Better performance of quantum batteries in different environments compared to closed batteries

We construct a central-spin battery model affected by noise or thermal bath and consider the impact of environment on battery performance. In a noisy environment, we show that the steady-state energy and ergotropy are related to the number of spins exposed to the noisy channel. In addition, we reveal the relationship between the steady-state energy and ergotropy with the quantum information lost in the battery. In a thermal bath, we find that the higher the mean occupation number of the thermal bath, the greater the steady-state energy. In particular, we find that, in two different environments, the steady-state energy and ergotropy are related to the effective magnetic-field strength of battery $B$, but not charger $h$. It is worth noting that, in the central-spin model, when $B$ is a constant and $|h/B-1|$ reaches a certain value, the steady-state energy and ergotropy with noise or thermal bath are always greater than or equal to the maximum energy and ergotropy of the corresponding closed battery. At the same time, we generalize the conclusions to the two-qubit, two-harmonic-oscillator, and harmonic-oscillator-qubit models.

Figure 1

Panel (a) is a schematic illustration of the closed central-spin model, showing the process of energy transfer from the charger in the excited state to the battery in the ground state through interaction. Panel (b) is a schematic illustration of the central-spin model in a noisy environment, showing the process in which the charger in the excited state transfers energy to the battery in the ground state through interaction, and both the battery and charger are affected by noise. Panel (c) shows the schematic illustration of the central-spin model in a thermal bath, showing that the thermal bath interacts with the charger and then transfers energy to the battery through the charger. Here, the green (red) arrow represents spin up (down), the blue dashed line represents the interaction between the battery and charger, the gray shadow represents noise, and the pink shadow represents a thermal bath.

Figure 2

Panels (a) and (b) show the changes of the stored energy $\mathrm{\Delta }{E}_B^P\left(t\right)$ and ergotropy ${\varepsilon }_B^P\left(t\right)$ over time $t$. The two figures show that different numbers of spins are exposed to phase-flip noise. Here $P=b$ (red solid line) indicates that the battery is exposed to phase-flip noise, $P=b+c$ (dark blue downward triangles) indicates that the battery and charger are exposed to the phase-flip noise, $P=1$, 2, 3, 4, 5 (from light blue dashed line to black upward triangles) indicates the number of spins in the charger exposed to phase-flip noise. Panel (c) shows the steady-state energy $E_B^P\left(\infty \right)$ (red dots), ergotropy ${\varepsilon }_B^P\left(\infty \right)$ (dark blue triangles), and the quantum information lost in the battery $\mathrm{\Delta }{S}_B^P\left(\infty \right)$ (purple squares) as a function of the number of spins exposed to the phase-flip noise $P$. The parameters $N_c=5, m=5, N_b=1, B=1, h=1, A=1, \mathrm{\Delta }=0, \kappa =1$.

Figure 3

Panel (a) shows the variation of steady-state energy $E_B^{b+c}\left(\infty \right)$ with the number of up spins of all charger spins $m$, here $N_b=1, N_c=5$. Panel (b) shows the variation of steady-state energy $E_B^{b+c}\left(\infty \right)$ with the spin numbers in charger $N_c$, here $N_b=1, m=3$. Panel (c) shows the variation of steady-state energy $E_B^{b+c}\left(\infty \right)$ with the spin numbers in battery $N_b$, here $N_c=5, m=5$. The other parameters are same as Fig. 2.

Figure 4

Panels (a) and (b) show the evolutions of the stored energy $\mathrm{\Delta }{E}_B^{b+c}\left(t\right)$ and ergotropy ${\varepsilon }_B^{b+c}\left(t\right)$ over time $t$ with the different strength of noise $\kappa =$ 0.1, 0.3, 0.5, 0.7, 0.9, respectively (from light blue solid line to dark blue upward triangles). Panel (c) shows the evolution of quantum information lost $\mathrm{\Delta }{S}_B^{b+c}\left(t\right)$ over time $t$ in the battery with the different strength of noise $\kappa =$ 0.1, 0.3, 0.5, 0.7, 0.9, respectively (from light blue solid line to dark blue upward triangles). The other parameters are same as Fig. 2.

