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Europhysics Letters (EPL)

Classical integrability and quantum velocity fields

Bambi Hu, Quanlin Jie, Jian-Zhong Ma, Jianzhong Gu

Published Mar 1, 2002• 1 citation
Publisher

Abstract

We study a stationary velocity field, defined via the probability current density for an energy eigenstate of a quantum system, and its classical analog determined by the Hamilton-Jacobi equation. For classically integrable systems, the quantum-classical correspondence can be properly established by specifying constants of motion in addition to the Hamiltonian. For classically chaotic systems, the correspondence is broken down; the manifestation of classical chaos in the quantum domain is shown numerically to be the appearance of turbulence in the quantum velocity field.

DOI
10.1209/epl/i2002-00580-y

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We study a stationary velocity field, defined via the probability current density for an energy eigenstate of a quantum system, and its classical analog determined by the Hamilton-Jacobi equation. For classically integrable systems, the quantum-classical correspondence can be properly established by specifying constants of motion in addition to the Hamiltonian. For classically chaotic systems, the correspondence is broken down; the manifestation of classical chaos in the quantum domain is shown numerically to be the appearance of turbulence in the quantum velocity field.
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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.
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