Tight Quantum Speed Limit for Ergotropy Charging in the N-Qubit Dicke Battery

We derive and analytically prove a tight quantum speed limit (QSL) for ergotropy charging in the $N$-qubit Dicke quantum battery: the first-passage time to normalised ergotropy $\epsilon$ satisfies ${\tau }^{\ast }(\epsilon )\ge \sqrt{N\epsilon }\mathrm{/}(2\lambda \sqrt{\bar{n}})$, where $\lambda$ is the coupling and $\bar{n}$ is the mean charger photon number. The bound follows from an exact perturbative identity $\epsilon (t)=A{\lambda }^2\bar{n}{t}^2+\mathcal{O}((\lambda t{)}^4)$, where $A=4\mathrm{/}N$ is the short-time ergotropy coefficient, combined with a global upper bound proved analytically for all $N$. The composite parameter ${\mathrm{\Gamma }}_N=2\lambda \sqrt{\bar{n}\mathrm{/}N}$ is the unique figure of merit for charging speed; all protocols collapse onto ${\mathrm{\Gamma }}_N{\tau }^{\ast }\ge \sqrt{\epsilon }$, with the bound saturated to within 1% at small $\epsilon$.