Virtual Resonant Emission and Oscillatory Long-Range Tails in van der Waals Interactions of Excited States: QED Treatment and Applications

We report on a quantum electrodynamic (QED) investigation of the interaction between a ground state atom with another atom in an excited state. General expressions, applicable to any atom, are indicated for the long-range tails that are due to virtual resonant emission and absorption into and from vacuum modes whose frequency equals the transition frequency to available lower-lying atomic states. For identical atoms, one of which is in an excited state, we also discuss the mixing term that depends on the symmetry of the two-atom wave function (these evolve into either the gerade or the ungerade state for close approach), and we include all nonresonant states in our rigorous QED treatment. In order to illustrate the findings, we analyze the fine-structure resolved van der Waals interaction for $nD\text{−}1S$ hydrogen interactions with $n=8$, 10, 12 and find surprisingly large numerical coefficients.

Figure 1

Ladder and crossed-ladder Feynman diagrams for the long-range interaction of atoms. The virtual state of atom $A$, labeled $|m_A⟩$, is assumed to have a lower energy than the reference state.

Figure 2

The upper panel (a) compares the Wick-rotated term $\mathcal{W}$ (dashed line) to the real part $\mathcal{P}=\mathrm{Re}\mathcal{Q}$ of the pole term (solid line) for the $12D\text{−}1S$ interaction (fine-structure average) in the intermediate range $a_0/\alpha \lesssim R\ll \hslash c/\mathcal{L}$. The interaction energy is $\mathrm{\Delta }E=\mathcal{Q}+\mathcal{W}$. The Wick-rotated term dominates despite parametric suppression [see Eq. (10)]. The lower panel (b) displays the logarithm of the modulus of the interaction energy (Wick-rotated term versus pole term) for very large interatomic distance; the pole term finally dominates. The oscillations of the pole term are dominated by virtual $2P$ state contribution; when the interaction energy changes sign, the logarithm diverges to $-\infty$.