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Chaos An Interdisciplinary Journal of Nonlinear Science

Quantum chaos in the Wigner representation

G. P. Berman, A. R. Kolovskii, Felix M. Izrailev, Alexander Iomin

Published Aug 1, 1991• 5 citations
Publisher

Abstract

The dynamics of a familiar model of stochastic behavior-the quantum kicked rotator-is analyzed in the Wigner representation. Exact nonlocal maps defined on a discrete phase space are derived. The basic dynamics of a quantum kicked rotator can be described satisfactorily by means of a simplified map that incorporates only the discrete nature of the phase space.

DOI
10.1063/1.165832

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The dynamics of a familiar model of stochastic behavior-the quantum kicked rotator-is analyzed in the Wigner representation. Exact nonlocal maps defined on a discrete phase space are derived. The basic dynamics of a quantum kicked rotator can be described satisfactorily by means of a simplified map that incorporates only the discrete nature of the phase space.
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We investigate a quantum algorithm that simulates efficiently the quantum kicked rotator model, a system that displays rich physical properties and enables to study problems of quantum chaos, atomic physics, and localization of electrons in solids. The effects of errors in gate operations are tested on this algorithm in numerical simulations with up to 20 qubits. In this way various physical quantities are investigated. Some of them, such as second moment of probability distribution and tunneling transitions through invariant curves, are shown to be particularly sensitive to errors. However, investigations of the fidelity and the Wigner and Husimi distributions show that these physical quantities are robust in presence of imperfections. This implies that the algorithm can simulate the dynamics of quantum chaos in presence of a moderate amount of noise.
A dissipative quantum system exhibiting chaos in its classical limit is constructed by coupling the quantum kicked rotator to a reservoir with exchange of action. A version of the quantized standard map that incorporates dissipation is derived from this model and iterated numerically to study the long-time behavior in various regions of the damping rate. It is found that even weak damping is capable of disrupting dynamical localization which suppresses chaotic motion in the conservative standard map, and thus restores diffusion in the action variable. On the time scale of classical relaxation, a steady state is reached which is the quantum analogue of a classical strange attractor. For strong dissipation, observable deviations from classical behavior both in the transients and in the steady state are due to quantum noise. They are reproduced by a classical stochastic map which is approached by the dissipative quantum map as its semi-classical limit.
We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function on Zd×Zd{\mathbb{Z}_d} \times {\mathbb{Z}_d}Zd​×Zd​ (ddd an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (d=3d=3d=3), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.