The influence of phonons is computed using the Ehrenfest mixed quantum-classical dynamics, and cavity loss through Lindblad dynamics using a stochastic approach, with details provided in the Supporting Information. The spatial distribution of the polariton is given by |ψ ± (x n , t)| 2 = |⟨±, n|ψ(t)⟩| 2 , with the real space polariton states expressed as