A Comprehensive Thesis on Quantum Chaos Theory

Journal of Mathematical Physics, 1994

A new definition of the entropy of a given dynamical system and of an instrument describing the measurement process is proposed within the operational approach to quantum mechanics. It generalizes other definitions of entropy, in both the classical and quantum cases. The Kolmogorov-Sinai (KS) entropy is obtained for a classical system and the sharp measurement instrument. For a quantum system and a coherent states instrument, a new quantity, coherent states entropy, is defined. It may be used to measure chaos in quantum mechanics. The following correspondence principle is proved: the upper limit of the coherent states entropy of a quantum map as h-+0 is less than or equal to the KS-entropy of the corresponding classical map.

Entropy, 2017

The research on quantum chaos finds its roots in the study of the spectrum of complex nuclei in the 1950s and the pioneering experiments in microwave billiards in the 1970s. Since then, a large number of new results was produced. Nevertheless, the work on the subject is, even at present, a superposition of several approaches expressed in different mathematical formalisms and weakly linked to each other. The purpose of this paper is to supply a unified framework for describing quantum chaos using the quantum ergodic hierarchy. Using the factorization property of this framework, we characterize the dynamical aspects of quantum chaos by obtaining the Ehrenfest time. We also outline a generalization of the quantum mixing level of the kicked rotator in the context of the impulsive differential equations.

Journal of Physics A: Mathematical and General, 1998

We use multi-time correlation functions of quantum systems to construct random variables with statistical properties that reflect the degree of complexity of the underlying quantum dynamics.

1996

—A critical analysis of the present-day concept of chaos in quantum systems as nothing but a " quantum signature " of chaos in classical mechanics is given. In contrast to the existing semi-intuitive guesses, a definition of classical and quantum chaos is proposed on the basis of the Liouville–Arnold theorem: a quantum chaotic system featuring N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) specified by the symmetry of the Hamiltonian of the system. Quantitative measures of quantum chaos that, in the classical limit, go over to the Lyapunov exponent and the classical stability parameter are proposed. The proposed criteria of quantum chaos are applied to solving standard problems of modern dynamical chaos theory.

Physica D: Nonlinear Phenomena, 1999

We consider the quantum evolution of classically chaotic systems in contact with surroundings. Based onh-scaling of an equation for time evolution of the Wigner's quasiprobability distribution function in presence of dissipation and thermal diffusion we derive a semiclassical equation for quantum fluctuations. This identifies an early regime of evolution dominated by fluctuations in the curvature of the potential due to classical chaos and dissipation. A stochastic treatment of this classical fluctuations leads us to a Fokker-Planck equation which is reminiscent of Kramers' equation for thermally activated processes. This reveals an interplay of three aspects of evolution of quantum noise in weakly dissipative open systems; the reversible Liouville flow, the irreversible chaotic diffusion which is characteristic of the system itself, and irreversible dissipation induced by the external reservoir. It has been demonstrated that in the dissipation-free case a competition between Liouville flow in the contracting direction of phase space and chaotic diffusion sets a critical width in the Wigner function for quantum fluctuations. We also show how the initial quantum noise gets amplified by classical chaos and ultimately equilibrated

Communications in Mathematical Physics, 1998

We review the notion of dynamical entropy by Connes, Narnhofer and Thirring and relate it to Quantum Chaos. A particle in a periodic potential is used as an example. This is worked out in the classical and the quantum mechanical framework, for the single particle as well as for the corresponding gas. The comparison does not only support the general assertion that quantum mechanics is qualitatively less chaotic than classical mechanics. More speci cally, the same dynamical mechanism by which a periodic potential leads to a positive dynamical entropy of the classical particle may reduce the dynamical entropy of the quantum gas in comparison to free motion.

Quantum dynamics of a classically chaotic 1D system in the presence of external noise is studied. Stability and reversibility properties of the motion (characterized by the Peres fidelity) as functions of the noise level σ are considered. We calculate fidelity analytically in the cases of weak and very strong noise and find critical value, σc(t), below which the effect of perturbation remains small. Decay of critical perturbation with time is found to be power-like after the Ehrenfest time tE. An estimation of the decoherence time t d (σ) is presented after which the averaged density matrix becomes diagonal and its evolution turns into a Markovian process.

We discuss how the concept of the quantum action can be used to characterize quantum chaos. As an example we study quantum mechanics of the inverse square potential in order to test some questions related to quantum action. Quantum chaos is discussed for the 2-D hamiltonian system of harmonic oscillators with anharmonic coupling.