Polariton Localization and Dispersion Properties of Disordered Quantum Emitters in Multimode Microcavities

Experiments have demonstrated that the strong light-matter coupling in polaritonic microcavities significantly enhances transport. Motivated by these experiments, we have solved the disordered multimode Tavis-Cummings model in the thermodynamic limit and used this solution to analyze its dispersion and localization properties. The solution implies that wave-vector-resolved spectroscopic quantities can be described by single-mode models, but spatially resolved quantities require the multimode solution. Nondiagonal elements of the Green’s function decay exponentially with distance, which defines the coherence length. The coherent length is strongly correlated with the photon weight and exhibits inverse scaling with respect to the Rabi frequency and an unusual dependence on disorder. For energies away from the average molecular energy $E_{\mathrm{M}}$ and above the confinement energy $E_C$, the coherence length rapidly diverges such that it exceeds the photon resonance wavelength ${\lambda }_0$. The rapid divergence allows us to differentiate the localized and delocalized regimes and identify the transition from diffusive to ballistic transport.

Figure 1

(a) One-dimensional microcavity of length $L$ containing $N$ molecules. (b) Sketch of the energy configuration of the cavity modes (red sine functions) and the molecules (blue circles, $E_j$ distributed around $E_{\mathrm{M}}$ with Gaussian width $\sigma$). The molecules are coupled with strength $g_{j,k}$ to the photonic modes, such that excitations can be transported via photons. (c) Wave-vector-resolved photon and molecule LDOSs for $E_C=0.4\mathrm{eV}$. (d),(e),(f) Average photon weight of the polaritons as a function of energy for $E_C=0.4$, 1.0, 1.3 eV. (g),(h),(i) Coherence length of the polaritons for the same $E_C$ values as in (d),(e),(f). Overall parameters are $L=125\mathrm{\mu }\mathrm{m}$, $N=5000$, $E_{\mathrm{M}}=1$, $\sigma =0.05$, and $g\sqrt{\rho }=0.14\mathrm{eV}$. The photonic cutoff energy is ${\omega }_{\text{cutoff}}=50\mathrm{eV}$, such that 5000 photonic modes are included in the simulations.

Figure 2

(a) Imaginary part of the Green’s function as a function of the relative position coordinate $r=r_j-r_{j^{\prime }}$ as defined in Eq. (5). The results are shown in red (photon contribution, $r>0$) and blue (molecule contribution, $r<0$). The black lines depict the amplitude decay predicted by the coherence length in Eq. (7). (b) Imaginary part of the Green’s function as a function of wave vector (i.e., the LDOS). The solid and dashed black lines depict the analytical predictions using Eq. (4). Parameters are the same as in Fig. 1. Each Green’s function has been averaged over an interval of width $\delta =0.005[\mathrm{eV}]$.

Figure 3

Coherence length as a function of Rabi splitting [(a),(c)] and disorder [(b),(d)]. Parameters are the same as in Fig. 1.