  We consider two three-dimensional isotropic harmonic oscillators interacting
with the quantum electromagnetic field in the Coulomb gauge and within dipole
approximation. Using a Bogoliubov-like transformation, we can obtain
transformed operators such that the Hamiltonian of the system, when expressed
in terms of these operators, assumes a diagonal form. We are also able to
obtain an expression for the energy shift of the ground state, which is valid
at all orders in the coupling constant. From this energy shift the
nonperturbative Casimir-Polder potential energy between the two oscillators can
be obtained. When approximated to the fourth order in the electric charge, the
well-known expression of the Casimir-Polder potential in terms of the
polarizabilities of the oscillators is recovered.
