e21030286 ◽

2019 ◽

Vol 21 (3) ◽

pp. 286

Author(s):

Thomas Dittrich

Keyword(s):

Phase Space ◽

Entropy Production ◽

Quantum Chaos ◽

Quantum Dynamics ◽

Chaotic Dynamics ◽

Quantum Systems ◽

Entropy Flow ◽

Classical Chaos ◽

Classical Models ◽

Closed Systems

Quantum chaos is presented as a paradigm of information processing by dynamical systems at the bottom of the range of phase-space scales. Starting with a brief review of classical chaos as entropy flow from micro- to macro-scales, I argue that quantum chaos came as an indispensable rectification, removing inconsistencies related to entropy in classical chaos: bottom-up information currents require an inexhaustible entropy production and a diverging information density in phase-space, reminiscent of Gibbs’ paradox in statistical mechanics. It is shown how a mere discretization of the state space of classical models already entails phenomena similar to hallmarks of quantum chaos and how the unitary time evolution in a closed system directly implies the “quantum death” of classical chaos. As complementary evidence, I discuss quantum chaos under continuous measurement. Here, the two-way exchange of information with a macroscopic apparatus opens an inexhaustible source of entropy and lifts the limitations implied by unitary quantum dynamics in closed systems. The infiltration of fresh entropy restores permanent chaotic dynamics in observed quantum systems. Could other instances of stochasticity in quantum mechanics be interpreted in a similar guise? Where observed quantum systems generate randomness, could it result from an exchange of entropy with the macroscopic meter? This possibility is explored, presenting a model for spin measurement in a unitary setting and some preliminary analytical results based on it.

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Quantum Chaos and Quantum Randomness—Paradigms of Entropy Production on the Smallest Scales

10.20944/preprints201902.0058.v1 ◽

2019 ◽

Author(s):

Thomas Dittrich

Keyword(s):

Phase Space ◽

Entropy Production ◽

Quantum Chaos ◽

Quantum Dynamics ◽

Chaotic Dynamics ◽

Quantum Systems ◽

Entropy Flow ◽

Classical Chaos ◽

Classical Models ◽

Closed Systems

Quantum chaos is presented as a paradigm of information processing by dynamical systems at the bottom of the range of phase-space scales. Starting with a brief review of classical chaos as entropy flow from micro- to macro-scales, I argue that quantum chaos came as an indispensable rectification, removing inconsistencies related to entropy in classical chaos: Bottom-up information currents require an inexhaustible entropy production and a diverging information density in phase space, reminiscent of Gibbs' paradox in Statistical Mechanics. It is shown how a mere discretization of the state space of classical models already entails phenomena similar to hallmarks of quantum chaos, and how the unitary time evolution in a closed system directly implies the “quantum death” of classical chaos. As complementary evidence, I discuss quantum chaos under continuous measurement. Here, the two-way exchange of information with a macroscopic apparatus opens an inexhaustible source of entropy and lifts the limitations implied by unitary quantum dynamics in closed systems. The infiltration of fresh entropy restores permanent chaotic dynamics in observed quantum systems. Could other instances of stochasticity in quantum mechanics be interpreted in a similar guise? Where observed quantum systems generate randomness, that is, produce entropy without discernible source, could it have infiltrated from the macroscopic meter? This speculation is worked out for the case of spin measurement.

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