Collective enhancement in dissipative quantum batteries

We study a quantum battery made out of $N$ nonmutually interacting qubits coupled to a dissipative single electromagnetic field mode in a resonator. We quantify the charging energy, ergotropy, transfer rate, and power of the system, showing that collective enhancements are still present despite losses, and can even increase with dissipation. Moreover, we observe that a performance deterioration due to dissipation can be reduced by scaling up the battery size. This is useful for experimental realizations when controlling the quality of the resonator and the number of qubits are limiting factors.

Figure 1

A generic QB consisting of a charger “$c$” and a holder “$h$” in its three operation stages: charging, storage, and discharging. The holder, initially empty, obtains energy from the charger interacting a time ${\tau }_c$ with the Hamiltonian ${\mathcal{H}}_{c-h}$. The energy transferred to the holder is stored for a time ${\tau }_s$ until we desire to use the energy. To discharge the holder in a time ${\tau }_d$ we can proceed in transducing mode (bottom) with external classical fields over the holder through a unitary operation ${\mathcal{U}}_t$, or in normal mode (top) through the charger where the dissipation channels might be used to finally extract the energy. We focus on timescales for which the charger dissipation ($\mathfrak{D}$) is relevant, but holder dissipation is negligible.

Figure 2

Collective (#) and parallel ($\parallel$) versions of a QB of size $N$. The collective QB has one charger “$c$” with $E_c$ initial energy and one holder “$h$” consisting in $N$ units cells “$h_1$”. The parallel QB has $N$ copies of “$c$” with $E_c/N$ initial energy, each interacting with a different unit cell “$h_1$” as holder. Hence, the $\parallel$ QB is just $N$ copies of the $N$-times down-scaled # QB.

Figure 3

Mean energy $E_h$ and ergotropy ${\mathcal{E}}_h$ charged on the holder (array of $N$ qubits) when charging from an initial Fock light state. The mean energy $E_c$ of the charger (resonator) and heat $Q$ from the QB are also shown. The dynamics seen correspond to Rabi-like oscillations, transferring energy from the charger to the holder. The holder oscillations for different $N$ and $\kappa$ are given in the Supplemental Material [50]. The segmented lines define $\overline{\tau }$ and ${\overline{E}}_h$.

Figure 4

Collective (a) enhancements ${\mathrm{\Gamma }}_{\overline{f}}$ and (b) performances for the first maximum charge when charging from Fock initial conditions for energy, transfer rate, power, and ergotropy. The results for coherent and thermal initial conditions are given in [50].

Figure 5

Distribution of the total energy over the QB. Collective (#) and parallel ($\parallel$) are compared for the three cases of initial conditions simulated.