Using Dark States to Charge and Stabilize Open Quantum Batteries

We introduce an open quantum-battery protocol using dark states to achieve both superextensive capacity and power density, with noninteracting spins coupled to a reservoir. Further, our power density actually scales with the number of spins $N$ in the battery. We show that the enhanced capacity and power are correlated with entanglement. While connected to the charger, the charged state of the battery is a steady state, stabilized through quantum interference in the open system.

Figure 1

The model. The spin-charged QB is modeled as an ensemble of spins in a reservoir. Initially, the QB is in thermal equilibrium with the reservoir. The charger is another ensemble of spins but in the excited state. The charging process is initiated by bringing the charger into the reservoir.

Figure 2

The superextensive capacity and the power density. (a) The energy density of the charger ${\mathcal{E}}_C(t)$ (dotted) and the battery ${\mathcal{E}}_B(t)$ (solid) during the charging process. ${\mathcal{E}}_C(t)$ decreases as ${\mathcal{E}}_B(t)$ correspondingly increases, indicating a transfer of energy from the charger to the battery. (b) The monotonic increase of the steady-state energy density ${\mathcal{E}}_B^{\mathrm{ss}}$ with $N_B$ shows the superextensive increase in the battery capacity. (c) ${\mathcal{E}}_B^{\mathrm{ss}}$ monotonically increases with $R$. (d) The power density of the battery ${\mathcal{P}}_B(t)$ during the charging process. (e) The peak power density ${\mathcal{P}}_B^{\max }$ scales superextensively with $N_B$. (f) ${\mathcal{P}}_B^{\max }$ monotonically increases with $R$. Parameters: (a),(d) $R$ = 5, $N_B$ = 1 (blue), 2 (orange), 3 (green); (b),(e) $R$ = 2 (blue), 5 (orange), 10 (green); (c),(f) $N_B$ = 1 (blue), 2 (orange), 3 (green). $\mathcal{E}$ and $\mathcal{P}$ are in units of $\hslash \omega$, with dimensionless $\gamma t$.

Figure 3

Ergotropy (solid line) and stored energy (dotted line) for: (a) various $N_B$ and $R=5$; (b) $R$ and $N_B=1$. For $N_B=1$, the ergotropy is zero until ${\mathcal{E}}_B>\frac{1}{2}$. As $N_B$ increases, the ergotropy approaches the stored energy. For $R=1$, the ergotropy is always zero, as the stored energy is always negative. As $R$ increases, the ergotropy approaches the stored energy. Parameters: (a) $N_B$ = 1 (blue), 2 (orange), 10 (green); (b) $R$ = 1 (blue), 5 (orange), 50 (green). The vertical axes are in units of $\hslash \omega$.

Figure 4

Entanglement and capacity. (a) The logarithmic negativity $S_B(t)$. A comparison of this plot with ${\mathcal{E}}_B(t)$ in Fig. 2 shows that higher entanglement corresponds to a higher energy density. (b) The relationship between entanglement and the energy density is shown in this parametrized plot of ${\mathcal{E}}_B(t)$ and ${\mathcal{S}}_B(t)$. (c) ${\mathcal{S}}_B^{\mathrm{ss}}$ scales positively with $N_B$. A comparison of this plot with ${\mathcal{E}}_B^{\mathrm{ss}}(N_B)$ in Fig. 2 shows that higher entanglement corresponds to a higher energy density, in the steady state. (d) ${\mathcal{E}}_B^{\mathrm{ss}}$ and ${\mathcal{S}}_B^{\mathrm{ss}}$ parametrized over $N_B$, showing the positive correlation between the capacity and entanglement. Parameters: (a),(b) $R$ = 5, $N_B=1$ (blue), 2 (orange), 3 (green); (c),(d) $R=2$ (blue), 3 (orange), 5 (green).

Figure 5

The entanglement rate and the power density. (a) The power density ${\mathcal{P}}_B(t)$. (b) The entanglement rate ${\dot{S}}_B(t)$. A comparison of (a) and (b) shows that periods of nonzero ${\mathcal{P}}_B(t)$ approximately correspond to periods of nonzero ${\dot{S}}_B(t)$. (c) The local maximum entanglement rate ${\dot{S}}_B^{\max }$ scales linearly with $N_B$. (d) The lag time between ${\mathcal{P}}^{\max }$ and ${\dot{S}}_B^{\max }$ decreases with $N_B$. Parameters: (a),(b) $R$ = 50, $N_B$ = 1 (blue), 2 (orange), 3 (green); (c),(d) $R$ = 3 (blue), 5 (orange), 10 (green).

Figure 6

The charger and battery performance at nonzero temperature. (a) The energy density of the charger ${\mathcal{E}}_C(t)$ and the battery ${\mathcal{E}}_B(t)$ during the charging process for various temperatures. As the temperature increases, less energy is transferred from the charger to the battery. (b) The steady-state energy density ${\mathcal{E}}_B^{\mathrm{SS}}(T)$ for various values of $R$. All states converge in the thermodynamic limit to ${\mathcal{E}}_B^{\mathrm{SS}}=\frac{1}{2}$. (c) A plot of the rate of change in the steady-state energy density against temperature $d{\mathcal{E}}_B^{\mathrm{SS}}/dT$. There is a decline in ${\mathcal{E}}_B^{\mathrm{SS}}$ at low temperatures, as thermal fluctuations destroy the dark states. This is followed by a deceleration in the loss of energy as the system thermalizes. The exception is for $R=1$, where the infusion of energy from the thermal reservoir more than compensates for the loss of energy from the destruction of the dark state. (d) A plot of ${\mathcal{E}}_B^{\mathrm{SS}}$ (solid) and ${\mathcal{W}}_B^{\mathrm{SS}}$ (dotted). At high temperatures, ${\mathcal{W}}_B^{\mathrm{SS}}\to 0$ as ${\mathcal{E}}_B^{\mathrm{SS}}\to \frac{1}{2}$. Parameters: (a) $T$ = 0 (blue), 2 (orange), 4 (green), $R$ = 10, $N_B$ = 1; (b),(c) $R$ = 1 (blue), 2 (orange), 3 (green), 4 (red), $N_B$ = 1; (d) $R=4$. The vertical axes are in units of $\hslash \omega$.