Optimal charging of open spin-chain quantum batteries via homodyne-based feedback control

We study the problem of charging a dissipative one-dimensional $XXX$ spin-chain quantum battery using local magnetic fields in the presence of spin decay. The introduction of quantum feedback control based on homodyne measurement contributes to improving various performances of the quantum battery, such as energy storage, ergotropy, and effective space utilization rate. For the zero-temperature environment, there is a set of optimal parameters to ensure that the spin-chain quantum battery can be fully charged and the energy stored in the battery can be fully extracted under the perfect measurement condition, which is found through the analytical calculation of a simple two-site spin-chain quantum battery and further verified by numerical simulation of a four-site spin-chain counterpart. For completeness, the adverse effects of imperfect measurement, anisotropic parameter, and finite temperature on the charging process of the quantum battery are also considered.

Figure 1

The spin-chain quantum battery. Each spin is coupled with an independent reservoir (light gray ellipse) and has a spontaneous emission rate $\mathrm{\Gamma }$. Initially, the battery is prepared in its ground state, and each spin is subjected to a homodyne measurement. After the detection, a corresponding homodyne photocurrent is produced. Then the feedback control is activated by applying local magnetic fields depending on the measured results.

Figure 2

(a) The stored energy $\mathrm{\Delta }E\left(\infty \right)/h$ of the battery at steady state is a function of $\alpha$ and $\chi$. (b) The maximum energy $\mathcal{E}\left(\infty \right)/h$ that can be extracted (ergotropy) from the battery; other parameters are $\eta =1$ and $J=h$.

Figure 3

The $y$-direction feedback control is applied to the system, i.e., $\alpha =\pi$. (a) The variation of the optimal stored energy $\mathrm{\Delta }E\left(\infty \right)/h$ of the battery at steady state with $J/h$. The inset describes the optimal feedback parameter $\chi$ at different $J$. (b) The variations of $R\left(\infty \right)$ and $\mathcal{E}\left(\infty \right)/\mathrm{\Delta }E\left(\infty \right)$ of the battery in the steady state with the coupling strength $J$. The measurement efficiency is given as $\eta =1$.

Figure 4

Stochastic charging dynamics of the battery under the optimal feedback condition. In the inset, the measuring efficiency is set to $\eta =0.8$. The shallow gray lines represent the random 200 trajectories of battery storage energy obtained by Eq. (8), the red solid line represents the average value of 200 energy trajectories, and the green dashed line represents the ensemble average energy obtained by Eq. (9).

Figure 5

(a) The stored energy $\mathrm{\Delta }E\left(\infty \right)/h$ (solid line) and the ergotropy (dash-dotted line) of the spin-chain quantum battery with different coupling strengths $J/h$. The inset shows the optimal feedback parameter $\chi$ required by the stored energy (solid line) and ergotropy (dash-dotted line). (b) The effective utilization rate $R\left(\infty \right)$ of the battery as a function of $J/h$ under the optimal feedback strength. The measurement efficiency is $\eta =0.8$.

Figure 6

The stored energy $\mathrm{\Delta }E\left(\infty \right)/h$ of the battery in steady state as a function of $J/h$ under different measurement efficiency $\eta$. The initial state of the battery is a spin-down state, and the feedback parameter $\chi$ at different $J$ values is optimal.

Figure 7

The effective space utilization rate $R\left(\infty \right)$ in steady state of the $XY$ spin-chain quantum battery ($\mathrm{\Delta }=0$) with spin numbers of $N=2$ (a) and $N=4$ (b) as a function of the anisotropic parameter $\gamma$ under different coupling strengths $J$. The insets in (a) and (b) describe the variations of the stored energy $\mathrm{\Delta }E\left(\infty \right)/h$ of the battery with $\gamma$ in steady state. The feedback strengths at different $\gamma$ values are optimal. Other parameters are $\eta =1, \alpha =\pi$, and $\mathrm{\Gamma }=h$. Parts (c), (d) show the variations of the effective space utilization rate with $\gamma$ of the battery with spin numbers of $N=2$ and 4 under different spontaneous emission rates $\mathrm{\Gamma }$ with $J=h$, respectively.

Figure 8

(a) The effective space utilization rate $R\left(\infty \right)$ of the $XYZ$ spin-chain quantum battery ($\mathrm{\Delta }=1$) in steady state as a function of the anisotropic parameter $\gamma$ for $N=2$. The inset describes the change of the effective space utilization rate $R$ of the battery with $\gamma$ in steady state. (b) The same as (a) but the spin number is $N=4$. Other conditions are the same as in Fig. 7. Parts (c) and (d) show the variations of the effective space utilization rate with $\gamma$ of the battery with spin numbers of $N=2$ and 4 under different spontaneous emission rates $\mathrm{\Gamma }$ with $J=h$, respectively.

Figure 9

Parts (a), (b) describe the optimal energy storage and ergotropy of the battery at steady state when the spin numbers are $N=2$ and 4, respectively. The corresponding optimal feedback parameter $\chi$ is shown in the inset. Part (c) describes the variation of the effective space utilization rate of the battery with the temperature. Part (d) shows the evolution of energy stored in the battery ($N=4$) with time $\mathrm{\Gamma }t$ at different temperatures, where the feedback strengths are adjusted to the optimal values, and the initial state of the battery is set to the corresponding ground state. Other parameters are ${\eta }_d={\eta }_c=0.8$ and $J=h$.

Figure 10

Parts (a), (b) describe the stored energy and ergotropy of the battery with a spin number of $N=2$ as functions of $\alpha$ and $\chi$ at steady state, respectively. Other parameters are $\eta =1$ and $J=h$. Parts (c), (d) are the same as (a), (b), but the spin number is $N=6$.

Figure 11

The value of $R\left(\infty \right)$ and $\mathcal{E}\left(\infty \right)/\mathrm{\Delta }E\left(\infty \right)$ of the battery with different spin number under the three groups of optimal feedback parameters, where the measurement efficiency is $\eta =1$.

Figure 12

(a) The critical value $J_c$ of spin-spin interaction in energy storage varies with the spin number. (b) The critical value $J_c$ of spin-spin interaction in energy extraction (ergotropy) varies with the spin number. Different curves correspond to different measuring efficiency $\eta$.