Precision and Work Fluctuations in Gaussian Battery Charging

Nicolai Friis1,2 and Marcus Huber1

1Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
2Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21a, 6020 Innsbruck, Austria

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


One of the most fundamental tasks in quantum thermodynamics is extracting energy from one system and subsequently storing this energy in an appropriate battery. Both of these steps, work extraction and charging, can be viewed as cyclic Hamiltonian processes acting on individual quantum systems. Interestingly, so-called passive states exist, whose energy cannot be lowered by unitary operations, but it is safe to assume that the energy of any not fully charged battery may be increased unitarily. However, unitaries raising the average energy by the same amount may differ in qualities such as their precision, fluctuations, and charging power. Moreover, some unitaries may be extremely difficult to realize in practice. It is hence of crucial importance to understand the qualities that can be expected from practically implementable transformations. Here, we consider the limitations on charging batteries when restricting to the feasibly realizable family of Gaussian unitaries. We derive optimal protocols for general unitary operations as well as for the restriction to easier implementable Gaussian unitaries. We find that practical Gaussian battery charging, while performing significantly less well than is possible in principle, still offers asymptotically vanishing relative charge variances and fluctuations.

► BibTeX data

► References

[1] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, The role of quantum information in thermodynamics — a topical review, J. Phys. A: Math. Theor. 49, 143001 (2016) [arXiv:1505.07835].

[2] J. Millen and A. Xuereb, Perspective on quantum thermodynamics, New J. Phys. 18, 011002 (2016) [arXiv:1509.01086].

[3] S. Vinjanampathy and J. Anders, Quantum Thermodynamics, Contemp. Phys. 57, 1 (2016) [arXiv:1508.06099].

[4] F. G. S. L. Brandão, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum thermodynamics, Proc. Natl. Acad. Sci. U.S.A. 112, 3275 (2015) [arXiv:1305.5278].

[5] F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, The Resource Theory of Quantum States Out of Thermal Equilibrium, Phys. Rev. Lett. 111, 250404 (2013) [arXiv:1111.3882].

[6] M. P. Müller, Correlating thermal machines and the second law at the nanoscale, e-print arXiv:1707.03451 [quant-ph] (2017).

[7] C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems, Rep. Prog. Phys. 79, 056001 (2016) [arXiv:1503.07538].

[8] W. Pusz and S. L. Woronowicz, Passive states and KMS states for general quantum systems, Commun. Math. Phys. 58, 273 (1978).

[9] M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, J. Tura, and A. Acín, Most energetic passive states, Phys. Rev. E 92, 042147 (2015) [arXiv:1502.07311].

[10] E. G. Brown, N. Friis, and M. Huber, Passivity and practical work extraction using Gaussian operations, New J. Phys. 18, 113028 (2016) [arXiv:1608.04977].

[11] C. Perry, P. Ć wikliński, J. Anders, M. Horodecki, and J. Oppenheim, A sufficient set of experimentally implementable thermal operations, e-print arXiv:1511.06553 [quant-ph] (2017).

[12] M. Lostaglio, Á. M. Alhambra, and C. Perry, Elementary Thermal Operations, Quantum 2, 52 (2018) [arXiv:1607.00394].

[13] P. Mazurek and M. Horodecki, Decomposability and Convex Structure of Thermal Processes, e-print arXiv:1707.06869 [quant-ph] (2017).

[14] F. Clivaz, R. Silva, G. Haack, J. Bohr Brask, N. Brunner, and M. Huber, Unifying paradigms of quantum refrigeration: resource-dependent limits, e-print arXiv:1710.11624 [quant-ph] (2017).

[15] M. Horodecki and J. Oppenheim, Fundamental limitations for quantum and nanoscale thermodynamics, Nat. Commun. 4, 2059 (2013) [arXiv:1111.3834].

[16] G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern, The resource theory of informational nonequilibrium in thermodynamics, Phys. Rep. 583, 1-58 (2015) [arXiv:1309.6586].

[17] J. Åberg, Catalytic Coherence, Phys. Rev. Lett. 113, 150402 (2014), [arXiv:1304.1060].

[18] A. S. L. Malabarba, A. J. Short, and P. Kammerlander, Clock-Driven Quantum Thermal Engines, New J. Phys. 17, 045027 (2015) [arXiv:1412.1338].

