  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.}
