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arXiv:quant-ph/0204040

Wave packet dynamics and factorization of numbers

Holger Mack, Marc Bienert, Florian Haug, F. Straub, Matthias Freyberger, Wolfgang P. Schleich

Published Apr 8, 2002• 2 citations
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Abstract

We connect three phenomena of wave packet dynamics: Talbot images, revivals of a particle in a box and fractional revivals. The physical origin of these effects is deeply rooted in phase factors which are quadratic in the quantum number. We show that the characteristic structures in the time evolution of these systems allow us to factorize large integers.

DOI
10.48550/arxiv.quant-ph/0204040

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We connect three phenomena of wave packet dynamics: Talbot images, revivals of a particle in a box and fractional revivals. The physical origin of these effects is deeply rooted in phase factors which are quadratic in the quantum number. We show that the characteristic structures in the time evolution of these systems allow us to factorize large integers.
Wave packet revivals and fractional revivals are striking quantum interference phenomena that can occur under suitable conditions in a system with a nonlinear spectrum. In the framework of a specific model (the propagation of an initially coherent wave packet in a Kerr-like medium), it is shown that distinctive signatures of these revivals and fractional revivals are displayed by the time evolution of the expectation values of physical observables and their powers, i.e., by experimentally measurable quantities. Moreover, different fractional revivals can be selectively identified by examining appropriate higher moments.
We present a generic treatment of wave-packet revivals for quantum-mechanical systems. This treatment permits a classification of certain ideal revival types. For example, wave packets for a particle in a one-dimensional box are shown to exhibit perfect revivals. We also examine the revival structure of wave packets for quantum systems with energies that depend on two quantum numbers. Wave packets in these systems exhibit quantum beats in the initial motion as well as new types of long-term revivals. As an example, we consider the revival structure of a particle in a two-dimensional box.
We examine the dynamical evolution of wave packets in a large class of quantum systems where two quantum numbers determine the energy spectrum and consequently the dynamical behavior. Using a generic Hamiltonian, we study the formation of coherent structures. The ratio of two time scales as well as the symmetry of the initial wave packet decisively determines the revival and fractional revivals of the system for which we give analytical results. The theory is applicable to a wide class of systems from diatomic molecules to ions in two-dimensional traps and two interacting Bose condensates.
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.
The general properties of the time-dependent characteristics of one-dimensional vibrational wave packets in an anharmonic system with the quadratic spectrum are discussed. The local behavior of the autocorrelation function and its square modulus (the survival function) is studied near the moments of fractional revival t=(p/q)Trev, where Trev is the period of the complete revival, p/q is an irreducible fraction, and q is the order of the revival. It is shown that all the wave packets have the same values of the survival function at the moments of fractional revival provided that the number of states in the packet exceeds the order of the revival q. It is also found that for the given packet, all the fractional revivals are identical to each other in the sense that their lifetimes do not depend on the order of the revival. The lifetimes of all fractional revivals are expressed in terms of the parameters of the initial packet (the effective number of states and the period of classical motion of the packet center) and the anharmonicity of the molecular system.