[19] P. Skrzypczyk, A. J. Short, and S. Popescu, Extracting work from quantum systems, e-print arXiv:1302.2811 [quant-ph] (2013).

[20] P. Skrzypczyk, A. J. Short, and S. Popescu, Work extraction and thermodynamics for individual quantum systems, Nat. Commun. 5, 4185 (2014) [arXiv:1307.1558].

[21] F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, Quantacell: Powerful charging of quantum batteries, New J. Phys. 17, 075015 (2015) [arXiv:1503.07005].

[22] F. Campaioli, F. A. Pollock, F. C. Binder, L. C. Céleri, J. Goold, S. Vinjanampathy, and K. Modi, Enhancing the charging power of quantum batteries, Phys. Rev. Lett. 118, 150601 (2017) [arXiv:1612.04991].

[23] D. Ferraro, M. Campisi, G. M. Andolina, V. Pellegrini, and M. Polini, High-Power Collective Charging of a Solid-State Quantum Battery, Phys. Rev. Lett. 120, 117702 (2018) [arXiv:1707.04930].

[24] P. P. Hofer, J.-R. Souquet, and A. A. Clerk, Quantum heat engine based on photon-assisted Cooper pair tunneling, Phys. Rev. B 93, 041418 (2016) [arXiv:1512.02165].

[25] P. P. Hofer, M. Perarnau-Llobet, J. Bohr Brask, R. Silva, M. Huber, and N. Brunner, Autonomous Quantum Refrigerator in a Circuit-QED Architecture Based on a Josephson Junction, Phys. Rev. B 94, 235420 (2016) [arXiv:1607.05218].

[26] M. T. Mitchison, M. Huber, J. Prior, M. P. Woods, and M. B. Plenio, Realising a quantum absorption refrigerator with an atom-cavity system, Quantum Sci. Technol. 1, 015001 (2016) [arXiv:1603.02082].

[27] G. Maslennikov, S. Ding, R. Hablutzel, J. Gan, A. Roulet, S. Nimmrichter, J. Dai, V. Scarani, and D. Matsukevich, Quantum absorption refrigerator with trapped ions, e-print arXiv:1702.08672 [quant-ph] (2017).

[28] J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, A single-atom heat engine, Science 352, 325 (2016) [arXiv:1510.03681].

[29] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012) [arXiv:1110.3234].

[30] M. Campisi, P. Hänggi, and P. Talkner, Colloquium. Quantum Fluctuation Relations: Foundations and Applications, Rev. Mod. Phys. 83, 771 (2011); Erratum: Rev. Mod. Phys. 83, 1653 (2011) [arXiv:1012.2268].

[31] Á. M. Alhambra, L. Masanes, J. Oppenheim, and C. Perry, The second law of quantum thermodynamics as an equality, Phys. Rev. X 6, 041017 (2016) [arXiv:1601.05799].

[32] J. G. Richens and L. Masanes, From single-shot to general work extraction with bounded fluctuations in work, Nat. Commun. 7, 13511 (2016) [arXiv:1603.02417].

[33] M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81, 1665 (2009) [arXiv:0811.3717].

[34] S. Olivares, Quantum optics in the phase space - A tutorial on Gaussian states, Eur. Phys. J. 203, 3 (2012) [arXiv:1111.0786].

[35] S. L. Braunstein, Squeezing as an irreducible resource, Phys. Rev. A 71, 055801 (2005) [arXiv:quant-ph/​9904002].

[36] S. Lloyd and S. L. Braunstein, Quantum computation over continuous variables, Phys. Rev. Lett. 82, 1784 (1999) [arXiv:quant-ph/​9810082].

[37] D. E. Bruschi, M. Perarnau-Llobet, N. Friis, K. V. Hovhannisyan, and M. Huber, The thermodynamics of creating correlations: Limitations and optimal protocols, Phys. Rev. E 91, 032118 (2015) [arXiv:1409.4647].

[38] D. E. Bruschi, N. Friis, I. Fuentes, and S. Weinfurtner, On the robustness of entanglement in analogue gravity systems, New J. Phys. 15, 113016 (2013) [arXiv:1305.3867].

[39] M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Acín, Extractable work from correlations, Phys. Rev. X 5, 041011 (2015) [arXiv:1407.7765].

