The maximum efficiency of nano heat engines depends on more than temperature

Mischa P. Woods1,2, Nelly Huei Ying Ng2,3, and Stephanie Wehner2

1Institute for Theoretical Physics, ETH Zurich, Switzerland
2QuTech, Delft University of Technology, Lorentzweg 1, 2611 CJ Delft, Netherlands
3Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

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

Abstract

Sadi Carnot's theorem regarding the maximum efficiency of heat engines is considered to be of fundamental importance in thermodynamics. This theorem famously states that the maximum efficiency depends only on the temperature of the heat baths used by the engine, but not on the specific structure of baths. Here, we show that when the heat baths are finite in size, and when the engine operates in the quantum nanoregime, a revision to this statement is required. We show that one may still achieve the Carnot efficiency, when certain conditions on the bath structure are satisfied; however if that is not the case, then the maximum achievable efficiency can reduce to a value which is strictly less than Carnot. We derive the maximum efficiency for the case when one of the baths is composed of qubits. Furthermore, we show that the maximum efficiency is determined by either the standard second law of thermodynamics, analogously to the macroscopic case, or by the non increase of the max relative entropy, which is a quantity previously associated with the single shot regime in many quantum protocols. This relative entropic quantity emerges as a consequence of additional constraints, called generalized free energies, that govern thermodynamical transitions in the nanoregime. Our findings imply that in order to maximize efficiency, further considerations in choosing bath Hamiltonians should be made, when explicitly constructing quantum heat engines in the future. This understanding of thermodynamics has implications for nanoscale engineering aiming to construct small thermal machines.

Nicolas Carnot famously showed that all heat engines, regardless of their heat source or working substance, could, in principle, achieve a theoretical maximum efficiency for converting heat into work; which only depended on the temperature of its thermal baths. This principle has underpinned a transport revolution from steam engines to jet planes. However, the principle was derived for macroscopic machines, and with the advent of quantum nano-devices, the question of whether this universal principle still holds is of great importance. Here, we show that in the realm of the very small where particles obey the laws of quantum physics, Carnot’s theoretical maximum still holds — but with a catch; that the quantized baths must have sufficiently small energy gaps. Baths with energy gaps that are too large will still allow for functioning heat engines, but with a reduced maximum efficiency. This imposes an important fundamental limitation on the future design of nanoscale heat engines.
When the system of interest is characterized by only a few quantum particles, the statistical arguments based on the law of large numbers break down. Therefore, in addition to the usual second law of thermodynamics, additional restrictions arise to govern the possibility of a thermodynamical state transition. It is due to these additional constraints that Carnot’s efficiency ceases to be universally achievable. This work is an important milestone in a growing body of work aiming to revolutionize our understanding of the nature of work and heat at the nanoscale.

► BibTeX data

► References

[1] S. Carnot ``Reflections on the Motive Power of Fire'' (1824).

[2] R. Clausius ``Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie'' Annalen der Physik 201, 353–400 (1865).

[3] W. Thompson (Lord Kelvin) ``On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam'' Transactions of the Royal Society of Edinburgh (1851).
https:/​/​doi.org/​10.1080/​14786445208647064

[4] N. Ngand M. P. Woods ``Resource Theory of Quantum Thermodynamics: Thermal Operations and Second Laws'' Springer International Publishing (2018).
https:/​/​doi.org/​10.1007/​978-3-319-99046-0_26

[5] F. Brandão, M. Horodecki, J. Oppenheim, J.M. Renes, and R.W. Spekkens, ``Resource theory of quantum states out of thermal equilibrium'' Physical Review Letters 111, 250404 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.250404

[6] M. Horodeckiand J. Oppenheim ``Fundamental limitations for quantum and nano thermodynamics'' Nature Communications 4 (2013).
https:/​/​doi.org/​10.1038/​ncomms3059

[7] F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, ``The second laws of quantum thermodynamics'' Proceedings of the National Academy of Sciences 112, 3275–3279 (2015).
https:/​/​doi.org/​10.1073/​pnas.1411728112

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

[9] O. Dahlsten, R. Renner, E. Rieper, and V. Vedral, ``Inadequacy of von Neumann entropy for characterizing extractable work'' New Journal of Physics 13, 053015 (2011).
https:/​/​doi.org/​10.1088/​1367-2630/​13/​5/​053015

[10] R. Uzdin, A. Levy, and R. 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

[11] R. Gallego, J. Eisert, and H. Wilming, ``Thermodynamic work from operational principles'' New Journal of Physics 18, 103017 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​10/​103017

