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Physical Review A

States of a dynamically driven spin. II. Classical model and the quantum-to-classical limit

L. E. Ballentine

Published Oct 1, 1991• 6 citations
Publisher

Abstract

A classical spin, dynamically driven by a polarized beam, is shown to exhibit a complex variety of behaviors, incuding stable fixed points, bifurcations, limit cycles, and chaos. In the classical (large-spin) limit of the corresponding quantum model, the probability distributions for the spin components converge to their classical counterparts, even though the classical motions are qualitatively different for different ranges of parameters. However, the nature of the quantum-to-classical limit has some unexpected features, which are discussed.

DOI
10.1103/physreva.44.4133

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A classical spin, dynamically driven by a polarized beam, is shown to exhibit a complex variety of behaviors, incuding stable fixed points, bifurcations, limit cycles, and chaos. In the classical (large-spin) limit of the corresponding quantum model, the probability distributions for the spin components converge to their classical counterparts, even though the classical motions are qualitatively different for different ranges of parameters. However, the nature of the quantum-to-classical limit has some unexpected features, which are discussed.
Classical and quantum models of a localized spin driven by a polarized-spin beam have been shown to exhibit a rich variety of dynamical behaviors. In the classical model these include stable limit points, limit cycles, and chaos; however, in the quantum model the density matrix of the localized spin generally approaches a steady state. This paradox is resolved by considering the autocorrelation functions of the spin components, which reveal the expected quantum-classical correspondence. The noncommuting of the limits j\ensuremath{\rightarrow}\ensuremath{\infty} and t\ensuremath{\rightarrow}\ensuremath{\infty} is explained in this model by the limiting behavior of certain eigenvalue distributions, and the significance of these limits is discussed. The study also highlights the unresolved problem of interpreting the complex correlation functions of noncommuting operators.
``Dynamical driving'' means driving a system by another dynamical system, in contrast to driving by an external force (which acts on the system but is not acted upon by the system). The model consists of a localized spin driven by a polarized beam of spins. A remarkably complex variety of behaviors is possible as a result of the competition between the aligning effect of interaction with the beam and the precession due to the external magnetic field. The state of the localized spin can evolve from pure to mixed, or from mixed to pure, and its motion is characterized by an attractor. The final state of the driven spin is usually independent of its initial state, so the model provides an example of quantum-state preparation. These effects cannot be produced by a prescribed external force.
We study the quantum-classical correspondence for systems with interacting spin particles that are strongly chaotic in the classical limit. This is done in the presence of constants of motion associated with the fixed angular momenta of individual spins. Our analysis of the Lyapunov spectra reveals that the largest Lyapunov exponent agrees with the Lyapunov exponent that determines the local instability of each individual spin moving under the influence of all other spins. Within this picture, we introduce a rigorous and simple test of ergodicity for the spin motion, and use it to identify when classical chaos is both strong and global in phase space. In the quantum domain, our analysis of the Hamiltonian matrix in a proper representation allows us to obtain the conditions for the onset of quantum chaos as a function of the model parameters. From the comparison between the quantum and classical domains, we demonstrate that quantum quantities, such as the local density of states (LDoS) and the shape of the chaotic eigenfunctions written in the noninteracting many-body basis, have well-defined classical counterparts. We also find a relationship between the Kolmogorov-Sinai entropy and the width of the LDoS, which is useful for studies of many-body dynamics.
This chapter presents an investigation of the dynamics of one anisotropic spin in an external, periodically time-dependent field. The simple quantum system displays chaotic behavior in the classical limit. Chaotic behavior in classical deterministic systems is characterized by an extremely sensitive dependence on initial conditions. The statistics of quasienergy levels are discussed for our driven spin system with a twofold purpose. The statistics of quasienergy levels is equivalent to the energy level statistics in a time-independent system and the dependence of the quasienergy level statistics on the quantum parameter is investigated. The statistical properties of the spectrum of a quantum problem in its classical limit are associated with the regularity properties of the corresponding classical system. Using the complementary approaches of phase space analysis and level statistics we have shown in both schemes that for large spins, i.e. in the semiclassical regime, stochasticity can be discussed using concepts developed for classical systems.
Dynamical chaos in quantum systems (quantum chaos) has been studied by a large number of physicists [1-7]. It has been shown that quantum chaos is possible only in the quasiclassical region, for example in the case of quantum states near the dissociation limit in molecules or near the ionization limit in atoms and molecules. However, the quantum equations of motion are linear and hence they cannot contain dynamical chaos. In the present paper we discuss the solution of quantum problems which are nonlinear in the classical limit. It will be show that in solving the quantum problem in this case it is necessary to use the solution of the nonlinear classical problem forming the classical limit of the quantum problem. i. We note first that here we are not interested in the usual field nonlinearities considered, for example, in quantum optics. This is because field nonlinearities cannot determine the bifurcation properties of the quantum equations of motion since the latter are linear in the wave function ~ or density matrix p. In other words, dynamical chaos, which is intimately connected with the bifurcation properties of the corresponding equations, is not determined by nonlinearities in the fields in quantum equations of motion linear in ~ or 0o
We study chaos in a classical limit of the Sachdev-Ye-Kitaev (SYK) model obtained in a suitably defined large-SSS limit. The low-temperature Lyapunov exponent is found to depend linearly on temperature, with a slope that is parametrically different than in the quantum case: It is proportional to N/SN/SN/S. The classical dynamics can be understood as the rotation of an NNN-dimensional body with a random inertia tensor, corresponding to the random couplings of the SYK Hamiltonian. This allows us to find an extensive number of fixed points, corresponding to the body's principal axes of rotation. The thermodynamics is mapped to the ppp-spin model with p=2p=2p=2, which exhibits a spin glass phase at low temperature whose presence does not preclude the existence of chaos.