Tight quantum speed limit for ergotropy charging in the N-qubit Dicke battery

Abstract We derive a quantum speed limit for ergotropy charging in the N-qubit Dicke battery. With the qubits initially in their ground state and the charger in a coherent state of mean photon number n̄, the charging time to a target fraction ε of the maximum extractable work obeys, in the classical-field limit, Γ_N τ*(ε) ≥ arcsin√ε, where Γ_N = 2λ√(n̄/N) is a single rate combining the light-matter coupling, the charger amplitude and the system size. The bound is geometric: although the ergotropy is defined by an optimisation over unitaries, it reduces to a fixed function of the angle swept by the collective spin on the Bloch sphere, and the drive bounds the rate at which that angle can grow, so the charging time is at least the required arc divided by the maximum angular velocity. The reduction rests on an exact statement about work extraction: only the coherent component of the drive delivers extractable work, while its vacuum fluctuations deposit energy that remains locked in the passive state. This makes the short-time coefficient independent of the charger occupation, and it is what separates a bound on extractable work from a bound on stored energy, which no instantaneous-power argument can supply. At finite charger occupation the inequality is tested rather than proved: numerical integration of the full Dicke Hamiltonian, with a dynamical charger and a mixed battery state, finds the bound respected at every admitted charging time for two to five qubits, and approached most closely in the limit of small target work. The familiar square-root estimate is the chord to this arc, recovered in the small-angle limit and weaker whenever a large fraction of the maximum work is required.