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arXiv:quant-ph/0603099 · Journal of Optics B Quantum and Semiclassical Optics

Quantum recurrences in periodically driven systems

Farhan Saif

Published Mar 1, 2005• 15 citations
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Abstract

We investigate the quantum recurrence phenomena in periodically driven systems. We calculate the classical period and the quantum recurrence time and develop their interdependence. We further predict the behavior of the recurrence phenomena for the power law potentials.
DOI
10.1088/1464-4266/7/3/015

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The Poincaré recurrence theorem shows that conservative systems in a bounded region of phase space eventually return arbitrarily close to their initial state after a finite amount of time. An analogous behavior occurs in certain quantum systems where quantum states can recur after sufficiently long unitary evolution, a phenomenon known as quantum recurrence. Periodically driven (i.e. Floquet) quantum systems in particular exhibit complex dynamics even in small dimensions, motivating the study of how interactions and Hamiltonian structure affect recurrence behavior. While most studies treat recurrence in an approximate, distance-based sense, here we address the problem of {exact}, state-independent recurrences in a broad class of finite-dimensional Floquet systems, spanning both integrable and non-integrable models. Leveraging techniques from algebraic field theory, we construct an arithmetic framework that identifies all possible recurrence times by analyzing the cyclotomic structure of the Floquet unitary's spectrum. This computationally tractable approach yields both positive results, enumerating all candidate recurrence times, and definitive negative results, rigorously ruling out candidate recurrence times for a given set of Hamiltonian parameters. We further prove that rational Hamiltonian parameters do not, in general, guarantee exact recurrences, revealing a subtle interplay between system parameters and long-time dynamics. Our findings sharpen the theoretical understanding of quantum recurrences, clarify their relationship to quantum chaos, and highlight parameter regimes of special interest for quantum metrology and control.
Periodically-driven systems are ubiquitous in science and technology. In quantum dynamics, even a small number of periodically-driven spins leads to complicated dynamics. Hence, it is of interest to understand what constraints such dynamics must satisfy. We derive a set of constraints for each number of cycles. For pure initial states, the observable being constrained is the recurrence probability. We use our constraints for detecting undesired coupling to unaccounted environments and drifts in the driving parameters. To illustrate the relevance of these results for modern quantum systems we demonstrate our findings experimentally on a trapped-ion quantum computer, and on various IBM quantum computers. Specifically, we provide two experimental examples where these constraints surpass fundamental bounds associated with known one-cycle constraints. This scheme can potentially be used to detect the effect of the environment in quantum circuits that cannot be classically simulated. Finally, we show that, in practice, testing an n -cycle constraint requires executing only O(n) cycles, which makes the evaluation of constraints associated with hundreds of cycles realistic.
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Periodically driven systems are nowadays a very powerful tool for the study of condensed quantum matter: indeed, they allow the observation of phenomena in huge contrast with the expected behavior of their classical counterparts. An outstanding example is dynamical localization: this phenomenon consists in the prevention of heating despite an external periodic perturbation. It has been defined for the first time in a single particle model, the kicked rotor. This chaotic system undergoes an unbounded heating in time in its classical limit; on the opposite, in the quantum regime, the kinetic energy grows until a saturation value is reached and the system stops heating. In my thesis I consider a chain of coupled quantum kicked rotors in order to investigate the fate of dynamical localization in presence of interactions. I remarkably show that a dynamically localized phase persists in the quantum system also in presence of interactions. This is an unexpected behavior since periodically driven, interacting, non-integrable quantum systems heat up to an infinite temperature state. Moreover, I find a genuine quantum dynamics also in the delocalized phase: the heating is not diffusive, as it happens in the classical system, but it follows a sub-diffusive power law. A focus on the properties of the Floquet eigenstates and operators matrix properties gives interesting hints, still under investigation, for a possible justification of the above mentioned slow heating.