Large-$N$ limit of Dicke superradiance

We investigate the thermodynamic limit of Dicke superradiance. We find an expression for the system's density matrix that we can prove is exact in the limit of large atom numbers $N$. This is in contrast to previously known solutions whose accuracy has only been established numerically and that are valid only for a range of times. We also introduce an asymptotically exact solution when the system is subject to additional incoherent decay of excitations as this is a common occurrence in experiments.

Figure 1

(a) Magnetization $\mu$ (upper blue solid line) and radiance $\rho$ (lower green solid line) as defined in Eq. (18), together with numerical data for $N=5,20,1000$ (light to dark) showing quick convergence. The largest radiance is $\rho \approx 0.196$ at $T\approx 1.391$, marked with a red dot. (b) Numerical evaluation of the one-norm error, given by Eq. (14), which vanishes as $N\to \infty$ by Theorem 2. The dashed line marks ${\tau }_1=1.6lnN$.