Dynamical Interference of Shannon Information Entropy: Identification of Wave-Packet Fractional Revivals
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Journal of Russian Laser Research, Volume 41, Number 5, September, 2020
DYNAMICAL INTERFERENCE OF SHANNON INFORMATION ENTROPY: IDENTIFICATION OF WAVE-PACKET FRACTIONAL REVIVALS Shahid Iqbal Department of Physics, School of Natural Sciences National University of Sciences and Technology Sector H-12, Islamabad, Pakistan E-mails:
siqbal @ sns.nust.edu.pk,
sic80 @ hotmail.com
Abstract Wave packet fractional revivals appear as manifestation of a multiple interference of various components of a wave packet during its long-time evolution in a bounded system. We study the phenomenon of fractional revivals by means of the time-dependent interference of information entropy density associated with a bound-state wave packet, which exhibits the formation of self-similar structures – quantum carpets woven by information entropy. The recurrences of self-similarity in the design of entropic carpet mimic the phenomena of quantum revivals and fractional revivals. We show that information theoretic measures are more sensitive to identify the wave-packet fractional revivals.
Keywords: Shannon entropy, dynamical interference, wave-packet quantum revivals, fractional revivals.
1.
Introduction
The notion of information entropy was initially introduced by Shannon, in 1948, while he was assigning probabilities to the outcome of uncertain events and to measure their uncertainty [1]. This early idea of Shannon information entropy, after passing through various development stages, has become a universal concept of statistical physics [2–6]. In quantum mechanics, the measurement uncertainty of an observable is an intrinsic feature, which traditionally is expressed quantitatively by the variance-based Heisenberg uncertainty relation. In 1975, Bialynicki-Birula, Beckner, and Mycielski (BBM) introduced entropic uncertainty relations as an alternative to the variance-based Heisenberg uncertainty relations [7–9]. These information theoretic measures have received a lot of interest in large variety of quantum systems [10–14]. Among various information theoretic measures, the Shannon entropy has played a particularly important role in measuring the uncertainty of different quantum systems [15–28]. The entropic uncertainty relations are also obtained in the context of quantum correlations [29, 30] and in the tomographic probability representations for various quantum states [31–34]. In dynamical systems, the phenomenon of wave-packet revivals is an important feature exhibited by a wide range of physical systems, that has received much attention over the last three decades [35]. Originally, it has been investigated theoretically in excited atomic and molecular systems [36, 37], and then observed experimentally in Rydberg wave packets and Bose–Einstein condensates [38, 39]. Later on, the phenomenon has extensively been studied in a large variety of contexts, such as Gaussian wave packets (WPs) in exactly solvable models systems [35], periodically driven systems [40–42], attosecond WP dynamics [43–45], revivals in coherent photon fields [46] and ph
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