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arXiv:2606.10054

Scaling law of asymptotic freedom in collective charging of quantum batteries

Gentaro Watanabe, Chunlin Chen, B. Prasanna Venkatesh

quant-phcond-mat.stat-mechSubmitted Jun 8, 2026• Announced Jun 10, 2026• 0 citations
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Abstract

We establish a universal scaling law for collective charging of quantum batteries, independent of microscopic details. We prove that the ergotropy-to-energy ratio approaches unity at least as fast as $\sim N^{-1}$ with the number of batteries $N$, implying generic asymptotic freedom. We further show how the universal $1/N$ scaling can be overcome: when the battery state becomes asymptotically pure, the convergence can be substantially faster, including $\sim N^{-b}$ with $b>1$ and even exponential scaling in $N^2$. Rigorous finite-$N$ upper and lower bounds on the ergotropy-to-energy ratio are further derived, providing nonasymptotic guarantees for the universal $1/N$ scaling.

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9+1 pages, 6+1 figures

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Quantum batteries, microscopic devices designed to address energy demands in quantum technologies, promise high power during charging and discharging processes. However, their performance depends crucially on reliability, quantified by the inverse of noise-to-signal ratios (NSRs), i.e., normalized fluctuations of work and power. We establish fundamental limits to this reliability: both work and power NSRs are universally bounded from below by a function of charging speed, imposing a reliability limit inherent to any quantum battery. More strikingly, we find that a quantum mechanical uncertainty relation forbids the simultaneous suppression of work and power fluctuations, revealing a fundamental trade-off that also limits the reliability of quantum batteries. We analyze the trade-off and limits, as well as their scaling behavior, across parallel, collective, and hybrid charging schemes for many-body quantum batteries, finding that increasing power by exploiting stronger entanglement comes at the cost of diminished reliability of power. Similar trends are also observed in the charging of quantum batteries utilizing Ising-like interactions. These suggest that achieving both high power and reliability requires neither parallel nor collective charging, but a hybrid charging scheme with an intermediate range of interactions. Therefore, our analysis shapes the practical and efficient design of reliable and high-performance quantum batteries.