Figure 5

Panel (a) shows the variations of the steady-state energy $E_B^{b+c}\left(\infty \right)$ (red solid line) and ergotropy ${\varepsilon }_B^{b+c}\left(\infty \right)$ (dark blue dashed line) with the effective magnetic-field strength of battery $B$. Panel (b) shows the variations of the steady-state energy $E_B^{b+c}\left(\infty \right)$ (red solid line) and ergotropy ${\varepsilon }_B^{b+c}\left(\infty \right)$ (dark blue dashed line) with the effective magnetic-field strength of charger $h$. The other parameters are same as Fig. 2.

Figure 6

Panel (a) shows the variations of maximum energy $E_{\max }$ (dark blue dashed line) in the closed central-spin system and steady-state energy $E_B^{b+c}\left(\infty \right)$ (red solid line) in the phase-flip noisy environment with $h/B$. Panel (b) shows the variations of maximum ergotropy ${\varepsilon }_{\max }$ (dark blue dashed line) in the closed central-spin system and steady-state ergotropy ${\varepsilon }_B^{b+c}\left(\infty \right)$ (red solid line) in the phase-flip noisy environment with $h/B$. Here $N_c=5, m=5, N_b=1, B=1, A=1, \mathrm{\Delta }=0, \kappa =1$.

Figure 7

Panel (a) shows that the maximum energy $E_{\max }$ (dark blue dashed line) in the closed two-qubit system and steady-state energy $E_B^{b+c}\left(\infty \right)$ (red solid line) in the phase-flip noisy environment vary with ${\omega }_c/{\omega }_b$ changes. Panel (b) shows that the maximum ergotropy ${\varepsilon }_{\max }$ (dark blue dashed line) in the closed two-qubit system and steady-state ergotropy ${\varepsilon }_B^{b+c}\left(\infty \right)$ (red solid line) in the phase-flip noisy environment vary with ${\omega }_c/{\omega }_b$ changes. Here ${\omega }_b=1, g=0.1, \kappa =1$.

Figure 8

The evolution of stored energy $\mathrm{\Delta }{E}_B\left(t\right)$ with time $t$, where the red solid line represents the global operation of thermal bath on the charger, and the dark blue dashed line represents the local operation of thermal bath on the charger. The parameters $N_c=5, m=1, N_b=1, B=1, h=1, A=1, \mathrm{\Delta }=0, \gamma =1, n_B=2$.

Figure 9

The variation of steady-state energy $E_B\left(\infty \right)$ with the mean occupation number of thermal bath $n_B$. The other parameters are same as Fig. 8.

Figure 10

Panel (a) shows the variation of steady-state energy $E_B\left(\infty \right)$ with the effective magnetic-field strength of battery $B$. Panel (b) shows the variation of steady-state energy $E_B\left(\infty \right)$ with the effective magnetic-field strength of charger $h$. The other parameters are same as Fig. 8.

Figure 11

Panel (a) represents the maximum energy $E_{\max }$ (dark blue dashed line) in the closed central-spin system and steady-state energy $E_B\left(\infty \right)$ (red solid line) in the thermal bath as a function of $h/B$. Panel (b) represents the maximum ergotropy ${\varepsilon }_{\max }$ (dark blue dashed line) in the closed central-spin system and steady-state ergotropy ${\varepsilon }_B\left(\infty \right)$ (red solid line) in the thermal bath as a function of $h/B$. Here $N_c=5, m=1, N_b=1, B=1, A=1, \mathrm{\Delta }=0, \gamma =1, n_B=2$.

Figure 12

Panel (a) shows that the maximum energy $E_{\max }$ (dark blue dashed line) in the closed two-qubit system and steady-state energy $E_B\left(\infty \right)$ (red solid line) in the thermal bath vary with ${\omega }_c/{\omega }_b$ changes. Panel (b) shows that the maximum ergotropy ${\varepsilon }_{\max }$ (dark blue dashed line) in the closed two-qubit system and steady-state ergotropy ${\varepsilon }_B\left(\infty \right)$ (red solid line) in the thermal bath vary with ${\omega }_c/{\omega }_b$ changes. Here ${\omega }_b=1, g=0.1, \gamma =1, n_B=2$.

