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arXiv:1609.09023 · Physical Review A

Poisson's spot and Gouy phase

I. G. da Paz, Rodolfo Reis Soldati, Luís A. Cabral, J. G. G. de Oliveira, Marcos Sampaio

Published Dec 12, 2016• 6 citations
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Abstract

Recently there have been experimental results on Poisson spot matter-wave interferometry followed by theoretical models describing the relative importance of the wave and particle behaviors for the phenomenon. We propose an analytical theoretical model for Poisson's spot with matter waves based on the Babinet principle, in which we use the results for free propagation and single-slit diffraction. We take into account effects of loss of coherence and finite detection area using the propagator for a quantum particle interacting with an environment. We observe that the matter-wave Gouy phase plays a role in the existence of the central peak and thus corroborates the predominantly wavelike character of the Poisson's spot. Our model shows remarkable agreement with the experimental data for deuterium (${\text{D}}_{2}$) molecules.

DOI
10.1103/physreva.94.063609

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Recently there have been experimental results on Poisson spot matter-wave interferometry followed by theoretical models describing the relative importance of the wave and particle behaviors for the phenomenon. We propose an analytical theoretical model for Poisson's spot with matter waves based on the Babinet principle, in which we use the results for free propagation and single-slit diffraction. We take into account effects of loss of coherence and finite detection area using the propagator for a quantum particle interacting with an environment. We observe that the matter-wave Gouy phase plays a role in the existence of the central peak and thus corroborates the predominantly wavelike character of the Poisson's spot. Our model shows remarkable agreement with the experimental data for deuterium (D2{\text{D}}_{2}D2​) molecules.
The Fresnel diffraction phenomenon referred to as Poisson's spot or spot of Arago has, beside its historical significance, become relevant in a number of fields. Among them are for example fundamental tests of the super-position principle in the transition from quantum to classical physics and the search for extra-solar planets using star shades. Poisson's spot refers to the positive on-axis wave interference in the shadow of any spherical or circular obstacle. While the spot's intensity is equal to the undisturbed field in the plane wave picture, its intensity in general depends on a number of factors, namely the size and wavelength of the source, the size and surface corrugation of the diffraction obstacle, and the distances between source, obstacle and detector. The intensity can be calculated by solving the Fresnel–Kirchhoff diffraction integral numerically, which however tends to be computationally expensive. We have therefore devised an analytical model for the on-axis intensity of Poisson's spot relative to the intensity of the undisturbed wave field and successfully validated it both using a simple light diffraction setup and numerical methods. The model will be useful for optimizing future Poisson-spot matter-wave diffraction experiments and determining under what experimental conditions the spot can be observed.
Matter–wave interferometry has been used extensively over the last few years to demonstrate the quantum-mechanical wave nature of increasingly larger and more massive particles. We have recently suggested the use of the historical Poisson spot setup to test the diffraction properties of larger objects. In this paper, we present the results of a classical particle van der Waals (vdW) force model for a Poisson spot experimental setup and compare these to Fresnel diffraction calculations with a vdW phase term. We include the effect of disc-edge roughness in both models. Calculations are performed with D 2 and with C 70 using realistic parameters. We find that the sensitivity of the on-axis interference/focus spot to disc-edge roughness is very different in the two cases. We conclude that by measuring the intensity on the optical axis as a function of disc-edge roughness, it can be determined whether the objects behave as de Broglie waves or classical particles. The scaling of the Poisson spot experiment to larger molecular masses is, however, not as favorable as in the case of near-field light-grating-based interferometers. Instead, we discuss the possibility of studying the Casimir–Polder potential using the Poisson spot setup.
The phase of de Broglie matter waves is a sensitive probe for small forces. In particular, the attractive van der Waals force experienced by polarizable atoms in the close vicinity of neutral surfaces is of importance in nanoscale systems. It results in a phase shift that can be observed in matter-wave diffraction experiments. Here, we observe Poisson spot diffraction of indium atoms at submillimeter distances behind spherical submicron silicon dioxide particles to probe the dispersion forces between atoms and the particle surfaces. We compare the measured relative intensity of Poisson's spot to theoretical results derived from first principles in an earlier communication and find a clear signature of the atom-surface interaction.
Diffraction of matter waves is an important demonstration of the fact that objects in nature possess a mixture of particlelike and wavelike properties. Unlike in the case of light diffraction, matter waves are subject to a vacuum-mediated interaction with diffraction obstacles. Here we present a detailed account of this effect through the calculation of the attractive Casimir-Polder potential between a dielectric sphere and an atomic beam. Furthermore, we use our calculated potential to make predictions about the diffraction patterns to be observed in an ongoing experiment where a beam of indium atoms is diffracted around a silicon-dioxide sphere. The result is an amplification of the on-axis bright feature, which is the matter-wave analog of the well-known ``Poisson spot'' from optics. Our treatment confirms that the diffraction patterns resulting from our complete account of the sphere Casimir-Polder potential are indistinguishable from those found via a large-sphere nonretarded approximation in the discussed experiments, establishing the latter as an adequate model.
In this thesis, we study theoretical and experimental aspects of the Gouy phase in matter waves. We show that this phase is related to the covariance between position and momentum and can be indirectly extracted from interferometric experiments. Next, we analyze a diffraction experiment with macro-molecules realized by Zeilingers group [1] and we show that the existence of a Gouy phase is compatible with those experimental data. Similar to the Gouy phase for classical electromagnetic waves, it is shown that a beam of matter waves suffers a phase anomaly around the focus of a quantum lens. An experimental proposal to measure this phase is presented with experimental parameters viable to current technology. We hope this work motivates studies of the theoretical problem, open a few years ago, and forexperiments that lead us to an understanding deeper and more realistic of the Gouy phase for mixed states of matter waves and light. At the end, we present our conclusions.
In the Poisson-spot experiment, waves emanating from a source are blocked by a circular obstacle. Due to their positive on-axis interference an image of the source (the Poisson spot) is observed within the geometrical shadow of the obstacle. In this paper we report the observation of Poisson's spot using a beam of neutral deuterium molecules. The wavelength independence and the weak constraints on angular alignment and position of the circular obstacle make Poisson's spot a promising candidate for applications ranging from the study of large molecule diffraction to patterning with molecules.