Impact of Casimir-Polder interaction on Poisson-spot diffraction at a dielectric sphere

Abstract

Diffraction of matter-waves is an important demonstration of the fact that objects in nature possess a mixture of particle-like and wave-like 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 analogue 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 non-retarded approximation in the discussed experiments, establishing the latter as an adequate model.

On the relative intensity of Poisson’s spot

“…The analytic model will be relevant as well in matter-wave studies as proposed in a recent article, where we discuss the possibility of determining Casimir-Polder forces by measuring I rel of Poisson's spot in indium matter-wave diffraction experiments [19]. For example, in the article effects from edge corrugation were table 3, for which the number and width of support bars keeping each disc in place was varied.…”

“…The effect of the Casimir-Polder potential was covered in detail for matter waves by Nimmrichter et al [16,17] and in one of our previous works [14]. A Poisson spot experiment realized for matter-waves utilizing a supersonic-expansion beam composed of deuterium molecules [18] did not show any noticeable effect in this respect, but is expected to become important in the case of more polarizable beam species like for example C 60 molecules and even indium atoms [19]. The Poisson spot therefore can be used to measure interaction potentials of this kind.…”

Poisson's spot and Gouy phase

“…In order to obtain the density matrix in the obstacle we have to take the limit when ǫ → 0 in the parameters B(t, ǫ), R(t, ǫ) and µ s (t, ǫ) of the wavefunction given by equation (7). After performing such limits using the expressions (11), (12) and (13), we obtain the following results lim ǫ→0 B(t, ǫ) = b 2 (t)β 2 β 2 +b 2 (t) , lim ǫ→0 R(t, ǫ) = r(t) and lim ǫ→0 µ s (t, ǫ) = µ f (t).…”

“…Thus multi-path interference leads to a bright spot at the center of the shadow region behind the obstacle. From the experimental viewpoint [6,7], it is believed that the diffraction pattern is significantly affected by the dispersive interaction between the matter-waves and the obstacle namely modifying the width and the height of the central Poisson spot, invalidating the Fresnel zone construction and Babinet principle. They argue that the spot could appear in the case of classical particles passing the obstacle following deflected trajectories due to the attractive force towards the obstacle (van-der-Walls).…”

“…In addition to that, we have the unavoidable edge-corrugation of the disc. In [7] it was studied the effect of Casimir-Polder/van der Waals (CP/vdW) dispersion forces on Poisson spot diffraction at a dielectric sphere which may obscure the distinction between particle and wave nature. Obviously these effects blemish the distinction between the quantum and the classical description for large enough interaction strengths, such that the appearance of the spot in itself is not exclusively due to wave-like behaviour of the particles.…”