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Chaos An Interdisciplinary Journal of Nonlinear Science

The quantized D-transformation

Marcos Saraceno, Raúl O. Vallejos

Published Jun 1, 1996• 31 citations
Publisher

Abstract

We construct a new example of a quantum map, the quantized version of the D-transformation, which is the natural extension to two dimensions of the tent map. The classical, quantum and semiclassical behavior is studied. We also exhibit some relationships between the quantum versions of the D-map and the parity projected baker's map. The method of construction allows a generalization to dissipative maps which includes the quantization of a horseshoe. (c) 1996 American Institute of Physics.

DOI
10.1063/1.166164

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We construct a new example of a quantum map, the quantized version of the D-transformation, which is the natural extension to two dimensions of the tent map. The classical, quantum and semiclassical behavior is studied. We also exhibit some relationships between the quantum versions of the D-map and the parity projected baker's map. The method of construction allows a generalization to dissipative maps which includes the quantization of a horseshoe. (c) 1996 American Institute of Physics.
The quantization of a completely chaotic system-the baker's transformation-is investigated using the fact that the relationship between classical and quantum mechanics is directly analogous to the one between ray and wave optics. A class of optical systems is constructed for which the ray dynamics is governed by the baker map. Applying wave-optical methods to these physical systems then gives, in the standard way, their corresponding quantum dynamics. The result is that in certain cases the quantum propagator is identical to that found previously using an ad hoc mathematical quantization procedure. However, this is not always so-for other systems with the same classical (ray) limit there are important differences. In particular, the propagator need not be block-diagonal in its q-p' representation, as had previously been assumed, although it always tends to this basic form in the semiclassical limit.
We study the behavior of an open quantum system, with an N-dimensional space of states, whose density matrix evolves according to a nonunitary map defined in two steps: A unitary step, where the system evolves with an evolution operator obtained by quantizing a classically chaotic map (baker's map and Harper's map are the two examples we consider). A nonunitary step where the evolution operator for the density matrix mimics the effect of diffusion in the semiclassical (large N) limit. The process of decoherence and the transition from quantum to classical behavior are analyzed in detail by means of numerical and analytic tools. The existence of a regime where the entropy grows with a rate that is independent of the strength of the diffusion coefficient is demonstrated. The nature of the processes that determine the production of entropy is analyzed.
Classical chaotic systems are distinguished by their sensitive dependence on initial conditions. The absence of this property in quantum systems has lead to a number of proposals for perturbation-based characterizations of quantum chaos, including linear growth of entropy, exponential decay of fidelity, and hypersensitivity to perturbation. All of these accurately predict chaos in the classical limit, but it is not clear that they behave the same far from the classical realm. We investigate the dynamics of a family of quantizations of the baker's map, which range from a highly entangling unitary transformation to an essentially trivial shift map. Linear entropy growth and fidelity decay are exhibited by this entire family of maps, but hypersensitivity distinguishes between the simple dynamics of the trivial shift map and the more complicated dynamics of the other quantizations. This conclusion is supported by an analytical argument for short times and numerical evidence at later times.
We define a class of dynamical systems on the sphere analogous to the baker map on the torus. The classical maps are characterized by dynamical entropy equal to ln 2. We construct and investigate a family of the corresponding quantum maps. In the simplest case of the model the system does not possess a time reversal symmetry and the quantum map is represented by real, orthogonal matrices of even dimension. The semiclassical ensemble of quantum maps, obtained by averaging over a range of matrix sizes, displays statistical properties characteristic of circular unitary ensemble. Time evolution of such systems may be studied with the help of the SU (2) coherent states and the generalized Husimi distribution.