Imperfect Thermalizations Allow for Optimal Thermodynamic Processes

Elisa Bäumer1, Martí Perarnau-Llobet2, Philipp Kammerlander1, Henrik Wilming1, and Renato Renner1

1Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland
2Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany

Optimal (reversible) processes in thermodynamics can be modelled as step-by-step processes, where the system is successively thermalized with respect to different Hamiltonians by an external thermal bath. However, in practice interactions between system and thermal bath will take finite time, and precise control of their interaction is usually out of reach. Motivated by this observation, we consider finite-time and uncontrolled operations between system and bath, which result in thermalizations that are only partial in each step. We show that optimal processes can still be achieved for any non-trivial partial thermalizations at the price of increasing the number of operations, and characterise the corresponding tradeoff. We focus on work extraction protocols and show our results in two different frameworks: A collision model and a model where the Hamiltonian of the working system is controlled over time and the system can be brought into contact with a heat bath. Our results show that optimal processes are robust to noise and imperfections in small quantum systems, and can be achieved by a large set of interactions between system and bath.

As we know from everyday live, when an object, such as a cup of coffee, is put in contact with a large body at a certain temperature, often called "heat bath", it will assume the temperature of the heat bath.
This is the very basic mechanism used in thermal machines, such as engines or power plants.
Recently, the study of thermal machines in the quantum regime has received great attention.
In this field of study it is customary to model a system, after it has been in contact with the heat bath, by a so-called canonical ensemble.
One says that the system "thermalizes" to the canonical ensemble.
Therefore, many recent results rest on the assumption that the system under consideration thermalizes exactly to the canonical ensemble due the interaction with a heat bath.
We show that a large class of results is still valid when only imperfect thermalization occurs due to the interaction and hence the system cannot be described exactly by the canonical ensemble.
We further show that not even the duration of certain protocols to run engines is modified significantly.
This shows that general arguments about thermodynamics in the quantum regime are not very sensitive to imperfect thermalization processes.

► BibTeX data

► References

[1] John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, and Paul Skrzypczyk. The role of quantum information in thermodynamics—a topical review. Journal of Physics A: Mathematical and Theoretical, 49 (14): 143001, Feb 2016. ISSN 1751-8121. 10.1088/​1751-8113/​49/​14/​143001. URL http:/​/​dx.doi.org/​10.1088/​1751-8113/​49/​14/​143001.
https:/​/​doi.org/​10.1088/​1751-8113/​49/​14/​143001

[2] Sai Vinjanampathy and Janet Anders. Quantum thermodynamics. Contemporary Physics, 57 (4): 545–579, Jul 2016. ISSN 1366-5812. 10.1080/​00107514.2016.1201896. URL http:/​/​dx.doi.org/​10.1080/​00107514.2016.1201896.
https:/​/​doi.org/​10.1080/​00107514.2016.1201896

[3] H.-P Breuer and Francesco Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 01 2006. 10.1093/​acprof:oso/​9780199213900.001.0001.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199213900.001.0001

[4] Märio Ziman, Peter Štelmachovič, and Vladimír Bužek. Description of quantum dynamics of open systems based on collision-like models. Open Systems & Information Dynamics, 12 (1): 81-91, Mar 2005. ISSN 1573-1324. 10.1007/​s11080-005-0488-0. URL https:/​/​doi.org/​10.1007/​s11080-005-0488-0.
https:/​/​doi.org/​10.1007/​s11080-005-0488-0

[5] Fernando G. S. L. Brandão, Michał Horodecki, Jonathan Oppenheim, Joseph M. Renes, and Robert W. Spekkens. Resource theory of quantum states out of thermal equilibrium. Phys. Rev. Lett., 111: 250404, Dec 2013. 10.1103/​PhysRevLett.111.250404. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.111.250404.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.250404

[6] H. Wilming, R. Gallego, and J. Eisert. Second law of thermodynamics under control restrictions. Phys. Rev. E, 93: 042126, Apr 2016. 10.1103/​PhysRevE.93.042126. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevE.93.042126.
https:/​/​doi.org/​10.1103/​PhysRevE.93.042126

