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arXiv:0907.4638

N-Slit Interference: Fractals in Near-Field Region, Bohmian Trajectories

Valeriy I. Sbitnev

Published Jul 27, 2009• 4 citations
arXivHTMLPDF

Abstract

Scattering cold particles on an $N$-slit grating is shown to reproduce an interference pattern, that manifests itself in the near-field region as the fractal Talbot carpet. In the far-field region the pattern is transformed to an ordinary diffraction, where principal beams are partitioned from each other by ($N-2$) weak ones. A probability density plot of the wave function, to be represented by a gaussian wavepacket, is calculated both in the near-field region and in the far-field one. Bohmian (geodesic) trajectories, to be calculated by a guidance equation, are superimposed on the probability density plot well enough. It means, that a particle, moving from a source to a detector, passes across the grating along a single bohmian trajectory through-passing one and only one slit. Keywords: Gaussian wavepacket, neutron scattering, guidance equation, bohmian trajectory, near-field interference, far-field diffraction, Talbot carpet, fractal

DOI
10.48550/arxiv.0907.4638

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Scattering cold particles on an NNN-slit grating is shown to reproduce an interference pattern, that manifests itself in the near-field region as the fractal Talbot carpet. In the far-field region the pattern is transformed to an ordinary diffraction, where principal beams are partitioned from each other by (N−2N-2N−2) weak ones. A probability density plot of the wave function, to be represented by a gaussian wavepacket, is calculated both in the near-field region and in the far-field one. Bohmian (geodesic) trajectories, to be calculated by a guidance equation, are superimposed on the probability density plot well enough. It means, that a particle, moving from a source to a detector, passes across the grating along a single bohmian trajectory through-passing one and only one slit. Keywords: Gaussian wavepacket, neutron scattering, guidance equation, bohmian trajectory, near-field interference, far-field diffraction, Talbot carpet, fractal
Path integrals give a possibility to compute in details routes of particles from particle sources through slit gratings and further to detectors. The path integral for a particle passing through the Gaussian slit results in the Gaussian wavepacket. The wavepackets prepared on N slits and superposed together give rise to interference pattern in the near-field zone. It transforms to diffraction in the far-field zone represented by divergent principal rays, at that all rays are partitioned from each other by (N-2) subsidiary rays. The Bohmian trajectories in the near-field zone of N-slit gratings show wavy behavior. And they become straight in the far-field zone. The trajectories show zigzag behavior on the interference Talbot carpet (ratio of particle wavelength to a distance between slits are much smaller than 1 and N>>1). Interference from the the N-slit gratings is simulated by scattering monochromatic neutrons (wavelength=0.5 nm). Also we have considered simulation of interference fringes arising at scattering on an N-slit grating of fullerene molecules (according to the real experiment stated in e-print 1001.0468).
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.
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.
Particle paths, emitted from distributed sources and passing out through slits of two gratings, G0 and G1, up to detectors, have been computed in detail by the path integral method. The particles under consideration are fullerene molecules with a De Broglie wavelength equal to 5 pm. The slits are Gaussian functions that simulate fuzzy edges of the slits. Waves of the matter computed by this method show perfect interference patterns both between the gratings and behind the second grating. Coherent and non-coherent distributed particle sources reproducing the interference patterns are discussed in detail. Paraxial approximation results from removing the distributed sources onto innity. This approximation gives a wave function reproducing an exact copy of the Talbot carpet. PACS numbers: 03.75.-b, 03.75.Dg, 42.25.Hz
Paths of particles, emitted from distributed sources and passing out through slits of two gratings, G0G_{0}G0​ and G1G_{1}G1​, up to detectors, have been computed in details by the path integral method. The slits are represented by Gaussian functions that simulate fuzzy edges of the slits. Waves of matter be computed by this method show perfect interference patterns both between the gratings and behind the second grating. Coherent and noncoherent the distributed particle sources reproducing the interference patterns are discussed in details. Paraxial approximation stems from the wave function when removing the distributed sources onto infinity. The more hard-edged slits of the grating G1G_{1}G1​ are examined by simulating those slits by a superposition of more hard the Gaussian functions. As for the particles here we consider fullerene molecules. De Broglie wavelength of the molecules is adopted equal to 5 pm.
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.