Thermodynamics of Minimal Coupling Quantum Heat Engines

Marcin Łobejko; Paweł Mazurek; Michał Horodecki

doi:10.22331/q-2020-12-23-375

Thermodynamics of Minimal Coupling Quantum Heat Engines

Marcin Łobejko¹, Paweł Mazurek^1,2, and Michał Horodecki^1,2

¹Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland
²International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland

Published:	2020-12-23, volume 4, page 375
Eprint:	arXiv:2003.05788v4
Doi:	https://doi.org/10.22331/q-2020-12-23-375
Citation:	Quantum 4, 375 (2020).

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Abstract

The minimal-coupling quantum heat engine is a thermal machine consisting of an explicit energy storage system, heat baths, and a working body, which alternatively couples to subsystems through discrete strokes --- energy-conserving two-body quantum operations. Within this paradigm, we present a general framework of quantum thermodynamics, where a work extraction process is fundamentally limited by a flow of non-passive energy (ergotropy), while energy dissipation is expressed through a flow of passive energy. It turns out that small dimensionality of the working body and a restriction only to two-body operations make the engine fundamentally irreversible. Our main result is finding the optimal efficiency and work production per cycle within the whole class of irreversible minimal-coupling engines composed of three strokes and with the two-level working body, where we take into account all possible quantum correlations between the working body and the battery. One of the key new tools is the introduced ``control-marginal state" --- one which acts only on a working body Hilbert space, but encapsulates all features regarding work extraction of the total working body-battery system. In addition, we propose a generalization of the many-stroke engine, and we analyze efficiency vs extracted work trade-offs, as well as work fluctuations after many cycles of the running of the engine.

► BibTeX data

@article{Lobejko2020thermodynamicsof,
  doi = {10.22331/q-2020-12-23-375},
  url = {https://doi.org/10.22331/q-2020-12-23-375},
  title = {Thermodynamics of {M}inimal {C}oupling {Q}uantum {H}eat {E}ngines},
  author = {{\L{}}obejko, Marcin and Mazurek, Pawe{\l{}} and Horodecki, Micha{\l{}}},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {375},
  month = dec,
  year = {2020}
}

► References

[1] Johannes Roßnagel, Samuel T. Dawkins, Karl N. Tolazzi, Obinna Abah, Eric Lutz, Ferdinand Schmidt-Kaler, and Kilian Singer, ``A single-atom heat engine'' Science 352, 325-329 (2016).
https://doi.org/10.1126/science.aad6320
arXiv:1510.03681

[2] Nathanaël Cottet, Sébastien Jezouin, Landry Bretheau, Philippe Campagne-Ibarcq, Quentin Ficheux, Janet Anders, Alexia Auffèves, Rémi Azouit, Pierre Rouchon, and Benjamin Huard, ``Observing a quantum Maxwell demon at work'' Proceedings of the National Academy of Sciences 114, 7561–7564 (2017).
https://doi.org/10.1073/pnas.1704827114
https://www.pnas.org/content/114/29/7561

[3] James Klatzow, Jonas N. Becker, Patrick M. Ledingham, Christian Weinzetl, Krzysztof T. Kaczmarek, Dylan J. Saunders, Joshua Nunn, Ian A. Walmsley, Raam Uzdin, and Eilon Poem, ``Experimental Demonstration of Quantum Effects in the Operation of Microscopic Heat Engines'' Phys. Rev. Lett. 122, 110601 (2019).
https://doi.org/10.1103/PhysRevLett.122.110601

[4] Daniel Goldwater, Benjamin A Stickler, Lukas Martinetz, Tracy E Northup, Klaus Hornberger, and James Millen, ``Levitated electromechanics: all-electrical cooling of charged nano- and micro-particles'' Quantum Science and Technology 4, 024003 (2019).
https://doi.org/10.1088/2058-9565/aaf5f3

[5] Christian Bergenfeldt, Peter Samuelsson, Björn Sothmann, Christian Flindt, and Markus Büttiker, ``Hybrid Microwave-Cavity Heat Engine'' Phys. Rev. Lett. 112, 076803 (2014).
https://doi.org/10.1103/PhysRevLett.112.076803

[6] Andreas Dechant, Nikolai Kiesel, and Eric Lutz, ``All-Optical Nanomechanical Heat Engine'' Phys. Rev. Lett. 114, 183602 (2015).
https://doi.org/10.1103/PhysRevLett.114.183602

