Boundary bound diffraction: a combined spectral and Bohmian analysis

Abstract

The diffraction-like process displayed by a spatially localized matter wave is here analyzed in a case where the free evolution is frustrated by the presence of hard-wall-type boundaries (beyond the initial localization region). The phenomenon is investigated in the context of a nonrelativistic, spinless particle with mass m confined in a one-dimensional box, combining the spectral decomposition of the initially localized wave function (treated as a coherent superposition of energy eigenfunctions) with a dynamical analysis based on the hydrodynamic or Bohmian formulation of quantum mechanics. Actually, such a decomposition has been used to devise a simple and efficient analytical algorithm that simplifies the computation of velocity fields (flows) and trajectories. As it is shown, the development of space-time patters inside the cavity depends on three key elements: the shape of the initial wave function, the mass of the particle considered, and the relative extension of the initial state with respect to the total length spanned by the cavity. From the spectral decomposition it is possible to identify how each one of these elements contribute to the localized matter wave and its evolution; the Bohmian analysis, on the other hand, reveals aspects connected to the diffraction dynamics and the subsequent appearance of interference traits, particularly recurrences and full revivals of the initial state, which constitute the source of the characteristic symmetries displayed by these patterns. It is also found that, because of the presence of confining boundaries, even in cases of increasingly large box lengths, no Fraunhofer-like diffraction features can be observed, as happens when the same wave evolves in free space. Although the analysis here is applied to matter waves, its methodology and conclusions are also applicable to confined modes of electromagnetic radiation (e.g., light propagating through optical fibers).

Decoherence in quantum cavities: Environmental erasure of carpet-type structures

“…It can be seen how the initial diffraction launches the trajectories in a relatively fast manner towards the boundaries of the cavity, thus covering the whole available space inside it. Although the appearance of recurrences and revivals is independent of the input signal, the initial boost strongly depends on it, as shown elsewhere [56]. The same behavior can be observed in both cases, symmetric and asymmetric, though with the difference that in the latter case trajectories on one side of the input signal have to travel a larger distance than those started on the other side.…”

“…Depending on whether a symmetric or an assymetric input signal is considered, we observe the presence of maxima at x = 0 or not, respectively. This maximum plays the role a certain effective barrier in the symmetric configurations: trajectories started on either side will never be able to cross the other side [56,68]. To some extent, the dynamics on either side of the central maximum is going to be ruled by this maximum and the borders of the cavity, in a similar fashion to an effective two-wave superposition (although with a more complex interference process in between, as it is seen in Fig.…”

“…Quantum carpets, the highly symmetric pattern displayed in both space and time by the probability density inside a cavity, arise as a consequence of a rather complex interference process involving a number of energy eigenstates (vibrational modes) of the cavity [47]. In brief, just pump a localized wave function, e.g., a single mode coming from a fiber or a neutron or electron matter wave crossing a whole, inside a larger cavity; its time-evolution makes the rest [56]. To better understand the process and also to be self-contained, let us start by briefly describing the process that leads to the appearance of quantum carpets as well as some of its most relevant properties in connection to this work.…”

“…The presence of revivals of the initial state as well as recurrences at fractional times gives rise to a pattern with space and time symmetries denoted as quantum carpet [47]. Typically the concept of quantum carpet is associated with the probability density, although similar symmetries can also be found in the amplitude and phase of the signal, or the quantum flux, which becomes more apparent when it is analyzed in terms of the associated streamlines or Bohmian trajectories [56]. In order to determine the effects of decoherence on the flow of probability inside the cavity, here we have proceeded similarly, i.e., considering the supplementary aid of a Bohmiantype description.…”

Position measurement-induced collapse states: Proposal of an experiment

“…The present work has some closely related antecedents in [8] and [9]. In the former, Berry has computed the probability density corresponding to a wave function of a particle in a box that evolves from an initial one with a discontinuity at its walls.…”