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Chaos An Interdisciplinary Journal of Nonlinear Science

Diffraction in phase space: Jumps and kinks in matter waves revealed by their dynamics

Gregor Finkbeiner, Maxim A. Efremov, Robert F. O’Connell, Wolfgang P. Schleich

Published Mar 1, 2025• 0 citations
Publisher

Abstract

We show that a discontinuity either in a wave function or its derivative, corresponding to a jump or a kink, causes ripples in Wigner phase space. In the free time evolution, these structures give rise to interference fringes in the probability density represented in spacetime.
DOI
10.1063/5.0260470

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Abstract Diffraction in time manifests itself as the appearance of probability-density fringes when a matter wave passes through an opaque screen with abrupt temporal variations of transmission properties. Here we analytically describe the phase-space structure of diffraction-in-time fringes for a class of smooth time gratings. More precisely, we obtain an analytic expression for the Husimi distribution representing the state of the system in the case of time gratings comprising a succession of Lorentzian-like slits. In particular, for a double-slit scenario, we derive a simple and intuitive expression that accurately captures the position of interference fringes in phase space.
When a stationary reflecting wall acting as a perfect mirror for an atomic beam with well-defined incident velocity is suddenly removed, the density profile develops during the time evolution an oscillatory pattern known as diffraction in time. The interference fringes are suppressed or their visibility is diminished by several effects such as averaging over a distribution of incident velocities, apodization of the aperture function, atom-atom interactions, imperfect reflection, or environmental noise. However, when the mirror moves with finite velocity along the direction of propagation of the beam, the visibility of the fringes is enhanced. For mirror velocities below beam velocity, as used for slowing down the beam, the matter wave splits into three regions separated by space-time points with classical analogs. For mirror velocities above beam velocity a visibility enhancement occurs without a classical counterpart. When the velocity of the beam approaches that of the mirror the density oscillations rise by a factor 1.8 over the stationary value.
The problem of Schrödinger propagation of a discontinuous wavefunction − diffraction in time − is studied under a new light. It is shown that the evolution map in phase space induces a set of affine transformations on discontinuous wavepackets, generating expansions similar to those of wavelet analysis. Such transformations are identified as the cause for the infinitesimal details in diffraction patterns. A simple case of an evolution map, such as SL(2) in a two-dimensional phase space, is shown to produce an infinite set of space-time trajectories of constant probability. The trajectories emerge from a breaking point of the initial wave. ar X iv
The problem of Schrödinger propagation of a discontinuous wavefunction – diffraction in time – is studied under a new light. It is shown that the evolution map in phase space induces a set of affine transformations on discontinuous wavepackets, generating expansions similar to those of wavelet analysis. Such transformations are identified as the cause for the infinitesimal details in diffraction patterns. A simple case of an evolution map, such as SL (2) in a two-dimensional phase space, is shown to produce an infinite set of space-time trajectories of constant probability. The trajectories emerge from a breaking point of the initial wave.
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.
The evolution of an initially smooth spatial inhomogeneity in the density of a one-dimensional Fermi gas is well described by classical mechanics. The classical evolution leads to the formation of a shock wave: the density develops kinks in its coordinate dependence. We show that quantum corrections to the shock wave produce density ripples which run off the kinks. Despite their quantum origin, the amplitude and period of the ripples are expressed only in terms of classical objects derived from a smooth density profile.