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arXiv:quant-ph/0702224 · The Journal of Chemical Physics

A causal look into the quantum Talbot effect

Ángel S. Sanz, Salvador Miret-Artés

Published Jun 18, 2007• 71 citations
arXivHTMLPDF

Abstract

A well-known phenomenon in both optics and quantum mechanics is the so-called Talbot effect. This near field interference effect arises when infinitely periodic diffracting structures or gratings are illuminated by highly coherent light or particle beams. Typical diffraction patterns known as quantum carpets are then observed. Here the authors provide an insightful picture of this nonlocal phenomenon as well as its classical limit in terms of Bohmian mechanics, also showing the causal reasons and conditions that explain its appearance. As an illustration, theoretical results obtained from diffraction of thermal He atoms by both N-slit arrays and weak corrugated surfaces are analyzed and discussed. Moreover, the authors also explain in terms of what they call the Talbot-Beeby effect how realistic interaction potentials induce shifts and distortions in the corresponding quantum carpets.

DOI
10.1063/1.2741555

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A well-known phenomenon in both optics and quantum mechanics is the so-called Talbot effect. This near field interference effect arises when infinitely periodic diffracting structures or gratings are illuminated by highly coherent light or particle beams. Typical diffraction patterns known as quantum carpets are then observed. Here the authors provide an insightful picture of this nonlocal phenomenon as well as its classical limit in terms of Bohmian mechanics, also showing the causal reasons and conditions that explain its appearance. As an illustration, theoretical results obtained from diffraction of thermal He atoms by both N-slit arrays and weak corrugated surfaces are analyzed and discussed. Moreover, the authors also explain in terms of what they call the Talbot-Beeby effect how realistic interaction potentials induce shifts and distortions in the corresponding quantum carpets.
Talbot and Talbot-Lau effects are frequently used in lensless imaging applications with light, ultrasound, x-rays, atoms and molecules--generally in situations where refractive optical elements are non-existent or not suitable. We here show an experimental visualization of the intriguing wave patterns that are associated with near-field interferometry behind a single periodic diffraction grating under plane wave illumination and which are often referred to as Talbot carpets or quantum carpets. We also show the patterns behind two separated diffraction gratings under nearly-monochromatic but spatially incoherent illumination that illustrate the nature of Talbot-Lau carpets.
Diffraction and interference of matter waves are key phenomena in quantum mechanics. Here we present some results on particle diffraction in a wide variety of situations, ranging from simple slit experiments to more complicated cases such as atom scattering by corrugated metal surfaces and metal surfaces with simple and isolated adsorbates. The principal novelty of our study is the use of the so-called Bohmian formalism of quantum trajectories. These trajectories are able to satisfactorily reproduce the main features of the experimental results and, more importantly, they provide a causal intuitive interpretation of the underlying dynamics. In particular, we will focus our attention on: (a) a revision of the concepts of near and far field in undulatory optics; (b) the transition to the classical limit, where it is found that although the quantum and classical diffraction patterns tend to be quite similar, some quantum features are maintained even when the quantum potential goes to zero; and (c) a qualitative description of the scattering of atoms by metal surfaces in the presence of a single adsorbate.
The concept of quantum superposition is reconsidered and discussed from the viewpoint of Bohmian mechanics, the hydrodynamic formulation of quantum mechanics, in order to elucidate some physical consequences that go beyond the simple mathematical idea of linearly combining vectors in a Hilbert space. Specifically, the discussion turns around the connection between symmetries characterizing the wave function and the behavior in time displayed by the quantum flux when the latter is analyzed in terms of streamlines (Bohmian trajectories). This is illustrated with a series of analytical results and numerical simulations, which include Young's two-slit experiment, counter-propagating wave packets, grating diffiraction and quantum carpets (e.g., Talbot carpets), and diffiraction under confinement conditions. From the analysis presented it follows that quantum paradoxes appear whenever symmetries related to interference are neglected in the interpretation and understanding of the corresponding phenomena.
We investigate the suppression of matter-wave Talbot interference under environmentally induced decoherence. The system is modeled as an atomic beam diffracted by a periodic grating whose transverse dynamics is described within the paraxial approximation. Environmental coupling is introduced through an effective open-system model that exponentially damps spatial coherences between diffracted components, allowing a continuous interpolation between the coherent Talbot regime and the incoherent far-field diffraction limit. In addition to the usual intensity and transverse-momentum distributions, we analyze the local probability flow associated with the diffracted matter wave. The corresponding Bohmian, or hydrodynamic, representation is used here as a diagnostic tool fully equivalent to the standard quantum description, with no additional assumptions beyond the probability current of the paraxial wave field. In the present Talbot geometry, this analysis shows how decoherence progressively suppresses the carpet structure and smooths the transverse-momentum distribution, while the flow may remain organized into channels determined by the grating periodicity. The results illustrate, in a periodic matter-wave Talbot geometry, that the loss of visible interference and the degradation of the grating-defined flow-domain organization need not occur simultaneously. In particular, flux-channel structures can persist in parameter regimes where multislit interference features have already been strongly reduced. This distinction provides a local characterization of decoherence in matter-wave Talbot interferometry and complements previous trajectory-based analyses of coherence loss in simpler interference and confined geometries.
Summary form only given.Recent developments in atom optics have led to different approaches to lithographic techniques with neutral atoms. One of these is the use of diffraction from free-standing masks, which is sometimes called atom holography. There, narrow structures arise due to matter-wave interference in the Fresnel regime. In the presented experiment, we demonstrate that the wave nature of atoms can be used to produce periodic structures with small periodicities. We used the Talbot effect in the Fresnel diffraction regime behind a microfabricated transmission grating. The Talbot effect is the self-imaging of a grating of periodicity d illuminated with plane coherent waves.
We introduce the superconducting Talbot effect---a macroscopic quantum self-imaging phenomenon occurring when proximitized Cooper pairs propagate through a ballistic two-dimensional electron gas. By configuring periodic superconducting leads into a phased-array Josephson junction with programmable phase differences, we demonstrate active steering of the resulting superconducting Talbot carpet. To overcome transport resolution limits, we design a Vernier-scale collector array that performs sub-wavelength sampling of the fractional Talbot pattern. This approach maps real-space quantum interference with high robustness to disorder, enabling direct extraction of Fermi wavelengths across helical, spin-degenerate, and spin-orbit-split Fermi surfaces. Tight-binding numerical calculations on a square lattice validate the real-space interference patterns. Our results establish a versatile framework for coherent wavefront engineering and quantum materials diagnostics in superconducting optics.