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arXiv:1511.03973

Casimir effect and quantum reflection

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

Published Nov 12, 2015• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1511.03973

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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.
In the GBAR experiment, cold antihydrogen atoms will be left to fall on an annihilation plate with the aim of measuring the gravitational acceleration of antimatter. Here, we study the quantum reflection of these antiatoms due to the Casimir-Polder potential above the plate. We give realistic estimates of the potential and quantum reflection amplitudes, taking into account the specificities of antihydrogen and the optical properties of the plate. We find that quantum reflection is enhanced for weaker potentials, for example above thin slabs, graphene and nanoporous media.
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 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.
We study the quantum reflection of ultracold antihydrogen atoms bouncing on the surface of a liquid helium film. The Casimir-Polder potential and quantum reflection are calculated for different thicknesses of the film supported by different substrates. Antihydrogen can be protected from annihilation for as long as 1.3 s on a bulk of liquid 4 He, and 1.7 s for liquid 3 He. These large lifetimes open interesting perspectives for spectroscopic measurements of the free fall acceleration of antihydrogen. Variation of the scattering length with the thickness of a film of helium shows interferences which we interpret through a Liouville transformation of the quantum reflection problem.
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.
The framework of this thesis is the GBAR collaboration at CERN, which aims to measure the free fall acceleration of antimatter . In this thesis, we study the quantum reflection of the antihydrogen on the detector, caused by the Casimir-Polder interaction that we calculate for different materials. We find a particularly high quantum reflection for an antihydrogen atom on a surface of liquid helium. We then present a complete description of the gravitational quantum states, mixing gravity and Casimir-Polder interaction. For this purpose, we revisit the theory of collisions in the case of the Casimir-Polder potential through a new "effective range theory", obtained after a Liouville transform. The knowledge of gravitational quantum states leads us to propose a new method of measuring free fall acceleration, by creating quantum interferences between these states. A statistical analysis of the interference pattern thus obtained is carried out, leading to an improvement in the accuracy until three orders of magnitude compared to the initial free-fall experiment. Finally, we study in detail the influence of the disorder at the level of the plate of detection, the latter being in fact not a perfect surface. We calculate the effect of this disorder on the fluctuations of the Casimir-Polder potential itself, and observe a different behavior in law for the conductivity models such as the plasma model and the Drude model.