# QUANTUM REFLECTION

**PEOPLE**

Pierre-Philippe Crépin

Marie-Pascale Gorza

Romain Guérout

Astrid Lambrecht

Serge Reynaud

Quantum reflection is a generic phenomenon for matter waves in 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.

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

Ultra cold anti-hydrogen atoms falling onto the detection plate in **GBAR** will suffer a partial reflection before touching the plate and this will affect the measurement of the free fall. 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)

##### 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 may be considered as upgrading options for 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 also when the absolute magnitude of the CP potential is decreased. The probability of quantum reflection is larger for atoms falling onto silica bulk than onto silicon bulks and 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 modified scattering potential. Scattering properties are invariant under the transformation although the semiclassical descriptions are completely different. We have used them to change 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 on the figure on the right column, for scattering problems corresponding to the same CP potential between an antihydrogen atom and a silica bulk at different energies.

We have also deduced 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)

##### Quantum levitation states

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 have analyzed this system by using a Liouville transformation with quantum reflection on the Casimir-Polder attractive well replaced by reflection on a repulsive wall, so that the problem is viewed as an ultracold atom trapped inside a cavity with gravity and Casimir-Polder potentials acting, respectively, as top and bottom mirrors.

We have calculated numerically Casimir-Polder shifts of the energies of the cavity resonances and proposed analytical expressions which are precise enough to discuss spectroscopy experiments aimed at tests of the weak equivalence principle on antihydrogen. We have also discussed the lifetimes by calculating complex energies associated with cavity resonances.

*Casimir-Polder shifts on quantum levitation states,*P.-P. Crépin, G. Dufour, R. Guérout, A. Lambrecht, S Reynaud, Phys. Rev. A**95**032501 (2017)

We have studied the quantum reflection of ultracold antihydrogen atoms bouncing on the surface of a liquid helium film, for different thicknesses of the film supported by different substrates. We have found that antihydrogen can be protected from annihilation for as long as 1.3 s on a bulk of liquid 4He, and 1.7 s for liquid 3He. 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 were interpreted through a Liouville transformation of the quantum reflection problem.

*Quantum reflection of antihydrogen from a liquid helium film*, P.-P. Crépin, E.A. Kupriyanova, R. Guérout, A. Lambrecht, V.V. Nesvizhevsky, S. Reynaud, S. Vasiliev, A.Yu. Voronin, EPL**119**33001 (2017)