Constructing mode-resolved quantum optical models for emitters in photonic crystals
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Abstract
Recent advances are enabling quantum emitters to interact with photonic crystals, whose electromagnetic modes exhibit complex dispersion relations, spatial mode structure, and polarization textures. However, modeling light-matter behavior in these systems faces a persistent trade-off: electromagnetic approaches based on Maxwell-equation solvers provide realistic vectorial descriptions but are difficult to integrate with quantum many-body and non-perturbative methods, whereas simplified quantum-optical lattice models are tractable but typically rely on scalar and spatially independent light-matter couplings that miss essential features of these structured photonic environments. Here, we introduce a constructive framework to derive quantum-optical lattice descriptions that overcome this trade-off. Combining symmetry-constrained tight-binding constructions with numerically computed photonic band structures and field profiles, our method yields minimal, symmetry-enforced lattice Hamiltonians that reproduce the target photonic dispersion while retaining the mode-resolved (position- and polarization-dependent) structure of the light-matter coupling. We show that these models recover Green's-function-based emitter dynamics in the perturbative regime, while providing access to non-perturbative quantum dynamical simulations beyond emitter-only descriptions. As a proof of principle, we apply the framework to a two-dimensional photonic crystal and show that it captures polarization-dependent directional emission inaccessible to scalar models, while enabling the analysis of non-Markovian light-matter dynamics and entanglement. Our results provide a practical bridge between classical electromagnetic simulation tools and quantum-optical many-body and non-Markovian modeling in photonic crystal settings.