[12] J. Gemmer.and J. Anders. ``From single-shot towards general work extraction in a quantum thermodynamic framework'' New Journal of Physics 17, 085006 (2015).
https:/​/​doi.org/​10.1038/​ncomms5185

[13] M. Lostaglio, D. Jennings, and T. Rudolph, ``Description of quantum coherence in thermodynamic processes requires constraints beyond free energy'' Nature Communications 6 (2015).
https:/​/​doi.org/​10.1038/​ncomms7383

[14] M. O. Scully, M. S. Zubairy, G. S. Agarwal, and H. Walther, ``Extracting work from a single heat bath via vanishing quantum coherence'' Science 299, 862–864 (2003).
https:/​/​doi.org/​10.1126/​science.1078955

[15] M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, ``Quantum Coherence, Time-Translation Symmetry, and Thermodynamics'' Physical Review X 5, 021001 (2015).
https:/​/​doi.org/​10.1103/​PhysRevX.5.021001

[16] F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, ``Quantum thermodynamics of general quantum processes'' Physical Review E 91, 032119 (2015).
https:/​/​doi.org/​10.1103/​PhysRevE.91.032119

[17] S. Salekand K. Wiesner ``Fluctuations in single-shot $ε$-deterministic work extraction'' Physical Review A 96, 052114 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.052114

[18] H. Tajimaand M. Hayashi ``Finite-size effect on optimal efficiency of heat engines'' Physical Review E 96, 012128 (2017).
https:/​/​doi.org/​10.1103/​PhysRevE.96.012128

[19] J. Gemmer, M. Michel, and G. Mahler, ``Quantum Thermodynamics: Emergence of Thermodynamic Behavior Within Composite Quantum Systems'' Springer Berlin Heidelberg (2009).

[20] J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and E. Lutz, ``Nanoscale Heat Engine Beyond the Carnot Limit'' Physical Review Letters 112, 030602 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.030602

[21] B. Gardasand S. Deffner ``Thermodynamic universality of quantum Carnot engines'' Physical Review E 92, 042126 (2015).
https:/​/​doi.org/​10.1103/​PhysRevE.92.042126

[22] M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Acín, ``Extractable Work from Correlations'' Physical Review X 5, 041011 (2015).
https:/​/​doi.org/​10.1103/​PhysRevX.5.041011

[23] A. M. Alhambra, J. Oppenheim, and C. Perry, ``Fluctuating States: What is the Probability of a Thermodynamical Transition?'' Physical Review X 6, 041016 (2016).
https:/​/​doi.org/​10.1103/​PhysRevX.6.041016

[24] A. M. Alhambra, L. Masanes, J. Oppenheim, and C. Perry, ``Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality'' Physical Review X 6, 041017 (2016).
https:/​/​doi.org/​10.1103/​PhysRevX.6.041017

[25] H. Spohnand J. L. Lebowitz ``Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs'' Adv. Chem. Phys 38, 109–142 (1978).
https:/​/​doi.org/​10.1002/​9780470142578.ch2

[26] M. T. Mitchisonand P. Potts ``Physical Implementations of Quantum Absorption Refrigerators'' Springer International Publishing (2018).
https:/​/​doi.org/​10.1007/​978-3-319-99046-0_6

[27] T. Batalhão, S. Gherardini, J. Santos, G. Landi, and M. Paternostro, ``Characterizing Irreversibility in Open Quantum Systems'' Springer International Publishing (2018).
https:/​/​doi.org/​10.1007/​978-3-319-99046-0_16

[28] P. Skrzypczyk, N. Brunner, N. Linden, and S. Popescu, ``The smallest refrigerators can reach maximal efficiency'' Journal of Physics A: Mathematical and Theoretical 44, 492002–492008 (2011).

