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=√2MC₃/(√C₄) which determines the relative importance of the -1/r³ and the -1/r⁴ parts of the potential. Near threshold, E=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 C₄ or the shape of the potential beyond the -1/r³ region. 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)].