Extracting ergotropy from nonequilibrium steady states of an $XXZ$ spin-chain quantum battery

Coupling of a quantum battery (QB) to two thermal reservoirs brings it to a nonequilibrium steady-state (NESS). In this situation, although, the QB can be charged through the NESS heat current, no work may be extracted from it through unitary cyclic processes. We exemplify this statement by studying the two- and three-qubit $XXZ$ spin-chain QBs coupled weakly to two different thermal reservoirs in a symmetric configuration wherein each end of chain interacts with an individual reservoir. We show that the work can be extracted at positive temperature bias when two ends of a $XXZ$ chain are collectively coupled to the hot reservoir.

Figure 1

The schematic diagram of the $XXZ$ spin-chain QB with two coupled qubits 1 and 2 interacting with a nonequilibrium environment consists of left and right reservoirs. The key $\varepsilon$ is used to switch on or of f the coupling between the qubit 2 and left reservoir, which allow us to control chain-reservoir coupling asymmetry.

Figure 2

Level structure of the QB asymmetrically coupled to the left and right reservoirs (the case $\varepsilon =1$). Red (blue) arrows indicate the allowed transitions in the interaction with the hot (cold) reservoir.

Figure 3

The stored energy $\mathrm{\Delta }{E}_B\left(\infty \right)$ in the NESS of a ferromagnetic $XXZ$ QB versus $\mathrm{\Delta }$ and $\mathrm{\Delta }T$ for different values of $T_M$ in the symmetric coupling setting. The uniform magnetic field is set as $B_0=2$. All parameters of QB are in units of interspin coupling $\left|J\right|=1$.

Figure 4

The stored energy $\mathrm{\Delta }{E}_B\left(\infty \right)$ in the NESS of an antiferromagnetic $XXZ$ QB versus $\mathrm{\Delta }$ and $\mathrm{\Delta }T$ for different values of $T_M$ in the symmetric coupling setting. The uniform magnetic field is set as $B_0=2$. All parameters of QB are in units of interspin coupling $\left|J\right|=1$.

Figure 5

The stored energy $\mathrm{\Delta }{E}_B\left(\infty \right)$ in the NESS of a ferromagnetic $XXZ$ QB versus $\mathrm{\Delta }$ and $\mathrm{\Delta }T$ for different values of $T_M$ in the asymmetric coupling setting. The uniform magnetic field is set as $B_0=2$. All parameters of QB are in units of interspin coupling $\left|J\right|=1$.

Figure 6

The stored energy $\mathrm{\Delta }{E}_B\left(\infty \right)$ in the NESS of an antiferromagnetic $XXZ$ QB versus $\mathrm{\Delta }$ and $\mathrm{\Delta }T$ for different values of $T_M$ in the asymmetric coupling setting. The uniform magnetic field is set as $B_0=2$. All parameters of QB are in units of interspin coupling $\left|J\right|=1$.

Figure 7

NESS population distribution of the ferromagnetic [panels (a) and (c)] and antiferromagnetic [panels (b) and (d)] spin-based QB versus $\mathrm{\Delta }T$ by setting $B_0=2$, $T_M=20$, and $\mathrm{\Delta }=\pm 0.5$. Panels (a) and (b) correspond to the symmetric spin-reservoir coupling setting, while panels (c) and (d) correspond to the asymmetric spin-reservoir coupling setting. All parameters of QB are in units of interspin coupling $\left|J\right|=1$.

Figure 8

NESS ergotropy $W\left({\rho }_{\mathrm{NESS}}\right)$ of the ferromagnetic spin-based QB versus $\mathrm{\Delta }$ and $\mathrm{\Delta }T$ for different values of $T_M$ in the asymmetric spin-reservoir coupling setting. The chain anisotropy is set as $B_0=2$. All parameters of QB are in units of interspin coupling $\left|J\right|=1$.

Figure 9

NESS ergotropy $W\left({\rho }_{\mathrm{NESS}}\right)$ of the antiferromagnetic spin-based QB versus $\mathrm{\Delta }$ and $\mathrm{\Delta }T$ for different values of $T_M$ in the asymmetric spin-reservoir coupling setting. The chain anisotropy is set as $B_0=2$. All parameters are in units of interspin coupling $\left|J\right|=1$.

Figure 10

NESS energy of (a) the antiferromagnetic and (b) the ferromagnetic $XXZ$ QB versus $\mathrm{\Delta }$ and $T_L$ in the symmetric coupling settings ($\varepsilon =0$). The magnetic field and inverse temperature of the reference heat reservoir are set as $B_0=2$ and ${\beta }_{\mathrm{eq}}=1/T_{\mathrm{eq}}=1/T_R=0.5$, respectively. All parameters of QB are in units of interspin coupling $\left|J\right|=1$.

Figure 11

The schematic diagram of the three-spin $XXZ$ spin-chain QB with two end qubits 1 and 3 interacting with a nonequilibrium environment consists of left and right reservoirs. The key $\varepsilon$ is used to switch on and off the coupling between the end qubit 3 and the left reservoir, which allows us to control chain-reservoir coupling asymmetry.

Figure 12

(a) The NESS stored energy $\mathrm{\Delta }{E}_B\left(\infty \right)$ and (b) the NESS ergotropy $W\left({\rho }_{\mathrm{NESS}}\right)$ of a three-spin antiferromagnetic $XXZ$ QB versus $\mathrm{\Delta }$ for different values of $T_M$ and $\mathrm{\Delta }T$ in the symmetric and asymmetric coupling settings. The chain anisotropy is set as $B_0=5$. All parameters of QB are in units of interspin coupling $\left|J\right|=1$.