Gauge-invariant theory of truncated quantum light-matter interactions in arbitrary media

The loss of gauge invariance in models of light-matter interaction which arises from material and photonic space truncation can pose significant challenges to conventional quantum optical models when matter and light strongly hybridize. In structured photonic environments, necessary in practice to achieve strong light-matter coupling, a rigorous model of field quantization within the medium is also needed. Here we use the framework of macroscopic QED by quantizing the fields in an arbitrary material system, with a spatially dependent dispersive and absorptive dielectric, starting from a fundamental light-matter action. We truncate the material and mode degrees of freedom while respecting the gauge principle by imposing a partial gauge-fixing constraint during canonical quantization, which admits a large number of gauges including the Coulomb and multipolar gauges commonly used in quantum optics. We also consider gauge conditions with explicit time dependence, enabling us to unambiguously introduce additional phenomenologically time-dependent light-matter interactions in any gauge. Our results allow one to derive rigorous nonrelativistic models of ultrastrong light-matter interactions in structured photonic environments with no gauge ambiguity. Results for two-level systems and the dipole approximation are discussed, as well as how to go beyond the dipole approximation for effective single-particle models. By comparing with the limiting case of an inhomogeneous dielectric, where dispersion and absorption can be neglected and the fields can be expanded in terms of the generalized transverse eigenfunctions of the dielectric, we show how lossy systems can introduce an additional gauge ambiguity, which we resolve and predict to have fundamental implications for open quantum system models. Finally, we show how observables in mode-truncated systems can be calculated without ambiguity by using a simple gauge-invariant model of photodetection.

Figure 1

Schematic representation of the gauge-invariant approach to light-matter interactions within a linear medium (see text for definition of mathematical symbols). The electromagnetic fields interact weakly with a passive material reservoir representing the dispersive and absorbing dielectric medium, allowing for the fields to be expressed as a linear functional of fundamental polariton operators $\widehat{\mathbf{b}}(\mathbf{x},\omega ), {\widehat{\mathbf{b}}}^{†}(\mathbf{x},\omega )$ (we have suppressed the $\omega$ functional dependence in the dielectric function, Green's function, and annihilation operator in the schematic for brevity). Light-matter interactions are introduced by a minimal coupling scheme of free “active” material particles with the (vacuum) electromagnetic fields. Gauge invariance under active material truncation is retained by expressing the Hamiltonian in terms of the unitary operators $\widehat{\mathcal{U}}\left({\widehat{\mathcal{r}}}_{\alpha }\right)$, where ${\widehat{\mathcal{r}}}_{\alpha }$ are the directly truncated position operators of the active material component. Similarly, gauge invariance is retained under photonic truncation by expressing the Hamiltonian in terms of $\widehat{\mathcal{V}}\left(\widehat{\mathcal{A}}\right)$, where $\widehat{\mathcal{A}}$ is the truncated vector potential. The approach is manifestly gauge invariant as the longitudinal component of the vector potential $\widehat{\mathcal{A}}{}_{\parallel }$ and transverse component of the polarization ${\widehat{\mathcal{P}}}_{\perp }$ are determined by the arbitrary transverse quantization function ${\mathbf{K}}_{\perp }(\mathbf{x},{\mathbf{x}}^{\prime })$.

Figure 2

Visualization of the photon and material truncation. An exemplary system consisting of a dielectric rod and a pointlike emitter can generally be described quantum mechanically by a multilevel electronic system interacting with a continuum of photon modes (left), which reflect hybridized states of the electromagnetic fields and the passive dielectric material. The electromagnetic field depends on a three-dimensional integration over the photon Green's function. The proper truncation of the material and photon degrees of freedom allows for a description of the quantum system with few energy levels (right), while still preserving gauge invariance. The electromagnetic field is obtained from the optical mode functions.