From superradiance to collective EIT in three-level ensembles

Abstract

We investigate the collective dynamics of a three-level ensemble under the Dicke limit, revealing a unified connection between superradiant emission and electromagnetically induced transparency (EIT). Our results show that the transient superradiant burst exhibits the expected peak intensity scaling $I_{\max}\!\sim\! N^2$, with a universal finite-size correction $|ξ(N)-2|\!\sim\! 1/\ln N$ that governs the apparent scaling exponent in realistic ensembles. In the stationary regime, collective broadening modifies the EIT response: although it typically enhances absorption, it counterintuitively increases the group velocity, leading to a relative scaling $v_g\!\propto\! N^2$, even while $v_g\!\ll\! c$. This effect suggests that cooperative interactions fundamentally limit the achievable slow-light delay in dense media. To achieve these results, we derive a representative-atom master equation that quantitatively reproduces both the superradiant and EIT regimes, in excellent agreement with the exact symmetric-subspace dynamics and correctly incorporating collective feedback and $N$-dependent broadening. This unified framework bridges transient superradiant emission and steady-state quantum interference, with direct implications for slow light, quantum memories, and precision metrology.

Figures (6)

• Figure 1: Three-level $\Lambda$ configuration relevant to electromagnetically induced transparency (EIT) and superradiance (SR). A weak probe field drives the $|1\rangle \leftrightarrow |3\rangle$ transition with frequency $\omega_{p}$ and detuning $\Delta_{1} = \omega_{31} - \omega_{p}$, while a strong control field couples $|2\rangle \leftrightarrow |3\rangle$ with frequency $\omega_{c}$ and detuning $\Delta_{2} = \omega_{32} - \omega_{c}$. $\Gamma_{31}$ and $\Gamma_{32}$ denote the spontaneous emission rates for the $3\!\to\!1$ and $3\!\to\!2$ channels. Superradiance occurs when both fields are turned off, whereas collective EIT emerges in the steady state when both fields are applied.
• Figure 2: Transient superradiant dynamics for asymmetric decay, $\Gamma_{31}=5\Gamma_{32}$. Left: Channel-resolved intensities $I_{13}(t)$ and $I_{23}(t)$ for $N=30$ with $\Omega_p=\Omega_c=0$ and $\gamma_2=0.01\Gamma$; mean-field (solid) vs exact (dashed). Right: Peak intensity $I_{\mathrm{peak}}$ versus $N$ with a power-law fit ($b\simeq 1.86$).
• Figure 3: Transient superradiant dynamics (symmetric decay channels). Left: channel-resolved emission for $N=30$ atoms with equal decay rates ($\Gamma_{31}=\Gamma_{32}=\Gamma$), $\gamma_2=0.01\Gamma$, and splittings $\omega_{31}=\omega$, $\omega_{32}=0.5\,\omega$ ($\omega/\Gamma=1000$), obtained in the absence of coherent drives ($\Omega_p=\Omega_c=0$) from the initial state $\ket{\psi(0)}=\sqrt{1-2\varepsilon^2}\ket{3}+\varepsilon\ket{1}+\varepsilon\ket{2}$ with $\varepsilon=0.1$. Solid lines with circular markers correspond to the mean-field result, while dashed lines with square markers denote the exact dynamics in the symmetric subspace. Right: scaling of the peak intensity $I_{\text{peak}}$ with atom number $N$, where a log--log fit yields $I_{\text{peak}}\sim N^{b}$ with $b\approx1.76$, confirming collective enhancement with finite-size corrections.
• Figure 4: Exact (solid) vs mean-field (dashed) susceptibility for $N=14$: (a) absorption $\mathrm{Im}\,\chi(\Delta_1)$ and (b) dispersion $\mathrm{Re}\,\chi(\Delta_1)$. The inset in panel (a) shows a magnified view of the narrow EIT transparency feature around $\Delta_1\simeq 0$.
• Figure 5: Susceptibility consistency check across solvers. Comparison of the steady-state probe susceptibility obtained from the exact collective solver (symmetric subspace), the representative-atom master equation (steady state), and the analytic linear-probe expression, as indicated in the legend. (a) Absorption plotted as a positive quantity, $-\mathrm{Im}\,\chi(\Delta_1)$. (b) Dispersion, $\mathrm{Re}\,\chi(\Delta_1)$. The detuning is shown in units of $\Gamma_{31}$, and the susceptibility is normalized per emitter (see text).