PRX Quantum•Jul 23, 2026•1 citation
We establish a universal scaling law for collective charging of quantum batteries, independent of microscopic details. We prove that the ergotropy-to-energy ratio approaches unity at least as fast as ∼N−1\sim N^{-1}∼N−1 with the number of batteries NNN, implying generic asymptotic freedom. We further show how the universal 1/N1/N1/N scaling can be overcome: when the battery state becomes asymptotically pure, the convergence can be substantially faster, including ∼N−b\sim N^{-b}∼N−b with b>1b>1b>1 and even exponential scaling in N2N^2N2. Rigorous finite-NNN upper and lower bounds on the ergotropy-to-energy ratio are further derived, providing nonasymptotic guarantees for the universal 1/N1/N1/N scaling.
Abstract Quantum batteries (QBs)—small-scale energy storage devices based on quantum systems—offer the potential for enhanced charging performance through quantum effects such as coherence and collectivity. In this work, we study the collective charging of QBs consisting of N qubits, coupled to a driven qubit charger in a star configuration, with controlled pure dephasing acting on the charger. We investigate how an ‘asymptotic freedom’-like behavior—in which all the energy deposited into the battery can be extracted as work, resulting in the ergotropy-to-energy ratio approaching unity—can emerge in the steady state of the battery. We show that the ergotropy-to-energy ratio increases with the number of qubits and approaches unity asymptotically as 1 − O ( 1 / N ) . In the large- N limit, the emergence of approximate ground-state degeneracy of the collective battery system leads to this asymptotic freedom behavior, despite the battery state remaining mixed. We also discuss the scaling behavior of the charging time of the battery with N .
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.
We derive a quantum speed limit for ergotropy charging in the NNN-qubit Dicke battery. With the qubits initially in their ground state and the charger in a coherent state of mean photon number nˉ\bar{n}nˉ, the charging time to a target fraction ε\varepsilonε of the maximum extractable work obeys, in the classical-field limit, ΓNτ∗(ε)≥arcsin⁡εΓ_N τ^{*}(\varepsilon) \geq \arcsin\sqrt{\varepsilon}ΓN​τ∗(ε)≥arcsinε​, where ΓN=2λnˉ/NΓ_N = 2λ\sqrt{\bar{n}/N}ΓN​=2λnˉ/N​ is a single rate combining the light-matter coupling, the charger amplitude and the system size. The bound is geometric: although the ergotropy is defined by an optimisation over unitaries, it reduces to a fixed function of the angle swept by the collective spin on the Bloch sphere, and the drive bounds the rate at which that angle can grow, so the charging time is at least the required arc divided by the maximum angular velocity. The reduction rests on an exact statement about work extraction: only the coherent component of the drive delivers extractable work, while its vacuum fluctuations deposit energy that remains locked in the passive state. This makes the short-time coefficient independent of the charger occupation, and it is what separates a bound on extractable work from a bound on stored energy, which no instantaneous-power argument can supply. At finite charger occupation the inequality is tested rather than proved: numerical integration of the full Dicke Hamiltonian, with a dynamical charger and a mixed battery state, finds the bound respected at every admitted charging time for two to five qubits, and approached most closely in the limit of small target work. The familiar square-root estimate is the chord to this arc, recovered in the small-angle limit and weaker whenever a large fraction of the maximum work is required.
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.
Quantum batteries, microscopic devices designed to address energy demands in quantum technologies, promise high power during charging and discharging processes. Yet their practical usefulness and performance depend critically on reliability, quantified by the noise-to-signal ratios (NSRs), i.e., normalized fluctuations of work and power, where reliability decreases inversely with increasing NSR. We establish fundamental limits to this reliability: both work and power NSRs are universally bounded from below by a function of charging speed, imposing a reliability limit inherent to any quantum battery. More strikingly, we find that a quantum mechanical uncertainty relation forbids the simultaneous suppression of work and power fluctuations, revealing a fundamental trade-off that also limits the reliability of quantum batteries. We analyze the trade-off and limits, as well as their scaling behavior, across parallel (local), collective {(fully non-local)}, and hybrid (semi-local) charging schemes for many-body quantum batteries, finding that increasing power by exploiting stronger entanglement comes at the cost of diminished reliability of power. Similar trends are also observed in the charging of quantum batteries utilizing transverse Ising-like interactions. These suggest that achieving both high power and reliability require neither parallel nor collective charging, but a hybrid charging scheme with an intermediate range of interactions. Therefore, our analysis shapes the practical and efficient design of reliable and high-performance quantum batteries.
Quantum batteries, microscopic devices designed to address energy demands in quantum technologies, promise high power during charging and discharging processes. However, their performance depends crucially on reliability, quantified by the inverse of noise-to-signal ratios (NSRs), i.e., normalized fluctuations of work and power. We establish fundamental limits to this reliability: both work and power NSRs are universally bounded from below by a function of charging speed, imposing a reliability limit inherent to any quantum battery. More strikingly, we find that a quantum mechanical uncertainty relation forbids the simultaneous suppression of work and power fluctuations, revealing a fundamental trade-off that also limits the reliability of quantum batteries. We analyze the trade-off and limits, as well as their scaling behavior, across parallel, collective, and hybrid charging schemes for many-body quantum batteries, finding that increasing power by exploiting stronger entanglement comes at the cost of diminished reliability of power. Similar trends are also observed in the charging of quantum batteries utilizing Ising-like interactions. These suggest that achieving both high power and reliability requires neither parallel nor collective charging, but a hybrid charging scheme with an intermediate range of interactions. Therefore, our analysis shapes the practical and efficient design of reliable and high-performance quantum batteries.