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arXiv:quant-ph/0403032

Multimode Interference: Identifying Channels and Ridges in Quantum Probability Distributions

Ross C. O’Connell, Will Loinaz A

Published Mar 3, 2004• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0403032

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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.
The dynamics of a quantum mechanical particle in a time-independent potential are found to contain many interesting phenomena. These are direct consequences of the (typical) existence of more than one time scale governing the problem. This gives rise to full revivals of initial wavepackets, fractional revivals (multiple wavepackets appearing at fractions of the revival time) and the striking quantum carpets. A variety of analytic techniques are used to consider the interference that gives rise to these phenomena while skirting calculations involving cross-terms. Novel results include a new theorem on the weighting coefficients ama_mam​ that govern fractional revivals, a demonstration that ΨclΨ_{cl}Ψcl​, the function that governs the distribution and features of these fractional revivals, really does behave classically, a treatment of the wavepacket dephasing in the infinite square well by means of the Poisson summation formula, and a correct analysis of the spatial distribution of intermode traces. Also, this work presents a coherent treatment of these phenomena, which before now did not exist.
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.
Generic channel and ridge structures are known to appear in the time-dependent position probability distribution of a one-dimensional quantum particle confined to a box. These structures are shown to have a detailed quantitative explanation in terms of a travelling-wave decomposition of the probability density, wherein each contributing term corresponds simultaneously to (i) a real wave propagating at a quantized velocity and (ii) to the time-averaged structure of the position distribution along a quantized direction in spacetime. The approach leads to new predictions of channel locations, widths and depths, and is able to provide more structural details than earlier approaches based on partial interference and Wigner functions. Results are also applicable to light diffracted by a periodic grating, and to the quantum rigid rotator.
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.