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Enhancing the Charging Power of Quantum Batteries
Open accessAbstract
Can collective quantum effects make a difference in a meaningful thermodynamic operation? Focusing on energy storage and batteries, we demonstrate that quantum mechanics can lead to an enhancement in the amount of work deposited per unit time, i.e., the charging power, when N batteries are charged collectively. We first derive analytic upper bounds for the collective quantum advantage in charging power for two choices of constraints on the charging Hamiltonian. We then demonstrate that even in the absence of quantum entanglement this advantage can be extensive. For our main result, we provide an upper bound to the achievable quantum advantage when the interaction order is restricted; i.e., at most k batteries are interacting. This constitutes a fundamental limit on the advantage offered by quantum technologies over their classical counterparts.
Overview
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TL;DR
The paper defines a collective quantum advantage for charging power when N quantum batteries are charged jointly rather than in parallel, and derives upper bounds on this advantage under energetic constraints. It shows that an extensive advantage is possible even without entanglement, but for physically realisable at-most-k-body interactions the advantage cannot scale with N and is bounded by a function quadratic in k.
Key findings
- For two constraints on the collective charging Hamiltonian, the quantum advantage is upper bounded as Γ_C1 ≤ √N and Γ_C2 ≤ N, with β = T‖/T_QSL^(1) quantifying inability to saturate the quantum speed limit in parallel charging.
- An extensive advantage can occur even for highly mixed states inside the separable ball: for N two-level thermal batteries, Γ_C1 = √N, Γ_C2 = N, and Γ_C0 = N, while the joint state need not be entangled during charging.
- For circuit-based charging with interaction order at most k, Theorem 2 gives Γ_C0 < γk, where γ does not scale with N; for ground-to-maximally-excited states γ = π/2.
- For generic time-dependent charging with interaction order k and maximum participation number m, Theorem 3 gives Γ_C0 < γ(k²(m−1)+k). Achievable comparisons are Γ_C1 = √k and Γ_C2 = k when N/k is an integer, and Γ_C0 = k in that same situation.
Methods
- Theoretical study of unitary charging of N quantum batteries, defining the quantum advantage as Γ = P♯/P‖ = T‖/T♯, where P♯ is collective charging power and P‖ is parallel charging power.
- Analytic derivation using the quantum speed limit, Bures angle L_N, Uhlmann fidelity, and three constraints C0, C1, and C2 on the collective Hamiltonian’s energy, standard deviation, or operator norm.
- Supplementary proofs use a piecewise unitary circuit model, Trotter-Suzuki decomposition, and a numerical search over Haar-random charging Hamiltonians with sample size ν = 10⁵ for (N,k) = (3,2), (4,2), (4,3), and (6,2).
Limitations
- The paper states that its bounds are not tight; Theorem 2’s strict inequality is only different by a constant factor from an achievable bound in a special case.
- Conjecture 4, extending Theorem 2 to any time-dependent k-body interaction Hamiltonian, is not proven; the numerical search found no counterexamples but “does not represent a proof,” and measure-zero counterexamples cannot be excluded.
- The paper states that k-body interactions are hard to engineer and that higher interaction orders are generally difficult; the exact radius of the separable ball is not known, though it has been bounded.
- Evident from the text: the work is theoretical and numerical, with no experimental demonstration of the predicted charging advantage.
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25 Own interpretation · 18 No basis given · 10 Not checked yet · 7 Own result
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“In contrast, the joint charging of N batteries is achieved in T ♯ = T ‖ /α ♯ using the global Hamiltonian H ♯ = α ♯ H ⊗ N , where the positive constant α ♯ is introduced to satisfy the chosen constraint.”
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“Proposition 1 An extensive quantum advantage can be attained even for highly mixed states, including those confined to the separable ball throughout the charging procedure.”
- Also stated in: § Enhancing the charging power of quantum batteries · p. 3 · plain
“no matter how mixed the battery is a quantum advantage that scales with the number of batteries involved is always achievable.”
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“We then go on to analytically prove that, for charging fields with finite interaction order, i.e. , involving at most k -body interaction terms, the quantum advantage is upper bounded by a quadratic function of k and cannot scale with the total number of batteries.”
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“The Hamiltonian used in the example above, and in Ref. [15], to saturate the bound for quantum advantage involves N -body interactions.”
What the sources say
- Not checked yetcites [15] F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New J. Phys. 17 , 075015 (2015). source
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“Theorem 2 For a circuit based charging procedure with interaction order of at most k , the achievable quantum advantage is upper bounded as Γ C0 < γk , where γ is a constant that does not scale with the number N of batteries.”