[7] J Lekscha, H Wilming, J Eisert, and R Gallego. Quantum thermodynamics with local control. Physical Review E, 97 (2): 022142, 2018. 10.1103/​PhysRevE.97.022142.
https:/​/​doi.org/​10.1103/​PhysRevE.97.022142

[8] Eric G Brown, Nicolai Friis, and Marcus Huber. Passivity and practical work extraction using gaussian operations. New Journal of Physics, 18 (11): 113028, 2016. 10.1088/​1367-2630/​18/​11/​113028. URL http:/​/​stacks.iop.org/​1367-2630/​18/​i=11/​a=113028.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​11/​113028
http:/​/​stacks.iop.org/​1367-2630/​18/​i=11/​a=113028

[9] Nicolai Friis and Marcus Huber. Precision and work fluctuations in gaussian battery charging. Quantum, 2: 61, apr 2018. 10.22331/​q-2018-04-23-61.
https:/​/​doi.org/​10.22331/​q-2018-04-23-61

[10] Matteo Lostaglio, Álvaro M Alhambra, and Christopher Perry. Elementary thermal operations. Quantum, 2: 52, 2018. 10.22331/​q-2018-02-08-52.
https:/​/​doi.org/​10.22331/​q-2018-02-08-52

[11] Christopher Perry, Piotr Ć wikliński, Janet Anders, Michał Horodecki, and Jonathan Oppenheim. A sufficient set of experimentally implementable thermal operations for small systems. Phys. Rev. X, 8: 041049, Dec 2018. 10.1103/​PhysRevX.8.041049. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.8.041049.
https:/​/​doi.org/​10.1103/​PhysRevX.8.041049

[12] J. Nulton, P. Salamon, B. Andresen, and Qi Anmin. Quasistatic processes as step equilibrations. The Journal of Chemical Physics, 83 (1): 334-338, jul 1985. 10.1063/​1.449774. URL https:/​/​doi.org/​10.1063/​1.449774.
https:/​/​doi.org/​10.1063/​1.449774

[13] Janet Anders and Vittorio Giovannetti. Thermodynamics of discrete quantum processes. New Journal of Physics, 15 (3): 033022, mar 2013. 10.1088/​1367-2630/​15/​3/​033022. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​3/​033022

[14] P. Skrzypczyk, A. J. Short, and S. Popescu. Extracting work from quantum systems. arXiv:1302.2811, 2013. URL https:/​/​arxiv.org/​abs/​1302.2811.
arXiv:1302.2811

[15] David Reeb and Michael M Wolf. An improved landauer principle with finite-size corrections. New Journal of Physics, 16 (10): 103011, oct 2014. 10.1088/​1367-2630/​16/​10/​103011. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​10/​103011

[16] Valerio Scarani, Mário Ziman, Peter Štelmachovič, Nicolas Gisin, and Vladimír Bužek. Thermalizing quantum machines: Dissipation and entanglement. Phys. Rev. Lett., 88: 097905, Feb 2002. 10.1103/​PhysRevLett.88.097905. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.88.097905.
https:/​/​doi.org/​10.1103/​PhysRevLett.88.097905

[17] P. Filipowicz, J. Javanainen, and P. Meystre. Theory of a microscopic maser. Phys. Rev. A, 34: 3077-3087, Oct 1986. 10.1103/​PhysRevA.34.3077. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.34.3077.
https:/​/​doi.org/​10.1103/​PhysRevA.34.3077

[18] Carlton M. Caves and G. J. Milburn. Quantum-mechanical model for continuous position measurements. Phys. Rev. A, 36: 5543-5555, Dec 1987. 10.1103/​PhysRevA.36.5543. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.36.5543.
https:/​/​doi.org/​10.1103/​PhysRevA.36.5543

