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Bulletin of the American Physical Society

Break-down of classical-quantum correspondence for time-dependent Hamiltonian systems

Michael Melles, Carlos O Reinhold, J.E. Burgdoerfer

Published May 1, 1993• 0 citations
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Abstract

We compare the classical and quantum evolution for the impulsively driven hydrogen atom which is subject to a sequence of periodic pulses ({open_quotes}kicks{close_quotes}). The classical system possesses a divided phase space with regions of regular and chaotic motion present. We solve the three-dimensional Schrodinger equation using a basis of {approx_equal}1000 states including pseudostates in the continuum. The dynamics is analyzed in terms of quasi-eigenstates and quasi-eigenenergies of the period-one evolution operator. We find a localization and stabilization in the quantum evolution due to the existence of resonances which are absent in the classical evolution. We show that the classical-quantum correspondence can be restored when quantum resonances are destroyed by a stochastic rather than a periodic sequence of kicks.

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•Aug 1, 1992•0 citations
We compare the classical and quantum evolution for the impulsively driven hydrogen atom which is subject to a sequence of periodic pulses ({open_quotes}kicks{close_quotes}). The classical system possesses a divided phase space with regions of regular and chaotic motion present. We solve the three-dimensional Schrodinger equation using a basis of {approx_equal}1000 states including pseudostates in the continuum. The dynamics is analyzed in terms of quasi-eigenstates and quasi-eigenenergies of the period-one evolution operator. We find a localization and stabilization in the quantum evolution due to the existence of resonances which are absent in the classical evolution. We show that the classical-quantum correspondence can be restored when quantum resonances are destroyed by a stochastic rather than a periodic sequence of kicks.
This thesis investigates quantum transport in the energy space of two paradigm systems of quantum chaos theory. These are highly excited hydrogen atoms subject to a microwave field, and kicked atoms which mimic the delta-kicked rotor model. Both of these systems show a complex dynamical evolution arising from the interaction with an external time-periodic driving force. In particular two quantum phenomena, which have no counterpart on the classical level, are studied: the suppression of classical diffusion, known as dynamical localisation, and quantum resonances as a regime of enhanced transport for the delta-kicked rotor. The first part of the thesis provides new support for the quantitative analogy between energy transport in strongly driven highly excited atoms and particle transport in Anderson-localised solids. A comprehensive numerical analysis of the atomic ionisation rates shows that they obey a universal power-law distribution, in agreement with Anderson localisation theory. This is demonstrated for a one-dimensional model as well as for the real three-dimensional atom. We also discuss the implications of the universal decay-rate distributions for the asymptotic time-decay of the survival probability of the atoms. The second part of the thesis clarifies the effect of decoherence, induced by spontaneous emission, on the quantum resonances which have been observed in a recent experiment with delta-kicked atoms. Scaling laws are derived, based on a quasi-classical approximation of the quantum evolution. These laws describe the shape of the resonance peaks in the mean energy of an experimental ensemble of kicked atoms. Our analytical results match perfectly numerical computations and explain the initially surprising experimental observations. Furthermore, they open the door to the study of the competing effects of decoherence and chaos on the stability of the time evolution of kicked atoms. This stability may be characterised by the overlap of two identical initial states which are subject to different time evolutions. This overlap, called fidelity, is investigated in an experimentally accessible situation.
We analyze the quantum dynamics of the one-dimensional periodically kicked Rydberg atom. The time-dependent Schr\"odinger equation is solved by employing a nonunitary representation of the period-one evolution operator in a finite basis set that accounts for the outbound probability flux. We find a direct correspondence between stable classical islands in phase space and the quantum Husimi distributions of stable Floquet states. These results explain the pronounced peak recently found experimentally in the frequency-dependent survival probability of Rydberg states subject to a sequence of half-cycle pulses.
We discuss a top undergoing constant precession around a magnetic field and suffering a periodic sequence of impulsive nonlinear kicks. The squared angular momentum being a constant of the motion the quantum dynamics takes place in a finite dimensional Hilbert space. We find a distinction between regular and irregular behavior for times exceeding the quantum mechanical quasiperiod at which classical behavior, whether chaotic or regular, has died out in quantum means. The degree of level repulsion depends on whether or not the top is endowed with a generalized time reversal invariance. Introduction The quantum treatment of systems capable of chaotic motion in the classical limit is interesting for several reasons. First, there is the desire to see how the classical distinguishability between regular and chaotic motion gradually arises as the system is turned more and more classical by changing a suitable parameter. By a controlled increase of quantum mechanical time scales such as wave packet spreading times or inverse level spacings, for instance, one would like to study the growth of the life time of effectively chaotic evolution of suitable observables. Perhaps an even greater incentive for quantum mechanical investigations lies in the question whether quantum chaos can be more than a mere transient mimicry of classical chaos. If so, we need intrinsically quantum mechanical criteria to distinguish regular an irregular behavior; the relation of such quantum criteria to the traditional ones for classical chaos would have to be clarified.
We study the three-dimensional (3D) unidirectionally kicked Rydberg atom. For parabolic initial states elongated in the direction of the kicks we show that the ionization of the quantum system is suppressed as compared to the classical counterpart and that the quantum wave function is localized along all degrees of freedom, whereas the classical system is globally diffusive. We discuss the connection to the previously studied one-dimensional (1D) model of the kicked Rydberg atom and verify that the 1D model is a good approximation to the 3D quantum case in the limiting case of the most elongated initial states. We further study the quantum phase-space distribution (Husimi distribution) of the eigenstates of the period-one time-evolution (Floquet) operator and show that the eigenstates are localized in phase space. For the most elongated parabolic initial state, we are able to identify the unstable periodic orbits around which Floquet states localize. We discuss the possibility of observing quantum localization in high Rydberg states in n>100.n>100.n>100.
The evolution of classically chaotic quantum systems is analyzed within the formalism of Quantum Pseudo-Probability Distributions. Due to the deep connections that a quantum system shows with its classical correspondent in this representation, the Pseudo-Probability formalism appears to be a useful method of investigation in the field of "Quantum Chaos." In the first part of the thesis we generalize this formalism to quantum systems containing spin operators. It is shown that a classical-like equation of motion for the pseudo-probability distribution ρw can be constructed, dρw/dt = (L_CL + L_QGD)ρw, which is rigorously equivalent to the quantum von Neumann-Liouville equation. The operator L_CL is undistinguishable from the classical operator that generates the semiclassical equations of motion. In the case of the spin-boson system this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spin-boson Hamiltonian the joint action of L_CL and L_QGD is illustrated. It is shown that the latter operator, L_QGD makes the spin system 'remember' its quantum nature, and competes with the irreversibility induced by the former operator. In the second part we test the idea of the enhancement of the quantum uncertainty triggered by the classical chaos by investigating the analogous effect of diffusive excitation in periodically kicked quantum systems. The classical correspondents of these quantum systems exhibit, in the chaotic region, diffusive behavior of the unperturbed energy. For the Quantum Kicked Harmonic Oscillator, in the case of quantum resonances, we provide an exact solution of the quantum evolution. This proves the existence of a deterministic drift in the energy increase over time of the system considered. More generally, this "superdiffusive" excitation of the energy is due to coherent quantum mechanical tunnelling between degenerate tori of the classical phase space. In conclusion we find that some of the quantum effects resulting from this fast increase do not have any classical counterpart, they are mainly tunnelling processes. This seems to be the first observation of an effect of this kind.