We outline a rather extraordinary series of similarities between classical and quantal behavior in the limit of adiabatic time changes. These include the power laws for the goodness of the respective invariants for isolated eigenstates and invariant tori for integrable systems, the nature of the breakdown of the invariances—level crossing in quantum systems and the role of ever present non-linear resonances is examined in the case of generically non-integrable classical dynamics—and the perhaps surprising relationship for fully chaotic systems where sufficiently slow switching in either classical or quantal systems precisely preserves the number of energy levels up to a given energy. For suitably small values of Planck's constant these similarities yield clear examples of the Bohr correspondence principle linking classical and quantum mechanics; for larger values the details in the classical picture are quenched in the quantum.
We outline a rather extraordinary series of similarities between classical and quantal behavior in the limit of adiabatic time changes. These include the power laws for the goodness of the respective invariants for isolated eigenstates and invariant tori for integrable systems, the nature of the breakdown of the invariances—level crossing in quantum systems and the role of ever present non-linear resonances is examined in the case of generically non-integrable classical dynamics—and the perhaps surprising relationship for fully chaotic systems where sufficiently slow switching in either classical or quantal systems precisely preserves the number of energy levels up to a given energy. For suitably small values of Planck's constant these similarities yield clear examples of the Bohr correspondence principle linking classical and quantum mechanics; for larger values the details in the classical picture are quenched in the quantum.
The famous Nils Bohr's quantum-classical correspondence principle states that the classical mechanics is a limiting case of the more general quantum mechanics. This implies that ``under certain conditions" quantum laws of motion become equivalent to classical laws. One of the conditions is fairly obvious: the corresponding classical action should be very large as compared with the Planck's constant. But this condition is not sufficient. Suppression of effects of quantum interference ("decoherence") is the phenomenon of primary importance for understanding of the Bohr's principle. Being, in essence, of quite general nature, this problem takes on special significance in the case of non-linear classically chaotic quantum systems. Whereas the rapid decay of phase correlations is an underlying feature of the classical dynamical chaos, the "quantum chaos" by itself is not capable of destroying the quantum phase coherence. Strictly speaking, any initially pure quantum state remains pure during an arbitrary long unitary evolution. Basically, formation of incoherent mixed states (decoherence) results only from the process of preparation of a mixed initial state or interaction with a noisy environment. Just the decoherence brigdes the quantum and classical worlds. Peculiarities of the time behavior of Peres fidelity, state purity, Shanon and von Neumann entropies are analyzed in detail. We demonstrate the ways the decoherence shows up in periodically driven systems that can be associated with Ramsey-type interferomentry experiments with ion traps. Finally, decoherence in ballistic electron quantum transport caused by interaction with a disordered environment is considered.
A connection between quantum mechanical and classical adiabatic invariance is shown for the special case of a charged particle moving in a homogeneous, time-dependent magnetic field. The relationship consists of the fact that the transition probability between energy eigenstates corresponding to the magnnetic field at two different times is a function of a certain classical adiabatic invariant alone. From this relation follows the well-known result that during sufficiently slow changes of the magnetic field the particle will remain in one quantum state which passes continuously through the different instantaneous energies. The adiabatic invariance of this classical quantity is shown directly by solving the equations of niotion when the magnetic field changes linearly with time. (auth)
A novel concept for quantum chaos in spin systems is proposed which differs distinctively from concepts currently in use. It is argued (a) that the definition of quantum chaos cannot be based on the correspondence principle and (b) that quantum chaos is not equivalent to statistical behavior. Quantum chaos in spin systems implies the existence of a new type of spectrum consisting of excitations which do not form regular patterns with a multiparameter continuum structure in the thermodynamic limit, thus removing the constraint imposed by van Hove singularities that time-dependent correlation functions for pure quantum states cannot decay more rapidly than as powers of t. The irregular spectrum is caused by strong level repulsion resulting from the lack of a sufficient number of conservation laws in nonintegrable quantum many-body systems. The similarities and differences between quantum and classical chaos and, more generally, between quantum (non)integrability and classical (non)integrability are discussed in detail.
Abstract The postulate of the existence of stationary states in multiply periodic dynamical systems requires that if the condition of such a system, when quantised, is changed in any way by the application of an external field or by the alteration of one of the internal constraints, the new state of the system must also be correctly quantised. It follows that the laws of classical mechanics cannot in general be true, even approximately, during the transition. There is one kind, of change, however, during which one may expect the classical laws to hold, namely, the so-called adiabatic change, which takes place infinitely slowly and regularly, so that the system practically remains multiply periodic all the time. In this case the quantum numbers cannot change, and it should be possible to deduce from the classical laws that the quantum integrals remain invariant. This was attempted by Burgers, who showed that they are invariant provided there are no linear relations of the type Σrmrωr = 0 (1) between the frequencies ωr, of the system, where the mr are integers. In general, however, the frequencies will alter during the adiabatic change, and in so doing will pass through an infinity of values for which relations such as (1) hold. A closer investigation is therefore necessary, as was pointed out by Burgers himself. In the following work, conditions which are rigorously sufficient to ensure the invariance of the quantum integrals, are obtained in such a form that it is possible for one to see whether they are satisfied or not without having to integrate the equations of adiabatic motion.
Abstract We present a unified formalism which identifies chaos in both quantum and classical systems in an equivalent manner by means of adiabatic transformations . The complexity of adiabatic transformations which preserve classical time-averaged trajectories (quantum eigenstates) in response to Hamiltonian deformations serves as a measure of chaos. This complexity is quantified by the (properly regularized) fidelity susceptibility or, more generally, by the geometric tensor. Physically this measure quantifies (i) long time instabilities of physical observables due to small changes in the Hamiltonian of the system and (ii) irregularity of physical observables contained in low frequency noise. Our exposition clearly showcases the common structures underlying quantum and classical chaos and allows us to distinguish integrable, chaotic but non-thermalizing, and ergodic/mixing regimes. We apply the fidelity susceptibility to a model of two coupled spins and demonstrate that it successfully predicts the universal onset of chaos, both for finite spin S and in the classical limit S → ∞ . Interestingly, we find that finite S effects are anomalously large close to integrability.
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.