Quantum reflection by Casimir-van der Waals potential tails

We study the reflectivity of Casimir-van der Waals potentials, which behave as -C4/r4 at large distances and as -C3/r3 at small distances. The overall behavior of the reflection amplitude R depends crucially on the parameter ρ=√(2M)C3/((ℎ/2π)√(C4)) which determines the relative importance of the -1/r3 and the -1/r4parts of the potential. Near threshold, E=(ℎ/2π)2k2/(2M)→0, the reflectivity is given by |R|∼exp(-2bk), with b depending on ρ and the shape of the potential at intermediate distances. In the limit of large energies, ln|R| is proportional to -k1/3 with a known constant of proportionality depending only on C3. For small values of ρ, the reflectivity behaves as for a homogeneous -1/r3 potential in the whole range of energies and does not depend on C4or the shape of the potential beyond the -1/r3region. For moderate and large values of ρ, the reflectivity depends on C4 and on the potential shape. For sufficiently large values of ρ, which are ubiquitous in realistic systems, there is a range of energies beyond the near-threshold region, where the reflectivity shows the high-energy behavior appropriate for a homogeneous -1/r4 potential, i.e., ln|R| is proportional to -√(k) with a proportionality constant depending only on C4. This conspicuous and model-independent signature of the Casimir effect is illustrated for the reflectivities of neon atoms scattered off a silicon surface, which were recently measured by Shimizu [Phys. Rev. Lett. 86, 987 (2001)]