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“The optimal local charging scheme is achieved in time T ‖ = π/ 2 by applying Hamiltonian H = | E 0 〉〈 E 1 | + | E 1 〉〈 E 0 | to each battery.”
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“We then demonstrate that even in the absence of quantum entanglement this advantage can be extensive.”
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“In the important case where ρ and σ are the ground and maximally excited states respectively, γ = π/ 2 .”
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“In this sense, it is a stricter constraint than C1 or C2.”
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“We first derive analytic upper bounds for the collective quantum advantage in charging power for two choices of constraints on the charging Hamiltonian.”
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“Firstly, for orthogonal pure initial and final states, the QSL can be saturated and β = 1 .”
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“In both cases (local and global) the deposited work is identical; thus, the quantum advantage is simply the ratio of T ‖ to T ♯ : Γ = α ♯ , which can be evaluated for the choice of constraint.”
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“Constraint C0 guarantees that both the time-averaged standard deviation and the time-averaged energy are bounded from above, as shown in Sec. A of SM.”
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“The two bounds in Eq. (4) are independent from each other, and constraint C1 is stronger than C2, as it leads to a stricter bound on the quantum advantage.”
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“The two bounds in Eq. (4) are independent from each other, and constraint C1 is stronger than C2, as it leads to a stricter bound on the quantum advantage.”
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“There, we show that E ♯ upper bounds both E ♯ / 2 and ∆ E ♯ .”
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“We find Γ C1 = √ N and Γ C2 = N (also Γ C0 = N for C0 given in Eq. (5)).”
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“Remarkably, neither T ‖ nor T ♯ depend on ϵ , while the total work done does.”
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“Here we show that ∆ E ♯ ≤ E ♯ and E ♯ ≤ 2 E ♯ by direct computation.”
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“We have proven two fundamental upper bounds for the quantum advantage, each corresponding to a different constraint on the charging Hamiltonian.”
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“Next, we show that attaining a quantum advantage requires entangling operations, but not entanglement itself.”
- Also stated in: § Enhancing the charging power of quantum batteries · p. 3 · plain
“Proposition 1 implies that, while a quantum advantage requires entangling operations, the joint state of N -batteries does not have to be entangled during the charging process.”
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“Since the distance from the maximally mixed state cannot change under unitary evolution, for a small enough choice of ϵ , the joint state of N batteries will lie within this ball throughout the evolution; yet, the quantum advantage remains extensive.”
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“Though the quantum advantage for power could be larger in other cases, including where the battery states are mixed [26], the improvement cannot grow with the number of batteries; i.e. , β is a constant function of N .”
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“For comparison, it has been shown elsewhere that Γ C1 = √ k and Γ C2 = k are achievable if the total number of batteries N can be divided by k , i.e. , if N/k = s ∈ N [15, 36].”
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“This numerical evidence does not represent a proof of our conjecture since there could be a measure zero set of Hamiltonians for which P > 1 .”
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In the paper
“The quantum advantage has been interpreted as the result of rapid evolution through the space of high-dimensional quantum states, typically obtained by means of global operations [15].”
What the sources say
- Not checked yetcites [15] F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New J. Phys. 17 , 075015 (2015). source
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In the paper
“Note that these bounds are not tight”
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In the paper
“There, an enhancement in sensing is still available for highly mixed states lying inside the separable ball [43].”
What the sources say
- Not checked yetcites [43] K. Modi, H. Cable, M. Williamson, and V. Vedral, Phys. Rev. X 1 , 021022 (2011). source
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In the paper
“While, in the case of pure states, entanglement is a necessary consequence of these global operations, a fully separable evolution is still accessible for those states that live in the separable ball.”
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In the paper
“Since each of the M terms at each time step must be implemented sequentially, U Trot takes M times longer to run than U , with a corresponding drop in power P ♯ = MP Trot .”
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“This Letter demonstrates that thermodynamic processes can indeed benefit from collective effects when time enters the picture, though physical limitations on the interaction order prevent us from utilising them.”
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In the paper
“For many physical systems, both k and m are limited: 2 or 3-body interactions are the norm for fundamental processes, and higher interaction orders are generally hard to engineer here [37-39].”
What the sources say
- Not checked yetcites [37] H. P. B¨ uchler, A. Micheli, and P. Zoller, Nat. Phys. 3 , 726 (2007). source
- Not checked yetcites [38] S. Boixo, S. T. Flammia, C. M. Caves, and J. Geremia, Phys. Rev. Lett. 98 , 090401 (2007). source
- Not checked yetcites [39] M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. Sewell, and M. W. Mitchell, Nature 471 , 486 (2011). source
plain- Asserted: passes — The authors put it forward as true.