[19] Johan Åberg. Catalytic coherence. Phys. Rev. Lett., 113: 150402, Oct 2014. 10.1103/​PhysRevLett.113.150402. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.113.150402.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.150402

[20] J. Åberg. Truly work-like work extraction via a single-shot analysis. Nat. Commun., 4 (1925): 1925, 2013. 10.1038/​ncomms2712. URL https:/​/​www.nature.com/​articles/​ncomms2712.
https:/​/​doi.org/​10.1038/​ncomms2712
https:/​/​www.nature.com/​articles/​ncomms2712

[21] M. Ziman, P. Štelmachovič, V. Bužek, M. Hillery, V. Scarani, and N. Gisin. Diluting quantum information: An analysis of information transfer in system-reservoir interactions. Phys. Rev. A, 65: 042105, Mar 2002. 10.1103/​PhysRevA.65.042105. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.65.042105.
https:/​/​doi.org/​10.1103/​PhysRevA.65.042105

[22] Giuseppe Gennaro, Giuliano Benenti, and G. Massimo Palma. Relaxation due to random collisions with a many-qudit environment. Phys. Rev. A, 79: 022105, Feb 2009. 10.1103/​PhysRevA.79.022105. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.79.022105.
https:/​/​doi.org/​10.1103/​PhysRevA.79.022105

[23] Stefano Cusumano, Vasco Cavina, Maximilian Keck, Antonella De Pasquale, and Vittorio Giovannetti. Entropy production and asymptotic factorization via thermalization: A collisional model approach. Phys. Rev. A, 98: 032119, Sep 2018. 10.1103/​PhysRevA.98.032119. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.98.032119.
https:/​/​doi.org/​10.1103/​PhysRevA.98.032119

[24] Zhong-Xiao Man, Yun-Jie Xia, and Rosario Lo Franco. Temperature effects on quantum non-markovianity via collision models. Phys. Rev. A, 97: 062104, Jun 2018. 10.1103/​PhysRevA.97.062104. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.97.062104.
https:/​/​doi.org/​10.1103/​PhysRevA.97.062104

[25] Lajos Diósi, Tova Feldmann, and Ronnie Kosloff. On the exact identity between thermodynamic and informatic entropies in a unitary model of friction. International Journal of Quantum Information, 4 (01): 99-104, 2006. 10.1142/​S0219749906001645. URL https:/​/​www.worldscientific.com/​doi/​abs/​10.1142/​S0219749906001645.
https:/​/​doi.org/​10.1142/​S0219749906001645

[26] Raam Uzdin and Ronnie Kosloff. The multilevel four-stroke swap engine and its environment. New Journal of Physics, 16 (9): 095003, 2014. 10.1088/​1367-2630/​16/​9/​095003. URL http:/​/​stacks.iop.org/​1367-2630/​16/​i=9/​a=095003.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​9/​095003
http:/​/​stacks.iop.org/​1367-2630/​16/​i=9/​a=095003

[27] Felipe Barra. The thermodynamic cost of driving quantum systems by their boundaries. Scientific reports, 5 (14873): 14873, 2015. 10.1038/​srep14873. URL https:/​/​www.nature.com/​articles/​srep14873.
https:/​/​doi.org/​10.1038/​srep14873
https:/​/​www.nature.com/​articles/​srep14873

[28] S. Lorenzo, R. McCloskey, F. Ciccarello, M. Paternostro, and G. M. Palma. Landauer's principle in multipartite open quantum system dynamics. Phys. Rev. Lett., 115: 120403, Sep 2015. 10.1103/​PhysRevLett.115.120403. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.115.120403.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.120403

[29] Marco Pezzutto, Mauro Paternostro, and Yasser Omar. Implications of non-markovian quantum dynamics for the landauer bound. New Journal of Physics, 18 (12): 123018, dec 2016. 10.1088/​1367-2630/​18/​12/​123018. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​12/​123018