[29] N. Brunner, N. Linden, S. Popescu, and P. Skrzypczyk, ``Virtual qubits, virtual temperatures, and the foundations of thermodynamics'' Physical Review E 85, 051117 (2012).
https:/​/​doi.org/​10.1103/​PhysRevE.85.051117

[30] M. T. Mitchison, M. P. Woods, J. Prior, and M. Huber, ``Coherence-assisted single-shot cooling by quantum absorption refrigerators'' New Journal of Physics 17, 115013 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​11/​115013

[31] N. Brunner, M. Huber, N. Linden, S. Popescu, R. Silva, and P. Skrzypczyk, ``Entanglement enhances cooling in microscopic quantum refrigerators'' Physical Review E 89, 032115 (2014).
https:/​/​doi.org/​10.1103/​PhysRevE.89.032115

[32] 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

[33] D. Gelbwaser-Klimovsky, W. Niedenzu, and G. Kurizki, ``Chapter Twelve - Thermodynamics of Quantum Systems Under Dynamical Control'' Academic Press (2015).
https:/​/​doi.org/​10.1016/​bs.aamop.2015.07.002

[34] R. Kosloff ``Quantum Thermodynamics: A Dynamical Viewpoint'' Entropy 15, 2100–2128 (2013).
https:/​/​doi.org/​10.1146/​annurev-physchem-040513-103724

[35] R. Kosloffand A. 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

[36] B. Bylicka, M. Tukiainen, D. Chruściński, J. Piilo, and S. Maniscalco, ``Thermodynamic power of non-Markovianity'' Scientific Reports 6 (2016).
https:/​/​doi.org/​10.1038/​srep27989

[37] P. Skrzypczyk, A.J. Short, and S. Popescu, ``Work extraction and thermodynamics for individual quantum systems'' Nature Communications 5 (2014).
https:/​/​doi.org/​10.1038/​ncomms5185

[38] L. Del Rio, J. Åberg, R. Renner, O. Dahlsten, and V. Vedral, ``The thermodynamic meaning of negative entropy'' Nature 474, 61–63 (2011).
https:/​/​doi.org/​10.1038/​nature10123

[39] P. Faist, F. Dupuis, J. Oppenheim, and R. Renner, ``The minimal work cost of information processing'' Nature Communications 6 (2015).
https:/​/​doi.org/​10.1038/​ncomms8669

[40] S. Deffnerand C. Jarzynski ``Information processing and the second law of thermodynamics: An inclusive, Hamiltonian approach'' Physical Review X 3, 041003 (2013).
https:/​/​doi.org/​10.1103/​PhysRevX.3.041003

[41] F. Campaioli, F. A. Pollock, F. C. Binder, L. Céleri, J. Goold, S. Vinjanampathy, and K. Modi, ``Enhancing the charging power of quantum batteries'' Physical Review Letters 118, 150601 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.118.150601

[42] W. Puszand S. L. Woronowicz ``Passive states and KMS states for general quantum systems'' Communications in Mathematical Physics 58, 273–290 (1978).
https:/​/​doi.org/​10.1007/​BF01614224

[43] M. Woods, R. Silva, and J. Oppenheim, ``Autonomous Quantum Machines and Finite-Sized Clocks'' Annales Henri Poincaré (2018).
https:/​/​doi.org/​10.1007/​s00023-018-0736-9

[44] H. Wilming, R. Gallego, and J. Eisert, ``Axiomatic Characterization of the Quantum Relative Entropy and Free Energy'' Entropy 19, 241 (2017).
https:/​/​doi.org/​10.3390/​e19060241

[45] M. Esposito, K. Lindenberg, and C. Van den Broeck, ``Entropy production as correlation between system and reservoir'' New Journal of Physics 12, 013013 (2010).
https:/​/​doi.org/​10.1088/​1367-2630/​12/​1/​013013

[46] J. Parrondo, J. M. Horowitz, and T. Sagawa, ``Thermodynamics of information'' Nature Physics 11, 131–139 (2015).
https:/​/​doi.org/​10.1038/​nphys3230

[47] M. P. Müller ``Correlating Thermal Machines and the Second Law at the Nanoscale'' Physical Review X 8, 041051 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.041051

[48] E. T. Jaynes ``Information theory and statistical mechanics'' Physical Review 106, 620 (1957).
https:/​/​doi.org/​10.1103/​PhysRev.106.620

[49] N. Ng, M. P. Woods, and S. Wehner, ``Surpassing the Carnot efficiency by extracting imperfect work'' New Journal of Physics 19, 113005 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa8ced

[50] R. Rennerand S. Wolf ``Smooth Rényi entropy and applications'' IEEE International Symposium on Information Theory 233–233 (2004).
https:/​/​doi.org/​10.1109/​ISIT.2004.1365269

[51] R. Konig, R. Renner, and C. Schaffner, ``The operational meaning of min-and max-entropy'' IEEE Transactions on Information theory 55, 4337–4347 (2009).
https:/​/​doi.org/​10.1109/​TIT.2009.2025545

