  By engineering the electromagnetic vacuum field, the induced Casimir-Polder
shift (also known as Lamb shift) and spontaneous emission rates of individual
atomic levels can be controlled. When the strength of these effects becomes
comparable to the energy difference between two previously uncoupled atomic
states, an environment-induced interaction between these states appears after
tracing over the environment. This interaction has been previously studied for
degenerate levels and simple geometries involving infinite, perfectly
conducting half-spaces or free space. Here, we generalize these studies by
developing a convenient description that permits the analysis of these
non-diagonal perturbations to the atomic Hamiltonian in terms of an accurate
non-Hermitian Hamiltonian. Applying this theory to a hydrogen atom close to a
dielectric nanoparticle, we show strong vacuum-field-induced state mixing that
leads to drastic modifications in both the energies and decay rates compared to
conventional diagonal perturbation theory. In particular, contrary to the
expected Purcell enhancement, we find a surprising decrease of decay rates
within a considerable range of atom-nanoparticle separations. Furthermore, we
quantify the large degree of mixing of the unperturbed eigenstates due to the
non-diagonal perturbation. Our work opens new quantum state manipulation
possibilities in emitters with closely spaced energy levels.
