Stochastic-Chebyshev Expansion Approach for the Simulation of Linear Polariton Spectroscopy under the Collective Coupling Regime

Molecular Polariton is becoming one of the leading directions to control a multitude of chemical and physical processes, such as charge transfer, selective bond breaking, and excited state dynamics. Accurately and efficiently simulating polariton properties under the collective coupling regimes (between $N$ molecules and the cavity mode) remains a central theoretical challenge. In this work, we use a stochastic resolution of the identity approach coupled with a Chebyshev expansion to compute various polariton photophysical properties, with a substantially reduced computational effort than would be needed for a direct diagonalization of the same Hamiltonian, which is often the bottleneck for such large dimensionality. Such quantities of interest are the total density of states (the eigenspectrum of the Hamiltonian) and the transmission spectrum (a probe of the photonic degrees of freedom), the latter of which is a direct observable in the experiment. We simulate the linear spectroscopy of molecule-cavity hybrid systems, specifically exploring the effects of the distribution and magnitude of molecular disorder for one, few, and many coupled molecules. We compare our numerical results to recent work, which formulated analytic expressions in the large-$N$ limit for the spectroscopic signals. We find that our results match those of the analytic results when $N= 100$, at which point we find that the collective effects for linear spectroscopy are converged.