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

Abstract

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 $\tau^{*}(\epsilon) \geq \sqrt{N\epsilon}/(2\lambda\sqrt{\bar{n}})$, 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 $A=4/N$ is the short-time ergotropy coefficient, combined with a global upper bound proved analytically for all $N$. The composite parameter $\Gamma_N = 2\lambda\sqrt{\bar{n}/N}$ is the unique figure of merit for charging speed; all protocols collapse onto $\Gamma_N \tau^{*} \geq \sqrt{\epsilon}$, with the bound saturated to within 1% at small $\epsilon$.