Dicke Superradiance Requires Interactions beyond Nearest Neighbors

Photon-mediated interactions within an excited ensemble of emitters can result in Dicke superradiance, where the emission rate is greatly enhanced, manifesting as a high-intensity burst at short times. The superradiant burst is most commonly observed in systems with long-range interactions between the emitters, although the minimal interaction range remains unknown. Here, we put forward a new theoretical method to bound the maximum emission rate by upper bounding the spectral radius of an auxiliary Hamiltonian. We harness this tool to prove that for an arbitrary ordered array with only nearest-neighbor interactions in all dimensions, a superradiant burst is not physically observable. We show that Dicke superradiance requires minimally the inclusion of next-nearest-neighbor interactions. For exponentially decaying interactions, the critical coupling is found to be asymptotically independent of the number of emitters in all dimensions, thereby defining the threshold interaction range where the collective enhancement balances out the decoherence effects. Our findings provide key physical insights to the understanding of collective decay in many-body quantum systems, and the designing of superradiant emission in physical systems for applications such as energy harvesting and quantum sensing.

Figure 1

Dynamics of the photon emission rate $R(t)$ for emitter arrays with only nearest-neighbor interactions of strength $\gamma$, normalized by the individual emitter decay rate ${\gamma }_0$. For $\gamma /{\gamma }_0<{\gamma }_s$, $\dot{R}(0)<0$ and the photon emission rate decays monotonically without a superradiant burst (blue). Superradiance occurs for $\gamma /{\gamma }_0>{\gamma }_s$ (red). The physically valid regime is defined by $0<\gamma /{\gamma }_0\le {\gamma }_p$. For nearest-neighbor interactions, ${\gamma }_p<{\gamma }_s$ (with a finite gap between ${\gamma }_p$ and ${\gamma }_s$) for any arbitrary emitter array in all dimensions, rendering Dicke superradiance physically impossible.

Figure 2

Region of superradiant burst in the ${\gamma }_2\text{−}{\gamma }_1$ plane. The physically valid (superradiant) regime is contained within the blue (red) boundary lines, with the conditions stated in the main text. Blue shaded region: physically valid, but not superradiant. Regions I, II, and III are defined in the main text. Red shaded region: physically valid with superradiant burst. Gray shaded region: unphysical regime. The red shaded region requires a minimum of ${\gamma }_2\approx 0.185$. All shaded regions here are obtained from numerical calculations for $N=100$, which agree very well with the analytical results obtained in the infinite-array limit.

Figure 3

Differential emission rate $\mathrm{\Delta }R=R(t)/R(0)-1$ against time (in units of emitter lifetime), for $N=9$ emitters. $\mathrm{\Delta }R>0$ indicates superradiance. (a) Dynamical behavior of $\mathrm{\Delta }R$ for the Dicke model (red), Next-nearest neighbor 1D ring (NNN, orange), nearest-neighbor 1D ring (NN, blue), and nearest-neighbor 2D square (NN, green) [see labels in (b)]. The coupling parameters are chosen to maximize $g^{(2)}(0)$. (b) Short-time behavior obtained by zooming into the gray region of (a). Only the Dicke and the next-nearest neighbor models exhibit superradiance. The curve for the Dicke model is scaled down by a factor of 10 for visualization purposes.

Figure 4

Critical coupling ${\gamma }_s$ for exponentially decaying interactions in a 1D chain, a 2D square array, and a 3D cubic array with $N$ emitters. Superradiance occurs for $\gamma >{\gamma }_s$. For all dimension $D$, ${\gamma }_s$ becomes independent of $N$ for large $N$. Inset: log-log plot of ${\gamma }_s$ against $D$ for $N\approx {10}^6$. ${\gamma }_s$ decreases as $D$ increases with a power-law scaling ${\gamma }_s\sim D^{-0.793}$.