[7] O. Fialkoand D. W. Hallwood ``Isolated Quantum Heat Engine'' Phys. Rev. Lett. 108, 085303 (2012).
https://doi.org/10.1103/PhysRevLett.108.085303
arXiv:1109.1589

[8] Jonatan Bohr Brask, Géraldine Haack, Nicolas Brunner, and Marcus Huber, ``Autonomous quantum thermal machine for generating steady-state entanglement'' New Journal of Physics 17, 113029 (2015).
https://doi.org/10.1088/1367-2630/17/11/113029
arXiv:1504.00187

[9] Jukka P. Pekola ``Towards quantum thermodynamics in electronic?circuits'' Nature Physics 11, 118–123 (2015).
https://doi.org/10.1038/nphys3169

[10] Keye Zhang, Francesco Bariani, and Pierre Meystre, ``Quantum Optomechanical Heat Engine'' Phys. Rev. Lett. 112, 150602 (2014).
https://doi.org/10.1103/PhysRevLett.112.150602
arXiv:1402.6746

[11] H. E. D. Scoviland E. O. Schulz-DuBois ``Three-Level Masers as Heat Engines'' Phys. Rev. Lett. 2, 262–263 (1959).
https://doi.org/10.1103/PhysRevLett.2.262

[12] R Alicki ``The quantum open system as a model of the heat engine'' Journal of Physics A: Mathematical and General 12, L103–L107 (1979).
https://doi.org/10.1088/0305-4470/12/5/007

[13] Marlan O. Scully ``Extracting Work from a Single Thermal Bath via Quantum Negentropy'' Phys. Rev. Lett. 87, 220601 (2001).
https://doi.org/10.1103/PhysRevLett.87.220601

[14] Marlan O. Scully ``Improving the Efficiency of an Ideal Heat Engine: The Quantum Afterburner'' Phys. Rev. Lett 88, 050602 (2002).
https://doi.org/10.1103/PhysRevLett.88.050602

[15] Paul Skrzypczyk, Anthony J. Short, and Sandu Popescu, ``Work extraction and thermodynamics for individual quantum systems'' Nature Communications 5 (2014).
https://doi.org/10.1038/ncomms5185

[16] D. Gelbwaser-Klimovsky, R. Alicki, and G. Kurizki, ``Work and energy gain of heat-pumped quantized amplifiers'' EPL (Europhysics Letters) 103, 60005 (2013).
https://doi.org/10.1209/0295-5075/103/60005
arXiv:1306.1472

[17] Raam Uzdin, Amikam Levy, and Ronnie Kosloff, ``Equivalence of Quantum Heat Machines, and Quantum-Thermodynamic Signatures'' Phys. Rev. X 5, 031044 (2015).
https://doi.org/10.1103/PhysRevX.5.031044

[18] Raam Uzdin, Amikam Levy, and Ronnie Kosloff, ``Quantum Heat Machines Equivalence, Work Extraction beyond Markovianity, and Strong Coupling via Heat Exchangers'' Entropy 18, 124 (2016).
https://doi.org/10.3390/e18040124
arXiv:1602.04925

[19] Arnab Ghosh, David Gelbwaser-Klimovsky, Wolfgang Niedenzu, Alexander I. Lvovsky, Igor Mazets, Marlan O. Scully, and Gershon Kurizki, ``Two-level masers as heat-to-work converters'' Proceedings of the National Academy of Science 115, 9941–9944 (2018).
https://doi.org/10.1073/pnas.1805354115
arXiv:1712.08936

[20] Janet Andersand Vittorio Giovannetti ``Thermodynamics of discrete quantum processes'' New Journal of Physics 15, 033022 (2013).
https://doi.org/10.1088/1367-2630/15/3/033022
arXiv:1211.0183

[21] Krzysztof Szczygielski, David Gelbwaser-Klimovsky, and Robert Alicki, ``Markovian master equation and thermodynamics of a two-level system in a strong laser field'' Phys. Rev. E 87, 012120 (2013).
https://doi.org/10.1103/PhysRevE.87.012120
arXiv:1211.5665

[22] Vasco Cavina, Andrea Mari, and Vittorio Giovannetti, ``Slow Dynamics and Thermodynamics of Open Quantum Systems'' Phys. Rev. Lett. 119, 050601 (2017).
https://doi.org/10.1103/PhysRevLett.119.050601
arXiv:1704.01509

