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arXiv:2411.01175 · Physical review. B./Physical review. B

Optimal energy storage and collective charging speedup in the central-spin quantum battery

Hui-Yu Yang, Kun Zhang, Xiao-Hui Wang, Hailong Shi

Published Feb 10, 2025• 20 citations
arXivHTMLPDF

Abstract

Quantum batteries (QBs) exploit principles of quantum mechanics to accelerate the charging process and aim to achieve optimal energy storage. However, analytical results for investigating these problems remain lacking due to the challenges associated with nonequilibrium dynamics. In this work we analytically investigate a central-spin QB model in which ${N}_{b}$ spin-$\frac{1}{2}$ battery cells interact with ${N}_{c}$ spin-$\frac{1}{2}$ charger units, using $m$ initially excited charger units as a resource. By employing the invariant subspace method and the shifted Holstein-Primakoff transformation, we identify four scenarios in which optimal energy storage can be achieved: (i) ${N}_{b}\ensuremath{\ll}m\ensuremath{\ll}{N}_{c}$, (ii) $m\ensuremath{\ll}{N}_{b}\ensuremath{\ll}{N}_{c}$, (iii) $m\ensuremath{\ll}{N}_{c}\ensuremath{\ll}{N}_{b}$, and (iv) ${N}_{b}\ensuremath{\ll}m=k{N}_{c} [k\ensuremath{\in}(0,1)]$. In these cases, optimal storage is ensured by the SU(2) symmetry emerging from the charging dynamics. The first three cases map the central-spin QB to different Tavis-Cummings (TC) QBs, while the fourth corresponds to the non-TC limit. We analytically determine the charging time and demonstrate that in the fully charging cases (i) and (iv), the collective charging exhibits an ${N}_{b}$-fold enhancement in speedup compared to the parallel charging scheme. Additionally, we numerically observe a unified charging behavior when $m={N}_{c}$, showing that asymptotically optimal energy storage is possible when ${N}_{b}=m={N}_{c}$. In this case, we find a collective charging enhancement scaling as ${N}_{b}^{0.8264}$. The origin of the collective charging advantage in central-spin quantum batteries is also analyzed through the quantum speed limit and a multipartite entanglement witness. Our results highlight the crucial role of dynamically emergent SU(2) symmetry in providing an analytical understanding of nonequilibrium charging dynamics in QBs.

