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Physical Review A

Multimode interference: Highly regular pattern formation in quantum wave-packet evolution

A. E. Kaplan, Irene Marzoli, Willis E. Lamb, Wolfgang P. Schleich

Published Feb 9, 2000• 83 citations
Publisher

Abstract

Highly regular spatiotemporal or multidimensional patterns in the quantum mechanical probability or classical field intensity distributions can appear due to pair interference between individual eigenmodes of the system, thus forming the so-called intermode traces. These patterns are strongly pronounced in any anharmonic potential, provided that the traces are multidegenerate; they may occur in many areas of wave physics.

DOI
10.1103/physreva.61.032101

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Highly regular spatiotemporal or multidimensional patterns in the quantum mechanical probability or classical field intensity distributions can appear due to pair interference between individual eigenmodes of the system, thus forming the so-called intermode traces. These patterns are strongly pronounced in any anharmonic potential, provided that the traces are multidegenerate; they may occur in many areas of wave physics.
Summary form only given. Wave packets (WPs) has recently emerged as a very active research field in quantum optics. We show here that the evolution of WPs may result in new class of large-scale interference effects: the formation of highly regular spatio-temporal patterns in the QM probability, |/spl Psi/|/sup 2/, with the key mechanism for this being pair interference between the eigen-modes of the system and the degeneracy of this interference. The new patterns have been discovered in the probability density |/spl Psi/|/sup 2/(x,t) of a particle in an infinitely deep square well, by numerical simulations of the evolution of a Gaussian WP; later work by a few groups used other initial conditions and media (e.g. Bose-Einstein condensate and angular WPs). In all these cases, the QM distribution shows a net of canals (minima), and ridges (maxima), running as straight lines in space-time (x,t). Similar patterns in the near field of a diffraction grating, and in matter waves in atom optics reveals a new aspects of the Talbot effect. So far the numerical exploration of these patterns has relied on the specific choice (i) of the initial conditions and (ii) of the square well; no physical explanation of the patterns has been suggested. Recently, we observed a quantum carpet under more general conditions, offered its universal explanation, and predicted distinct curved patterns in general case of an anharmonic potential.
The multimode interference technique is a simple way to study the interference patterns found in many quantum probability distributions. We demonstrate that this analysis not only explains the existence of so-called "quantum carpets," but can explain the spatial distribution of channels and ridges in the carpets. With an understanding of the factors that govern these channels and ridges we have a limited ability to produce a particular pattern of channels and ridges by carefully choosing the weighting coefficients c_{n} . We also use these results to demonstrate why fractional revivals of initial wavepackets are themselves composed of many smaller packets.
One of the key features of quantum mechanics is the interference of probability amplitudes. The reason for the appearance of interference is mathematically very simple. It is the linear structure of the Hilbert space which is used for the description of quantum systems. In terms of physics we usually talk about the superposition principle valid for individual and composed quantum objects. So, while the source of interference is understandable it leads in fact to many counter-intuitive physical phenomena which puzzle physicists for almost hundred years. The present thesis studies interference in two seemingly disjoint fields of physics. However, both have strong links to quantum information processing and hence are related. In the first part we study the intriguing properties of quantum walks. In the second part we analyze a sophisticated application of wave packet dynamics in atoms and molecules for factorization of integers. The main body of the thesis is based on the original contributions listed separately at the end of the thesis. The more technical aspects and brief summaries of used methods are left for appendices.
Localized quantum wave packets can be produced in a variety of physical systems and are the subject of much current research in atomic, molecular, chemical, and condensed-matter physics. They are particularly well suited for studying the classical limit of a quantum-mechanical system. The motion of a localized quantum wave packet initially follows the corresponding classical motion. However, in most cases the quantum wave packet spreads and undergoes a series of collapses and revivals. We present a generic treatment of wave-packet evolution, and we provide conditions under which various types of revivals occur in ideal form. The discussion is at a level appropriate for an advanced undergraduate or first-year graduate course in quantum mechanics. Explicit examples of different types of revival structure are provided, and physical applications are discussed.