Logarithmic terms in atom-surface potentials: Limited applicability of rational approximations for intermediate distance

It is usually assumed that interaction potentials, in general, and atom-surface potential, in particular, can be expressed in terms of an expansion involving integer powers of the distance between the two interacting objects. Here, we show that, in the short-range expansion of the interaction potential of a neutral atom and a dielectric surface, logarithms of the atom-wall distance appear. These logarithms are accompanied with logarithmic sums over virtual excitations of the atom interacting with the surface in analogy to Bethe logarithms in quantum electrodynamics. We verify the presence of the logarithmic terms in the short-range expansion using a model problem with realistic parameters. By contrast, in the long-range expansion of the atom-surface potential, no logarithmic terms appear, and the interaction potential can be described by an expansion in inverse integer powers of the atom-wall distance. Several subleading terms in the large-distance expansion are obtained as a byproduct of our investigations. Our findings explain why the use of simple interpolating rational functions for the description of the atom-wall interaction in the intermediate regions leads to significant deviations from exact formulas.