Polaritonic Bloch's Theorem beyond the Long-Wavelength Approximation

TL;DR

This work shows that Bloch's theorem remains valid for crystalline solids inside cavities when strong light–matter coupling is considered, by introducing a global translation symmetry that jointly acts on electrons and photons to yield polaritonic Bloch states with preserved lattice periodicity. It develops a general multimode framework, weighting oblique cavity modes by Planck statistics, and demonstrates that multimode corrections to energetics are finite at low frequencies and temperatures, with oblique contributions effectively producing a uniform in-plane field in the single-photon limit. The dominant interaction is governed by the longitudinal cavity mode, while multimode effects are subleading, thereby formally justifying the widely used single-mode and long-wavelength approximations in molecular polaritonics and enabling ab initio cavity QED approaches for predictive materials design. Overall, the results establish a rigorous theoretical foundation for describing polaritonic states in crystalline solids and provide a pathway to quantify temperature, geometry, and mode-dependent effects in cavity-modified materials.

Abstract

Cavity quantum electrodynamics offers a powerful route to manipulate material properties. However, it is unclear whether and how quantized fields affect crystals periodicity. Here, we extend Bloch's theorem to crystals under the strong light-matter coupling, showing that polariton quasiparticles preserve lattice periodicity. We formulate a general framework to incorporate the effect of multimode cavity fields in a simple and tractable way. We find that the additional modes contribute to the system's energy by small modifications that become noticeable only at low frequencies. Within the single-photon approximation the multimode contribution manifests as a spatially uniform effective field in the crystal's plane. This provides a formal justification for the single-mode and long-wavelength approximations commonly used in molecular polaritonics. This work establishes a rigorous theoretical framework that clarifies how polaritonic states in crystalline solids should be described.

Figures (3)

• Figure 1: a) Two-dimensional periodic material in a Fabry–Pérot cavity with resonance frequency $\omega_{\mathbf{r}}=c\sqrt{\mathbf{k_x}^2+\mathbf{k_y}^2+\mathbf{k_z}^2}$. Along the cavity axis ($z$, lower left inset), the field is confined by the cavity, which justifies the single-mode approximation. Besides, its wavelength $\lambda_{z}=\frac{\pi}{k_z}$ is much longer than the material thickness, allowing the use of the long-wavelength approximation. In the cavity plane ($x$, $y$, lower right inset), the cavity is open, so that multiple in-plane photon modes must be considered and the long-wavelength approximation does not hold. b) Cross-sectional views along the cavity axis (top) and in the cavity plane (bottom) with the hexagonal unit cell indicated.
• Figure 2: a) Representation of the components of the vector potential inside the cavity. The principal resonant component ($\hat{A}_{\mathbf{k}_z}$) lies at the center of a cone made of all the other possible oblique components $\hat{A}_{\mathbf{k}_{\mathrm{obl}}}$. b) Scheme of the in-plane ($\mathbf{k}_{\parallel}$) and out-of-plane ($\mathbf{k}_{z}$) components of the oblique modes of the cavity field.
• Figure 3: Trend of the bare multimode integral (prior to multiplication by the prefactor $2\Tilde{A}_{0}\cos(k_{z}z)\,p_{z}\,P_{vac}$) of Equation \ref{['eq:MET_3']} in section \ref{['subsec1_met']} of Methods \ref{['sec:MET']} as function of resonance frequency (Panel a) and temperature (Panel b).