- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Experimental extraction of coherent ergotropy and its energetic cost in a superconducting qubit

Li Li1,2,*, Silu Zhao1,2, Yun-Hao Shi1, Kai Xu1,3,4, Heng Fan1,2,3,4, Dongning Zheng1,2,4, and Zhongcheng Xiang1,4,†

  • *Contact author: lilisc05@iphy.ac.cn
  • †Contact author: zcxiang@iphy.ac.cn
PDF

Phys. Rev. Research 7, 043236 – Published 1 December, 2025

DOI: https://doi.org/10.1103/ptr1-59tv

Abstract

Quantum coherence, encoded in the off-diagonal elements of a system's density matrix, is a key resource in quantum thermodynamics, fundamentally limiting the maximum extractable work known as ergotropy. While previous experiments have isolated coherence-related contributions to work extraction, it remains unclear how coherence can be harnessed in a controllable and energy-efficient manner. Here, we experimentally investigate the role of initial-state coherence in work extraction from a superconducting transmon qubit. By preparing a variety of pure states and implementing three complementary extraction protocols, we reveal how coherence governs the partitioning of ergotropy. We find that the choice of initial state depends on the dominant decoherence channel-energy relaxation or dephasing. By further accounting for thermodynamic costs, we identify optimal initial states that maximize the efficiency. Our results demonstrate that the initial-state design provides a scalable approach to coherence control and advances the development of efficient quantum thermodynamic devices.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (69)

