Constructing mode-resolved quantum optical models for emitters in photonic crystals

Quantum emitters interacting with structured photonic environments, such as photonic crystals, exhibit complex phenomena like directional emission, non-Markovian dynamics, and collective entanglement. Modeling these systems presents a fundamental trade-off: classical Maxwell-equation solvers provide high-fidelity spatial and polarization data but are incompatible with non-perturbative quantum many-body methods. Conversely, standard quantum-optical lattice models are computationally efficient but typically rely on oversimplified, scalar light-matter couplings that ignore the nuanced spatial and polarization textures of the photonic environment.

This paper introduces a constructive methodology to derive minimal quantum-optical lattice Hamiltonians that retain the mode-resolved (position- and polarization-dependent) structure of light-matter interactions. The approach combines three key components:

1. Symmetry-Constrained Tight-Binding: Using band representation theory, the authors identify the minimal set of localized photonic orbitals required to reproduce the target photonic band structure.
2. Mode-Resolved Coupling: By projecting the electromagnetic field profiles obtained from classical solvers onto this symmetry-enforced basis, the framework constructs an interaction Hamiltonian that explicitly accounts for the emitter's position and dipole orientation.
3. Consistency Verification: The authors demonstrate that in the perturbative regime, this model recovers the standard results of macroscopic quantum electrodynamics (QED), specifically the Green's-function-based emitter dynamics.

By providing a systematic way to map classical electromagnetic data onto a tractable quantum-optical lattice, this framework enables the study of regimes previously inaccessible to either approach. It allows researchers to perform non-perturbative quantum dynamical simulations—such as those involving strong light-matter coupling or non-Markovian effects—while maintaining the physical realism of the underlying photonic crystal. The authors demonstrate this utility by showing that their model captures polarization-dependent directional emission in a 2D photonic crystal, a feature that scalar models fail to resolve.