Quantum steering probes energy transfer in quantum batteries

TL;DR

This work addresses how Einstein-Podolsky-Rosen (EPR) steering informs energy transfer in quantum batteries coupled to a shared reservoir. Using a two-qubit QB model with coherent coupling $g$, drive $F$, and dissipation $\Gamma$, evolved under a Lindblad master equation, the authors quantify energy via $E_B(t)=\mathrm{Tr}[H_B \rho_B(t)]$ and extractable work via ergotropy $W_B(t)=E_B(t)-\mathrm{Tr}[H_B \rho_p]$, while detecting steering with local orthogonal observables (LOOs) and a steering function $S_{A\to B}$. They show that steering is stored during energy growth and later consumed to boost final energy and ergotropy, with steering peaking near energy equilibrium and serving as a witness to battery population balance; steering remains a reliable energy indicator across symmetric/asymmetric reservoir couplings and various temperatures, including extended wireless charging with driving fields. The findings identify steering as a practical diagnostic and resource for achieving high-performance quantum batteries and suggest applicability to broader open QB protocols. The results have implications for designing QB charging schemes where steering enhances predictability and control of energy storage and work extraction.

Abstract

This study investigates the role of EPR steering in characterizing the energy dynamics of quantum batteries (QBs) within \textcolor{black}{a charging system that features shared reservoirs. After optimizing parameter configurations to achieve high-energy systems, we observe across a variety of charging scenarios with low-dissipation regimes that steering serves as a vital resource: it is initially stored until the system reaches energy equilibrium, and then subsequently utilized to sustain the enhancement of energy storage. Furthermore, steering acts as a witness to battery population balance and a consumable that enhances extractable work. Additionally, we discuss the contribution of the steering potential to energy upon high-dissipation charging in details. These findings establish a novel indicator for monitoring QB energy variations, which will be beneficial to achieve the high-performance quantum batteries.

Figures (7)

• Figure 1: As is shown, a charger ($A$) coherently interacts with a battery ($B$) with a coupling rate $g$, where the frequencies of the charger and battery are $\omega_{A}$ and $\omega_{B}$, respectively. The charger and battery form a two-level system, where $\left| g \right\rangle$ represents the ground state and $\left| e \right\rangle$ the excited state. Both the charger and the battery simultaneously dissipate at a rate of $\Gamma$ into a common reservoir. And the charging system is powered by an external pump with a frequency of $\omega_{L}$ and an amplitude of $F$. The charging period is $t \in \left[ {0,\tau } \right]$, meaning that the battery reaches its maximum energy storage capacity at $t=\tau$.
• Figure 2: The steady-state energy ${E_{A,B}}\left( \infty \right)$ (units of $\omega_0$) of the system as a function of pump $F$, dissipation rate $\Gamma$, and interaction strength $g$. Graph (a): the charger's energy ${E_{A}}\left( \infty \right)$, and Graph (b): the battery's energy ${E_{B}}\left( \infty \right)$. Here we have set up a resonant pump and a reservoir at zero-temperature, i.e., $\Delta = T = 0$. The black dashed lines represent contour lines, while the white line in (b) depict energy pathway resulting from adjusting the coupling strength $g$ when the $F=\Gamma$. Specifically, the continuously increasing coupling strength will lead to the decrease of $\Gamma/g$ and $F/g$.
• Figure 3: The system's energy ${E_{A,B}}\left( \infty \right)$ (units of $\omega_0$) as a function of the average particle number $n_{b,f}$ and the detuning $\Delta$. Here we have considered different types of non-zero-temperature reservoirs and the effect of detuning on energy. Here we have examined different types of non-zero-temperature reservoirs and the impact of detuning on energy. Since temperature elevation directly improves the average particle number $n(T)$ in the reservoir, we intuitively demonstrate how variations in $n(T)$ affect energy. Note that symmetric driving, dissipation, and coherent coupling are set here (i.e, $F=\Gamma=g$) with charger energy plotted in (a) and (c), and energy storage depicted in (b) and (d).
• Figure 4: The system's energy ${E_{A,B}}/ \omega_0$ and steering function $S_{A\to B,(B\to A)}$ as functions of time. Here, (a) and (b) account for resonance-driven processes and zero-temperature reservoirs, while (b) emphasizes stronger system dissipation. (c) and (d) set corresponding parameters for reservoirs at different temperatures to achieve maximum system energy. Since (a)–(d) all belong to the no-pumping charging scenario ($F=\Delta=0, g=1$), we selected a fully charged initial charger and a completely depleted battery, i.e., $\left| {\psi \left( 0 \right)} \right\rangle = \left| {eg} \right\rangle$. Note that the magenta dots and black dots represent the maxima of steering functions $S_{A\to B}$ and $S_{B\to A}$, respectively, while the corresponding dashed lines indicate the time points at which these maxima occur.
• Figure 5: The time evolution of the energy $E_{A,B}/\omega_0$ and steering function $S_{A\to B}$ and $S_{B\to A}$ are plotted here. We considered resonance-driven charging scenarios ($F=3,\Gamma=0.001,\Delta=0,g=1$), including different initial conditions and the influence of reservoir temperature on energy. Note that the magenta dots and black dots represent the maxima of steering functions $S_{A\to B}$ and $S_{B\to A}$, respectively, while the corresponding dashed lines indicate the time points at which these maxima occur.