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 will suffer a quantum 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 atoms or 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 thinner and thinner silica slabs leads to the paradoxical result that quantum reflection increases although the Casimir-Polder potential decreases (see the figure on the right column).
Quantum reflection of anti-hydrogen atoms from nanoporous media shows a similar 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)
The best quantum reflection calculated up to now corresponds to the case of liquid helium films. 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)
Understanding the paradoxes in quantum reflection
Paradoxes appear in the study of quantum reflection (QR) from the Casimir-Polder (CP) interaction above a surface. Reflection is produced by a rapidly varying attractive potential, with its probability increasing when the energy of the incident atom is decreased, or 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 liquid helium bulk. Liouville transformations allow one to shed light on these paradoxical behaviors.
Liouville transformations map in a rigorous manner one Schrödinger equation into another, with a modified potential. Scattering properties are invariant under the transformation although the semiclassical descriptions are completely different. For example quantum reflection of an atom on an attractive CP well can be changed 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 paradoxes in the initial problem become intuitive predictions of the transformed problem. 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 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)
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. 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 the trap formed by quantum reflection and gravity.
- 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. This dispersion could be improved for shaping the distribution of the vertical velocities due to elastic and specular bouncing of anti-atoms on a bottom mirror disk, associated with absorption of atoms on a top rough disk. This method has however a drawback due to large absorption loss.
- 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 74 2731 (2014)
We have proposed a much wiser solution, using quantum interferences between the quantum states in the trap to measure the value of the free fall acceleration. In this method all atoms are used (there is no absorption loss) and interferences between quantum states are read without inducing transitions between them. Monte-Carlo simulations predict an improvement of precision by three orders of magnitude with respect to the classical timing method.
- Quantum interference test of the equivalence principle on antihydrogen, P.-P. Crépin, C. Christen, R. Guérout, V. V. Nesvizhevsky, A.Yu. Voronin, and S. Reynaud1, Phys. Rev. A 99 042119 (2019)
This method is considered as an upgrading option for the GBAR experiment.
Casimir-Polder shifts on quantum levitation states
Quantum reflection from the attractive Casimir-Polder interaction holds atoms against gravity and leads to quantum levitation states, which can be used for high-precision measurement of gravity or other forces. These states can be studied by using Liouville transformations, 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.
Casimir-Polder shifts of the energies of the cavity resonances and associated lifetimes can be calculated numerically by this method. They can also be represented by analytical expressions (improved effective range expansion) which are precise enough for quantum techniques aimed at tests of the weak equivalence principle on antihydrogen.