[52] M. Tomamichel, C. Schaffner, A. Smith, and R. Renner, ``Leftover hashing against quantum side information'' IEEE Transactions on Information Theory 57, 5524–5535 (2011).
https:/​/​doi.org/​10.1109/​TIT.2011.2158473

[53] L. Wang, R. Colbeck, and R. Renner, ``Simple channel coding bounds'' 2009 IEEE International Symposium on Information Theory 1804–1808 (2009).
https:/​/​doi.org/​10.1109/​ISIT.2009.5205312

[54] I. Csiszár ``Generalized cutoff rates and Renyi's information measures'' IEEE Transactions on information theory 41, 26–34 (1995).
https:/​/​doi.org/​10.1109/​18.370121

[55] O. Shayevitz ``On Rényi measures and hypothesis testing'' ISIT 894–898 (2011).
https:/​/​doi.org/​10.1109/​ISIT.2011.6034266

[56] N. Datta ``Min-and max-relative entropies and a new entanglement monotone'' IEEE Transactions on Information Theory 55, 2816–2826 (2009).
https:/​/​doi.org/​10.1109/​TIT.2009.2025545

[57] F. G. S. L. Brandãoand N. Datta ``One-shot rates for entanglement manipulation under non-entangling maps'' IEEE Transactions on Information Theory 57, 1754–1760 (2011).
https:/​/​doi.org/​10.1109/​TIT.2011.2104531

[58] M. Espositoand C. Van den Broeck ``Second law and Landauer principle far from equilibrium'' EPL (Europhysics Letters) 95, 40004 (2011).
https:/​/​doi.org/​10.1209/​0295-5075/​95/​40004

[59] M. Lostaglio, M. P. Müller, and M. Pastena, ``Stochastic Independence as a Resource in Small-Scale Thermodynamics'' Physical Review Letters 115, 150402 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.150402

[60] F. Reif ``Fundamentals of statistical and thermal physics'' McGraw-Hill (1965).

[61] R. Uzdin, S. Gasparinetti, R. Ozeri, and R. Kosloff, ``Markovian heat sources with the smallest heat capacity'' New Journal of Physics 20, 063030 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aac932

[62] R. Uzdin, A. Levy, and R. Kosloff, ``Equivalence of Quantum Heat Machines, and Quantum-Thermodynamic Signatures'' Physical Review X 5, 031044 (2015).
https:/​/​doi.org/​10.1103/​PhysRevX.5.031044

[63] V. Shoup ``A Computational Introduction to Number Theory and Algebra'' Cambridge University Press (2009).

[64] T. M. Coverand J. A. Thomas ``Elements of Information Theory 2nd Edition (Wiley Series in Telecommunications and Signal Processing)'' Wiley-Interscience (2006).

[65] C. Shannon ``A Mathematical Theory of Communication'' Bell Labs Technical Journal 27, 379 (1948).
https:/​/​doi.org/​10.1002/​j.1538-7305.1948.tb01338.x

[66] S. P. Boydand L. Vandenberghe ``Convex Optimization'' Cambridge University Press (2004).

Cited by

[1] Marek Gluza, João Sabino, Nelly H.Y. Ng, Giuseppe Vitagliano, Marco Pezzutto, Yasser Omar, Igor Mazets, Marcus Huber, Jörg Schmiedmayer, and Jens Eisert, "Quantum Field Thermal Machines", PRX Quantum 2 3, 030310 (2021).

[2] Mischa P. Woods and Michał Horodecki, "Autonomous Quantum Devices: When Are They Realizable without Additional Thermodynamic Costs?", Physical Review X 13 1, 011016 (2023).

[3] Carlo Sparaciari, Marcel Goihl, Paul Boes, Jens Eisert, and Nelly Huei Ying Ng, "Bounding the resources for thermalizing many-body localized systems", Communications Physics 4 1, 3 (2021).

[4] Marcin Łobejko, Paweł Mazurek, and Michał Horodecki, "Thermodynamics of Minimal Coupling Quantum Heat Engines", Quantum 4, 375 (2020).

[5] Federico Cerisola, Facundo Sapienza, and Augusto J. Roncaglia, "Heat engines with single-shot deterministic work extraction", Physical Review E 106 3, 034135 (2022).

[6] Naoto Shiraishi and Takahiro Sagawa, "Quantum Thermodynamics of Correlated-Catalytic State Conversion at Small Scale", Physical Review Letters 126 15, 150502 (2021).