[30] Philipp Strasberg, Gernot Schaller, Tobias Brandes, and Massimiliano Esposito. Quantum and information thermodynamics: A unifying framework based on repeated interactions. Phys. Rev. X, 7: 021003, Apr 2017. 10.1103/​PhysRevX.7.021003. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.7.021003.
https:/​/​doi.org/​10.1103/​PhysRevX.7.021003

[31] P. Skrzypczyk, A. J. Short, and S. Popescu. Work extraction and thermodynamics for individual quantum systems. Nature Comm., 5 (4185): 4185, 2014. 10.1038/​ncomms5185. URL https:/​/​www.nature.com/​articles/​ncomms5185.
https:/​/​doi.org/​10.1038/​ncomms5185
https:/​/​www.nature.com/​articles/​ncomms5185

[32] M. Horodecki and J. Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics. Nature Comm., 4 (2059): 033022, 2013. 10.1038/​ncomms3059. URL https:/​/​www.nature.com/​articles/​ncomms3059.
https:/​/​doi.org/​10.1038/​ncomms3059
https:/​/​www.nature.com/​articles/​ncomms3059

[33] Gilad Gour, Markus P. Muller, Varun Narasimhachar, Robert W. Spekkens, and Nicole Yunger Halpern. The resource theory of informational nonequilibrium in thermodynamics. Physics Reports, 583: 1 - 58, 2015. ISSN 0370-1573. 10.1016/​j.physrep.2015.04.003. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S037015731500229X.
https:/​/​doi.org/​10.1016/​j.physrep.2015.04.003
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S037015731500229X

[34] Markus P. Müller. Correlating thermal machines and the second law at the nanoscale. Phys. Rev. X, 8: 041051, Dec 2018. 10.1103/​PhysRevX.8.041051. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.8.041051.
https:/​/​doi.org/​10.1103/​PhysRevX.8.041051

[35] Kamil Korzekwa, Matteo Lostaglio, Jonathan Oppenheim, and David Jennings. The extraction of work from quantum coherence. New Journal of Physics, 18 (2): 023045, feb 2016. 10.1088/​1367-2630/​18/​2/​023045. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​023045

[36] Robert Alicki, Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Thermodynamics of quantum information systems — hamiltonian description. Open Systems Information Dynamics, 11 (03): 205-217, 2004. 10.1023/​B:OPSY.0000047566.72717.71. URL http:/​/​www.worldscientific.com/​doi/​abs/​10.1023/​B3AOPSY.0000047566.72717.71.
https:/​/​doi.org/​10.1023/​B:OPSY.0000047566.72717.71
http:/​/​www.worldscientific.com/​doi/​abs/​10.1023/​B3AOPSY.0000047566.72717.71

[37] M. Esposito and C. Van den Broeck. Second law and landauer principle far from equilibrium. EPL (Europhysics Letters), 95 (4): 40004, aug 2011. 10.1209/​0295-5075/​95/​40004. URL https:/​/​doi.org/​10.1209.
https:/​/​doi.org/​10.1209/​0295-5075/​95/​40004

[38] Peter Talkner, Eric Lutz, and Peter Hänggi. Fluctuation theorems: Work is not an observable. Phys. Rev. E, 75: 050102, May 2007. 10.1103/​PhysRevE.75.050102. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevE.75.050102.
https:/​/​doi.org/​10.1103/​PhysRevE.75.050102

[39] M. Perarnau-Llobet, H. Wilming, A. Riera, R. Gallego, and J. Eisert. Strong coupling corrections in quantum thermodynamics. Physical Review Letters, 120 (12), mar 2018. 10.1103/​physrevlett.120.120602.
https:/​/​doi.org/​10.1103/​physrevlett.120.120602

[40] Matteo Scandi and Martí Perarnau-Llobet. Thermodynamic length in open quantum systems. arXiv preprint arXiv:1810.05583, 2018. URL https:/​/​arxiv.org/​abs/​1810.05583.
arXiv:1810.05583

