Fig. 2.

Calculation methods of the interaction energy. (A) QED perturbation theory: one of 12 possible processes (diagrams) that contribute to the energy correction of the state |G〉 = |g₁, g₂, 0〉, Eq. 3. Here the intermediate states are $|I_1\rangle =|n_1,g_2,1_{kj}\rangle ,\hspace{0.5em}\hspace{0.17em}|I_2\rangle =|g_1,g_2,1_{kj}{1}_{k^{\prime }{j}^{\prime }}\rangle$, and $|I_3\rangle =|g_1,n_2,1_{k^{\prime }{j}^{\prime }}\rangle$, where $|1_{kj}\rangle ={\widehat{a}}_{kj}^{†}|0\rangle$. (B) Scattering of vacuum fluctuations: the vacuum field ${\widehat{\mathbf{E}}}_0\left(\mathbf{r}\right)$ exists in all space and interacts with both dipoles at their positions r₁ and r₂; hence it is also scattered by the dipoles. The scattered field from dipole 1, ${\widehat{\mathbf{E}}}_{sc}^{\left(1\right)}\left(\mathbf{r}\right)$, arrives at dipole 2, resulting in an interaction interpreted as the vdW/Casimir interaction U.