[7] Tanmoy Biswas, A. de Oliveira Junior, Michał Horodecki, and Kamil Korzekwa, "Fluctuation-dissipation relations for thermodynamic distillation processes", Physical Review E 105 5, 054127 (2022).

[8] Feng-Jui Chan, Yi-Te Huang, Jhen-Dong Lin, Huan-Yu Ku, Jui-Sheng Chen, Hong-Bin Chen, and Yueh-Nan Chen, "Maxwell's two-demon engine under pure dephasing noise", Physical Review A 106 5, 052201 (2022).

[9] 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).

[10] Naoto Shiraishi, "Two constructive proofs on d-majorization and thermo-majorization", Journal of Physics A: Mathematical and Theoretical 53 42, 425301 (2020).

[11] Mohit Lal Bera, Maciej Lewenstein, and Manabendra Nath Bera, "Attaining Carnot efficiency with quantum and nanoscale heat engines", npj Quantum Information 7 1, 31 (2021).

[12] Mohit Lal Bera, Sergi Julià-Farré, Maciej Lewenstein, and Manabendra Nath Bera, "Quantum heat engines with Carnot efficiency at maximum power", Physical Review Research 4 1, 013157 (2022).

[13] Philipp Strasberg, Gernot Schaller, Tobias Brandes, and Massimiliano Esposito, "Quantum and Information Thermodynamics: A Unifying Framework Based on Repeated Interactions", Physical Review X 7 2, 021003 (2017).

[14] Yelena Guryanova, Sandu Popescu, Anthony J. Short, Ralph Silva, and Paul Skrzypczyk, "Thermodynamics of quantum systems with multiple conserved quantities", Nature Communications 7, 12049 (2016).

[15] Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry, "Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality", Physical Review X 6 4, 041017 (2016).

[16] Jonatan Bohr Brask and Nicolas Brunner, "Small quantum absorption refrigerator in the transient regime: Time scales, enhanced cooling, and entanglement", Physical Review E 92 6, 062101 (2015).

[17] Markus P. Müller, "Correlating Thermal Machines and the Second Law at the Nanoscale", Physical Review X 8 4, 041051 (2018).

[18] Johan Åberg, "Fully Quantum Fluctuation Theorems", Physical Review X 8 1, 011019 (2018).

[19] Matteo Lostaglio, Markus P. Müller, and Michele Pastena, "Stochastic Independence as a Resource in Small-Scale Thermodynamics", Physical Review Letters 115 15, 150402 (2015).

[20] Carlo Sparaciari, Jonathan Oppenheim, and Tobias Fritz, "Resource theory for work and heat", Physical Review A 96 5, 052112 (2017).

[21] Raam Uzdin, "Coherence-Induced Reversibility and Collective Operation of Quantum Heat Machines via Coherence Recycling", Physical Review Applied 6 2, 024004 (2016).

[22] Nicole Yunger Halpern, Christopher David White, Sarang Gopalakrishnan, and Gil Refael, "Quantum engine based on many-body localization", Physical Review B 99 2, 024203 (2019).

[23] Raam Uzdin, Amikam Levy, and Ronnie Kosloff, "Quantum Heat Machines Equivalence, Work Extraction beyond Markovianity, and Strong Coupling via Heat Exchangers", Entropy 18 4, 124 (2016).

[24] Ralph Silva, Gonzalo Manzano, Paul Skrzypczyk, and Nicolas Brunner, "Performance of autonomous quantum thermal machines: Hilbert space dimension as a thermodynamical resource", Physical Review E 94 3, 032120 (2016).

[25] Carlo Sparaciari, David Jennings, and Jonathan Oppenheim, "Energetic instability of passive states in thermodynamics", Nature Communications 8, 1895 (2017).

[26] Adrian Chapman and Akimasa Miyake, "How an autonomous quantum Maxwell demon can harness correlated information", Physical Review E 92 6, 062125 (2015).

[27] Philippe Faist and Renato Renner, "Fundamental Work Cost of Quantum Processes", Physical Review X 8 2, 021011 (2018).

[28] Jonathan G. Richens, Álvaro M. Alhambra, and Lluis Masanes, "Finite-bath corrections to the second law of thermodynamics", Physical Review E 97 6, 062132 (2018).

[29] Alejandro Pozas-Kerstjens, Eric G. Brown, and Karen V. Hovhannisyan, "A quantum Otto engine with finite heat baths: energy, correlations, and degradation", New Journal of Physics 20 4, 043034 (2018).