[41] Massimiliano Esposito, Katja Lindenberg, and Christian Van den Broeck. Entropy production as correlation between system and reservoir. New Journal of Physics, 12 (1): 013013, jan 2010a. 10.1088/​1367-2630/​12/​1/​013013. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​12/​1/​013013

[42] R Gallego, A Riera, and J Eisert. Thermal machines beyond the weak coupling regime. New Journal of Physics, 16 (12): 125009, dec 2014. 10.1088/​1367-2630/​16/​12/​125009. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​12/​125009

[43] Kurt Jacobs. Second law of thermodynamics and quantum feedback control: Maxwell's demon with weak measurements. Phys. Rev. A, 80: 012322, Jul 2009. 10.1103/​PhysRevA.80.012322. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.80.012322.
https:/​/​doi.org/​10.1103/​PhysRevA.80.012322

[44] 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, Feb 2017. 10.1103/​PhysRevLett.118.070601. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.118.070601.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.070601

[45] Massimiliano Esposito, Ryoichi Kawai, Katja Lindenberg, and Christian Van den Broeck. Efficiency at maximum power of low-dissipation carnot engines. Phys. Rev. Lett., 105: 150603, Oct 2010b. 10.1103/​PhysRevLett.105.150603. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.105.150603.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.150603

[46] Bernard Gaveau and LS Schulman. Master equation based formulation of nonequilibrium statistical mechanics. Journal of Mathematical Physics, 37 (8): 3897-3932, 1996. 10.1063/​1.531608. URL https:/​/​aip.scitation.org/​doi/​10.1063/​1.531608.
https:/​/​doi.org/​10.1063/​1.531608

[47] Vasco Cavina, Andrea Mari, and Vittorio Giovannetti. Slow dynamics and thermodynamics of open quantum systems. Phys. Rev. Lett., 119: 050601, Aug 2017. 10.1103/​PhysRevLett.119.050601. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.119.050601.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.050601

[48] Martí Perarnau-Llobet, Arnau Riera, Rodrigo Gallego, Henrik Wilming, and Jens Eisert. Work and entropy production in generalised gibbs ensembles. New Journal of Physics, 18 (12): 123035, dec 2016. 10.1088/​1367-2630/​aa4fa6. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​aa4fa6

[49] C. Jarzynski. Nonequilibrium equality for free energy differences. Phys. Rev. Lett., 78: 2690-2693, Apr 1997. 10.1103/​PhysRevLett.78.2690. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.78.2690.
https:/​/​doi.org/​10.1103/​PhysRevLett.78.2690

Cited by

[1] Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry, "Entanglement fluctuation theorems", Physical Review A 100 1, 012317 (2019).

[2] Steve Campbell, Francesco Ciccarello, G. Massimo Palma, and Bassano Vacchini, "System-environment correlations and Markovian embedding of quantum non-Markovian dynamics", Physical Review A 98 1, 012142 (2018).

[3] Georgios Styliaris, Álvaro M. Alhambra, and Paolo Zanardi, "Mixing of quantum states under Markovian dissipation and coherent control", Physical Review A 99 4, 042333 (2019).

[4] Angeline Shu, Yu Cai, Stella Seah, Stefan Nimmrichter, and Valerio Scarani, "Almost thermal operations: inhomogeneous reservoirs", arXiv:1904.08736.

[5] Stella Seah, Stefan Nimmrichter, and Valerio Scarani, "Nonequilibrium dynamics with finite-time repeated interactions", Physical Review E 99 4, 042103 (2019).

[6] 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 3, 032119 (2018).

[7] Angeline Shu, Yu Cai, Stella Seah, Stefan Nimmrichter, and Valerio Scarani, "Violation of all the second laws of thermal operations by inhomogeneous reservoirs", arXiv:1806.08108.

[8] 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).

The above citations are from Crossref's cited-by service (last updated 2019-09-22 05:49:43) and SAO/NASA ADS (last updated 2019-09-22 05:49:44). The list may be incomplete as not all publishers provide suitable and complete citation data.