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Quantum revivals, geometric phases and circle map recurrences

Seshadri, S R, Lakshmibala, S, Balakrishnan, V

Published Jan 1, 1999• 6 citations
OpenAlex

Abstract

Revivals of the coherent states of a deformed, adiabatically and cyclically varying oscillator Hamiltonian are examined. The revival time distribution is exactly that of Poincaré recurrences for a rotation map: only three distinct revival times can occur, with specified weights. A link is thus established between quantum revivals and recurrences in a coarse-grained discrete-time dynamical system.

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The theory of quantal revivals is extended to Hamiltonians that are time dependent through the slow noncyclic change of some parameters. It is shown by generalizing the cyclic geometric phase (Berry's phase) to noncyclic Hamiltonians that there is a noncyclic geometric displacement for all quantum revivals. The treatment of the classical mechanical geometric angle is also extended to open paths in parameter space and its semiclassical relation with the noncyclic geometric phase derived. The displacement of the revivals is then shown to be given by the total angle shift on the final torus sum of the dynamical angle and the noncyclic geometric angle.

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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.

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Quantum-fractal revival structure in CN quadratic spectra: Base-Nquantum computer registers

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A closer examination of Eberly ``revivals,'' Berry ``quantum fractals,'' and Schleich ``carpets'' exposes a simple unitary-cyclic group structure in the dynamics of wave packets for rotorlike systems with quadratic spectra. Quantum quasifractal patterns are revealed to be permeated by self-similar ``X-braid'' structures seen by Hofstadter in Zeeman-perturbed band structures and in other spectra related to circle maps. The patterns are shown to correspond to revival dynamics in the flow of energy or information through these N-level systems with X-braids being directly related to quantum beats or elementary U(2) resonance transfer. ${C}_{N}$-group analysis and Farey arithmetic discretely catalog X-braid hierarchy and suggest the possibility of homocyclic nanostructures or ${C}_{N}$ circuits that are base-N digital quantum computing registers. The phase dynamics corresponds to a fairly sophisticated integer arithmetic that includes an output of all rational numbers resolvable by a given energy input. ${C}_{N}$ circuits automatically give all factors of N. The dynamics of a ${C}_{6}$ device is discussed in detail.

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The study of wave-packet revivals is extended to the case of Hamiltonians which are made time dependent through the adiabatic cycling of some parameters. It is shown that the quantal geometric phase (Berry's phase) causes the revived packet to be displaced along the classical trajectory by an amount equal to the classical geometric phase (Hannay's angle), in one degree of freedom. A physical example illustrating this effect in three degrees of freedom is mentioned.

arXiv:solv-int/9501009•Physical Review Letters•Feb 20, 1995•16 citations
Revivals of the coherent states of a deformed, adiabatically and cyclically varying oscillator Hamiltonian are examined. The revival time distribution is exactly that of Poincaré recurrences for a rotation map: only three distinct revival times can occur, with specified weights. A link is thus established between quantum revivals and recurrences in a coarse-grained discrete-time dynamical system.
The theory of quantal revivals is extended to Hamiltonians that are time dependent through the slow noncyclic change of some parameters. It is shown by generalizing the cyclic geometric phase (Berry's phase) to noncyclic Hamiltonians that there is a noncyclic geometric displacement for all quantum revivals. The treatment of the classical mechanical geometric angle is also extended to open paths in parameter space and its semiclassical relation with the noncyclic geometric phase derived. The displacement of the revivals is then shown to be given by the total angle shift on the final torus sum of the dynamical angle and the noncyclic geometric angle.
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.
Wave packet revivals and fractional revivals are hallmark quantum interference phenomena that arise in systems with nonlinear energy spectra, and their signatures in expectation values of observables have been studied extensively in earlier work. In this article, we build on these studies and extend the analysis in two important directions. First, we investigate fractional revival dynamics in angular momentum observables, deriving explicit expressions for the time evolution of their moments and demonstrating that higher-order angular momentum moments provide clear and selective signatures of fractional revivals. Second, we examine classical analogs of quantum revival phenomena and elucidate structural similarities between quantum fractional revivals and recurrence behavior in representative classical systems. Using the Kerr-type nonlinear Hamiltonian as a paradigmatic model, we analyze the autocorrelation function, moment dynamics, and phase-space structures, supported by visualizations such as quantum carpets. Our results broaden the range of experimentally accessible diagnostics of fractional revivals and provide a unified perspective on revival phenomena across quantum and classical dynamical systems.
A closer examination of Eberly ``revivals,'' Berry ``quantum fractals,'' and Schleich ``carpets'' exposes a simple unitary-cyclic group structure in the dynamics of wave packets for rotorlike systems with quadratic spectra. Quantum quasifractal patterns are revealed to be permeated by self-similar ``X-braid'' structures seen by Hofstadter in Zeeman-perturbed band structures and in other spectra related to circle maps. The patterns are shown to correspond to revival dynamics in the flow of energy or information through these N-level systems with X-braids being directly related to quantum beats or elementary U(2) resonance transfer. CN{C}_{N}CN​-group analysis and Farey arithmetic discretely catalog X-braid hierarchy and suggest the possibility of homocyclic nanostructures or CN{C}_{N}CN​ circuits that are base-N digital quantum computing registers. The phase dynamics corresponds to a fairly sophisticated integer arithmetic that includes an output of all rational numbers resolvable by a given energy input. CN{C}_{N}CN​ circuits automatically give all factors of N. The dynamics of a C6{C}_{6}C6​ device is discussed in detail.
Analytical eigenfunctions and eigenvalues for the Morse oscillator were applied to investigate the quantum resonant beats and revivals of wave packet propagation. A concise way for exact prediction of the complete revival period of the Morse oscillator was given for the first time. It was suggested that any complete period was made of integer numbers of the minimum or fundamental period. Within the fundamental period, the anharmonicity of this oscillator appeared to cause interesting space-time phenomena that include relatively simple Farey-sum revival structures. In addition, a simple sum of two Morse oscillators led to a double-Morse well whose geometric symmetry provided analytical eigenfunctions and eigenvalues for certain low-lying energy levels. The quantum tunneling between the double-Morse well significantly affected the resonant beats and revivals local to each well, and gave rise to interesting tsunami-like waves in the middle of the double well. Furthermore, quantum rotor wave functions based upon Wigner-D matrix were applied to investigate the quantum resonant beats and revivals that occur in experimentally accessible spin systems. Interesting physical effects in quantum rotors between half-integer spin and integer spin systems were observed to show effects of symmetry. Essentially, the quantum revivals in these quantum systems exhibited number-information aspects of surprisingly simple Farey-sum and Ford circles geometry. Such quantum dynamics will provide a physical insight to further develop matter wave packet technology, and might have applications for quantum information processing and quantum computing.