Fluctuations in extractable work bound the charging power of quantum batteries

We study the connection between the charging power of quantum batteries and the fluctuations of the extractable work. We prove that in order to have a non-zero rate of change of the extractable work, the state $\rho_\mathcal{W}$ of the battery cannot be an eigenstate of a `\emph{free energy operator}', defined by $\mathcal{F}~\equiv~H_\mathcal{W}~+~\beta^{-1}\log(\rho_\mathcal{W})$, where $H_\mathcal{W}$ is the Hamiltonian of the battery and $\beta$ is the inverse temperature of a reference thermal bath with respect to which the extractable work is calculated. We do so by proving that fluctuations in the free energy operator upper bound the charging power of a quantum battery. Our findings also suggest that quantum coherence in the battery enhances the charging process, which we illustrate on a toy model of a heat engine. \ \emph{Note: this version includes our Reply to a Comment by Cusumano and Rudnicki, both published in Phys. Rev. Lett.}