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arXiv:1409.5581 · Physical Review A

Fractional revivals through Rényi uncertainty relations

E. Romera, F. de los Santos

Published Jul 24, 2008• 41 citations
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Abstract

We show that the R\'enyi uncertainty relations give a good description of the dynamical behavior of wave packets and constitute a sound approach to revival phenomena by analyzing three model systems: the simple harmonic oscillator, the infinite square well, and the quantum bouncer. We prove the usefulness of entropic uncertainty relations as a tool for identifying fractional revivals by providing a comparison in different contexts with the usual Heisenberg uncertainty relation and with the common approach in terms of the autocorrelation function.

DOI
10.1103/physreva.78.013837

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We show that the R'enyi uncertainty relations give a good description of the dynamical behavior of wave packets and constitute a sound approach to revival phenomena by analyzing three model systems: the simple harmonic oscillator, the infinite square well, and the quantum bouncer. We prove the usefulness of entropic uncertainty relations as a tool for identifying fractional revivals by providing a comparison in different contexts with the usual Heisenberg uncertainty relation and with the common approach in terms of the autocorrelation function.
Wave-packet fractional revivals is a relevant feature in the long time-scale evolution of a wide range of physical systems, including atoms, molecules, and nonlinear systems. We show that the sum of information entropies in both position and momentum conjugate spaces is an indicator of fractional revivals by analyzing three different model systems: (i) the infinite square well, (ii) a particle bouncing vertically against a wall in a gravitational field, and (iii) the vibrational dynamics of hydrogen iodide molecules. This description in terms of information entropies complements the usual one in terms of the autocorrelation function.
We show that the time frequency analysis of the autocorrelation function is, in many ways, a more appropriate tool to resolve fractional revivals of a wave packet than the usual time domain analysis. This advantage is crucial in reconstructing the initial state of the wave packet when its coherent structure is short-lived and decays before it is fully revived. Our calculations are based on the model example of fractional revivals in a Rydberg wave packet of circular states. We end by providing an analytical investigation which fully agrees with our numerical observations on the utility of time-frequency analysis in the study of wave packet fractional revivals.
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.
Fractional revivals of wave packets in an infinite square well are scrutinized with a viewpoint rooted in Fourier analysis, and a compact relation, expressing the wavefunction Ψ( x , t ) for certain values of t in terms of spatially displaced copies of Ψ( x ,0), is derived without appealing to a classical analogy ; conditions for the appearance of an interference pattern in a plot of the position probability density of a wave packet are deduced, along with related results obtained previously from different considerations, with minimal effort. The specific case of a packet of Gaussian shape is analysed in greater detail to provide concrete illustrations of fractional revivals with or without interference between overlapping copies of wave packets, depending on whether a finite value of the so-called classical time can or cannot be assigned to the wave packet.
We investigate the short-, medium-, and long-term time dependence of wave packets in the infinite square well. In addition to emphasizing the appearance of wave packet revivals, i.e., situations where a spreading wave packet reforms with close to its initial shape and width, we also examine in detail the approach to the collapsed phase where the position-space probability density is almost uniformly spread over the well. We focus on visualizing these phenomena in both position- and momentum-space as well as by following the time-dependent expectation values of and uncertainties in position and momentum. We discuss the time scales for wave packet collapse, using both an autocorrelation function analysis as well as focusing on expectation values, and find two relevant time scales which describe different aspects of the decay phase. In an Appendix, we briefly discuss wave packet revival and collapse in a more general, one-dimensional power-law potential given by V(k)(x)=V0|x/a|k which interpolates between the case of the harmonic oscillator (k=2) and the infinite well (k=∞).
The harmonic oscillator and the infinite square well are two of the simplest one-dimensional quantum systems which exhibit wave packet revivals (trivially so in the case of the oscillator.) These two potentials can be thought of as special cases of the general one-dimensional power-law potential given by V(k)(x)=V0|x/a|k (with k=2 and k→∞ for the oscillator and square well, respectively.) Using an autocorrelation function approach and the WKB approximation for the quantized energy levels in such potentials, we exhibit numerical evidence for wave packet revivals in the case of arbitrary k>0 which agree well with more analytic results. We derive expressions for the revival and collapse time scales in terms of the physical quantities of the system as well as the wave packet parameters. We find that both times scale with the power-law exponent k as |(k+2)/(k−2)|. In this way, we can explicitly exhibit the approach to the two familiar limiting cases. We also briefly consider the case of a “half” well where an infinite wall in added at the origin.