[40] M. Huber, M. Perarnau-Llobet, K. V. Hovhannisyan, P. Skrzypczyk, C. Klöckl, N. Brunner, and A. Acín, Thermodynamic cost of creating correlations, New J. Phys. 17, 065008 (2015) [arXiv:1404.2169].

[41] N. Friis, M. Huber, and M. Perarnau-Llobet, Energetics of correlations in interacting systems, Phys. Rev. E 93, 042135 (2016) [arXiv:1511.08654].

[42] M. Brunelli, M. G. Genoni, M. Barbieri, and M. Paternostro, Detecting Gaussian entanglement via extractable work, Phys. Rev. A 96, 062311 (2017) [arXiv:1702.05110].

Cited by

[1] Si-Yuan Bai and Jun-Hong An, "Floquet engineering to reactivate a dissipative quantum battery", Physical Review A 102 6, 060201 (2020).

[2] A Crescente, M Carrega, M Sassetti, and D Ferraro, "Charging and energy fluctuations of a driven quantum battery", New Journal of Physics 22 6, 063057 (2020).

[3] Asadullah Khalid, Shahid Tufail, and Arif I. Sarwat, SoutheastCon 2021 1 (2021) ISBN:978-1-6654-0379-5.

[4] Sergi Julià-Farré, Tymoteusz Salamon, Arnau Riera, Manabendra N. Bera, and Maciej Lewenstein, "Bounds on the capacity and power of quantum batteries", Physical Review Research 2 2, 023113 (2020).

[5] Francesco Campaioli, Felix A. Pollock, and Sai Vinjanampathy, Fundamental Theories of Physics 195, 207 (2018) ISBN:978-3-319-99045-3.

[6] Wei Chang, Tian-Ran Yang, Hui Dong, Libin Fu, Xiaoguang Wang, and Yu-Yu Zhang, "Optimal building block of multipartite quantum battery in the driven-dissipative charging", New Journal of Physics 23 10, 103026 (2021).

[7] Philip Taranto, Faraj Bakhshinezhad, Andreas Bluhm, Ralph Silva, Nicolai Friis, Maximilian P.E. Lock, Giuseppe Vitagliano, Felix C. Binder, Tiago Debarba, Emanuel Schwarzhans, Fabien Clivaz, and Marcus Huber, "Landauer Versus Nernst: What is the True Cost of Cooling a Quantum System?", PRX Quantum 4 1, 010332 (2023).

[8] Charles Andrew Downing and Muhammad Shoufie Ukhtary, "Hyperbolic enhancement of a quantum battery", Physical Review A 109 5, 052206 (2024).

[9] Benjamin Yadin, Hyejung H Jee, Carlo Sparaciari, Gerardo Adesso, and Alessio Serafini, "Catalytic Gaussian thermal operations", Journal of Physics A: Mathematical and Theoretical 55 32, 325301 (2022).

[10] Francesco Caravelli, Bin Yan, Luis Pedro García-Pintos, and Alioscia Hamma, "Energy storage and coherence in closed and open quantum batteries", Quantum 5, 505 (2021).

[11] Luis Pedro García-Pintos, Alioscia Hamma, and Adolfo del Campo, "Fluctuations in Extractable Work Bound the Charging Power of Quantum Batteries", Physical Review Letters 125 4, 040601 (2020).

[12] Yunxiu Jiang, Tianhao Chen, Chu Xiao, Kaiyan Pan, Guangri Jin, Youbin Yu, and Aixi Chen, "Quantum Battery Based on Hybrid Field Charging", Entropy 24 12, 1821 (2022).

[13] Barış Çakmak, "Ergotropy from coherences in an open quantum system", Physical Review E 102 4, 042111 (2020).

[14] Ju-Yeon Gyhm, Dario Rosa, and Dominik Šafránek, "Minimal time required to charge a quantum system", Physical Review A 109 2, 022607 (2024).

[15] Elisa Bäumer, Martí Perarnau-Llobet, Philipp Kammerlander, Henrik Wilming, and Renato Renner, "Imperfect Thermalizations Allow for Optimal Thermodynamic Processes", Quantum 3, 153 (2019).

[16] Varun Narasimhachar, Syed Assad, Felix C. Binder, Jayne Thompson, Benjamin Yadin, and Mile Gu, "Thermodynamic resources in continuous-variable quantum systems", npj Quantum Information 7 1, 9 (2021).

