Ergotropy in Quantum Batteries

TL;DR

This work addresses how ergotropy evolves in quantum batteries by decomposing it into coherent $\mathcal{E}_c$ and incoherent $\mathcal{E}_i$ contributions and linking their behavior to population dynamics and quantum resources. It develops a universal, model-independent framework based on random sampling of states and Hamiltonians to map how coherence $\mathcal{C}$, diagonal entropy $S_{\mathrm{diag}}$, participation ratio $PR$, and purity govern both ergotropy and charging efficiency $\mathcal{R}$. Key results include bounds where the lower ergotropy bound is $\mathcal{E}_i$, the upper bound is the stored energy $E(\rho_B)$, and $\mathcal{E}_c$ is bounded by pure/delocalized limits; purity generally reduces locked energy and enhances charging efficiency. The authors validate the framework with Jaynes–Cummings and Tavis–Cummings QB paradigms and discuss implications for designing high-performance QBs by tuning quantum resources.

Abstract

Ergotropy--a key figure of merit for quantum battery (QB) performance--plays a crucial role. However, the dynamics and physical mechanisms governing ergotropy evolution remain open challenges. Here, we investigate the ergotropy of a general QB model and find that the charging process is accompanied by the variation and inversion of the energy level populations. In the absence of population inversion, the ergotropy is fully consistent with coherent ergotropy; in local and global population inversion, it is determined by both coherent and incoherent ergotropy. Via random sampling of quantum states and Hamiltonians, we show that coherence and the participation ratio enhance coherent ergotropy, whereas incoherent ergotropy--whether enhanced, unchanged, or suppressed--depends on diagonal entropy, the participation ratio, and energy level population ordering. We demonstrate that the ergotropy lower bound is incoherent ergotropy, the upper bound is the QB stored energy, and enhanced QB purity suppresses locked energy and boosts charging efficiency. Furthermore, we use the Tavis-Cummings (TC) and Jaynes-Cummings (JC) batteries as paradigms to validate our findings. Our work elucidates ergotropy underlying mechanisms in general QBs and establishes a rigorous framework for optimizing ergotropy and charging efficiency, paving the way for high-performance quantum energy-storage devices.

Figures (10)

• Figure 1: (a) Schematic diagram of the charging process, which consists of three stages: Stage $\mathrm{\@slowromancap i@}$, no population inversion; Stage $\mathrm{\@slowromancap ii@}$, local population inversion; Stage $\mathrm{\@slowromancap iii@}$, global population inversion. The other parameters satisfy $0\leq t_1\leq t_2\leq T$ and $T>0$. (b) The ergotropy as a function of the excited energy level population $p_2$ for two-level QBs. (c),(d) The incoherent ergotropy and the stored energy as a function of the population of $2$th and $3$th energy level, $p_2,p_3$ for three-level QBs. In (c), the region of local population inversion contains different population ordering: In $\mathrm{\@slowromancap ii@}_1$, the population ordering is $p_1 \geq p_3 > p_2$; in $\mathrm{\@slowromancap ii@}_2$, $p_2 > p_1 \geq p_3$; in $\mathrm{\@slowromancap ii@}_3$, $p_2 > p_3 \geq p_1$ or $p_2 \geq p_3 > p_1$; and in $\mathrm{\@slowromancap ii@}_4$, $p_3 \geq p_1 > p_2$ or $p_3 > p_1 \geq p_2$. Here, the Hamiltonian $H_B$ has equally spaced energy levels for two-level and three-level QBs.
• Figure 2: (a),(b) The coherent ergotropy $\mathcal{E}_c$ as a function of the coherence $\mathcal{C}$ and participation ratio $PR$ for randomly sampled states $\rho_B$, Hamiltonian $H_B$ for two-level and three-level QBs. Here, the Hamiltonian $H_B$ is normalized. (c)-(f) For the cases of the no, local and global population inversion the incoherent ergotropy as a function of the diagonal entropy and participation ratio for randomly sampled states $\rho_B$ and Hamiltonian $H_B$. The symbols $\mathrm{\@slowromancap i@}$, $\mathrm{\@slowromancap ii@}_{1}$, $\mathrm{\@slowromancap ii@}_{2}$, $\mathrm{\@slowromancap ii@}_{3}$, $\mathrm{\@slowromancap ii@}_{4}$ and $\mathrm{\@slowromancap iii@}$ denote the same configurations as in Fig. \ref{['fig1']}(c). The orange dots in (e) and (f) correspond to the critical value of diagonal entropy, $\log 2$. We focus on Hamiltonians with equally spaced and normalized energy levels. Panels (a) and (b) contain $10^{5}$ randomly sampled states, while panels (c)-(f) contain $10^{7}$ states.
• Figure 3: The dependence of both (a),(b) the locked energy $E(\tilde{\rho}_B)$ and (c),(d) the charging efficiency $\mathcal{R}$ on purity $\mathcal{P}$ for two-level and three-level QBs. In (d), the symbols $\mathrm{\@slowromancap ii@}_{1}$, $\mathrm{\@slowromancap ii@}_{2}$, $\mathrm{\@slowromancap ii@}_{3}$, $\mathrm{\@slowromancap ii@}_{4}$ and $\mathrm{\@slowromancap iii@}$ denote the same configurations as in Fig. \ref{['fig1']}(c). Here, the Hamiltonian $H_B$ is normalized. Each panel contains $10^5$ randomly sampled states.
• Figure 4: Time evolution of (a),(b) ergotropy, coherent ergotropy, incoherent ergotropy, (c),(d) energy level populations, (e),(f) coherence, diagonal entropy, participation ratio, (g),(h) locked energy, purity and charging efficiency. (a), (c), (e) and (g) correspond to the case of JC QBs, and (b), (d), (f) and (h) correspond to the case of TC QBs containing two atoms ($N_B=2$), which is analogous to the three-level QBs. The symbols $\mathrm{\@slowromancap i@}$, $\mathrm{\@slowromancap ii@_2}$, $\mathrm{\@slowromancap ii@}_3$, $\mathrm{\@slowromancap iii@}$ denote the same setting as in Fig. \ref{['fig1']}(c). Other parameters are set to $\omega=1$, $g=0.1$ and $N_c=4$.
• Figure S1: Distribution of the diagonal entropy $\mathcal{S}_{\text{diag}}$ for the two random sampling method for $d=3$. The red histogram originates from HSRS, while the blue one is generated by FERS algorithm with $N/N_{HS}=3/2$. Each histogram contains $10^7$ points.