Aging of a quantum battery

We study the use of ensembles of quantum two-level systems as batteries to store and release radiative energy. Collective coupling to a tuneable cavity mode leads to enhanced charging power and a means to recover the energy on demand in a controllable manner. Individual decay and decoherence of the emitters occur on a much slower timescale, but they gradually cause population of states with lower symmetry and similar to the aging of rechargeable household batteries, they reduce the efficiency of the battery with time.

Figure 1

Left panel: The quantum battery consists of $N$ two-level atoms located in a lossy cavity. The atoms are coherently pumped with a resonant classical (laser or microwave) field and they release their energy by collective coupling to the cavity output field (see main text). Right panel: Battery charging and discharging cycles consist of a rapid excitation pulse, a storage time ${\tau }_s$ in which the atoms decay with individual decay rates $\gamma$ and a discharging time ${\tau }_d$ in which the atoms are resonant with the cavity mode and experience collective superradiant emission with the Purcell enhanced emission rate $\mathrm{\Gamma }$.

Figure 2

(a) The mean value of the collective spin component $M=\langle S_z\rangle$ in units of $N$ is shown as a function of time. The heights of the six almost vertical drops show the release of useful energy in units of $N\hslash {\omega }_s$ if the cavity is brought into resonance with the atoms at the corresponding instants of time. (b) The spin dynamics is represented by time-dependent trajectories in the space of collective spin Dicke quantum numbers $S$ and $M$, where $S(S+1)=〈{\vec{S}}^2〉$. The rightmost vertical green line shows the rapid classical excitation, the red curve shows the evolution under the slow individual decay, and the left blue vertical lines show the rapid collective decay, followed by (dashed lines) slow individual decay of the spins. The calculations were done for $N=140$, $\mathrm{\Gamma }=1600\gamma$ (see main text for details).

Figure 3

(a) Excitation of the battery $M=\langle S_z\rangle$ as a function of time during multiple subsequent cycles of charging, storage, and discharging of $N=140$ emitters with $\mathrm{\Gamma }=1600\gamma$ and ${\tau }_s={\tau }_d=0.0187{\gamma }^{-1}$. (b) The variation of $M$ is constrained by the length of the spin $S$ $[S(S+1)=〈{\vec{S}}^2〉]$, which shrinks due to the decay of individual emitters. The results suggest that the system reaches a limit cycle of constant charging capacity, where the decrease in $S$ during the storage is balanced by the increase of $S$ at the end of each discharging interval (see text). In panels (c) and (d), the discharging intervals are longer, ${\tau }_s=0.0187{\gamma }^{-1}$ and ${\tau }_d=0.0375{\gamma }^{-1}$, leading to a larger limit cycle value of $S$ and a higher charging capacity of the battery.