DOI
10.1103/physrevb.111.085410

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Quantum batteries (QBs) exploit principles of quantum mechanics to accelerate the charging process and aim to achieve optimal energy storage. However, analytical results for investigating these problems remain lacking due to the challenges associated with nonequilibrium dynamics. In this work we analytically investigate a central-spin QB model in which Nb{N}_{b}Nb​ spin-12\frac{1}{2}21​ battery cells interact with Nc{N}_{c}Nc​ spin-12\frac{1}{2}21​ charger units, using mmm initially excited charger units as a resource. By employing the invariant subspace method and the shifted Holstein-Primakoff transformation, we identify four scenarios in which optimal energy storage can be achieved: (i) Nb\ensuremath≪m\ensuremath≪Nc{N}_{b}\ensuremath{\ll}m\ensuremath{\ll}{N}_{c}Nb​\ensuremath≪m\ensuremath≪Nc​, (ii) m\ensuremath≪Nb\ensuremath≪Ncm\ensuremath{\ll}{N}_{b}\ensuremath{\ll}{N}_{c}m\ensuremath≪Nb​\ensuremath≪Nc​, (iii) m\ensuremath≪Nc\ensuremath≪Nbm\ensuremath{\ll}{N}_{c}\ensuremath{\ll}{N}_{b}m\ensuremath≪Nc​\ensuremath≪Nb​, and (iv) Nb\ensuremath≪m=kNc[k\ensuremath∈(0,1)]{N}_{b}\ensuremath{\ll}m=k{N}_{c} [k\ensuremath{\in}(0,1)]Nb​\ensuremath≪m=kNc​[k\ensuremath∈(0,1)]. In these cases, optimal storage is ensured by the SU(2) symmetry emerging from the charging dynamics. The first three cases map the central-spin QB to different Tavis-Cummings (TC) QBs, while the fourth corresponds to the non-TC limit. We analytically determine the charging time and demonstrate that in the fully charging cases (i) and (iv), the collective charging exhibits an Nb{N}_{b}Nb​-fold enhancement in speedup compared to the parallel charging scheme. Additionally, we numerically observe a unified charging behavior when m=Ncm={N}_{c}m=Nc​, showing that asymptotically optimal energy storage is possible when Nb=m=Nc{N}_{b}=m={N}_{c}Nb​=m=Nc​. In this case, we find a collective charging enhancement scaling as Nb0.8264{N}_{b}^{0.8264}Nb0.8264​. The origin of the collective charging advantage in central-spin quantum batteries is also analyzed through the quantum speed limit and a multipartite entanglement witness. Our results highlight the crucial role of dynamically emergent SU(2) symmetry in providing an analytical understanding of nonequilibrium charging dynamics in QBs.
The Tavis-Cummings (TC) model, which serves as a natural physical realization of a quantum battery, comprises Nb{N}_{b}Nb​ atoms as battery cells that collectively interact with a shared photon field, functioning as the charger, initially containing n0{n}_{0}n0​ photons. In this paper, we introduce the invariant subspace method to effectively represent the quantum dynamics of the TC battery. Our findings indicate that in the limiting case of ${n}_{0}\ensuremath{\gg}{N}_{b}$ or ${N}_{b}\ensuremath{\gg}{n}_{0}$, a distinct SU(2) symmetry emerges in the dynamics, thereby ensuring the realization of optimal energy storage. We also establish a negative relationship between the battery-charger entanglement and the energy storage capacity. As a result, we demonstrate that asymptotically optimal energy storage can be achieved in the scenario where ${N}_{b}={n}_{0}\ensuremath{\gg}1$. Our approach not only enhances our comprehension of the algebraic structure inherent in the TC model but also contributes to the broader theoretical framework of quantum batteries. Furthermore, it provides crucial insights into the relation between energy transfer and quantum correlations.
Collective quantum batteries (QBs) demonstrate remarkable acceleration in charging dynamics compared to their individual counterparts, underscoring the pivotal contribution of quantum correlations to advanced energy storage paradigms. A fundamental challenge lies in identifying QBs that exhibit genuine quantum advantages derived from multipartite entanglement. In this Letter, based on numerical and analytical evidence, we conjecture a universal bound on the charging rate for fully charging schemes, which is determined by the maximum entanglement depth arising during the charging dynamics. Here, the charging rate quantifies the intrinsic evolution speed of the charging process, appropriately normalized against the quantum speed limit (QSL). We analytically validate this conjecture in three distinct scenarios: (i) fully charging schemes saturating the QSL, (ii) fully parallel charging schemes, and (iii) the SU(2) fully charging schemes. Moreover, we establish a novel lower bound for entanglement depth detection, facilitating numerical verification of our proposed conjecture. By defining the genuine quantum charging advantage as the ratio between entanglement-enhanced charging rates and the maximum achievable non-entangling charging rate, we demonstrate that the charging rate constitutes a robust indicator of genuine quantum advantages.
