  The ability to achieve ultra-strong coupling between light and matter
promises to bring about new means to control material properties, new concepts
for manipulating light at the atomic scale, and fundamentally new insights into
quantum electrodynamics (QED). Thus, there is a need to develop quantitative
theories of QED phenomena in complex electronic and photonic systems. In this
Letter, we develop a variational theory of general non-relativistic QED systems
of coupled light and matter. Essential to our ansatz is the notion of an
effective photonic vacuum whose modes are different than the modes in the
absence of light-matter coupling. This variational formulation leads to a set
of general equations that can describe the ground state of multi-electron
systems coupled to many photonic modes in real space. As a first step towards a
new ab initio approach to ground and excited state energies in QED, we apply
our ansatz to describe a multi-level emitter coupled to many optical modes, a
system with no analytical solution. We find a compact semi-analytical formula
which describes ground and excited state energies to less than 1% error in all
regimes of coupling parameters allowed by sum rules. Additionally, our
formulation provides essentially a non-perturbative theory of Lamb shifts and
Casimir-Polder forces, as well as suggesting new physical concepts such as the
Casimir energy of a single atom in a cavity. Our method should give rise to
highly accurate descriptions of phenomena in general QED systems, such as
Casimir forces, Lamb shifts, spontaneous emission, and other fluctuational
electrodynamical effects.
