Early-stage memory effect on the dephasing charger-mediated quantum battery

TL;DR

This work analyzes a two-qubit charger-mediated quantum battery (CmB) coupled to a Lorentzian reservoir, deriving a time-local Lindblad master equation with a time-dependent dephasing rate $\gamma_0(t)$ that can become negative at early times, signaling non-Markovian memory. It shows that such early-stage memory can boost the maximal ergotropy relative to the Markovian case with asymptotic rate $\gamma_0(\infty)$, explained by information backflow and non-Markovian quantum jumps. The dynamics are unraveled using the non-Markovian quantum-jump method, revealing that a small number of trajectory classes suffices to capture the enhancement, and motivating a discrete-time, measurement-enhanced charging scheme realized by a quantum circuit. The findings highlight the importance of short-time memory effects for quantum battery performance and suggest practical protocols to exploit them for faster charging.

Abstract

We investigate the performance of the charger-mediated quantum battery modeled by a two-qubit system. One of the qubits acts as the battery and the other acts as the charger which is subjected to a reservoir. We derived the time-local master equation in Lindblad form with a time-dependent dephasing rate. The dephasing rate may be negative in the early-stage of the charging process and thus indicate the presence of the memory effect. We find that such early-stage memory effect could increase the maximal ergotropy of the battery compared with the one under Markovian approximation with the corresponding asymptotic dephase rate. The enhancement of the performance is explained by means of the non-Markovian quantum jumps. Moreover, a discrete time scheme of the measurement-enhanced quantum battery is proposed in a quantum circuit with global and random local operations.

Figures (5)

• Figure 1: (a) The model. The charger qubit is subjected to the reservoir while the battery qubit is isolated. The spectral densities of the reservoirs in both models are Lorentzian. The charger is driven by an external field as the energy supplier and interacts with the battery. (b) The quantum circuit diagram for the measurement-enhanced quantum battery. The global unitary operations $\hat{U}$ acting on the charger (A) and the battery (B), alternate with the random local operations $\hat{\sigma}^x_A$ acting on the charger (A), in the $\gamma_0>0$ period of the early stage. The local operation is suspended on during the $\gamma_0<0$ period (red shaded region).
• Figure 2: (a) The time-dependence of the decay rates of the charger. The solid line denotes the behavior of $\gamma_0$. In the early-stage $\lambda t\lesssim 4$, the positive $\gamma_0$ is punctuated by a segment with negative $\gamma_0$. For $\lambda t\gtrsim 4$, the $\gamma_0$ remains positive and reach to the asymptotic value at large $\lambda t$. The decay rates $\gamma_+(t)$ and $\gamma_-(t)$ oscillates between the positive and negative values. The amplitude is so small ($<10^{-2}$) compared with $\gamma_0$ that can be neglected. The inset shows the coarse-grained value of $\gamma_0$. The parameters are chosen as $p=100$ and $s=6$. (b) The concurrence of the CmB system with $g=0$. The initial state is the maximally entangled state. A revival of entanglement can be observed during the period of negative $\gamma_0$ indicating the back flow of information. The shaded regions in red indicate the period of negative $\gamma_0$.
• Figure 3: The time-evolution of the ergotropy of the charger-mediated battery with $\omega_A=\omega_B=\omega_L=100\lambda$, $\Omega=0.1\omega_A$ and $g=4$. The coupling strength between the charger and the reservoir in Eq. (\ref{['eq_gamma_t']}) are $\eta^2=0.5$, $1$, $2$ and $3$ for panels (a)-(d). The initial state is $\left| \psi_{AB}(0) \right>=\left| \uparrow^y_A\downarrow^z_B \right>$. The solid, dashed, and dash-dotted lines represents the cases (i), (ii), and (iii), respectively. The shaded region in red indicates the period of $\gamma_0<0$.
• Figure 4: (a) The existence probabilities of the no-jump, one-jump and two-jump trajectory classes. During the period of positive decay rate the $K_o^{\o}$ decreases due the actions of the normal jumps while the $H_1^\alpha$ and $H_2^\alpha$ trajectories are created. The inset shows the sum of the existence probabilities up to $n=2$ which is close to one throughout the time window. (b) The thick line represents the ergotropy under unitary time evolution. The solid, dashed and dash-dotted (thin) lines represent the ergotropy with respect to the density matrix computed by the NMQJ method with the summation up to $n=0$, $1$, and $2$, respectively. The initial state is $\left| \uparrow_A^y\downarrow_B^z \right>$. The coupling between the charger and the battery is $g=4$ and the time interval is $\delta t=5\times10^{-4}\lambda^{-1}$. Other parameters are chosen the same as in Fig. \ref{['fig3_ergotropy']}.
• Figure 5: (a) The early-stage time evolution of the ergotropy. The thick solid line represents the results computed by the forth-order Runge-Kutta method, while the thin solid, dashed, and dot-dashed line represent the NMQJ results with time interval $\delta t=2\times10^{-4}\lambda^{-1}$, $5\times10^{-4}\lambda^{-1}$ and $10^{-3}\lambda^{-1}$, respectively. (b) The long-time evolution of the ergotropy. The coupling strength between the charger and the battery is chosen as $g=0.2$ and other parameters are chosen the same as in Fig. \ref{['fig3_ergotropy']}.