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arXiv:1502.06119 · Journal of Physics B Atomic Molecular and Optical Physics

Liouville transformations and quantum reflection

Gabriel Dufour, R. Guérout, Astrid Lambrecht, Serge Reynaud

Published Jun 26, 2015• 8 citations
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Abstract

Abstract Liouville transformations of Schrödinger equations preserve the scattering amplitudes while changing the effective potential. We discuss the properties of these gauge transformations and introduce a special Liouville gauge which allows one to map the problem of quantum reflection of an atom on an attractive Casimir–Polder well into that of reflection on a repulsive wall . We deduce a quantitative evaluation of quantum reflection probabilities in terms of the universal probability which corresponds to the solution of the far-end Casimir–Polder potential.

DOI
10.1088/0953-4075/48/15/155002

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Abstract Liouville transformations of Schrödinger equations preserve the scattering amplitudes while changing the effective potential. We discuss the properties of these gauge transformations and introduce a special Liouville gauge which allows one to map the problem of quantum reflection of an atom on an attractive Casimir–Polder well into that of reflection on a repulsive wall . We deduce a quantitative evaluation of quantum reflection probabilities in terms of the universal probability which corresponds to the solution of the far-end Casimir–Polder potential.
The GBAR experiment will time the free fall of cold antihydrogen atoms dropped onto an annihilation plate to test the universality of free fall on antimatter. In this contribution, we study the quantum reflection of the anti-atom resulting from the Casimir-Polder attraction to the plate. We evaluate the Casimir-Polder potential and the associated quantum reflection amplitudes and find that reflection is enhanced for weaker potentials. A Liouville transformation of the Schrödinger equation is used to map the quantum reflection problem onto an equivalent problem of scattering on a barrier, leading to an intuitive understanding of the phenomenon.
An ultracold atom above a horizontal mirror experiences quantum reflection from the attractive Casimir-Polder interaction, which holds it against gravity and leads to quantum levitation states. We analyze this system by using a Liouville transformation of the Schr\"odinger equation and a Langer coordinate adapted to problems with a classical turning point. Reflection on the Casimir-Polder attractive well is replaced by reflection on a repulsive wall, and the problem is then viewed as an ultracold atom trapped inside a cavity with gravity and Casimir-Polder potentials acting, respectively, as top and bottom mirrors. We calculate numerically Casimir-Polder shifts of the energies of the cavity resonances and propose an approximate treatment which is precise enough to discuss spectroscopy experiments aimed at tests of the weak-equivalence principle on antihydrogen. We also discuss the lifetimes by calculating complex energies associated with cavity resonances.
Collisions between ultracold atoms and material surfaces are characterized by the reflection of the atomic matter wave from the attractive Casimir-Polder potential. This quantum reflection is particularly relevant to experiments such as GBAR, which will determine the gravitational acceleration of a cold antihydrogen atom by timing its fall onto a detection plate. In this thesis, the Casimir-Polder potential is computed from the electromagnetic scattering properties of the atom and surface and it is found to depend notably on the dielectric response, thickness and density of the medium. We show that reflection on this potential is associated with a breakdown of the semiclassical approximation and that it is enhanced for slow atoms and weak potentials. Liouville transformations relate Schrödinger equations with different potential landscapes but identical scattering properties. We gain new insights on the problem of quantum reflection on a potential well by mapping it onto an equivalent problem of tunneling through a wall. We also discuss the effect of gravity on the atomic wavepacket and its implications for free fall experiments with atoms. When combined with quantum reflection from a horizontal mirror, gravity can be used to trap particles in long lived states with promising applications for metrology. In particular, we suggest a scheme to improve the precision of the GBAR experiment by reducing the velocity dispersion of the falling atoms.
Abstract Among the fundamental quantum effects, quantum reflection (QR) is one of the most notable phenomena. Approximating arbitrary potentials in the Schrödinger equation as multistep potentials allows us to determine the reflection coefficient by means of a Riccati equation. We introduce a measure for the wave function's characteristic reflection distance and derive an analytical expression for this reflection point in an arbitrary power-law potential. We study the associated QR rate in the context of Casimir-Polder atom surface interactions, revealing its strong dependence on critical parameters such as velocity and interaction duration. As an application, we demonstrate how the matter-wave diffraction of a low-velocity metastable argon beam impinging on a rotated grating facilitates high-precision potential analysis.
We present a scheme to generate an all-range long repulsive Casimir-Polder potential between a perfect negative reflecting surface and a ground-state atom. The repulsive potential is stable and does not decay with time. The Casimir-Polder potential is proportional to ${z}^{\ensuremath{-}2}$ at short atom-surface distances and to ${z}^{\ensuremath{-}4}$ at long atom-surface distances. Because of these advantages, this potential can help in building quantum reflectors, quantum levitating devices, and waveguides for matter waves.