[17] Stella Seah, Martí Perarnau-Llobet, Géraldine Haack, Nicolas Brunner, and Stefan Nimmrichter, "Quantum Speed-Up in Collisional Battery Charging", Physical Review Letters 127 10, 100601 (2021).

[18] Francesco Campaioli, Stefano Gherardini, James Q. Quach, Marco Polini, and Gian Marcello Andolina, "Colloquium : Quantum batteries", Reviews of Modern Physics 96 3, 031001 (2024).

[19] Jie Chen, Liyao Zhan, Lei Shao, Xingyu Zhang, Yuyu Zhang, and Xiaoguang Wang, "Charging Quantum Batteries with a General Harmonic Driving Field", Annalen der Physik 532 4, 1900487 (2020).

[20] Patryk Lipka-Bartosik, Henrik Wilming, and Nelly H. Y. Ng, "Catalysis in quantum information theory", Reviews of Modern Physics 96 2, 025005 (2024).

[21] Uttam Singh, Jarosław K. Korbicz, and Nicolas J. Cerf, "Gaussian work extraction from random Gaussian states is nearly impossible", Physical Review Research 5 3, L032010 (2023).

[22] Emma McKay, Nayeli A. Rodríguez-Briones, and Eduardo Martín-Martínez, "Fluctuations of work cost in optimal generation of correlations", Physical Review E 98 3, 032132 (2018).

[23] Fang Zhao, Fu-Quan Dou, and Qing Zhao, "Quantum battery of interacting spins with environmental noise", Physical Review A 103 3, 033715 (2021).

[24] Saikat Mondal and Sourav Bhattacharjee, "Periodically driven many-body quantum battery", Physical Review E 105 4, 044125 (2022).

[25] Pharnam Bakhshinezhad, Beniamin R. Jablonski, Felix C. Binder, and Nicolai Friis, "Trade-offs between precision and fluctuations in charging finite-dimensional quantum batteries", Physical Review E 109 1, 014131 (2024).

[26] Alan C. Santos, Barış Çakmak, Steve Campbell, and Nikolaj T. Zinner, "Stable adiabatic quantum batteries", Physical Review E 100 3, 032107 (2019).

[27] Niels Lörch, Christoph Bruder, Nicolas Brunner, and Patrick P Hofer, "Optimal work extraction from quantum states by photo-assisted Cooper pair tunneling", Quantum Science and Technology 3 3, 035014 (2018).

[28] Mark T. Mitchison, John Goold, and Javier Prior, "Charging a quantum battery with linear feedback control", Quantum 5, 500 (2021).

[29] Dong-Lin Yang, Fang-Mei Yang, and Fu-Quan Dou, "Three-level Dicke quantum battery", Physical Review B 109 23, 235432 (2024).

[30] Raffaele Salvia and Vittorio Giovannetti, "Energy upper bound for structurally stableN-passive states", Quantum 4, 274 (2020).

[31] Lu Wang, Shu-Qian Liu, Feng-lin Wu, Hao Fan, and Si-Yuan Liu, "Two-mode Raman quantum battery dependent on coupling strength", Physical Review A 108 6, 062402 (2023).

[32] Srijon Ghosh, Titas Chanda, and Aditi Sen(De), "Enhancement in the performance of a quantum battery by ordered and disordered interactions", Physical Review A 101 3, 032115 (2020).

[33] Dario Ferraro, Gian Marcello Andolina, Michele Campisi, Vittorio Pellegrini, and Marco Polini, "Quantum supercapacitors", Physical Review B 100 7, 075433 (2019).

[34] Satoya Imai, Otfried Gühne, and Stefan Nimmrichter, "Work fluctuations and entanglement in quantum batteries", Physical Review A 107 2, 022215 (2023).

[35] Giuseppe Vitagliano, Claude Klöckl, Marcus Huber, and Nicolai Friis, Fundamental Theories of Physics 195, 731 (2018) ISBN:978-3-319-99045-3.

[36] Vahid Shaghaghi, Varinder Singh, Giuliano Benenti, and Dario Rosa, "Micromasers as quantum batteries", Quantum Science and Technology 7 4, 04LT01 (2022).

[37] Kai Xu, Hong-Guo Li, Han-Jie Zhu, and Wu-Ming Liu, "Inhibiting the self-discharging process of quantum batteries in non-Markovian noises", Physical Review E 109 5, 054132 (2024).

