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Out-of-Time-order Correlator in billiard systems
Keywords: Quantum Chaos, Out-of-time-Order Correlators, Billiards
Recently, Out-of-Time-order Correlator (OTOC) has turned out to be relevant to measure quantum chaos, in the sense that the classical limit corresponds to the definition of Lyaponov exponent. We calculate the OTOC in both non- relativistic and relativistic quantum billiard systems, prototypical toy models for the quantum chaos study, governed by the Schrodinger and Dirac equations, respectively. We obtain the following key results. First, when classical dynamics of the billiards is chaotic, there is no exponential growth of the OTOC, even when temperature is very high. Notably, the OTOC will oscillate at late time t>>t_E when the classical dynamics is integrable, while it quickly saturate in the classically chaotic case. This saturation can be associated with quantum localization, which is different from the classical case. Second, we made relevant comparisons in terms of different types of commutation definitions, i.e. < [p (t), p (0)]> and < [p (t), x (0)]>, which show the similar results. The concrete calculations show that, in general, the growth rate of OTOC and the Classical Lyaponov exponent are two distinct quantities. For the same shape of billiards with integrable dynamics, it is found that the oscillation magnitude of OTOC for the non-relativistic case governed by the Schrodinger equation is higher than that for the relativistic case, whose quantum dynamics is described by the Dirac equation. Moreover, in the relativistic Dirac billiards the growth rate difference between integrable and classically chaotic cases is much larger. Third, using a family of quantum billiards proposed by Robnik, we study the order-to-chaos transition in terms of the OTOC. As the billiard shape goes to nonconvex, the classical dynamics has finite Kolmogorov entropy, which has its quantum manifestation of ceasing oscillation in the OTOC. This phenomenon is also identified in the one-dimensional case with random potentials.
Chendi Han,
School of Electrical, Computer, and Energy Engineering, Arizona State University
Tempe, Arizona

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