  1. F. Campaioli, S. Gherardini, J. Q. Quach, M. Polini, and G. M. Andolina, Colloquium: Quantum batteries, Rev. Mod. Phys. 96, 031001 (2024).
  2. R. Alicki and M. Fannes, Entanglement boost for extractable work from ensembles of quantum batteries, Phys. Rev. E 87, 042123 (2013).
  3. P. Skrzypczyk, A. J. Short, and S. Popescu, Work extraction and thermodynamics for individual quantum systems, Nat. Commun. 5, 4185 (2014).
  4. F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, Quantacell: Powerful charging of quantum batteries, New J. Phys. 17, 075015 (2015).
  5. D. E. Bruschi, M. Perarnau-Llobet, N. Friis, K. V. Hovhannisyan, and M. Huber, Thermodynamics of creating correlations: Limitations and optimal protocols, Phys. Rev. E 91, 032118 (2015).
  6. F. Campaioli, F. A. Pollock, F. C. Binder, L. Céleri, J. Goold, S. Vinjanampathy, and K. Modi, Enhancing the charging power of quantum batteries, Phys. Rev. Lett. 118, 150601 (2017).
  7. 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).
  8. G. M. Andolina, M. Keck, A. Mari, M. Campisi, V. Giovannetti, and M. Polini, Extractable work, the role of correlations, and asymptotic freedom in quantum batteries, Phys. Rev. Lett. 122, 047702 (2019).
  9. J.-X. Liu, H.-L. Shi, Y.-H. Shi, X.-H. Wang, and W.-L. Yang, Entanglement and work extraction in the central-spin quantum battery, Phys. Rev. B 104, 245418 (2021).
  10. L. F. C. Moraes, A. Saguia, A. C. Santos, and M. S. Sarandy, Charging power and stability of always-on transitionless driven quantum batteries, Europhys. Lett. 136, 23001 (2021).
  11. F.-Q. Dou, Y.-Q. Lu, Y.-J. Wang, and J.-A. Sun, Extended dicke quantum battery with interatomic interactions and driving field, Phys. Rev. B 105, 115405 (2022).
  12. F.-Q. Dou, H. Zhou, and J.-A. Sun, Cavity Heisenberg-spin-chain quantum battery, Phys. Rev. A 106, 032212 (2022).
  13. F. Zhao, F.-Q. Dou, and Q. Zhao, Charging performance of the Su-Schrieffer-Heeger quantum battery, Phys. Rev. Res. 4, 013172 (2022).
  14. B. Mojaveri, R. Jafarzadeh Bahrbeig, M. A. Fasihi, and S. Babanzadeh, Enhancing the direct charging performance of an open quantum battery by adjusting its velocity, Sci. Rep. 13, 19827 (2023).
  15. J.-Y. Gyhm and U. R. Fischer, Beneficial and detrimental entanglement for quantum battery charging, AVS Quantum Science 6, 012001 (2024).
  16. P.-R. Lai, J.-D. Lin, Y.-T. Huang, H.-C. Jan, and Y.-N. Chen, Quick charging of a quantum battery with superposed trajectories, Phys. Rev. Res. 6, 023136 (2024).
  17. W.-L. Song, H.-B. Liu, B. Zhou, W.-L. Yang, and J.-H. An, Remote charging and degradation suppression for the quantum battery, Phys. Rev. Lett. 132, 090401 (2024).
  18. S. Elghaayda, A. Ali, S. Al-Kuwari, A. Czerwinski, M. Mansour, and S. Haddadi, Performance of a superconducting quantum battery, Adv. Quantum Technol. 8, 2400651 (2025).
  19. D. Rossini, G. M. Andolina, D. Rosa, M. Carrega, and M. Polini, Quantum advantage in the charging process of Sachdev-Ye-Kitaev batteries, Phys. Rev. Lett. 125, 236402 (2020).
  20. A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhys. Lett. 67, 565 (2004).
  21. E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
  22. A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium : Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
  23. K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jennings, The extraction of work from quantum coherence, New J. Phys. 18, 023045 (2016).
  24. G. Francica, J. Goold, and F. Plastina, Role of coherence in the nonequilibrium thermodynamics of quantum systems, Phys. Rev. E 99, 042105 (2019).
  25. B. Çakmak, Ergotropy from coherences in an open quantum system, Phys. Rev. E 102, 042111 (2020).
  26. G. Francica, F. C. Binder, G. Guarnieri, M. T. Mitchison, J. Goold, and F. Plastina, Quantum coherence and ergotropy, Phys. Rev. Lett. 125, 180603 (2020).
  27. H.-L. Shi, S. Ding, Q.-K. Wan, X.-H. Wang, and W.-L. Yang, Entanglement, coherence, and extractable work in quantum batteries, Phys. Rev. Lett. 129, 130602 (2022).
  28. Z. Niu, Y. Wu, Y. Wang, X. Rong, and J. Du, Experimental investigation of coherent ergotropy in a single spin system, Phys. Rev. Lett. 133, 180401 (2024).
