Variational Theory of Nonrelativistic Quantum Electrodynamics

The ability to achieve ultrastrong 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 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 multielectron systems coupled to many photonic modes in real space. As a first step toward a new ab initio approach to ground and excited state energies in QED, we apply our Ansatz to describe a multilevel emitter coupled to many optical modes, a system with no analytical solution. We find a compact semianalytical formula which describes ground and excited state energies very well in all regimes of coupling parameters allowed by sum rules. Our formulation provides a nonperturbative theory of Lamb shifts and Casimir-Polder forces, as well as suggest new physical concepts such as the Casimir energy of a single atom in a cavity. Our method thus give rise to highly accurate nonperturbative descriptions of many other phenomena in general QED systems.

Figure 1

Ground-state Ansatz applied to matter in a cavity: effectively decoupled matter and photons. (Left) Bare description of the coupled light-matter ground state in terms of many virtual excitations of the emitter state and the bare cavity photons. (Right) Quasiparticle description of the coupled system as a factorizable state of an effective emitter in its ground state and the vacuum of an effective photonic degree of freedom.

Figure 2

Variational theory of ground and excited states in nonperturbative QED. (a) Lowest few energy levels of a two- (top), three- (middle), and four- (bottom) level system embedded in the middle of a one-dimensional cavity. The results of the variational method (blue) are compared to perturbation theory (orange), as well as numerical diagonalization (red) with the Fock space truncated to 50 cavity modes with no more than four photons. (Inset) The fourth and fifth energy levels show a weak anticrossing behavior which is reproduced by the variational theory. (b) Mechanism of overestimation of couplings and resonances in perturbation theory: modes derived from the variational theorem are suppressed in the vicinity of the emitter, self-consistently decreasing light-matter coupling.

Figure 3

Expectation value of the correlated observable $⟨A·p⟩$ as a function of coupling. Parameters are identical to those of the top panel of Fig. 2. Despite correlations being treated perturbatively, this observable is in excellent agreement with exact diagonalization, while in poor agreement with perturbation theory in the bare photonic modes.