Lower and upper bounds of quantum battery power in multiple central spin systems

We study the energy-transfer process in quantum battery systems consisting of multiple central spins and bath spins. Here with ``quantum battery'' we refer to the central spins, whereas the bath serves as the ``charger.'' For the single-central-spin battery, we analytically derive the time evolutions of the energy transfer and the charging power with arbitrary number of bath spins. For the case of multiple central spins in the battery, we find the scaling-law relation between the maximum power ${P}_{\mathrm{max}}$ and the number of central spins ${N}_{B}$. It approximately satisfies a scaling law relation ${P}_{\mathrm{max}}\ensuremath{\propto}{N}_{B}^{\ensuremath{\alpha}}$, where scaling exponent $\ensuremath{\alpha}$ varies with the bath spin number $N$ from the lower bound $\ensuremath{\alpha}=1/2$ to the upper bound $\ensuremath{\alpha}=3/2$. The lower and upper bounds correspond to the limits $N\ensuremath{\rightarrow}1$ and $N\ensuremath{\gg}{N}_{B}$, respectively. In thermodynamic limit, by applying the Holstein-Primakoff transformation, we rigorously prove that the upper bound is ${P}_{\mathrm{max}}=0.72BA\sqrt{N}{N}_{B}^{3/2}$, which shows the same advantage in scaling of a recent charging protocol based on the Tavis-Cummings model. Here $B$ and $A$ are the external magnetic field and coupling constant between the battery and the charger.