General Theory of the van der Waals Interaction: A Model-Independent Approach

Abstract

We study the van der Waals interaction ${V}_{2\ensuremath{\gamma}}^{\mathrm{AB}}(R)$ arising from two-photon exchange between neutral spinless systems $A$ and $B$. By using the analytic properties of the two-photon contribution to the scattering amplitude for $A+B\ensuremath{\rightarrow}A+B$ and of the full amplitudes for $\ensuremath{\gamma}+A\ensuremath{\rightarrow}\ensuremath{\gamma}+A$ and $\ensuremath{\gamma}+B\ensuremath{\rightarrow}\ensuremath{\gamma}+B$, we show that it is possible to express ${V}_{2\ensuremath{\gamma}}^{\mathrm{AB}}(R)$ entirely in terms of measurable quantities, the elastic scattering amplitudes for photons of various frequencies $\ensuremath{\omega}$. This approach includes relativistic corrections, higher multipoles, and retardation effects from the outset and thus avoids any $\frac{v}{c}$ expansion or any direct reference to the detailed structure of the systems involved. We obtain a generalized form of the Casimir-Polder potential, which includes both electric and magnetic effects, and, correspondingly, a generalized asympotic form ${V}_{2\ensuremath{\gamma}}^{\mathrm{AB}}(r)\ensuremath{\sim}\ensuremath{-}\frac{D}{{R}^{7}}$, where $D=\frac{[23({\ensuremath{\alpha}}_{E}^{A}{\ensuremath{\alpha}}_{E}^{B}+{\ensuremath{\alpha}}_{M}^{A}{\ensuremath{\alpha}}_{M}^{B})\ensuremath{-}7({\ensuremath{\alpha}}_{E}^{A}{\ensuremath{\alpha}}_{M}^{B}+{\ensuremath{\alpha}}_{M}^{A}{\ensuremath{\alpha}}_{E}^{B})]}{4\ensuremath{\pi}}$ and the $\ensuremath{\alpha}$'s denote static polarizabilities. In addition, we show that the potential may be written as a single integral over $\ensuremath{\omega}$, involving products of the dynamical polarizabilities ${\ensuremath{\alpha}}_{X}(\ensuremath{\omega})$ evaluated at real frequencies, in contrast to the familiar integral over imaginary frequencies; for the case of interacting atoms, the domain of applicability of the various formulas is clarified, and the problem of evaluating ${V}_{2\ensuremath{\gamma}}^{\mathrm{AB}}(R)$ from present experimental information is discussed. Some simple interpolation formulas are presented, which may accurately describe ${V}_{2\ensuremath{\gamma}}^{\mathrm{AB}}(R)$ in terms of a few constants.