  29. N. P. de Leon, K. M. Itoh, D. Kim, K. K. Mehta, T. E. Northup, H. Paik, B. S. Palmer, N. Samarth, S. Sangtawesin, and D. W. Steuerman, Materials challenges and opportunities for quantum computing hardware, Science 372, eabb2823 (2021).
  30. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys. 82, 1155 (2010).
  31. Y. Zheng, S. Campbell, G. De Chiara, and D. Poletti, Cost of counterdiabatic driving and work output, Phys. Rev. A 94, 042132 (2016).
  32. S. Campbell and S. Deffner, Trade-off between speed and cost in shortcuts to adiabaticity, Phys. Rev. Lett. 118, 100601 (2017).
  33. O. Abah, R. Puebla, A. Kiely, G. De Chiara, M. Paternostro, and S. Campbell, Energetic cost of quantum control protocols, New J. Phys. 21, 103048 (2019).
  34. J. Monsel, M. Fellous-Asiani, B. Huard, and A. Auffèves, The energetic cost of work extraction, Phys. Rev. Lett. 124, 130601 (2020).
  35. T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
  36. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A 76, 042319 (2007).
  37. P. M. Harrington, E. J. Mueller, and K. W. Murch, Engineered dissipation for quantum information science, Nat. Rev. Phys. 4, 660 (2022).
  38. X. Mi, A. A. Michailidis, S. Shabani, K. C. Miao, P. V. Klimov, J. Lloyd, E. Rosenberg, R. Acharya, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, J. C. Bardin, A. Bengtsson, G. Bortoli, A. Bourassa, J. Bovaird et al., Stable quantum-correlated many-body states through engineered dissipation, Science 383, 1332 (2024).
  39. L. Li, T. Liu, X.-Y. Guo, H. Zhang, S. Zhao, Z.-A. Wang, Z. Xiang, X. Song, Y.-X. Zhang, K. Xu, H. Fan, and D. Zheng, Observation of multiple steady states with engineered dissipation, npj Quantum Inf. 11, 2 (2025).
  40. Y. Liang, C. Xie, Z. Guo, P. Huang, W. Huang, Y. Liu, J. Qiu, X. Sun, Z. Wang, X. Yang, J. Zhang, J. Zhang, L. Zhang, J. Chu, W. Guo, J. Jiang, X. Linpeng, S. Liu, J. Niu, Y. Zhou et al., Dephasing-assisted diffusive dynamics in superconducting quantum circuits, Appl. Phys. Lett. 127, 154003 (2025).
  41. Z. Tao, F. Schmolke, C.-K. Hu, W. Huang, Y. Zhou, J. Zhang, J. Chu, L. Zhang, X. Sun, Z. Guo, J. Niu, W. Weng, S. Liu, Y. Zhong, D. Tan, D. Yu, and E. Lutz, Noise-Induced Quantum Synchronization with Entangled Oscillations, Nat. Commun. 16, 8457 (2025).
  42. J. Han, W. Cai, L. Hu, X. Mu, Y. Ma, Y. Xu, W. Wang, H. Wang, Y. P. Song, C.-L. Zou, and L. Sun, Experimental simulation of open quantum system dynamics via trotterization, Phys. Rev. Lett. 127, 020504 (2021).
  43. J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, An open-system quantum simulator with trapped ions, Nature (London) 470, 486 (2011).
  44. K. W. Murch, U. Vool, D. Zhou, S. J. Weber, S. M. Girvin, and I. Siddiqi, Cavity-assisted quantum bath engineering, Phys. Rev. Lett. 109, 183602 (2012).
  45. T. Brown, E. Doucet, D. Ristè, G. Ribeill, K. Cicak, J. Aumentado, R. Simmonds, L. Govia, A. Kamal, and L. Ranzani, Trade off-free entanglement stabilization in a superconducting qutrit-qubit system, Nat. Commun. 13, 3994 (2022).
  46. T. Liu, S. Liu, H. Li, H. Li, K. Huang, Z. Xiang, X. Song, K. Xu, D. Zheng, and H. Fan, Observation of entanglement transition of pseudo-random mixed states, Nat. Commun. 14, 1971 (2023).
  47. R. Simmonds, K. Lang, D. Hite, S. Nam, D. Pappas, and J. Martinis, Decoherence in Josephson phase qubits from junction resonators, Phys. Rev. Lett. 93, 077003 (2004).
  48. J. Lisenfeld, C. Müller, J. H. Cole, P. Bushev, A. Lukashenko, A. Shnirman, and A. V. Ustinov, Measuring the temperature dependence of individual two-level systems by direct coherent control, Phys. Rev. Lett. 105, 230504 (2010).
  49. J. Lisenfeld, C. Müller, J. H. Cole, P. Bushev, A. Lukashenko, A. Shnirman, and A. V. Ustinov, Rabi spectroscopy of a qubit-fluctuator system, Phys. Rev. B 81, 100511 (2010).
  50. J. Burnett, L. Faoro, I. Wisby, V. L. Gurtovoi, A. V. Chernykh, G. M. Mikhailov, V. A. Tulin, R. Shaikhaidarov, V. Antonov, P. J. Meeson, A. Ya. Tzalenchuk, and T. Lindström, Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator, Nat. Commun. 5, 4119 (2014).