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The authors state this as an assumption.
Each hµ acts on a completely different set of k batteries, i.e., [hµ, hµ'] = 0 for all µ, µ'.
“assuming that each h µ acts on a completely different set of k batteries, i.e. , [ h µ , h µ ′ ] = 0 ∀ µ, µ ′ .”
The authors state this as an assumption.
The reach of the interaction should be limited.
“This restriction is motivated by the idea that the reach of the interaction should be limited.”
The paper relies on this without support.
The effective participation number m tends to be constrained by the spatial arrangement of systems and the fact that interaction strength often drops off with distance.
“The effective participation number, or reach, m tends to be constrained by the spatial arrangement of systems and the fact that interaction strength often drops off with distance.”
The paper relies on this without support.
Collective behaviour has been demonstrated to provide an advantage in performing many information theoretic tasks, but such distinctions from classical behaviour are few and far between in thermodynamics.
“While collective behaviour has been demonstrated to provide an advantage in performing many information theoretic tasks, such distinctions from classical behaviour are few and far between in thermodynamics.”
Claims other papers make about this one
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In the citing paper
“In addition, similar techniques have been employed to optimally charge a quantum battery [110,111].”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [111]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Appendix C: Proof of Theorem 3 · p. 7
“( k = 3 , m = 2) Now the first 3-partition (1 , 2 , 3) is followed by (1 , 4 , 5) , (2 , 4 , 6) and (3 , 5 , 6) .”
§ Enhancing the charging power of quantum batteries · p. 4
“each battery is simultaneously interacting with at most m others.”
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Superabsorption in an organic microcavity: Toward a quantum battery
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Bounds on the capacity and power of quantum batteries
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Quantum Coherence and Ergotropy
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Self-citationQuantum Charging Advantage Cannot Be Extensive without Global Operations
Gyhm, J.-Y., Šafránek, D., & Rosa, D. (2022) · Physical Review Letters · doi:10.1103/physrevlett.128.140501 · Cited by 220
Quantum Technologies Need a Quantum Energy Initiative
Auffèves, A. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.020101 · Cited by 211
Using Dark States to Charge and Stabilize Open Quantum Batteries
Quach, J. Q., & Munro, W. J. (2020) · Physical Review Applied · doi:10.1103/physrevapplied.14.024092 · Cited by 202
Quantum versus classical many-body batteries
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Stable adiabatic quantum batteries
Santos, A. C., Çakmak, B., Campbell, S., & Zinner, N. T. (2019) · Physical review. E · doi:10.1103/physreve.100.032107 · Cited by 206
Molecular Polaritonics: Chemical Dynamics Under Strong Light–Matter Coupling
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Quantum thermal machines and batteries
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Ultrafast charging in a two-photon Dicke quantum battery
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Powerful harmonic charging in a quantum battery
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Many-body localized quantum batteries
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Entanglement, Coherence, and Extractable Work in Quantum Batteries
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Quantum Speed Limits across the Quantum-to-Classical Transition
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Experimental investigation of a quantum battery using star-topology NMR spin systems
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Entanglement, coherence, and charging process of quantum batteries
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Tightening Quantum Speed Limits for Almost All States
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Self-citationThe Energetic Cost of Work Extraction
Monsel, J., Fellous-Asiani, M., Huard, B., & Auffèves, A. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.130601 · Cited by 149
Role of coherence in the nonequilibrium thermodynamics of quantum systems
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Self-citationEnhancement in the performance of a quantum battery by ordered and disordered interactions
