Comparison between two models of absorption of matter waves by a thin time-dependent barrier

Abstract

We report a quantitative, analytical and numerical, comparison between two models of the interaction of a non-relativistic quantum particle with a thin time-dependent absorbing barrier. The first model represents the barrier by a set of time-dependent discontinuous matching conditions, which are closely related to Kottler boundary conditions used in stationary wave optics as a mathematical basis for Kirchhoff diffraction theory. The second model mimics the absorbing barrier with an off-diagonal δ-potential with a time-dependent amplitude. We show that the two models of absorption agree in their predictions in a semiclassical regime -the regime readily accessible in modern experiments with ultracold atoms.

Phase-space representation of diffraction in time: Analytic results

“…The aperture function model then allows one to express the particle's wave function in the transmission region as an integral involving the initial state Ψ(x, 0) and the aperture function χ(t). This transmitted wave function constructed from the aperture function model has been explicitly shown to be consistent with the wave function obtained from a first-principle analysis of physically relevant atom-optics systems [31].…”

“…Indeed, t c precisely corresponds to the time needed for a classical free particle initially located at position x 0 and moving with the velocity v 0 to reach the barrier at x = 0. We then focus on the transmitted state Ψ(x, t) of the particle after it has crossed the barrier: indeed, the aperture function model that we consider in the sequel has been explicitly shown to be physically relevant in the transmission region x > 0 [31]. To this end, we assume that the final time t is, on the one hand, large enough so as to fulfil the condition…”

“…Though the assumptions ( 12)-( 13) have been drawn from the freely evolved Gaussian wave packet ψ σ, x 0 , v 0 (x, t), the latter has actually been explicitly shown [31,34] to be representative of the spatial region where the actual transmitted state Ψ(x, t) of the particle in the presence of the barrier admits non-negligible values. Therefore, the dynamics of the particle in the presence of the barrier can be adequately described within the particular regime that can be summarized by the set of assumptions…”

“…For this reason, here and in the sequel χ is referred to as the aperture function of the barrier, and thus the model itself as the aperture function model. The function χ has been explicitly connected to the (time-dependent) intensity of a laser beam in [31]. It can then be shown [27] that the propagator K(x, x , t), unique solution to the well-posed problem formed by ( 16)- (20), is given by…”

“…∞ along the upper half circle to vanish. Combining (44) with the definitions (31) and (38) of the functions ϕ and f froz , respectively, and assuming that we can interchange the limit and the integral hence yields for…”