[38] Martí Perarnau-Llobet and Raam Uzdin, "Collective operations can extremely reduce work fluctuations", New Journal of Physics 21 8, 083023 (2019).

[39] Vahid Shaghaghi, Varinder Singh, Matteo Carrega, Dario Rosa, and Giuliano Benenti, "Lossy Micromaser Battery: Almost Pure States in the Jaynes–Cummings Regime", Entropy 25 3, 430 (2023).

[40] Dario Ferraro, Michele Campisi, Gian Marcello Andolina, Vittorio Pellegrini, Marco Polini, and E. Puppin, "Quantum resources for energy storage", EPJ Web of Conferences 230, 00003 (2020).

[41] Dario Rosa, Davide Rossini, Gian Marcello Andolina, Marco Polini, and Matteo Carrega, "Ultra-stable charging of fast-scrambling SYK quantum batteries", Journal of High Energy Physics 2020 11, 67 (2020).

[42] Francesco Caravelli, Ghislaine Coulter-De Wit, Luis Pedro García-Pintos, and Alioscia Hamma, "Random quantum batteries", Physical Review Research 2 2, 023095 (2020).

[43] Li Peng, Wen-Bin He, Stefano Chesi, Hai-Qing Lin, and Xi-Wen Guan, "Lower and upper bounds of quantum battery power in multiple central spin systems", Physical Review A 103 5, 052220 (2021).

[44] Fang Zhao, Fu-Quan Dou, and Qing Zhao, "Charging performance of the Su-Schrieffer-Heeger quantum battery", Physical Review Research 4 1, 013172 (2022).

[45] Kornikar Sen and Ujjwal Sen, "Local passivity and entanglement in shared quantum batteries", Physical Review A 104 3, L030402 (2021).

[46] Sourav Bhattacharjee and Amit Dutta, "Quantum thermal machines and batteries", The European Physical Journal B 94 12, 239 (2021).

[47] Mir Alimuddin, Tamal Guha, and Preeti Parashar, "Structure of passive states and its implication in charging quantum batteries", Physical Review E 102 2, 022106 (2020).

[48] Bin-Yuan Huang, Zhi He, and Yu Chen, "Charging performance of quantum batteries based on intensity-dependent Dicke model", Acta Physica Sinica 72 18, 180301 (2023).

[49] Stefano Gherardini, Francesco Campaioli, Filippo Caruso, and Felix C. Binder, "Stabilizing open quantum batteries by sequential measurements", Physical Review Research 2 1, 013095 (2020).

[50] Salvatore Tirone, Raffaele Salvia, and Vittorio Giovannetti, "Quantum Energy Lines and the Optimal Output Ergotropy Problem", Physical Review Letters 127 21, 210601 (2021).

[51] A. Serafini, M. Lostaglio, S. Longden, U. Shackerley-Bennett, C.-Y. Hsieh, and G. Adesso, "Gaussian Thermal Operations and The Limits of Algorithmic Cooling", Physical Review Letters 124 1, 010602 (2020).

[52] Kai Xu, Han-Jie Zhu, Hao Zhu, Guo-Feng Zhang, and Wu-Ming Liu, "Charging and self-discharging process of a quantum battery in composite environments", Frontiers of Physics 18 3, 31301 (2023).

[53] Tiago Debarba, Gonzalo Manzano, Yelena Guryanova, Marcus Huber, and Nicolai Friis, "Work estimation and work fluctuations in the presence of non-ideal measurements", New Journal of Physics 21 11, 113002 (2019).

[54] Ju-Yeon Gyhm, Dominik Šafránek, and Dario Rosa, "Quantum Charging Advantage Cannot Be Extensive without Global Operations", Physical Review Letters 128 14, 140501 (2022).

[55] Raffaele Salvia, Martí Perarnau-Llobet, Géraldine Haack, Nicolas Brunner, and Stefan Nimmrichter, "Quantum advantage in charging cavity and spin batteries by repeated interactions", Physical Review Research 5 1, 013155 (2023).

[56] Alba Crescente, Matteo Carrega, Maura Sassetti, and Dario Ferraro, "Ultrafast charging in a two-photon Dicke quantum battery", Physical Review B 102 24, 245407 (2020).

[57] Fu-Quan Dou, You-Qi Lu, Yuan-Jin Wang, and Jian-An Sun, "Extended Dicke quantum battery with interatomic interactions and driving field", Physical Review B 105 11, 115405 (2022).