[23] M. Perarnau-Llobet, H. Wilming, A. Riera, R. Gallego, and J. Eisert, ``Strong Coupling Corrections in Quantum Thermodynamics'' Phys. Rev. Lett. 120, 120602 (2018).
https://doi.org/10.1103/PhysRevLett.120.120602
arXiv:1704.05864

[24] Ronnie Kosloffand Amikam Levy ``Quantum Heat Engines and Refrigerators: Continuous Devices'' Annual Review of Physical Chemistry 65, 365–393 (2014) PMID: 24689798.
https://doi.org/10.1146/annurev-physchem-040513-103724

[25] E. B. Davies ``Markovian master equations'' Commun. Math. Phys. 39, 91–110 (1974).
https://doi.org/10.1007/BF01608389

[26] Vittorio Gorini, Andrzej Kossakowski, and E. C. G. Sudarshan, ``Completely positive dynamical semigroups of N-level systems'' Journal of Mathematical Physics 17, 821–825 (1976).
https://doi.org/10.1063/1.522979

[27] G. Lindblad ``On the generators of quantum dynamical semigroups'' Communications in Mathematical Physics 48, 119–130 (1976).
https://doi.org/10.1007/BF01608499

[28] R Alickiand K Lendi ``Quantum dynamical semigroups and applications'' Springer (2007).
https://doi.org/10.1007/3-540-70861-8
https://cds.cern.ch/record/1105909

[29] Raam Uzdinand Ronnie Kosloff ``The multilevel four-stroke swap engine and its environment'' New Journal of Physics 16, 095003 (2014).
https://doi.org/10.1088/1367-2630/16/9/095003

[30] Philipp Strasberg, Gernot Schaller, Tobias Brandes, and Massimiliano Esposito, ``Quantum and Information Thermodynamics: A Unifying Framework Based on Repeated Interactions'' Physical Review X 7 (2017).
https://doi.org/10.1103/physrevx.7.021003

[31] Marco Pezzutto, Mauro Paternostro, and Yasser Omar, ``An out-of-equilibrium non-Markovian quantum heat engine'' Quantum Science and Technology 4, 025002 (2019).
https://doi.org/10.1088/2058-9565/aaf5b4

[32] Stefano Cusumano, Vasco Cavina, Maximilian Keck, Antonella De Pasquale, and Vittorio Giovannetti, ``Entropy production and asymptotic factorization via thermalization: A collisional model approach'' Physical Review A 98 (2018).
https://doi.org/10.1103/physreva.98.032119

[33] Franklin L.S. Rodrigues, Gabriele De Chiara, Mauro Paternostro, and Gabriel T. Landi, ``Thermodynamics of Weakly Coherent Collisional Models'' Physical Review Letters 123 (2019).
https://doi.org/10.1103/physrevlett.123.140601

[34] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and Th. Beth, ``Thermodynamic Cost of Reliability and Low Temperatures: Tightening Landauer's Principle and the Second Law'' International Journal of Theoretical Physics 39, 2717–2753 (2000).
https://doi.org/10.1023/A:1026422630734

[35] R. Streater ``Statistical Dynamics: A Stochastic Approach to nonequilibrium Thermodynamics'' Imperial College Press, London, UK (1995).
https://doi.org/10.1007/BF02174220

[36] Ernst Ruchand Alden Mead ``The principle of increasing mixing character and some of its consequences'' Theoretica chimica acta 41, 95–117 (1976).
https://doi.org/10.1007/BF01178071

[37] Michał Horodeckiand Jonathan Oppenheim ``Fundamental limitations for quantum and nanoscale thermodynamics'' Nature Communications 4 (2013).
https://doi.org/10.1038/ncomms3059

[38] Mischa P. Woods, Nelly Huei Ying Ng, and Stephanie Wehner, ``The maximum efficiency of nano heat engines depends on more than temperature'' Quantum 3, 177 (2019).
https://doi.org/10.22331/q-2019-08-19-177
arXiv:1506.02322

[39] Nelly Huei Ying Ng, Mischa Prebin Woods, and Stephanie Wehner, ``Surpassing the Carnot efficiency by extracting imperfect work'' New Journal of Physics 19, 113005 (2017).
https://doi.org/10.1088/1367-2630/aa8ced

[40] H. T. Quan, Yu-xi Liu, C. P. Sun, and Franco Nori, ``Quantum thermodynamic cycles and quantum heat engines'' Phys. Rev. E 76, 031105 (2007).
https://doi.org/10.1103/PhysRevE.76.031105

