Casimir-Polder shifts on quantum levitation states
Résumé
Ultracold atoms can be held in quantum levitation states by the joint effect of
gravity and quantum reflection from the attractive Casimir-Polder interaction
above a horizontal mirror. We calculate the Casimir-Polder shifts of the
energies of these states with a precision sufficient to discuss spectroscopy
experiments aiming at tests of the weak equivalence principle on antihydrogen.
Gravitationally bound quantum states have been observed with ultracold neutrons
1,2 , but not with neutral atoms. At the low energies corresponding to such
bound states, an efficient quantum reflection is produced by the Casimir-Polder
(CP) tail above a surface 3,4,5,6 , so that quantum levitation states of atoms
can be created, where quantum reflection above a horizontal mirror balances
gravity 7. It has recently be proposed to test the weak equivalence principle
with antihydrogen by timing its free fall from a trapping device 8. The
precision of this test, of the order of 1%, could be improved by spectroscopic
measurements of quantum levitation states of antihydrogen above a material
surface, since the properties of these states are essentially determined by
gravity 9,10. However an accurate determination of the acceleration requires a
precise evaluation of Casimir-Polder shifts on quantum levitation states 11. We
consider an atom of mass m and energy E bouncing in the gravity potential mgz
and reflected on a plane horizontal mirror due to quantum reflection on the CP
potential V CP (z). The wavefunction ψ(z) obeys the one-dimensional Schrödinger
equation − 2 2m d 2 ψ dz 2 (z) + V (z)ψ(z) = Eψ(z) , V (z) = mgz + V CP (z) ,
(1) with the potential V depending only on the altitude z of the particle above
the mirror sitting at z = 0. We also suppose that atoms are absorbed when
touching the surface, which corresponds to the physical boundary condition for
antihydrogen annihilated at contact with matter. Length and energy scales
associated with the Casimir-Polder potential are respectively much smaller and
much larger than those associated with the gravitational potential g = 2 2m 2 g
1/3 ≈ 5.87 µm , g = mg g ≈ 0.602 peV , (for g ≈ 9.81 m.s −2). (2) An approximate
solution of the problem is thus found by decoupling the effects of gravity and
CP interactions. For quantum levitation states with low values of the quantum
number n, the scattering amplitudes are mainly given by the scattering length a.
It follows that the energies E 0 n