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American Journal of Physics

Quantum interference in a driven washboard potential

Quentin Thommen, Jean Claude Garreau, Véronique Zehnlé

Published Jul 14, 2004• 6 citations
Publisher

Abstract

The dynamics of a quantum particle in a one-dimensional, tilted (“washboard”) and time-dependent lattice is studied. The approach uses quantum mechanics at the undergraduate level and leads to analytical results that show a rich variety of dynamical behavior and illustrate the fundamental role of interference in quantum systems.
DOI
10.1119/1.1757446

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Interference is one of the most fundamental features which characterizes quantum systems. Here we provide an exhaustive analysis of the interfere dynamics associated with wave-packet superpositions from both the standard quantum-mechanical perspective and the Bohmian one. From this analysis, clear and insightful pictures of the physics involved in these kind of processes are obtained, which are of general validity (i.e., regardless of the type of wave packets considered) in the understanding of more complex cases where interference is crucial (e.g., scattering problems, slit diffraction, quantum control scenarios or, even, multipartite interactions). In particular, we show how problems involving wave-packet interference can be mapped onto problems of wave packets scattered off potential barriers.
Tilted washboard potentials are periodic functions of potential energy whose graphs remind the profiles of inclined washboards for scrubbing clothes. Practically unknown until the 1970, washboard potentials are at present ubiquitous seminal landscapes across diverse physics researches and technologies now receiving significant attention. However, washboard potentials are still rarely presented to science and engineering undergraduates; perhaps they could hear about those when first studying Josephson junctions. Aiming to promote the presentation of such potentials to undergraduates, we here consider cases of washboard potentials in: classical mechanics, quantum physics and optics. The quantum case is the superconducting Josephson junction whose phase quasi-particle has a tilted washboard potential leading to important applications, e.g. the phase qubit in quantum computation. Optical cases are electro-optical potentials generated with laser beams in which thousands of atom electric dipoles may become confined or evolve, potentials being generated either with laser Bessel beams, or with laser standing waves, whose periodic potential profile is implicitly tilted or may be experimentally adjusted to become tilted. Electro-optical potentials are presently used in fast-paced fundamental research, e.g. Brownian particle motors, atom trapping in optical lattices, optical atomic clocks, protein transport in biological cells, and in condensed matter physics to name a few.
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.