Charge-Preserving Operations in Quantum Batteries

TL;DR

This work introduces isoergotropic states and ergotropy-preserving operations as a framework to redistribute a quantum battery’s stored work without altering total ergotropy. It provides concrete realizations for both discrete two-level systems and continuous-variable Gaussian states, showing how internal charge can be reallocated between incoherent/coherent or displacement/squeezing components using beam-splitter-type interactions with an auxiliary system. The authors derive CPTP maps and selective measurement schemes (POVMs) that implement isoergotropic transformations, and extend the analysis to multi-cell batteries, offering strategies to mitigate charge loss in open quantum batteries. They further demonstrate how this formalism informs charging protocols and robustness against dissipation, including a CV analogue of the Mpemba effect, highlighting practical pathways for experimental realization. Overall, the paper lays groundwork for a resource-theoretic view of ergotropy and suggests concrete tools for dynamically and autonomously reshaping internal charge distributions in quantum batteries.

Abstract

Ergotropy provides a fundamental measure of the extractable work from a quantum system and, consequently, of the maximal useful energy, or charge, stored within it. Understanding how this quantity can be manipulated and transformed efficiently is crucial for advancing quantum energy management technologies. Here, we introduce and formalize the concepts of isoergotropic states and ergotropy-preserving operations, which reorganize the internal structure of ergotropy while keeping its total value unchanged. These ideas are illustrated for both discrete (two-level systems) and continuous-variable systems (single-mode Gaussian states). In each case, we show how ergotropy-preserving operations redistribute the respective coherent-incoherent and displacement-squeezing components. We further examine the thermodynamic exchanges accompanying ergotropy-preserving operations, including variations in energy and entropy, and demonstrate that these transformations can be dynamically implemented through standard beam-splitter-type interactions with an auxiliary system. Finally, we discuss the practical implications of isoergotropic states and operations in optimizing charging protocols and mitigating charge loss in open quantum batteries.

Figures (15)

• Figure 1: Isoergotropic states for (a) a single two-level quantum battery and (b) a single-mode Gaussian oscillator. Each colored surface at the (a) Bloch sphere (b) parameter space, corresponds to states with fixed charge but distinct internal configurations of ergotropy $\mathcal{R}$, with red and green being respectively low and high charge. The Wigner functions $W(\alpha)$ represent Gaussian states with the same charge and thermal occupation $N$ but different displacement $\mu$ and squeezing $\xi$ components of $\mathcal{R}$Serafini.
• Figure 2: (a) Isoergotropic surfaces at the Bloch sphere for distinct values of $\bar{p}$. (b) Ergotropy profile in terms of $p$ and $\mathcal{C}^{2}\in[0,4p(1-p)]$. The isoergotropic states within $\mathcal{L}_{\bar{p}}$ are represented by the colored (a) surfaces within the Bloch sphere (b) lines $\mathcal{C}_{\bar{p}}^{2}(p)=8(\bar{p}-p)P_{\bar{p}}$. The black curve in (b) characterize the isoergotropic pure states $\hat{\rho}_{\bar{p}}(P_{\bar{p}},\theta)=|\Psi_{\bar{p},\theta}\rangle\langle\Psi_{\bar{p},\theta}|$ (surface of the Bloch sphere) relative to the incoherent states $\hat{\varrho}_{\bar{p}}=\left(1-p\right)|g\rangle\langle g|+p|e\rangle\langle e|$ (at $s_z$ axis). (c) Top: Ergotropy distribution between the internal (in)coherent components along the path described by $\mathcal{C}_{0.8}^{2}(p)$. Bottom: Ergotropy distribution along the pure states $\hat{\rho}_{\bar{p}}(P_{\bar{p}},\theta)$ represented by the black curve.
• Figure 3: (a) Ratio $\mathcal{R}/\langle H \rangle = (2\bar{p} - 1)/p$ on the $(p, \bar{p})$-plane. For each $\bar{p}$, the ratio decreases monotonically with $p$ from $1$ at the pure point $p = P_{\bar{p}}$ to $(2\bar{p} - 1)/\bar{p}$ at the incoherent point $p = \bar{p}$. (b) Energy-entropy diagram on $\mathcal{L}_{\bar{p}}$: $\langle \hat{H} \rangle = \omega p$ and $S_{vN}(p) = \mathcal{H}_2(2\bar{p} - p)$ increase monotonically with $p$. Arrows indicate heat $Q = \omega(p' - p)$ absorbed (right, red) or released (left, blue) when moving along $\mathcal{L}_{\bar{p}}$. Black dashed: $p = P_{\bar{p}}$ (pure, $S_{vN} = 0$); blue solid: $p = \bar{p}$ (incoherent, $S_{vN} = \mathcal{H}_2(\bar{p})$).
• Figure 4: Isoergotropic operations. The continuous and dashed lines indicate two distinct ergotropy-preserving trajectories along $\mathcal{L}_{0.7}$ from the black to the red points.
• Figure 5: Top: Bloch sphere representation of the (a) battery and (b) auxiliary TLS dynamics induced by $\hat{\mathcal{U}}(t)$, assuming $\hat{\rho}_{B}(0)=\hat{\rho}_{\bar{p}}(P_{\bar{p}},\theta)$ and $\hat{\rho}_A(0)=\hat{\rho}_{\bar{p}}(\bar{p},\phi)$ (colored dots) as initial states, and $\omega=\eta=1$. The isoergotropic curves $\mathcal{L}_{\bar{p}}$ for different values of $\bar{p}$ are represented. Bottom: Ergotropy and internal components dynamics for the (c) battery and (d) auxiliary system for $\bar{p}=0.8$.

Theorems & Definitions (2)

• Definition 1
• Definition 2