Work and Fluctuations: Coherent vs. Incoherent Ergotropy Extraction

Marcin Łobejko

Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdansk, Poland
International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland

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We consider a quasi-probability distribution of work for an isolated quantum system coupled to the energy-storage device given by the ideal weight. Specifically, we analyze a trade-off between changes in average energy and changes in weight's variance, where work is extracted from the coherent and incoherent ergotropy of the system. Primarily, we reveal that the extraction of positive coherent ergotropy can be accompanied by the reduction of work fluctuations (quantified by a variance loss) by utilizing the non-classical states of a work reservoir. On the other hand, we derive a fluctuation-decoherence relation for a quantum weight, defining a lower bound of its energy dispersion via a dumping function of the coherent contribution to the system's ergotropy. Specifically, it reveals that unlocking ergotropy from coherences results in high fluctuations, which diverge when the total coherent energy is unlocked. The proposed autonomous protocol of work extraction shows a significant difference between extracting coherent and incoherent ergotropy: The former can decrease the variance, but its absolute value diverges if more and more energy is extracted, whereas for the latter, the gain is always non-negative, but a total (incoherent) ergotropy can be extracted with finite work fluctuations. Furthermore, we present the framework in terms of the introduced quasi-probability distribution, which has a physical interpretation of its cumulants, is free from the invasive nature of measurements, and reduces to the two-point measurement scheme (TPM) for incoherent states. Finally, we analytically solve the work-variance trade-off for a qubit, explicitly revealing all the above quantum and classical regimes.

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[1] Stefan Aimet and Hyukjoon Kwon, "Engineering a heat engine purely driven by quantum coherence", Physical Review A 107 1, 012221 (2023).

[2] V. Stepanyan and A.E. Allahverdyan, "Energy densities in quantum mechanics", Quantum 8, 1223 (2024).

[3] Gourab Bhanja, Devvrat Tiwari, and Subhashish Banerjee, "Impact of non-Markovian quantum Brownian motion on quantum batteries", Physical Review A 109 1, 012224 (2024).

[4] Federico Cerisola, Franco Mayo, and Augusto J. Roncaglia, "A Wigner Quasiprobability Distribution of Work", Entropy 25 10, 1439 (2023).

[5] Dominik Šafránek, "Ergotropic interpretation of entanglement entropy", arXiv:2306.08987, (2023).

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