Quantum reflection by Casimir-van der Waals potential tails

We study the reflectivity of Casimir-van der Waals potentials, which behave as -C-4/r(4) at large distances and as -C-3/r(3) at small distances. The overall behavior of the reflection amplitude R depends crucially on the parameter rho=root2MC(3)/((h) over bar rootC(4)) which determines the relative importance of the -1/r(3) and the -1/r(4) parts of the potential. Near threshold, E=(h) over bar (2)k(2)/(2M)-->0, the reflectivity is given by \R\similar toexp(-2bk), with b depending on rho and the shape of the potential at intermediate distances. In the limit of large energies, ln\R\ is proportional to -k(1/3) with a known constant of proportionality depending only on C-3. For small values of rho, the reflectivity behaves as for a homogeneous -1/r(3) potential in the whole range of energies and does not depend on C-4 or the shape of the potential beyond the -1/r(3) region. For moderate and large values of rho, the reflectivity depends on C-4 and on the potential shape. For sufficiently large values of rho, 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/r(4) potential, i.e., ln\R\ is proportional to -rootk with a proportionality constant depending only on C-4. 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)].