  51. P. V. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant, B. Burkett, R. Barends, K. Arya, B. Chiaro, Y. Chen, A. Dunsworth, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, T. Huang, E. Jeffrey, E. Lucero, J. Y. Mutus et al., Fluctuations of energy-relaxation times in superconducting qubits, Phys. Rev. Lett. 121, 090502 (2018).
  52. S. M. Meißner, A. Seiler, J. Lisenfeld, A. V. Ustinov, and G. Weiss, Probing individual tunneling fluctuators with coherently controlled tunneling systems, Phys. Rev. B 97, 180505 (2018).
  53. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Measurement of qubits, Phys. Rev. A 64, 052312 (2001).
  54. R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, Qudit quantum-state tomography, Phys. Rev. A 66, 012303 (2002).
  55. J. Johansson, P. Nation, and F. Nori, Qutip: An open-source python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 183, 1760 (2012).
  56. J. Johansson, P. Nation, and F. Nori, Qutip 2: A python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 184, 1234 (2013).
  57. G. S. Uhrig, Keeping a quantum bit alive by optimized π-pulse sequences, Phys. Rev. Lett. 98, 100504 (2007).
  58. Ł. Cywiński, R. M. Lutchyn, C. P. Nave, and S. D. Sarma, How to enhance dephasing time in superconducting qubits, Phys. Rev. B 77, 174509 (2008).
  59. J.-M. Cai, B. Naydenov, R. Pfeiffer, L. P. McGuinness, K. D. Jahnke, F. Jelezko, M. B. Plenio, and A. Retzker, Robust dynamical decoupling with concatenated continuous driving, New J. Phys. 14, 113023 (2012).
  60. X. Xu, Z. Wang, C. Duan, P. Huang, P. Wang, Y. Wang, N. Xu, X. Kong, F. Shi, X. Rong, and J. Du, Coherence-protected quantum gate by continuous dynamical decoupling in diamond, Phys. Rev. Lett. 109, 070502 (2012).
  61. Q. Guo, S.-B. Zheng, J. Wang, C. Song, P. Zhang, K. Li, W. Liu, H. Deng, K. Huang, D. Zheng, X. Zhu, H. Wang, C.-Y. Lu, and J.-W. Pan, Dephasing-insensitive quantum information storage and processing with superconducting qubits, Phys. Rev. Lett. 121, 130501 (2018).
  62. K. C. Miao, J. P. Blanton, C. P. Anderson, A. Bourassa, A. L. Crook, G. Wolfowicz, H. Abe, T. Ohshima, and D. D. Awschalom, Universal coherence protection in a solid-state spin qubit, Science 369, 1493 (2020).
  63. Z. Huang, P. S. Mundada, A. Gyenis, D. I. Schuster, A. A. Houck, and J. Koch, Engineering dynamical sweet spots to protect qubits from 1/f noise, Phys. Rev. Appl. 15, 034065 (2021).
  64. S. Deffner, Energetic cost of Hamiltonian quantum gates, Europhys. Lett. 134, 40002 (2021).
  65. C.-K. Hu, J. Qiu, P. J. P. Souza, J. Yuan, Y. Zhou, L. Zhang, J. Chu, X. Pan, L. Hu, J. Li, Y. Xu, Y. Zhong, S. Liu, F. Yan, D. Tan, R. Bachelard, C. J. Villas-Boas, A. C. Santos, and D. Yu, Optimal charging of a superconducting quantum battery, Quantum Sci. Technol. 7, 045018 (2022).
  66. Y. Ge, X. Yu, W. Xin, Z. Wang, Y. Zhang, W. Zheng, S. Li, D. Lan, and Y. Yu, Efficient charging and discharging of a superconducting quantum battery through frequency-modulated driving, Appl. Phys. Lett. 123, 154002 (2023).
  67. Z. Lin, L. Zhang, X. Long, Y.-a. Fan, Y. Li, K. Tang, J. Li, X. Nie, T. Xin, X.-J. Liu, and D. Lu, Experimental quantum simulation of non-Hermitian dynamical topological states using stochastic Schrödinger equation, npj Quantum Inf. 8, 77 (2022).
  68. R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O'Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, Coherent Josephson qubit suitable for scalable quantum integrated circuits, Phys. Rev. Lett. 111, 080502 (2013).
  69. L. Li, S.-L. Zhao, Y.-H. Shi, B.-J. Chen, X. Ruan, G.-H. Liang, W.-P. Yuan, J.-C. Song, C.-L. Deng, Y. Liu, T.-M. Li, Z.-H. Liu, X.-Y. Guo, X. Song, K. Xu, H. Fan, Z. Xiang, and D. Zheng, Stable and efficient charging of superconducting C-shunt flux quantum batteries, Phys. Rev. Appl. 24, 054033 (2025).

Outline

Information

Phys. Rev. Research 7, 043236– Published 1 December, 2025
  • Received 23 June 2025
  • Revised 4 October 2025
  • Accepted 8 November 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation