Quantum battery with non-Hermitian charging

We propose a design of a quantum battery exploiting the non-Hermitian Hamiltonian as a charger. In particular, starting with the ground or the thermal state of the interacting (noninteracting) Hamiltonian as the battery, the charging of the battery is performed via parity-time ($\mathcal{PT}$)- and rotational-time ($\mathcal{RT}$)-symmetric Hamiltonians to store energy. We report that such a quenching with a non-Hermitian Hamiltonian leads to an enhanced power output compared to a battery with a Hermitian charger. We identify the region in the parameter space which provides the gain in performance. We also demonstrate that the improvements persist with the increase of system size for batteries with both $\mathcal{PT}$- and $\mathcal{RT}$-symmetric chargers. In the $\mathcal{PT}$-symmetric case, although the anisotropy of the $\mathrm{XY}$ model does not help in the performance, we show that the $\mathrm{XXZ}$ model as a battery with a non-Hermitian charger performs better than that of the $\mathrm{XX}$ model having certain interaction strengths. We also exhibit that the advantage of non-Hermiticity remains valid even at finite temperatures in the initial states.

Figure 1

(Color online.) Advantage of non-Hermitian charging over Hermitian ones in terms of the maximum power. Map plot of ${\delta }_{P_{max}}^{\mathcal{PT}}$, defined in Eq. (11), with respect to the parameters, $\alpha /\pi$ ($x$ axis) and $J/\left|h\right|$ ($y$ axis). In the entire parameter regime considered here, a nonvanishing advantage is obtained, thereby establishing the benefit of a non-Hermitian $\mathcal{PT}$ charger in Eq. (8) over the Hermitian one [Eq. (11)]. The initial state is the ground state of the battery Hamiltonian. Note that for the non-Hermitian charger, $\alpha ={\alpha }_i$ while $\alpha ={\alpha }_r$ for the Hermitian one. All the axes are dimensionless.

Figure 2

Role of non-Hermiticity. Maximum power output, $P_{max}$ vs non-Hermiticity parameter ${\alpha }_i$ in the charger. The effect of increase in system size is also depicted by taking different $N$ values. The initial state of the QB is again taken to be the ground state of the battery. Here $J/\left|h\right|=1$ and $\mathrm{\Delta }=0$ in the battery Hamiltonian, $H_B$ in Eq. (7). All the axes are dimensionless.

Figure 3

Interaction dependence. $P_{max}$ (ordinate) against $J/\left|h\right|$ (abscissa). Different curves represent different non-Hermiticity and Hermiticity parameters, ${\alpha }_i$, and ${\alpha }_r$ in Eqs. (8) and (11) respectively. The $\mathcal{PT}$-symmetric local charger, given in Eq. (8), is applied at each site of the battery in the non-Hermitian case while the Hermitian charging Hamiltonian in Eq. (11) is used in the Hermitian scenario. In both the situations, the $\mathrm{XX}$ model acts as the QB. Solid lines represent $P_{max}$ via non-Hermitian chargers, having higher power than that of the Hermitian ones (the dashed lines). Here $N=8$. All the axes are dimensionless.

Figure 4

Scaling: $P_{max}$ (ordinate) as a function of $N$ (abscissa). All other specifications are the same as in Fig. 2. Fitting the data shows that $P_{max}\propto \sqrt{N}$. Both the axes are dimensionless.

Figure 5

Dependence on anisotropy. $P_{max}$ (vertical axis) with $\gamma$ (horizontal axis) of the QB Hamiltonian for different values of ${\alpha }_i$. ${\alpha }_i=0$ represents the battery with a Hermitian charger. Other specifications are the same as in Fig. 2. All the axes are dimensionless.

Figure 6

Importance of the $\mathrm{XXZ}$ model as battery. Trends of maximum power output, $P_{max}$, with respect to the variation of interaction strength in the $z$ direction, $\mathrm{\Delta }/\left|h\right|$, for different values of ${\alpha }_i$. Clearly, we observe that the $\mathrm{XXZ}$ model as a battery has some beneficial role over the $\mathrm{XX}$ model with $J/\left|h\right|=1.0$. All other specifications are the same as in Fig. 2. Both the axes are dimensionless.

Figure 7

Temperature dependence of the initial state: $P_{max}$ (ordinate) vs $\beta =1/k_{B}T$ (abscissa) for different values of ${\alpha }_i$. The initial state of the QB is prepared as the canonical equilibrium state of the $\mathrm{XX}$ model while the charger is the $\mathcal{PT}$-symmetric one with ${\alpha }_i\ne 0$. Notice that the decreasing behavior of the power with the increase of temperature is the same as typically observed in the Hermitian domain. However, close to high temperature, in the framework of non-Hermitian systems, we find some different behavior than the one in the Hermitian paradigm. Inset: $W\left(t\right)$ with $t$ for a fixed $\beta =1.0$. It shows that $W\left(t\right)$ goes negative which is responsible for a nonmonotonic behavior of $P_{max}$ at high temperature. All the axes are dimensionless.

Figure 8

Temperature dependence of $P_{max}$ and minimum time taken to achieve $P_{max}$, denoted by $t_{min}$. $P_{max}$ [dashed line (left ordinate)] and $t_{min}$ [solid line (right ordinate)] vs $\beta =1/k_{B}T$ (abscissa) for ${\alpha }_i=2\pi /3$. The initial state of the QB is prepared as the canonical equilibrium state of the $\mathrm{XX}$ model while the charger is the $\mathcal{PT}$-symmetric one with ${\alpha }_i=2\pi /3$. Notice that the decreasing behavior of the $P_{max}$ as well as $t_{min}$ with the increase of temperature is the same as typically observed in the Hermitian domain. However, close to high temperature, in the framework of the non-Hermitian systems, nonmonotonic behaviors for both the quantities emerge with $\beta$. All the axes are dimensionless.

Figure 9

Non-Hermitian effects on the QB. Map plot of ${\delta }_{P_{max}}^{\mathcal{RT}}$ with the variation of parameters in the charging Hamiltonian, ${\gamma }^{\prime }$ (horizontal axis) and $h$ (vertical axis). The initial state is the ground state of the noninteracting battery Hamiltonian, $H_B^{\mathrm{nonint}}$ given in Eq. (15). Note that in Fig. 1, the difference was plotted with respect to the battery Hamiltonian. However, both the plots manifest some advantage in presence of the non-Hermitian charger over the Hermitian one. Here $N=2$. Both the axes are dimensionless.

Figure 10

Comparison between Hermitian and non-Hermitian chargers: $P_{max}$ (ordinate) vs ${\gamma }^{\prime }$ (abscissa). Solid lines represent non-Hermitian charger in Eq. (16) ($\gamma ={\gamma }^{\prime }$) while dashed lines represent the Hermitian ones ($\gamma =-i{\gamma }^{\prime }$), representing the $\mathrm{XY}$ model. The ground state of the noninteracting battery Hamiltonian represents the initial state of the QB as in Fig. 9. Here $h_i$ and $h_r$ represent the non-Hermitian and Hermitian chargers respectively. The system size is taken to be 8, i.e., $N=8$. Both the axis are dimensionless.

Figure 11

Scaling of the QB with $\mathcal{RT}$-symmetric charger: $P_{max}$ ($y$ axis) vs $N$ ($x$ axis) for ${\gamma }^{\prime }=0.8$. All other specifications are the same as in Fig. 10. Both the axes are dimensionless.

Figure 12

Effects of thermal fluctuations on the non-Hermitian battery. The maximal power (ordinate) against $\beta ={\beta }^{\prime }/\left|J\right|$ (abscissa) where the initial state is prepared as the thermal state of the battery. The charging is again by the $\mathcal{RT}$-symmetric Hamiltonian with ${\gamma }^{\prime }=0.8$. All other specifications are the same as in Fig. 10. Both the axes are dimensionless.