Restoring Bloch's Theorem for Cavity Exciton Polaron-Polaritons

TL;DR

This work addresses the breakdown of Bloch's theorem in hybrid photon–exciton–phonon QED under strong coupling, which complicates calculations for periodic systems. They construct a symmetry-informed representation by transforming to a center-of-mass/relative frame and applying unitary boosts $ \hat{U}_\mathrm{ph}$ and $ \hat{U}_\mathrm{pn}$ to remove $ e^{i {\bf q}\cdot \hat{\mathbf X}}$ factors, producing a block-diagonal Hamiltonian $ \tilde{H}_\mathrm{LM}$ and a momentum-resolved family $ \hat{\mathcal{H}}(K)$. Applied to a 2D exciton model with a periodic potential and Fröhlich-type phonon coupling, the method yields exciton, polariton, and polaron-polariton dispersions with multiple avoided crossings and allows direct calculation of the dielectric function from linear response using the polaron-polariton eigenstates: $ \epsilon(\omega) = 1 - v(\mathbf{q}) P(\mathbf{q},\omega)$ where $ P(\mathbf{q},\omega) = \sum_{nm} \frac{|\langle \Psi_n | \hat{\rho}(\mathbf{q}) | \Psi_m \rangle|^2}{\omega - (E_n - E_m) + i\eta} (f_m - f_n)$. Finally, the framework restores translational invariance without long-wavelength approximations, enabling exact, scalable simulations for Moiré and van der Waals heterostructures and enabling tuning of coherent transport and symmetry-forbidden transitions.

Abstract

We introduce a symmetry-informed representation for hybrid photon--exciton--phonon quantum electrodynamics Hamiltonians to restore Bloch's theorem. The interchange of momenta between fermions and bosons breaks crystalline excitons' translational symmetry under strong coupling. Restoring said symmetry, we efficiently compute experimentally accessible observables without introducing approximations to the Hamiltonian, enabling investigations that elucidate material properties in strong coupling with applications enhancing coherent transport and unlocking symmetry-forbidden matter transitions.

Figures (3)

• Figure 1: Dispersion relation of an exciton in a 2D cosine potential. Bands are color-coded based on the expectation value of the relative quasiparticle's principal quantum number for each state. Note how each band is duplicated and shifted for each hydrogenic state. Throughout this letter, the exciton parameters are $m_\mathrm{e} = m_\mathrm{h} = 0.3$au, $2 \pi / |{\bf b}_1| = 2 \pi / |{\bf b}_2| = 9$au, $w_{\boldsymbol{\kappa} \in \{\pm {\bf b}_1, \pm {\bf b}_2 \} }= 0.05$au, and all other $w_{\boldsymbol \kappa} = 0$.
• Figure 2: Dispersion relation of exciton-polariton system without any long-wavelength approximation. Bands color-coded based on photonic character. Due to the large number of modes, the transparency of the bands with a photonic character greater than 0.9 linearly increases to 95% at pure photonic character. Photonic parameters: 2D Cartesian grid of $N = 529$ Fabry-Pérot TE modes with $\omega_0 = 0.15$au, $\sqrt{N}A_0 = 0.015$au and a maximum $q = 0.01$au.
• Figure 3: Imaginary component of the dielectric function for 256 photonic modes under the single photon limit and a phonon mode with 16 Fock states. Note that imaginary component is purely negative, so its absolute value is plotted on a logarithm scale. Other exciton and photonic parameters match Fig. \ref{['fig:polariton_disp']}. Phonon parameters: $\omega_{\bf k} = 500\mathrm{cm}^{-1}$, $|{\bf k}| = 1 \mathrm{nm}^{-1}$, $\gamma = 1\mathrm{cm}^{-1}$. Additionally, we set $\eta = 10\mathrm{cm}^{-1}$