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arXiv:quant-ph/0411162 · Physical Review E

Quantum fidelity decay in quasi-integrable systems

Yaakov S. Weinstein, C. Stephen Hellberg

Published Jan 12, 2005• 27 citations
arXivHTMLPDF

Abstract

We show, via numerical simulations, that the fidelity decay behavior of quasi-integrable systems is strongly dependent on the location of the initial coherent state with respect to the underlying classical phase space. In parallel to classical fidelity, the quantum fidelity generally exhibits Gaussian decay when the perturbation affects the frequency of periodic phase space orbits and power-law decay when the perturbation changes the shape of the orbits. For both behaviors the decay rate also depends on initial state location. The spectrum of the initial states in the eigenbasis of the system reflects the different fidelity decay behaviors. In addition, states with initial Gaussian decay exhibit a stage of exponential decay for strong perturbations. This elicits a surprising phenomenon: a strong perturbation can induce a higher fidelity than a weak perturbation of the same type.

DOI
10.1103/physreve.71.016209

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We show, via numerical simulations, that the fidelity decay behavior of quasi-integrable systems is strongly dependent on the location of the initial coherent state with respect to the underlying classical phase space. In parallel to classical fidelity, the quantum fidelity generally exhibits Gaussian decay when the perturbation affects the frequency of periodic phase space orbits and power-law decay when the perturbation changes the shape of the orbits. For both behaviors the decay rate also depends on initial state location. The spectrum of the initial states in the eigenbasis of the system reflects the different fidelity decay behaviors. In addition, states with initial Gaussian decay exhibit a stage of exponential decay for strong perturbations. This elicits a surprising phenomenon: a strong perturbation can induce a higher fidelity than a weak perturbation of the same type.
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We demonstrate that a system's rate of fidelity decay under repeated perturbations may be measured efficiently on a quantum information processor, and analyze the conditions under which this indicator is a reliable probe of quantum chaos. The type and rate of the decay are not dependent on the eigenvalue statistics of the unperturbed system, but depend on the system's eigenvector statistics in the eigenbasis of the perturbation. For random eigenvector statistics, the decay is exponential with a rate fixed by the variance of the perturbation's energy spectrum. Hence, even classically regular models can exhibit an exponential fidelity decay under generic quantum perturbations. These results clarify which perturbations can distinguish classically regular and chaotic quantum systems.
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We derive a simple and general relation between the fidelity of quantum motion, characterizing the stability of quantum dynamics with respect to arbitrary static perturbation of the unitary evolution propagator, and the integrated time auto-correlation function of the generator of perturbation. Surprisingly, this relation predicts the slower decay of fidelity the faster the decay of correlations. In particular, for non-ergodic and non-mixing dynamics, where asymptotic decay of correlations is absent, a qualitatively different and faster decay of fidelity is predicted on a timescale ∝ 1/δ as opposed to mixing dynamics where the fidelity is found to decay exponentially on a timescale ∝ 1/δ 2 , where δ is the strength of perturbation. A detailed discussion of a semiclassical regime of small effective values of Planck constant is given where classical correlation functions can be used to predict quantum fidelity decay. Note that the correct and intuitively expected classical stability behaviour is recovered in the classical limit → 0, as the two limits δ → 0 and → 0 do not commute. In addition, we also discuss non-trivial dependence on the number of degrees of freedom. All the theoretical results are clearly demonstrated numerically on the celebrated example of a quantized kicked top.