  We study, in the presence of an external electrostatic field, the interatomic
interaction between two ground-state 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.

Quantum Physics