Symbolic Quantum-Trajectory Method for Multichannel Dicke Superradiance

Abstract

We develop and solve a Dicke superradiant model with two or more competing collective decay channels of tunable rates. Recent work analyzed stationary properties of multichannel Dicke superradiance using hydrodynamic mean-field approximations as shown by Mok et al. [Phys. Rev. Res. 7, L022015 (2025)]. We extend this with a symbolic quantum-trajectory method, providing a simple route to analytic solutions. For two channels, the behavior of the stationary ground-state distribution resembles a first-order phase transition at the point where the channel-rate ratio is equal to unity. For $d$ competing channels, we obtain scaling laws for the superradiant peak time and intensity. These results unify and extend single-channel Dicke dynamics to multilevel emitters and provide a compact tool for cavity and waveguide experiments, where permutation-symmetric reservoirs engineer multiple collective decay paths.

Figures (7)

• Figure 1: (a) Lossy cavity generated two-channel Dicke superradiance with two cavity modes, acting as permutation-symmetric reservoirs. Each mode is coupled on resonance with a transition of $N$$\Lambda$-type systems with coupling strengths $\tilde{g}_\alpha$ and decay rates $\kappa_\alpha$, engineering cavity-tuned collective decay rates $\Gamma_\alpha=4\tilde{g}_\alpha^2/\kappa_\alpha$ in the bad cavity limit $\kappa_\alpha\gg \tilde{g}_\alpha$. (b) This creates a competition between the ground state manifold spanned by $\{|g_1\rangle,|g_2\rangle \}$, which becomes increasingly sensitive to the decay rate ratio $r\!=\!\Gamma_2/\Gamma_1$ with larger $N$. Our symbolic quantum-trajectory method yields the full time dynamics from the initially inverted state through the superradiant burst to the steady state with a ground state distribution (Eq. \ref{['eq:Multinomial']}) in stark contrast to the weighted binomial distribution for independently decaying emitters, where $|N \! -\!x,x\rangle$ refers instead to the product states $\ket{g_1}^{\otimes(N-x)} \! \otimes \!\ket{g_2}^{\otimes x}$. (c) The limit $N\!\rightarrow\! \infty$ there is a sharp transition in the occupation of the two ground states around $r\!=\!1$, shown as the fraction of emitters in $|g_2\rangle$.
• Figure 2: (a) Single-channel Dicke superradiance. Decay cascade starting from the fully inverted state, non-Hermitian time evolution with rates $\Lambda_m$ and successive jumps ($\hat{S}$) lead to the state $|m\rangle$ (Eq. \ref{['eq:trajSingle']}). The dynamics reside on the surface of the collective Bloch sphere at all times. (b) Two-channel Dicke superradiance. An inverted ensemble of $N$ three-level systems with a two‑fold ground manifold decays through two collective channels at rates $\Gamma_1$, $\Gamma_2$. Starting from the fully excited state ($|0,0\rangle$), a possible path $(j)$ with non-Hermitian time evolution (decay rate $\Lambda_{k}^{(j)}$) and successive jumps ($\hat{S}_1$, $\hat{S}_2$) through states $|u_1^{(j)},u_2^{(j)}\rangle$ lead to the final state $|n_1,n_2\rangle$ (Eq. \ref{['eq:two-channel']}). The ground state of the system is formed by a distribution of states with $n_1+n_2=N$.
• Figure 3: (a) Time evolution of the emitted intensity for increasing numbers of collective decay channels $d$ with balanced rates $\Gamma_1= \!\cdots \!=\Gamma_d\!=\! \Gamma/d$ for $N\!= \!150$. As $d$ grows, the superradiant burst weakens and shifts to later times, reflecting reduced collective enhancement and slower emission. Crosses mark the analytical predictions from Eqs. (\ref{['eq:theory']}). (b) The total emitted peak intensity (decreasing linearly with $d$) and peak time as a function of $N$ show excellent agreement with Eqs. (\ref{['eq:theory']}) (dashed lines). (c) Final population fraction $\bar{n}_2$ in the second ground state versus the decay-rate ratio $r$. The order parameter changes sharply near $r\!=\! 1$, with its slope $\partial_r \bar{n}_2|_{r=1}\!\sim\!\ln N$ pointing towards a dissipative first-order phase transition in the large-$N$ limit.
• Figure A1: Dicke superradiance realized and tuned by a single mode cavity in the bad cavity limit $\kappa \gg g$. Far from the bad cavity limit for $\kappa \sim g$ oscillations between the emitters and cavity mode emerge, differing strongly from the Dicke superradiance model. The two models converge as the cavity decay rate $\kappa$ is increased and become indistinguishable for $\kappa \gg g$. For the simulations, we chose $N=5$ emitters and a cut-off for the cavity mode occupation of $10$. The time is shown in units of the inverse of the cavity coupling rate $g$.
• Figure A2: Comparison of numerical results for the steady state from rate equation calculation with steady state formula in Eq. \ref{['eq:main_formula']}. The two solutions show perfect agreement for representative values of $N$ and $r$.