Quantum Chaos Versus Classical Chaos: Why is Quantum Chaos Weaker?

We discuss the questions: How to compare quantitatively classical chaos with quantum chaos? Which one is stronger? What are the underlying physical reasons

Exact field ionization rates in the barrier suppression-regime from numerical TDSE calculations

Numerically determined ionization rates for the field ionization of atomic hydrogen in strong and short laser pulses are presented. The laser pulse intensity reaches the so-called "barrier suppression ionization" regime where field ionization occurs within a few half laser cycles. Comparison of our numerical results with analytical theories frequently used shows poor agreement. An empirical formula for the "barrier suppression ionization"-rate is presented. This rate reproduces very well the course of the numerically determined ground state populations for laser pulses with different length, shape, amplitude, and frequency. Number(s): 32.80.RmComment: Enlarged and newly revised version, 22 pages (REVTeX) + 8 figures in ps-format, submitted for publication to Physical Review A, WWW: http://www.physik.tu-darmstadt.de/tqe

UNIFIED THEORY OF LYAPUNOV EXPONENTS AND A POSITIVE EXAMPLE OF DETERMINISTIC QUANTUM CHAOS

Faisal F, SCHWENGELBECK U. UNIFIED THEORY OF LYAPUNOV EXPONENTS AND A POSITIVE EXAMPLE OF DETERMINISTIC QUANTUM CHAOS. PHYSICS LETTERS A. 1995;207(1-2):31-36.A unified theory of quantum Lyapunov exponents, based on the Hamilton-Jacobi formulation of quantum mechanics, is applied to Weigert's quantum cat map. It is shown to provide a definite positive example of deterministic quantum chaos in terms of extreme sensitivity on initial conditions and a positive definite Lyapunov exponent

IONIZATION OF THE ONE-DIMENSIONAL COULOMB ATOM IN AN INTENSE LASER FIELD

SCHWENGELBECK U, Faisal F. IONIZATION OF THE ONE-DIMENSIONAL COULOMB ATOM IN AN INTENSE LASER FIELD. PHYSICAL REVIEW A. 1994;50(1):632-640.Interaction of the one-dimensional (1D) Coulomb atom with an intense radiation field is analyzed using the ''hard'' Coulomb potential V(c)(x)=-1/\x\ and the ''soft'' Coulomb potential V(x)=-1/square-root x2 + 1. Evolution of the probability wave packet and the photoelectron spectra is simulated numerically. In strong fields it is found (a) that the electron wave packet can break up into individual subpackets, (b) that there is a one-to-one correlation between the subpackets in space and the above-threshold-ionization peaks in energy, and (c) that a spatial confinement of the probability density can occur at a very high intensity. Finally, an example is given which indicates that the 1D ''hard'' Coulomb atom is less stable than the 1D ''soft'' Coulomb atom

Transition to deterministic chaos in a periodically driven quantum system and breaking of the time-reversal symmetry

Schwengelbeck U, Faisal F. Transition to deterministic chaos in a periodically driven quantum system and breaking of the time-reversal symmetry. PHYSICAL REVIEW E. 1997;55(5):6260-6263.The phenomenon of transition from regular to chaotic dynamics in a periodically driven quantum system is demonstrated. The associated quantum Lyapunov numbers are determined analytically. A numerical experiment is made to test the nature of time evolution predicted by the theory. In contrast to the evolution in the regular domain, the passage to deterministic quantum chaos is found to break the time-reversal symmetry of the quantum dynamics, whenever the latter cannot be followed with infinite precision

DEFINITION OF LYAPUNOV EXPONENTS AND KS ENTROPY IN QUANTUM DYNAMICS

SCHWENGELBECK U, Faisal F. DEFINITION OF LYAPUNOV EXPONENTS AND KS ENTROPY IN QUANTUM DYNAMICS. PHYSICS LETTERS A. 1995;199(5-6):281-286.We propose a definition of quantum Lyapunov exponents and the associated KS entropy, which allows a rigorous characterization of regular or irregular quantum dynamics in the same terms as in classical dynamics. An illustration of the use of this concept is given for the quantum standard map

Eigenstate expansion method for simulations of non-perturbative multiphoton processes

Nurhuda M, Faisal F, Schwengelbeck U. Eigenstate expansion method for simulations of non-perturbative multiphoton processes. COMPUTER PHYSICS COMMUNICATIONS. 2001;134(3):291-306.A method to solve the time-dependent Schrodinger equation of the hydrogen atom and hydrogen-like ions in intense laser pulses, based on an expansion of the total wavefunction in bound and continuum eigenstates of the unperturbed system, is presented. The problem arising from the well-known singular continuum-continuum dipole matrix elements is fully analyzed and resolved by a combination of analytical integration of the singular contribution and a numerical integration of the singularity-free difference contribution. Unlike in usual finite-difference methods using a spatial grid, in the present approach, probability amplitudes of bound-bound, bound-free and free-free transitions are obtained without additional calculations. The efficacy and accuracy of the present method is demonstrated by calculating above-threshold ionization spectra (ATI) and high harmonic generation spectra (HHG) for different values of laser parameters. Finally, the results obtained by using the present method are compared with that of a finite-difference method involving a spatial grid. The present method proves to be an efficient alternative to the usual spatial grid methods for the solution of the time-dependent Schrodinger equation and the estimation of ATI and HHG spectra in case of intense laser pulses. (C) 2001 Elsevier Science B.V. All rights reserved

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

Faisal F, Schwengelbeck U. de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance. In: Pramana. PRAMANA-JOURNAL OF PHYSICS. Vol 51. INDIAN ACADEMY SCIENCES; 1998: 585-595.A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking 'wave function collapse' or 'measurements'