Quantum batteries provide a useful setting for exploring nonequilibrium many-body effects in energy storage. Here we investigate the optimization of a quantum battery based on the central-spin model. We identify two complementary structural indicators associated with the effective charging dynamics: one yields an upper bound on the average charging power, while the other characterizes the buildup of stored energy. We show that these two indicators are jointly optimized at a distinguished initial charger excitation number, which selects a particular Dicke sector of the model. At this common optimal point, the effective charging Hamiltonian becomes exactly mirror symmetric, suggesting mirror symmetry as a useful structural indicator for optimizing the performance of quantum batteries in terms of both charging power and energy storage. We further show that the corresponding optimal dynamics can be closely approximated by product initial states, in particular by spin coherent states whose excitation-number distribution is centered at the symmetry-selected point. Our results establish a direct connection between charging performance, optimal-state structure, and emergent symmetry in the central-spin quantum battery, and suggest symmetry as a useful organizing principle for efficient charging in interacting many-body quantum systems.
We study the energy-transfer process in quantum battery systems consisting of multiple central spins and bath spins. Here with ``quantum battery'' we refer to the central spins, whereas the bath serves as the ``charger.'' For the single-central-spin battery, we analytically derive the time evolutions of the energy transfer and the charging power with arbitrary number of bath spins. For the case of multiple central spins in the battery, we find the scaling-law relation between the maximum power Pmax{P}_{\mathrm{max}}Pmax​ and the number of central spins NB{N}_{B}NB​. It approximately satisfies a scaling law relation ${P}_{\mathrm{max}}\ensuremath{\propto}{N}_{B}^{\ensuremath{\alpha}}$, where scaling exponent $\ensuremath{\alpha}$ varies with the bath spin number NNN from the lower bound $\ensuremath{\alpha}=1/2$ to the upper bound $\ensuremath{\alpha}=3/2$. The lower and upper bounds correspond to the limits $N\ensuremath{\rightarrow}1$ and $N\ensuremath{\gg}{N}_{B}$, respectively. In thermodynamic limit, by applying the Holstein-Primakoff transformation, we rigorously prove that the upper bound is Pmax=0.72BANNB3/2{P}_{\mathrm{max}}=0.72BA\sqrt{N}{N}_{B}^{3/2}Pmax​=0.72BAN​NB3/2​, which shows the same advantage in scaling of a recent charging protocol based on the Tavis-Cummings model. Here BBB and AAA are the external magnetic field and coupling constant between the battery and the charger.
Abstract In this paper, we study the performance of a quantum battery composed of three non-interacting central qubits, which are charged through interactions with three distinct thermal baths, each bath consisting of interacting qubits. We utilize the Holstein-Primakoff transformation and thermodynamic limits to map the collective spin operators of the baths to bosonic modes. Additionally, we employ the finite trio coherent state as the initial state for the charging process to explore its impact on the overall system performance. The study focuses on evaluating the stored energy efficiency and the entropy of the charger modes to assess the overall charging performance. By examining the system parameters such as anisotropy, dimension and shift parameters, and the intensity of the trio coherent state, the optimal conditions for efficient energy storage are identified. Our results provided the role of thermodynamic processes and quantum correlations in enhancing the performance of our quantum battery model.
We consider a central-spin battery where Nb{N}_{b}Nb​ central spins serve as battery cells and Nc{N}_{c}Nc​ bath spins serve as charging units. It is shown that the energy stored in the battery that can be extractable is quantified by the ergotropy, and that battery-charger entanglement is quantified via the Von Neumann entropy. By using an exact approach to a one-cell and two-cell battery, our analytical results suggest that, during the charging process, the extractable work slowly increases before the battery-charger entanglement reaches its maximum and then it will rapidly increase when the entanglement begins to decrease. In particular, we rigorously show that there is an inverse relationship between the extractable work and the entanglement at the end of the charging process. Moreover, we investigate different approaches to realize optimal work extraction without wasted energy. Among them a central-spin battery with an unpolarized Dicke state as the charger possesses a universal charging time $\ensuremath{\propto}1/{N}_{c}$, large extractable work and Nc\sqrt{{N}_{c}}Nc​​-improvement of charging power compared with the battery in the Tavis-Cummings limit. The above-mentioned results have also been numerically verified in multicell batteries. Our results pave the way to improve extractable work storage in the central-spin battery and highlight a competitive relation between the extractable work and the battery-charger entanglement.