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Journal of Modern Optics

Correlations in phase space and the creation of focusing wave packets

R. Mack, Valery P. Yakovlev, Wolfgang P. Schleich

Published Jun 10, 2010• 6 citations
Publisher

Abstract

We consider the time evolution of a special class of wave packets representing the motion of a free particle along a line. In order to make contact with an earlier work on s-waves in two or higher dimensions we confine ourselves to wave packets with a probability density that is symmetric with respect to the origin. Here we concentrate on the phenomenon that an appropriately prepared wave packet is focusing before it displays its familiar behavior of spreading. We derive the condition on the wave packet for this effect to occur: the probability density at the origin has to be concave. The translation of this requirement into Wigner phase space reads: the average of the kinetic energy along the momentum axis has to be negative. In this way we identify the origin of this focusing effect of the wave packet as quantum interference.

DOI
10.1080/09500340.2010.486873

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We consider the time evolution of a special class of wave packets representing the motion of a free particle along a line. In order to make contact with an earlier work on s-waves in two or higher dimensions we confine ourselves to wave packets with a probability density that is symmetric with respect to the origin. Here we concentrate on the phenomenon that an appropriately prepared wave packet is focusing before it displays its familiar behavior of spreading. We derive the condition on the wave packet for this effect to occur: the probability density at the origin has to be concave. The translation of this requirement into Wigner phase space reads: the average of the kinetic energy along the momentum axis has to be negative. In this way we identify the origin of this focusing effect of the wave packet as quantum interference.
We discuss a class of solutions of the time-dependent Schrödinger equation such that the position uncertainty temporarily decreases. This self-focusing or contractive behavior is a consequence of the anti-correlation of the position and momentum observables. Since the associated position density satisfies a continuity equation, upon contraction the probability current at a given fixed point may flow in the opposite direction of the group velocity of the wave packet. For definiteness, we consider a free particle incident from the left of the origin, and establish a condition for the initial position-momentum correlation such that a negative probability current at the origin is possible. This implies a decrease in the particle's detection probability in the region x > 0, and we calculate how long this occurs. Analogous results are obtained for a particle subject to a uniform gravitational force if we consider the particle approaching the turning point. We show that position-momentum anti-correlation may cause a negative probability current at the turning point, leading to a temporary decrease in the particle's detection probability in the classically forbidden region.
We examine the free time evolution of a rectangular one dimensional Schrödinger wave packet of constant phase during the early stage which in the paraxial wave approximation is identical to the diffraction of a scalar field from a single slit. Our analysis, based on numerics and the Cornu spiral reveals considerable intricate detail behavior in the density and phase of the wave. We also point out a concentration of the intensity that occurs on axis and propose a new measure of width that expresses this concentration.
Asymptotic time evolution of a wave packet describing a non-relativistic particle incident on a potential barrier is considered, using the Wigner phase-space distribution. The distortion of the trasmitted wave packet is determined by two time-like parameters, given by the energy derivative of the complex transmission amplitude. The result is consistent with various definitions of the tunneling time (e.g. the Büttiker-Landauer time, the complex time and Wigner's phase time). The speed-up effect and the negative dispersion are discussed, and new experimental implications are considered.
Much of the discussion of decoherence has been in terms of a particle moving in one dimension that is placed in an initial superposition state (a Schrödinger "cat" state) corresponding to two widely separated wave packets. Decoherence refers to the destruction of the interference term in the quantum probability function. Here, we stress that a quantitative measure of decoherence depends not only on the specific system being studied but also on whether one is considering coordinate, momentum or phase space. We show that this is best illustrated by considering Wigner phase space where the measure is again different. Analytic results for the time development of the Wigner distribution function for a two-Gaussian Schrodinger "cat" state have been obtained in the high-temperature limit (where decoherence can occur even for negligible dissipation) which facilitates a simple demonstration of our remarks.
A quantum particle (represented by a Gaussian wave packet) under uniform gravity, projected upward (the conventional positive direction) along the vertical degree of freedom x, may have a negative probability current on a set of points q. This imposes the condition that the position and momentum observables of the state are negatively correlated. We calculate the effect of the negative correlation on the individual contributions to the probability current by the positive and negative momentum amplitudes of the state using an integral representation of the probability current. Negative position-momentum correlation causes a transient attenuation of all contributions to the probability flow, and wave packet spreading is temporarily mitigated.