[58] Dengke Qu, Xiang Zhan, Haiqing Lin, and Peng Xue, "Experimental optimization of charging quantum batteries through a catalyst system", Physical Review B 108 18, L180301 (2023).

[59] Andrew R Hogan and Andy M Martin, "Quench dynamics in the Jaynes-Cummings-Hubbard and Dicke models", Physica Scripta 99 5, 055118 (2024).

[60] Yu-Yu Zhang, Tian-Ran Yang, Libin Fu, and Xiaoguang Wang, "Powerful harmonic charging in a quantum battery", Physical Review E 99 5, 052106 (2019).

[61] Wei-Xi Guo, Fang-Mei Yang, and Fu-Quan Dou, "Analytically solvable many-body Rosen-Zener quantum battery", Physical Review A 109 3, 032201 (2024).

[62] Federico Centrone, Luca Mancino, and Mauro Paternostro, "Charging batteries with quantum squeezing", Physical Review A 108 5, 052213 (2023).

[63] Ludovico Lami, Bartosz Regula, Xin Wang, Rosanna Nichols, Andreas Winter, and Gerardo Adesso, "Gaussian quantum resource theories", Physical Review A 98 2, 022335 (2018).

[64] Felipe Barra, "Efficiency Fluctuations in a Quantum Battery Charged by a Repeated Interaction Process", Entropy 24 6, 820 (2022).

[65] Alba Crescente, Dario Ferraro, Matteo Carrega, and Maura Sassetti, "Analytically Solvable Model for Qubit-Mediated Energy Transfer between Quantum Batteries", Entropy 25 5, 758 (2023).

[66] Domingos S. P. Salazar and Gabriel T. Landi, "Nonlinear Onsager relations for Gaussian quantum maps", Physical Review Research 2 3, 033090 (2020).

[67] G. Francica, F. C. Binder, G. Guarnieri, M. T. Mitchison, J. Goold, and F. Plastina, "Quantum Coherence and Ergotropy", Physical Review Letters 125 18, 180603 (2020).

[68] Anna Delmonte, Alba Crescente, Matteo Carrega, Dario Ferraro, and Maura Sassetti, "Characterization of a Two-Photon Quantum Battery: Initial Conditions, Stability and Work Extraction", Entropy 23 5, 612 (2021).

[69] Raffaele Salvia and Vittorio Giovannetti, "Zero-fluctuation quantum work extraction", Physical Review A 110 1, 012213 (2024).

[70] Fu-Quan Dou, Yuan-Jin Wang, and Jian-An Sun, "Closed-loop three-level charged quantum battery", EPL (Europhysics Letters) 131 4, 43001 (2020).

[71] Fu-Quan Dou, Hang Zhou, and Jian-An Sun, "Cavity Heisenberg-spin-chain quantum battery", Physical Review A 106 3, 032212 (2022).

[72] Charles Andrew Downing and Muhammad Shoufie Ukhtary, "A quantum battery with quadratic driving", Communications Physics 6 1, 322 (2023).

[73] Charles Andrew Downing and Muhammad Shoufie Ukhtary, "Energetics of a pulsed quantum battery", Europhysics Letters 146 1, 10001 (2024).

[74] Arnab Ghosh, Wolfgang Niedenzu, Victor Mukherjee, and Gershon Kurizki, Fundamental Theories of Physics 195, 37 (2018) ISBN:978-3-319-99045-3.

[75] Thao P. Le, Jesper Levinsen, Kavan Modi, Meera M. Parish, and Felix A. Pollock, "Spin-chain model of a many-body quantum battery", Physical Review A 97 2, 022106 (2018).

[76] Francesco Campaioli, Felix A. Pollock, and Sai Vinjanampathy, "Quantum Batteries - Review Chapter", arXiv:1805.05507, (2018).

[77] Tiago Debarba, Gonzalo Manzano, Yelena Guryanova, Marcus Huber, and Nicolai Friis, "Work estimation and work fluctuations in the presence of non-ideal measurements", arXiv:1902.08568, (2019).

[78] Raffaele Salvia and Vittorio Giovannetti, "Zero-Fluctuation Quantum Work Extraction", arXiv:2402.16964, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-07-15 11:21:25) and SAO/NASA ADS (last updated successfully 2024-07-15 11:21:26). The list may be incomplete as not all publishers provide suitable and complete citation data.