Ghosh, S., Chanda, T., & Sen, A. (2020) · Physical Review A · doi:10.1103/physreva.101.032115 · Cited by 154
Extended Dicke quantum battery with interatomic interactions and driving field
Dou, F.-Q., Lu, Y.-Q., Wang, Y., & Sun, J. (2022) · Physical review. B./Physical review. B · doi:10.1103/physrevb.105.115405 · Cited by 149
Precision and Work Fluctuations in Gaussian Battery Charging
Friis, N., & Huber, M. (2018) · Quantum · doi:10.22331/q-2018-04-23-61 · Cited by 143
Stabilizing open quantum batteries by sequential measurements
Gherardini, S., Campaioli, F., Caruso, F., & Binder, F. C. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.013095 · Cited by 135
Self-citationFluctuations in Extractable Work Bound the Charging Power of Quantum Batteries
García-Pintos, L. P., Hamma, A., & del Campo, A. (2020) · Physical Review Letters · doi:10.1103/physrevlett.125.040601 · Cited by 146
Non-Markovian effects on charging and self-discharging process of quantum batteries
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Nonreciprocal Quantum Batteries
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Quantum Speed-Up in Collisional Battery Charging
Seah, S., Perarnau-Llobet, M., Haack, G., Brunner, N., & Nimmrichter, S. (2021) · Physical Review Letters · doi:10.1103/physrevlett.127.100601 · Cited by 131
Pirmoradian, F., & Mølmer, K. (2019) · Physical Review A · doi:10.1103/physreva.100.043833 · Cited by 123
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Coherent and dissipative dynamics at quantum phase transitions
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Fast charging of a quantum battery assisted by noise
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Ultra-stable charging of fast-scrambling SYK quantum batteries
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Battery Capacity of Energy-Storing Quantum Systems
Yang, X., Yang, Y., Alimuddin, M., Salvia, R., Fei, S.-M., Zhao, L.-M., Nimmrichter, S., & Luo, M. (2023) · Physical Review Letters · doi:10.1103/physrevlett.131.030402 · Cited by 126
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Tabesh, F. T., Kamin, F. H., & Salimi, S. (2020) · Physical Review A · doi:10.1103/physreva.102.052223 · Cited by 115
Charging and energy fluctuations of a driven quantum battery
Crescente, A., Carrega, M., Sassetti, M., & Ferraro, D. (2020) · New Journal of Physics · doi:10.1088/1367-2630/ab91fc · Cited by 109
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Cavity Heisenberg-spin-chain quantum battery
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Loss-Free Excitonic Quantum Battery
Liu, J., Segal, D., & Hanna, G. (2019) · The Journal of Physical Chemistry C · doi:10.1021/acs.jpcc.9b06373 · Cited by 100
Charging a quantum battery with linear feedback control
Mitchison, M. T., Goold, J., & Prior, J. (2021) · Quantum · doi:10.22331/q-2021-07-13-500 · Cited by 101
Self-citationFloquet engineering to reactivate a dissipative quantum battery
Bai, S., & An, J. (2020) · Physical Review A · doi:10.1103/physreva.102.060201 · Cited by 99
Entanglement and work extraction in the central-spin quantum battery
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Remote Charging and Degradation Suppression for the Quantum Battery
Song, W.-L., Liu, H., Zhou, B., Yang, W., & An, J. (2024) · Physical Review Letters · doi:10.1103/physrevlett.132.090401 · Cited by 107
Beneficial and detrimental entanglement for quantum battery charging
Gyhm, J.-Y., & Fischer, U. R. (2024) · AVS Quantum Science · doi:10.1116/5.0184903 · Cited by 97
Geometric Speed Limit of Accessible Many-Body State Preparation
Bukov, M., Sels, D., & Polkovnikov, A. (2019) · Physical Review X · doi:10.1103/physrevx.9.011034 · Cited by 97
Cooperative many-body enhancement of quantum thermal machine power
Niedenzu, W., & Kurizki, G. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aaed55 · Cited by 92
Ergotropy from coherences in an open quantum system
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Quantum engines and refrigerators
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Tight, robust, and feasible quantum speed limits for open dynamics
Campaioli, F., Pollock, F. A., & Modi, K. (2019) · Quantum · doi:10.22331/q-2019-08-05-168 · Cited by 92
“Such bounds are typically applied to estimate the speed of computational gates [5, 6], the precision in quantum metrology [7–11], the performance in quantum optimal control [4, 12–19], and the charging power in quantum thermodynamics [20, 21].”
