  In a recent paper we have formulated a theory of non-relativistic quantum
electrodynamics in the presence of an inhomogeneous Huttner-Barnett dielectric.
Here we generalize the formalism to anisotropic materials and show how it may
be modified to include conducting surfaces. We start with the derivation of the
photon propagator for a slab of material and use it to work out the
energy-level shift near a medium whose conductivity in the direction parallel
to the surface far exceeds that in the direction perpendicular to the surface.
We investigate the influence of the anisotropy of the material's
electromagnetic response on the Casimir-Polder shifts, both analytically and
numerically, and show that it may have a significant impact on the atom-surface
interaction, especially in the non-retarded regime, i.e. for small atom-surface
separations. Our results for the energy shift may be used to estimate the
Casimir-Polder force acting on quantum objects close to multilayers of graphene
or graphite. They are particularly important for the case of trapped cold
molecules whose dispersive interactions with surfaces often fall within the
non-retarded regime where the anisotropy of the material strongly influences
the Casimir-Polder force. We also give a formula for the change in the
spontaneous decay rate of an excited atom or molecule near an anisotropically
conducting surface.
