Probing Dynamical Sensitivity of a Non-KAM System Through Out-of-Time-Order Correlators

Abstract

Non-KAM (Kolmogorov-Arnold-Moser) systems, when perturbed by weak time-dependent fields, offer a fast route to classical chaos through an abrupt breaking of invariant phase space tori. In this work, we employ out-of-time-order correlators (OTOCs) to study the dynamical sensitivity of a perturbed non-KAM system in the quantum limit as the parameter that characterizes the $\text{resonance}$ condition is slowly varied. For this purpose, we consider a quantized kicked harmonic oscillator (KHO) model, which displays stochastic webs resembling Arnold's diffusion that facilitate large-scale diffusion in the phase space. Although the Lyapunov exponent of the KHO at resonances remains close to zero in the weak perturbative regime, making the system weakly chaotic in the conventional sense, the classical phase space undergoes significant structural changes. Motivated by this, we study the OTOCs when the system is in resonance and contrast the results with the non-resonant case. At resonances, we observe that the long-time dynamics of the OTOCs are sensitive to these structural changes, where they grow quadratically as opposed to linear or stagnant growth at non-resonances. On the other hand, our findings suggest that the short-time dynamics remain relatively more stable and show the exponential growth found in the literature for unstable fixed points. The numerical results are backed by analytical expressions derived for a few special cases. We will then extend our findings concerning the non-resonant cases to a broad class of near-integrable KAM systems.

17 pages, 6 figures. Close to the accepted version in Phys. Rev. E

References in corpus (13)

• Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System
• Slow scrambling in disordered quantum systems
• Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves
• Scrambling and Complexity in Phase Space
• Quantum Chaos and the Correspondence Principle
• Timescales in the quench dynamics of many-body quantum systems: Participation ratio vs out-of-time ordered correlator
• Out-of-time-order correlations and Floquet dynamical quantum phase transition
• Trotter errors from dynamical structural instabilities of Floquet maps in quantum simulation
• Statistical and dynamical properties of the quantum triangle map
• Dynamical transition from localized to uniform scrambling in locally hyperbolic systems
• Witnessing quantum chaos using observational entropy
• Out-of-time-ordered correlators and quantum walks
• KAM-Stability for Conserved Quantities in Finite-Dimensional Quantum Systems

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