[30] Nelly Huei Ying Ng, Mischa Prebin Woods, and Stephanie Wehner, "Surpassing the Carnot efficiency by extracting imperfect work", New Journal of Physics 19 11, 113005 (2017).

[31] V. Mukherjee, W. Niedenzu, A. G. Kofman, and G. Kurizki, "Speed and efficiency limits of multilevel incoherent heat engines", Physical Review E 94 6, 062109 (2016).

[32] Chris Perry, Piotr Ćwikliński, Janet Anders, Michał Horodecki, and Jonathan Oppenheim, "A sufficient set of experimentally implementable thermal operations", arXiv:1511.06553, (2015).

[33] Jonathan G. Richens and Lluis Masanes, "Work extraction from quantum systems with bounded fluctuations in work", Nature Communications 7, 13511 (2016).

[34] Kosuke Ito and Masahito Hayashi, "Optimal performance of generalized heat engines with finite-size baths of arbitrary multiple conserved quantities beyond independent-and-identical-distribution scaling", Physical Review E 97 1, 012129 (2018).

[35] Remco van der Meer, Nelly Huei Ying Ng, and Stephanie Wehner, "Smoothed generalized free energies for thermodynamics", Physical Review A 96 6, 062135 (2017).

[36] Christopher T. Chubb, Marco Tomamichel, and Kamil Korzekwa, "Beyond the thermodynamic limit: finite-size corrections to state interconversion rates", Quantum 2, 108 (2018).

[37] Nicole Yunger Halpern, Christopher David White, Sarang Gopalakrishnan, and Gil Refael, "MBL-mobile: Quantum engine based on many-body localization", arXiv:1707.07008, (2017).

[38] Christopher Perry, Piotr Ćwikliński, Janet Anders, Michał Horodecki, and Jonathan Oppenheim, "A Sufficient Set of Experimentally Implementable Thermal Operations for Small Systems", Physical Review X 8 4, 041049 (2018).

[39] Álvaro M. Alhambra, Stephanie Wehner, Mark M. Wilde, and Mischa P. Woods, "Work and reversibility in quantum thermodynamics", Physical Review A 97 6, 062114 (2018).

[40] Max F. Frenzel, David Jennings, and Terry Rudolph, "Quasi-autonomous quantum thermal machines and quantum to classical energy flow", New Journal of Physics 18 2, 023037 (2016).

[41] Johan Aberg, "Fully quantum fluctuation theorems", arXiv:1601.01302, (2016).

[42] Jonas F. G. Santos, "Gravitational quantum well as an effective quantum heat engine", European Physical Journal Plus 133 8, 321 (2018).

[43] Amikam Levy and David Gelbwaser-Klimovsky, "Quantum Features and Signatures of Quantum Thermal Machines", Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions 195, 87 (2018).

[44] Raam Uzdin, "Additional energy-information relations in thermodynamics of small systems", Physical Review E 96 3, 032128 (2017).

[45] Angeline Shu, Jibo Dai, and Valerio Scarani, "Power of an optical Maxwell's demon in the presence of photon-number correlations", Physical Review A 95 2, 022123 (2017).

[46] Pharnam Bakhshinezhad, Beniamin R. Jablonski, Felix C. Binder, and Nicolai Friis, "Trade-offs between precision and fluctuations in charging finite-dimensional quantum systems", arXiv:2303.16676, (2023).

[47] Kosuke Ito and Masahito Hayashi, "Optimal performance of generalized heat engines with finite-size baths of arbitrary multiple conserved quantities beyond i.i.d. scaling", arXiv:1612.04047, (2016).

[48] Akihito Kato and Yoshitaka Tanimura, "Hierarchical Equations of Motion Approach to Quantum Thermodynamics", Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions 195, 579 (2018).

[49] Jonatan Bohr Brask and Nicolas Brunner, "Small quantum absorption refrigerator in the transient regime: time scales, enhanced cooling and entanglement", arXiv:1508.02025, (2015).

[50] Nicole Yunger Halpern, "Toward physical realizations of thermodynamic resource theories", arXiv:1509.03873, (2015).

[51] Momir Arsenijevic, Jasmina Jeknic-Dugic, and Miroljub Dugic, "Collective versus individual classicality for a pair of interacting qubits immersed in independent local environments", arXiv:1605.09270, (2016).

The above citations are from Crossref's cited-by service (last updated successfully 2023-11-29 11:52:53) and SAO/NASA ADS (last updated successfully 2023-11-29 11:52:54). The list may be incomplete as not all publishers provide suitable and complete citation data.