For molecular polaritons, disorder and phonon timescales control the activation of dark states in the thermodynamic limit

Abstract

Collective light-matter systems host an extensive manifold of dark states whose role in the emergence of thermodynamic behavior remains poorly understood, especially in the presence of disorder and structured environments. Here, we develop a hybrid matrix product state-hierarchical equations of motion (MPS-HEOM) approach that enables numerically exact simulations of polariton dynamics from a few emitters to the thermodynamic limit under both static and dynamic disorder. This allows us, for the first time, to provide a quantitative and operational answer to the long-standing question of what is the minimum system size required to reach the thermodynamic limit in collective polaritonic systems. By introducing a convergence scale, $N_{T}$, i.e., the number of molecules required for the photonic dynamics to reach the thermodynamic limit, we show that dynamic disorder generally poses a greater computational challenge than static disorder. We attribute this behavior to the suppression of collective light-matter dynamics by disorder, which dynamically activates non-collective degrees of freedom. We further find that $N_{T}$ exhibits a turnover behavior as the bath becomes more Markovian, as the bath timescales regulate bright-to-dark energy transfer and the involvement of dark and gray states. Hence, phonon timescales control both the breakdown of collective behavior and the growth of $N_{T}$. Our results establish the suppression of collective behavior as the key mechanism governing thermodynamic convergence in disordered light-matter systems.

Figures (3)

• Figure 1: (a) Schematic of a molecular ensemble in an optical microcavity. Each molecule is modeled as a TLS coupled to a vibrational environment characterized by spectral density, $J(\omega)$. (b) MPS representation of the ADOs for the HTC model with two TLSs. Red and blue circles represent the cavity and TLS indices, respectively, while green circles track HEOM indices for the two effective vibrational modes.
• Figure 2: (a) Threshold system size, $N_T$, as a function of disorder strength $\sigma$ for frequency versus light–matter coupling disorder. (b) Comparison of $N_T$ obtained for static (SD) and dynamic (DD) frequency disorder for three bath characteristic frequency, $\gamma=0.02$, $\gamma=0.04$ and $\gamma \to \infty$. (c) Time evolution of the average photon number $\langle a^\dagger a\rangle$ for different TLS numbers $N$ at fixed $\eta = 0.05$($\sigma = 0.0354$), corresponding to the red star in panel (b).
• Figure 3: (a) Evolution of the total exciton population for SD in the coupling versus frequency. (b) Evolution of the dark state population at $\sigma = 0.0316$ for static disorder and dynamic disorder with different bath characteristic frequencies ($\gamma$). (c)The bright--to--dark state transfer rate versus $\gamma$ for different disorder strengths calculated within perturbative rate theory. The red-shaded region highlights the breakdown of the perturbative theory. (d) Time dependence of the threshold system size $N_T$ for SD and DD.