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

We study the van der Waals interaction $V_{2\gamma }^{\mathrm{AB}}\mathrm{}\left(R\right)\mathrm{}$ 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 $A+B\to A+B$ and of the full amplitudes for $\gamma +A\to \gamma +A$ and $\gamma +B\to \gamma +B$, we show that it is possible to express $V_{2\gamma }^{\mathrm{AB}}\mathrm{}\left(R\right)\mathrm{}$ entirely in terms of measurable quantities, the elastic scattering amplitudes for photons of various frequencies $\omega$. 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 $V_{2\gamma }^{\mathrm{AB}}\mathrm{}\left(r\right)\mathrm{}\sim -\frac{D}{R^7}$, where $D=\frac{\left[23\right({\alpha }_E^A{\alpha }_E^B+{\alpha }_M^A{\alpha }_M^B)-7({\alpha }_E^A{\alpha }_M^B+{\alpha }_M^A{\alpha }_E^B\left)\right]}{4\pi }$ and the $\alpha$'s denote static polarizabilities. In addition, we show that the potential may be written as a single integral over $\omega$, involving products of the dynamical polarizabilities ${\alpha }_X\left(\omega \right)$ 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 $V_{2\gamma }^{\mathrm{AB}}\mathrm{}\left(R\right)\mathrm{}$ from present experimental information is discussed. Some simple interpolation formulas are presented, which may accurately describe $V_{2\gamma }^{\mathrm{AB}}\mathrm{}\left(R\right)\mathrm{}$ in terms of a few constants.