Quantum corrections to the classical electrostatic interaction between induced dipoles

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