[41] Alexandre Roulet, Stefan Nimmrichter, Juan Miguel Arrazola, Stella Seah, and Valerio Scarani, ``Autonomous rotor heat engine'' Phys. Rev. E 95, 062131 (2017).
https://doi.org/10.1103/PhysRevE.95.062131
arXiv:1609.06011

[42] Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry, ``Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality'' Phys. Rev. X 6, 041017 (2016).
https://doi.org/10.1103/PhysRevX.6.041017

[43] Johan Åberg ``Fully Quantum Fluctuation Theorems'' Phys. Rev. X 8, 011019 (2018).
https://doi.org/10.1103/PhysRevX.8.011019

[44] Patryk Lipka-Bartosik, Paweł Mazurek, and Michał Horodecki, ``Second law of thermodynamics for batteries with vacuum state'' arXiv:1905.12072 (2019).
arXiv:1905.12072

[45] W. Puszand S. L. Woronowicz ``Passive states and KMS states for general quantum systems'' Comm. Math. Phys. 58, 273–290 (1978).
https://doi.org/10.1007/BF01614224
https://projecteuclid.org:443/euclid.cmp/1103901491

[46] A. E Allahverdyan, R Balian, and Th. M Nieuwenhuizen, ``Maximal work extraction from finite quantum systems'' Europhysics Letters (EPL) 67, 565–571 (2004).
https://doi.org/10.1209/epl/i2004-10101-2

[47] Robert Alickiand Mark Fannes ``Entanglement boost for extractable work from ensembles of quantum batteries'' Physical Review E 87 (2013).
https://doi.org/10.1103/physreve.87.042123

[48] Peter Talkner, Eric Lutz, and Peter Hänggi, ``Fluctuation theorems: Work is not an observable'' Phys. Rev. E 75, 050102 (2007).
https://doi.org/10.1103/PhysRevE.75.050102

[49] Peter Talknerand Peter Hänggi ``Aspects of quantum work'' Phys. Rev. E 93, 022131 (2016).
https://doi.org/10.1103/PhysRevE.93.022131

[50] Martí Perarnau-Llobet, Elisa Bäumer, Karen V. Hovhannisyan, Marcus Huber, and Antonio Acin, ``No-Go Theorem for the Characterization of Work Fluctuations in Coherent Quantum Systems'' Phys. Rev. Lett. 118, 070601 (2017).
https://doi.org/10.1103/PhysRevLett.118.070601

[51] Johan Åberg ``Truly work-like work extraction via a single-shot analysis'' Nature Communications 4, 1925 (2013).
https://doi.org/10.1038/ncomms2712

[52] Masahito Hayashiand Hiroyasu Tajima ``Measurement-based formulation of quantum heat engines'' Phys. Rev. A 95, 032132 (2017).
https://doi.org/10.1103/PhysRevA.95.032132

[53] R. Sampaio, S. Suomela, T. Ala-Nissila, J. Anders, and T. G. Philbin, ``Quantum work in the Bohmian framework'' Phys. Rev. A 97, 012131 (2018).
https://doi.org/10.1103/PhysRevA.97.012131

[54] Lluís Masanesand Jonathan Oppenheim ``A general derivation and quantification of the third law of thermodynamics'' Nature Communications 8, 14538 (2017).
https://doi.org/10.1038/ncomms14538

[55] A. E Allahverdyan, R Balian, and Th. M Nieuwenhuizen, ``Maximal work extraction from finite quantum systems'' Europhysics Letters (EPL) 67, 565–571 (2004).
https://doi.org/10.1209/epl/i2004-10101-2

[56] Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry, ``Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality'' Phys. Rev. X 6, 041017 (2016).
https://doi.org/10.1103/PhysRevX.6.041017

[57] Piotr Ć wikliński, Michał Studziński, Michał Horodecki, and Jonathan Oppenheim, ``Limitations on the Evolution of Quantum Coherences: Towards Fully Quantum Second Laws of Thermodynamics'' Phys. Rev. Lett. 115, 210403 (2015).
https://doi.org/10.1103/PhysRevLett.115.210403

[58] F. L. Curzonand B. Ahlborn ``Efficiency of a Carnot engine at maximum power output'' American Journal of Physics 43, 22–24 (1975).
https://doi.org/10.1119/1.10023

[59] Matteo Lostaglio, Álvaro M. Alhambra, and Christopher Perry, ``Elementary Thermal Operations'' Quantum 2, 52 (2018).
https://doi.org/10.22331/q-2018-02-08-52

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