Theory of dynamical superradiance in organic materials

Abstract

We develop the theory of dynamical superradiance -- the collective exchange of energy between an ensemble of initially excited emitters and a single-mode cavity -- for organic materials where electronic states are coupled to vibrational modes. We consider two models to capture the vibrational effects: first, vibrations treated as a Markovian bath for two-level emitters, via a pure dephasing term in the Lindblad master equation for the system; second, vibrational modes directly included in the system via the Holstein--Tavis--Cummings Hamiltonian. By exploiting the permutation symmetry of the emitters and weak U(1) symmetry, we develop a numerical method capable of exactly solving the Tavis-Cummings model with local dissipation for up to 140 emitters. Using the exact method, we validate mean-field and second-order cumulant approximations and use them to describe macroscopic numbers of emitters. We analyse the dynamics of the average cavity photon number, electronic coherence, and Bloch vector length, and show that the effect of vibrational mode coupling goes beyond simple dephasing. Our results show that superradiance is possible in the presence of vibrational mode coupling; for negative cavity detunings, the vibrational coupling may even enhance superradiance. We identify asymmetry of the photon number rise time as a function of the detuning of the cavity frequency as an experimentally accessible signature of such vibrationally assisted superradiance.

Figures (10)

• Figure 1: (a) Schematic of the dynamical SR system. Emitters (yellow circles) are placed inside a single-mode cavity, which is subject to photon losses at a rate $\kappa$. The internal structure of the emitters follows either (i) the TC model or (ii) the HTC model. For the TC model, the emitters are ideal two-level systems with energy splitting $\omega_0$, and are subject to non-radiative decay at rate $\gamma$ and pure dephasing at rate $\gamma_\phi$. For the HTC model, the emitters are dressed by vibrational levels with spacing $\omega_\nu$, with additional vibrational thermalization (rates $\gamma_\uparrow$ and $\gamma_\downarrow$). (b) Illustration of SF and SR initial conditions according to the polar angle $\theta$ of the state of each two-level system.
• Figure 2: Comparison of exact dynamics of the average number of photons in the cavity calculated with PIBS and cumulant approaches. Panels (a) and (c) show the dynamics for SF initial conditions for $N=10$ and $N=140$. Panels (b) and (d) show the dynamics for SR initial conditions ($\theta=\pi/4$) for $N=10$ and $N=100$. Other parameters: $g\sqrt{N}=0.4\,\mathrm{eV},\,\Delta=\omega_\mathrm{c}-\omega_0=-0.35\,\mathrm{eV},\,\kappa=0.01\,\mathrm{eV},\,\gamma=0.001\,\mathrm{eV},\,\gamma_\phi=0.0075\,\mathrm{eV}$.
• Figure 3: Comparison of symmetry-broken second-order cumulant approximation to mean-field approximation for (a) $N=100$ [same as Fig. \ref{['fig:pibs_c2']}(d)], (b) $N=1000$, and (c) $N=10000$. The SR angle is set to $\theta=\pi/4$, all other parameters are the same as in Fig. \ref{['fig:pibs_c2']}.
• Figure 4: (a), (b) and (c) Average number of photons in the cavity mode, electronic coherence, and magnitude of the Bloch vector, respectively, as a function of time for different coupling strengths to the vibrational mode. (c), (d), (f) Same as (a), (b), (c) but with the vibrational mode replaced by a pure dephasing term with rate $\gamma_\phi$. The curves $S=0$ in (a), (b), (c) are the same as the curves with $\gamma_\phi=0$ in (d), (e), (f). Note that the values of $S$ and $\gamma_\phi$ are not equidistantly spaced. Parameters: $N=10^8,\,g\sqrt{N}=0.2\,\mathrm{eV},\,\Delta=0,\,\kappa=0.01\,\mathrm{eV},\,\gamma=10^{-6}\,\mathrm{eV},\,\theta=10^{-3}\pi, \,\omega_\nu=0.15\,\mathrm{eV},\,\gamma_\nu=0.01\,\mathrm{eV},\,T=0.026\,\mathrm{eV}$. For (a), (b), (c): $\gamma_\phi=0$, for (d), (e), (f): $S=0$.
• Figure 5: Average number of photons in the cavity mode as a function of time for different coupling strengths to the vibrational mode for (a) $\Delta=-0.15\,\mathrm{eV}$ and (b) $\Delta = -0.3\,\mathrm{eV}$. All other parameters are the same as in Fig. \ref{['fig:htc_phot']}, and we specifically note that the vibrational frequency is set to $\omega_\nu=0.15\,\mathrm{eV}$.