Quantum corrections to the classical electrostatic interaction between induced dipoles

We study, in the presence of an external electrostatic field, the interatomic interaction between two groundstate atoms coupled with vacuum electromagnetic fluctuations within the dipole coupling approximation based on the perturbation theory. We show that, up to the fourth order, the electrostatic-field-induced interatomic interaction is just the classical dipole-dipole interaction, which disagrees with the recent result from Fiscelli et al. [G. Fiscelli et al., Phys. Rev. Lett. 124, 013604 (2020)]. However, to higher orders, there exist external-field-related quantum corrections to the induced classical electrostatic dipole-dipole interaction. In the sixth order, the external field effectively modifies the atomic polarizability to give rise to a two-photon-exchange quantum correction, while in the eighth order, the external field enables an additional process of three-photon exchange which is not allowed in the absence of the external field, and this process generates an r(-11) term in the interaction potential in the far regime, where r is the interatomic separation. Numerical estimations show that these external-field-related quantum corrections are much smaller than the two-photon-exchange Casimir-Polder interaction.