Fluctuations in extractable work bound the charging power of quantum batteries

Abstract: 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.}