# QUANTUM REFLECTION

**PEOPLE**

Pierre-Philippe Crépin

Romain Guérout

Astrid Lambrecht

Serge Reynaud

Quantum reflection is a generic phenomenon for matter waves in a rapidly varying potential. It can be observed in particular for atoms experiencing an attractive Casimir-Polder potential in the vicinity of a solid surface.

Quantum reflection on the Casimir-Polder interaction is an important question for the project **GBAR** (*Gravitational Behaviour of Antihydrogen at Rest*) which aims at measuring the free fall acceleration of neutral anti-hydrogen atoms in the terrestrial gravitational field. **GBAR** is an international collaboration including twelve institutes from France, Germany, Japan, Poland, Russia, United-Kingdom and Switzerland and several groups of our laboratory are involved in it. The project, submitted to CERN in September 2011 and approved in May 2012, is described in an article of the collaboration.

Accurate evaluations of the Casimir-Polder potential, as well as of the resulting quantum reflection, are necessary to take this effect into account in the data analysis for GBAR. The phenomenon of quantum reflection can also be used for designing new measurement strategies which will lead in the future to improved accuracies.

##### Quantum reflection of cold antihydrogen atoms in GBAR

Quantum reflection is a generic phenomenon for matter waves experiencing a rapidly varying potential. It has been observed in particular for atoms experiencing an attractive Casimir-Polder potential in the vicinity of a solid surface. A fraction of the cold anti-hydrogen atoms falling onto the detection plate in **GBAR** will be reflected before touching the plate and this will affect the free fall measurement. In collaboration with Valery Nesvizhevskyat (Institut Laue-Langevin, Grenoble) and Alexei Voronin (Lebedev Institute, Moscow), our group has carried out accurate evaluations of the Casimir potential for (anti-)atoms above surfaces, and of the associated quantum reflection.

*Quantum reflection of antihydrogen from the Casimir potential above matter slabs*, G. Dufour, A. Gérardin, R. Guérout, A. Lambrecht, V.V. Nesvizhevsky, S. Reynaud, A.Yu. Voronin Phys. Rev. A**87**012901 (2013)

Studying quantum reflection probability on thinner and thinner silica slabs, we have observed the paradoxical result that quantum reflection increases although the Casimir-Polder potential decreases (see the figure on the right column).

We have then investigated quantum reflection of anti-hydrogen atoms from nanoporous media due to the Casimir-Polder potential. Using a simple effective medium model, we have shown a dramatic increase of the probability of quantum reflection when the porosity of the medium increases, that is again when the Casimir-Polder potential decreases.

*Quantum reflection of antihydrogen from nanoporous media*, G. Dufour, R. Guérout, A. Lambrecht, V.V. Nesvizhevsky, S. Reynaud, A.Yu. Voronin Phys. Rev. A**87**022506 (2013)

In the limiting case of antihydrogen atoms impinging the surface with a very low energy, quantum reflection over the Casimir potential becomes so efficient that it allows one to trap and manipulate antimatter thanks to the vicinity of a material surface ! In future generations of the GBAR project, this idea can in principle be used to measure the free fall acceleration with a largely improved accuracy.

##### Using quantum reflection for improving accuracy

Anti-hydrogen atoms with a very low energy above the surface experience an extremely efficient quantum reflection over the Casimir potential. This should allow one to trap anti-hydrogen bouncing on the material surface under the effect of the Earth gravity field. In future generations of the **GBAR** project, this idea can in principle be used to measure the free fall acceleration with an improved accuracy though a spectroscopic measurement of the quasi-stationary quantum states in this trap.

*Interference of the whispering gallery states of antihydrogen*, A.Yu. Voronin, V.V. Nesvizhevsky, S Reynaud, J. Phys. B**45**165007 (2012)*Quantum ballistic experiment on antihydrogen fall*, AY Voronin, VV Nesvizhevsky, G Dufour, S Reynaud, J. Phys. B**49**054001 (2016)

In the present design of the **GBAR** project, the precision of the timing experiment is limited by the dispersion of initial vertical velocities of the atoms as well as on the reliable control of their distribution. These two factors are improved simultaneously by a new method that we have proposed for shaping the distribution of the vertical velocities. The method is based on quantum reflection of elastically and specularly bouncing anti-hydrogen atoms with small initial vertical velocity on a bottom mirror disk, and absorption of atoms with large initial vertical velocities on a top rough disk (see the figure on the right column).

*Shaping the distribution of vertical velocities of antihydrogen in GBAR,*G. Dufour, P. Debu, A. Lambrecht, V.V. Nesvizhevsky, S. Reynaud, A.Yu. Voronin*,**Eur. Phys. J. C*(2014)**74**2731

These methods are currently considered as upgrading options for the design of the **GBAR** experiment.

*Prospects for Studies of the Free Fall and Gravitational Quantum States of Antimatter,*G. Dufour, D.B. Cassidy, P. Crivelli, P. Debu, A. Lambrecht, V.V. Nesvizhevsky, S. Reynaud, A.Yu. Voronin, T.E. Wall*,**Adv. High Energy Phys. 379642*(2015)

##### Understanding the paradoxes in quantum reflection through Liouville transformations

Paradoxical phenomena appear in the study of quantum reflection (QR) from the Casimir-Polder (CP) interaction above a surface. First reflection is produced by a rapidly varying attractive potential. Then, the probability of reflection increases when the energy of the incident atom is decreased, and increases as well when the absolute magnitude of the CP potential is decreased. For example, the probability of quantum reflection is larger for atoms falling onto silica bulk than onto metallic or silicon bulks and it is even larger

on nano-porous silica than on silica bulk. We have shed light on these QR paradoxical behaviors by using Liouville transformations.

Liouville transformations map in a rigorous manner one Schrödinger equation into another, with a changed scattering potential. Remarkably, scattering properties are invariant under the Liouville transformation although the corresponding semiclassical descriptions are completely different. We have used them to transform quantum reflection of an atom on an attractive CP *well* into reflection of the atom on a repulsive *wall*.

*Quantum reflection and Liouville transformations from wells to walls,*G. Dufour, R. Guérout, A. Lambrecht, S Reynaud, EPL**110**30007 (2015)

The paradoxical features of the initial QR problem become intuitive predictions of the transformed problem of reflection on the repulsive wall. This is illustrated for example on the figure on the right column, for scattering problems corresponding to the same CP potential between an (anti-)hydrogen atom and a silica bulk and different energies.

We also deduce from this method a quantitative evaluation of quantum reflection probabilities in terms of the universal probability calculated from the solution of the far-end Casimir–Polder potential The dashed black curve on the figure is the universal transformed function calculated for this power law.

*Liouville transformations and quantum reflection*, G. Dufour, R. Guérout, A. Lambrecht, S. Reynaud, J. Phys. B**48**155002 (2015)