BackgroundSelf-citationMicromasers as quantum batteries
Shaghaghi, V., Singh, V., Benenti, G., & Rosa, D. (2022) · Quantum Science and Technology · doi:10.1088/2058-9565/ac8829 · Cited by 87
Lu, Z.-G., Tian, G., Lü, X., & Shang, C. (2025) · Physical Review Letters · doi:10.1103/physrevlett.134.180401 · Cited by 107
Single-atom energy-conversion device with a quantum load
Horne, N. V., Yum, D., Dutta, T., Peter Hänggi, A., Gong, J., Poletti, D., & Mukherjee, M. (2020) · npj Quantum Information · doi:10.1038/s41534-020-0264-6 · Cited by 88
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Strong long-range spin-spin coupling via a Kerr magnon interface
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Optimal control methods for quantum batteries
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Collective enhancement in dissipative quantum batteries
Carrasco, J., Maze, J. R., Hermann-Avigliano, C., & Barra, F. (2022) · Physical review. E · doi:10.1103/physreve.105.064119 · Cited by 85
Unifying Quantum and Classical Speed Limits on Observables
García-Pintos, L. P., Nicholson, S. B., Green, J. R., del Campo, A., & Gorshkov, A. V. (2022) · Physical Review X · doi:10.1103/physrevx.12.011038 · Cited by 96
IBM Quantum Platforms: A Quantum Battery Perspective
Gemme, G., Grossi, M., Ferraro, D., Vallecorsa, S., & Sassetti, M. (2022) · Batteries · doi:10.3390/batteries8050043 · Cited by 93
Charging Quantum Batteries via Indefinite Causal Order: Theory and Experiment
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Superconducting transmon qubit-resonator quantum battery
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Operator delocalization in quantum networks
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Bounds on charging power of open quantum batteries
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Quantum advantage in charging cavity and spin batteries by repeated interactions
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Quantum batteries: The future of energy storage?
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Collective effects and quantum coherence in dissipative charging of quantum batteries
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Campaioli, F., Pollock, F. A., & Vinjanampathy, S. (2018) · Fundamental Theories of Physics · doi:10.1007/978-3-319-99046-0_8 · Cited by 71
Self-citationQuantum Statistical Enhancement of the Collective Performance of Multiple Bosonic Engines
Gentaro Watanabe, A., Venkatesh, B. P., Peter Talkner, A., Hwang, M.-J., & del Campo, A. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.210603 · Cited by 68
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Dou, F.-Q., Wang, Y., & Sun, J. (2021) · Frontiers of Physics · doi:10.1007/s11467-021-1130-5 · Cited by 66
Characterization of a Two-Photon Quantum Battery: Initial Conditions, Stability and Work Extraction
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Dimensional enhancements in a quantum battery with imperfections
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Three-level Dicke quantum battery
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Universal Bound on Energy Cost of Bit Reset in Finite Time
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Self-citationControlling Energy Storage Crossing Quantum Phase Transitions in an Integrable Spin Quantum Battery
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Lower and upper bounds of quantum battery power in multiple central spin systems
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Quantum battery based on quantum discord at room temperature
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Charging a quantum battery in a non-Markovian environment: a collisional model approach
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Environment-mediated entropic uncertainty in charging quantum batteries
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Enhancing the performance of an open quantum battery via environment engineering
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Closed-loop three-level charged quantum battery
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Arrachea, L. (2023) · Reports on Progress in Physics · doi:10.1088/1361-6633/acb06b · Cited by 63
Catalysis in charging quantum batteries
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Optimal quantum control of charging quantum batteries
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Enhancing self-discharging process with disordered quantum batteries
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Collective performance of a finite-time quantum Otto cycle
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Refrigeration beyond weak internal coupling
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Energy storage and coherence in closed and open quantum batteries
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Metadata
- Authors
- Campaioli, F., Pollock, F. A., Binder, F. C., Céleri, L., Goold, J., Vinjanampathy, S., & Modi, K.
- Affiliations
- Monash University (AU)
- Nanyang Technological University (SG)
- Universidade Federal de Goiás (BR)
- The Abdus Salam International Centre for Theoretical Physics (ICTP) (IT)
- Centre for Quantum Technologies (SG)
- ORCID
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2017-04-12
- Volume
- 118
- Issue
- 15
- Pages
- 150601-150601
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2561264561
- Cited by
- 531
- References
- 50
- Citations by year
- 2026: 108 · 2025: 107 · 2024: 77 · 2023: 51 · 2022: 51 · 2021: 43 · 2020: 39 · 2019: 30 · 2018: 23 · 2017: 2
- FWCI
- 13.30
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- PubMed (publishedVersion)
- RMIT Research Repository (RMIT University Library) (submittedVersion)
- DR-NTU (Nanyang Technological University) (submittedVersion)
- DSpace (IIT Bombay) (submittedVersion)
- Figshare (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Research Foundation Singapore (NRF-CRP14-2014-02, NRF-NRFF2016-02)
- European Cooperation in Science and Technology (MP1209)
- Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq: 401230/2014-7, CNPq: 305086/2013-8, CNPq: 445516/2014-3)
- IITB-IRCC (16IRCCSG019)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Engineering › Mechanical Engineering › Advanced Thermodynamic Systems and Engines
- Keywords
- quantum batteries, charging power, quantum advantage, energy storage, work rate, upper bounds, quantum entanglement, interaction order, quantum thermodynamics
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