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 $ε$ satisfies $τ^{*}(ε) \geq \sqrt{Nε}/(2λ\sqrt{\bar{n}})$, where $λ$ is the coupling and $\bar{n}$ is the mean charger photon number. The bound follows from an exact perturbative identity $ε(t) = Aλ^2\bar{n}t^2 + \mathcal{O}((λt)^4)$, where $A=4/N$ is the short-time ergotropy coefficient, combined with a global upper bound proved analytically for all $N$. The composite parameter $Γ_N = 2λ\sqrt{\bar{n}/N}$ is the unique figure of merit for charging speed; all protocols collapse onto $Γ_N τ^{*} \geq \sqrtε$, with the bound saturated to within 1% at small $ε$.

Figures (1)

• Figure 1: (Color online) Quantum speed limit for ergotropy charging in the $N=2$ Dicke quantum battery. (a) Charging trajectories $\varepsilon(t)$ for five representative $(\lambda/\omega_0, \bar{n})$ pairs. Filled circles: optimal time $\tau^{*}$. Dotted parabolas: analytical upper bound $(4/N)\lambda^2\bar{n}\,t^2$ (Theorem \ref{['prop:global']}). Every trajectory lies strictly below its bound. (b) Universal collapse in the $(\varepsilon, X)$ plane, $X=\Gamma_N\cdot\tau^{*}$, $\Gamma_N=2\lambda\sqrt{\bar{n}/N}$, for all 2032 points ($N=2$). Black curve: QSL bound $X = \sqrt{\varepsilon}$ (Corollary \ref{['cor:universal']}). Blue diamonds: numerical lower envelope $X_{\min}(\varepsilon)$. Shaded region: forbidden by the QSL. (c) Optimal charging time $\tau^{*}(\varepsilon=0.5)$ versus $1/\Gamma_N=\sqrt{N}/(2\lambda\sqrt{\bar{n}})$. Black line: QSL bound (slope $\sqrt{0.5}=0.707$). Red dashed line: linear fit (slope $1.86$). Parameters: $\omega_c = \omega_0$, $\hbar=1$, $N=2$; $N_{\rm Fock}=40$, $N_T=2000$.

Theorems & Definitions (6)

• Lemma 1: Short-time ergotropy expansion
• proof
• Theorem 2: Global ergotropy bound
• proof
• proof
• Corollary 4: Universal collapse