Figure 13

Panel (a) shows that the maximum energy $E_{\max }$ (dark blue dashed line) in the closed two-harmonic-oscillator system and steady-state energy $E_B^{b+c}\left(\infty \right)$ (red solid line) in the phase-flip noisy environment vary with ${\omega }_c/{\omega }_b$ changes. Panel (b) shows that the maximum ergotropy ${\varepsilon }_{\max }$ (dark blue dashed line) in the closed two-harmonic-oscillator system and steady-state ergotropy ${\varepsilon }_B^{b+c}\left(\infty \right)$ (red solid line) in the phase-flip noisy environment vary with ${\omega }_c/{\omega }_b$ changes. The other parameters are same as Fig. 7.

Figure 14

Panel (a) shows that the maximum energy $E_{\max }$ (dark blue dashed line) in the closed harmonic-oscillator-qubit system and steady-state energy $E_B^{b+c}\left(\infty \right)$ (red solid line) in the phase-flip noisy environment vary with ${\omega }_c/{\omega }_b$ changes. Panel (b) shows that the maximum ergotropy ${\varepsilon }_{\max }$ (dark blue dashed line) in the closed harmonic-oscillator-qubit system and steady-state ergotropy ${\varepsilon }_B^{b+c}\left(\infty \right)$ (red solid line) in the phase-flip noisy environment vary with ${\omega }_c/{\omega }_b$ changes. The other parameters are same as Fig. 7.

Figure 15

Panel (a) shows that the maximum energy $E_{\max }$ (dark blue dashed line) in the closed two-harmonic-oscillator system and steady-state energy $E_B^{b+c}\left(\infty \right)$ (red solid line) in the thermal bath vary with ${\omega }_c/{\omega }_b$ changes. Panel (b) shows that the maximum ergotropy ${\varepsilon }_{\max }$ (dark blue dashed line) in the closed two-harmonic-oscillator system and steady-state ergotropy ${\varepsilon }_B^{b+c}\left(\infty \right)$ (red solid line) in the thermal bath vary with ${\omega }_c/{\omega }_b$ changes. The other parameters are same as Fig. 12.

Figure 16

Panel (a) shows that the maximum energy $E_{\max }$ (dark blue dashed line) in the closed harmonic-oscillator-qubit system and steady-state energy $E_B^{b+c}\left(\infty \right)$ (red solid line) in the thermal bath vary with ${\omega }_c/{\omega }_b$ changes. Panel (b) shows that the maximum ergotropy ${\varepsilon }_{\max }$ (dark blue dashed line) in the closed harmonic-oscillator-qubit system and steady-state ergotropy ${\varepsilon }_B^{b+c}\left(\infty \right)$ (red solid line) in the thermal bath vary with ${\omega }_c/{\omega }_b$ changes. The other parameters are same as Fig. 12.

Figure 17

Panel (a) shows the variations of maximum energy $E_{\max }$ (dark blue dashed line) in the closed central-spin system and steady-state energy $E_B^{b+c}\left(\infty \right)$ (red solid line) in the bit-flip noisy environment with $h/B$. Panel (b) shows the variations of maximum ergotropy ${\varepsilon }_{\max }$ (dark blue dashed line) in the closed central-spin system and steady-state ergotropy ${\varepsilon }_B^{b+c}\left(\infty \right)$ (red solid line) in the bit-flip noisy environment with $h/B$. The other parameters are same as Fig. 6.

Figure 18

Panel (a) shows that the maximum energy $E_{\max }$ (dark blue dashed line) in the closed two-qubit system and steady-state energy $E_B^{b+c}\left(\infty \right)$ (red solid line) in the bit-flip noisy environment vary with ${\omega }_c/{\omega }_b$ changes. Panel (b) shows that the maximum ergotropy ${\varepsilon }_{\max }$ (dark blue dashed line) in the closed two-qubit system and steady-state ergotropy ${\varepsilon }_B^{b+c}\left(\infty \right)$ (red solid line) in the bit-flip noisy environment vary with ${\omega }_c/{\omega }_b$ changes. The other parameters are same as Fig. 7.