Timing quantum emission: coherence, superradiance, and entanglement in order

Abstract

We investigate the short-term temporal dynamics of superradiance in closely spaced quantum emitters. Building on Dicke's 1954 framework, we analyze the sequential emergence of coherence, superradiance, and entanglement, revealing a distinct temporal hierarchy in their extremal values: relative coherence develops first, followed by the peak of correlated emission, then minimal entanglement, and finally correlated dephasing. These findings suggest that enhanced relative coherence initiates correlated emission and when correlated dephasing is negligible, entanglement and correlated emission become tightly linked in time.

Figures (5)

• Figure 1: (Color online) Schematics of spontaneous radiation and superradiant emission. On the left, emitters are far apart and decay individually following a characteristic rate, while on the right the emitters are influenced by each other's radiation field, hence are constructively coupled.
• Figure 2: (Color online) Correlated emission, relative entropy of coherence, and entanglement witness are shown for the $N=2$ emitter subsystem as functions of dimensionless time. Correlated emission and relative entropy of coherence are measured on the left ordinate, while entanglement witness is measured on the right ordinate (see colors). Vertical lines indicate those moments in time, where each quantity reaches its extremal value. In this setup $C_{\text{rel}}$, reaches its maximum first, then $C_{0}$ and $\braket{W}$ reach their extrema simultaneously. The corresponding time differences are $\tau_{\text{cw}}$ (coherence vs. witness) and $\tau_{\text{ew}}$ (emission vs. witness). Other parameters: $g=1$, $\gamma/g=0.1$, $\kappa/g=0.1$, and $\gamma_{\phi}/g=0.0225$.
• Figure 3: (Color online) Time evolution of $C_{\text{rel}}$, $C_{0}$, and $\braket{W}$, are plotted for $N=2$, 4, 6, and 8 emitters. The number of emitters are shown in the top figure as labels and different colors. The colors are common in all subplots. Vertical dashed lines indicate the moments when these quantities reach their extrema for $N=6$. Other parameters are as in Fig. \ref{['fig:ExampleForTwoEmitters']}.
• Figure 4: (Color online) Time-delay, $\tau_{C_{\text{rel}}}$ is depicted as a function of $N$. The shaded area, $N \le 9$, is calculated using two different approaches: exact time-evolution in computational basis and a permutational invariant quantum solver (PIQS). These approaches lead to similar values. The four sets of data, distinguished by different markers and colors, correspond to different decay parameters, $\gamma = 0$, 0.1, 1, 2. Qualitatively, $\tau_{C_{\text{rel}}}$ diminishes as $\ln{\!(N)}/N^{\alpha}$, where $\alpha \approx 1$ for $\gamma=0$, and $\alpha \approx 1.2$ for $\gamma = 2$, as shown by the dashed lines. Similar results apply to $\tau_{C_{0}}$ and $\tau_{C_{W}}$, but left out for clarity.
• Figure 5: (Color online) Time gaps, $\Delta \tau = \tau_{C_{0}} - \tau_{C_{\text{rel}}}$ and $\Delta \tau = \tau_{W} - \tau_{C_{\text{rel}}}$ are displayed for $N=2$ as functions of $g$. The vertical dashed line ($g_W \approx 0.20$) represents the interaction strength above which entanglement is detected. Other parameters: $n_{\text{p}} = 1$, $\gamma = 0.1$, $\kappa = 0.1$, and $\gamma_{\phi} = 0.0225$.