Thermodynamic length in open quantum systems

Matteo Scandi and Martí Perarnau-Llobet

Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany

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


The dissipation generated during a quasistatic thermodynamic process can be characterised by introducing a metric on the space of Gibbs states, in such a way that minimally-dissipating protocols correspond to geodesic trajectories. Here, we show how to generalize this approach to open quantum systems by finding the thermodynamic metric associated to a given Lindblad master equation. The obtained metric can be understood as a perturbation over the background geometry of equilibrium Gibbs states, which is induced by the Kubo-Mori-Bogoliubov (KMB) inner product. We illustrate this construction on two paradigmatic examples: an Ising chain and a two-level system interacting with a bosonic bath with different spectral densities.

Any thermodynamic protocol performed in finite time produces dissipation, and it is a crucial question how to minimise it. A particularly powerful approach to address this question is by means of differential geometry: one can associate a metric in the thermodynamic space in such a way geodesics correspond to minimally dissipative processes. In this paper we apply this idea (which started in the 80s for macroscopic systems) to open quantum systems described by a Lindblad equation, so that the problem of finding optimal thermodynamic protocols between two Hamiltonians reduces to the one of solving the geodesics equation. Differences with previous classical works appear in the presence of coherences in the energy basis, which are created when the Hamiltonian does not commute with itself at different times. We illustrate these ideas for a qubit coupled to a bath with different spectral densities, and an Ising chain in a controllable transverse field.

► BibTeX data

► References

[1] F. Weinhold. Metric geometry of equilibrium thermodynamics. The Journal of Chemical Physics, 63: 2479, 1975a. URL https:/​/​​10.1063/​1.431689.

[2] F. Weinhold. Metric geometry of equilibrium thermodynamics. III. Elementary formal structure of a vector-algebraic representation of equilibrium thermodynamics. The Journal of Chemical Physics, 63 (6): 2488–2495, 1975b. 10.1063/​1.431636. URL https:/​/​​10.1063/​1.431636.

[3] F. Schlögl. Thermodynamic metric and stochastic measures. Zeitschrift für Physik B Condensed Matter, 59 (4): 449–454, Dec 1985. ISSN 1431-584X. 10.1007/​BF01328857. URL https:/​/​​10.1007/​BF01328857.

[4] P. Salamon and R. S. Berry. Thermodynamic length and dissipated availability. Phys. Rev. Lett., 51: 1127–1130, Sep 1983. 10.1103/​PhysRevLett.51.1127. URL https:/​/​​doi/​10.1103/​PhysRevLett.51.1127.

[5] P. Salamon, B. Andresen, P. D. Gait, and R. S. Berry. The significance of Weinhold's length. The Journal of Chemical Physics, 73 (2): 1001–1002, 1980. URL https:/​/​​10.1063/​1.440217.

[6] J. Nulton, P. Salamon, B. Andresen, and Q. Anmin. Quasistatic processes as step equilibrations. The Journal of Chemical Physics, 83: 334, 1985. URL https:/​/​​10.1063/​1.449774.

[7] B. Andresen, R. S. Berry, R. Gilmore, E. Ihrig, and P. Salamon. Thermodynamic geometry and the metrics of Weinhold and Gilmore. Phys. Rev. A, 37: 845–848, Feb 1988. 10.1103/​PhysRevA.37.845. URL https:/​/​​doi/​10.1103/​PhysRevA.37.845.

[8] L. Diosi, K. Kulacsy, B. Lukacs, and A. Racz. Thermodynamic length, time, speed, and optimum path to minimize entropy production. The Journal of chemical physics, 105 (24): 11220–11225, 1996. URL https:/​/​​doi/​abs/​10.1063/​1.472897. 10.1063/​1.472897.

[9] G. E. Crooks. Measuring thermodynamic length. Phys. Rev. Lett., 99: 100602, Sep 2007. 10.1103/​PhysRevLett.99.100602. URL https:/​/​​doi/​10.1103/​PhysRevLett.99.100602.

[10] P. R. Zulkowski, D. A. Sivak, G. E. Crooks, and M. R. DeWeese. Geometry of thermodynamic control. Phys. Rev. E, 86: 041148, Oct 2012. 10.1103/​PhysRevE.86.041148. URL https:/​/​​doi/​10.1103/​PhysRevE.86.041148.

[11] E. H. Feng and G. E. Crooks. Far-from-equilibrium measurements of thermodynamic length. Phys. Rev. E, 79: 012104, Jan 2009. 10.1103/​PhysRevE.79.012104. URL https:/​/​​doi/​10.1103/​PhysRevE.79.012104.

[12] D. A. Sivak and G. E. Crooks. Thermodynamic metrics and optimal paths. Phys. Rev. Lett., 108: 190602, May 2012. 10.1103/​PhysRevLett.108.190602. URL https:/​/​​doi/​10.1103/​PhysRevLett.108.190602.

[13] D. A. Sivak and G. E. Crooks. Thermodynamic geometry of minimum-dissipation driven barrier crossing. Phys. Rev. E, 94: 052106, Nov 2016. 10.1103/​PhysRevE.94.052106. URL https:/​/​​doi/​10.1103/​PhysRevE.94.052106.

[14] P. R. Zulkowski, D. A. Sivak, and M. R. DeWeese. Optimal control of transitions between nonequilibrium steady states. PloS one, 8 (12): e82754, 2013. URL https:/​/​​plosone/​article?id=10.1371/​journal.pone.0082754. 10.1371/​journal.pone.0082754.

[15] P. R. Zulkowski and M. R. DeWeese. Optimal control of overdamped systems. Phys. Rev. E, 92: 032117, Sep 2015a. 10.1103/​PhysRevE.92.032117. URL https:/​/​​doi/​10.1103/​PhysRevE.92.032117.

[16] D. Petz and G. Toth. The Bogoliubov inner product in quantum statistics. Letters in Mathematical Physics, 27 (3): 205–216, Mar 1993. ISSN 1573-0530. 10.1007/​BF00739578. URL https:/​/​​10.1007/​BF00739578.

[17] P. W. Michor, D. Petz, and A. Andai. On the curvature of a certain Riemannian space of matrices. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 03 (02): 199–212, 2000. 10.1142/​S0219025700000145. URL https:/​/​​10.1142/​S0219025700000145.

[18] D. Petz. Covariance and Fisher information in quantum mechanics. Journal of Physics A: Mathematical and General, 35 (4): 929, 2002. URL http:/​/​​0305-4470/​35/​i=4/​a=305. 10.1088/​0305-4470/​35/​4/​305.

[19] D. Petz and C. Ghinea. Introduction to quantum Fisher information, pages 261–281. World Scientific, 2011. 10.1142/​9789814338745_0015. URL https:/​/​​doi/​abs/​10.1142/​9789814338745_0015.

[20] R. Balian. The entropy-based quantum metric. Entropy, 16 (7): 3878–3888, 2014. ISSN 1099-4300. 10.3390/​e16073878. URL http:/​/​​1099-4300/​16/​7/​3878.

[21] S. Deffner and E. Lutz. Generalized clausius inequality for nonequilibrium quantum processes. Phys. Rev. Lett., 105: 170402, Oct 2010. 10.1103/​PhysRevLett.105.170402. URL https:/​/​​doi/​10.1103/​PhysRevLett.105.170402.

[22] S. Deffner and E. Lutz. Thermodynamic length for far-from-equilibrium quantum systems. Phys. Rev. E, 87: 022143, Feb 2013. 10.1103/​PhysRevE.87.022143. URL https:/​/​​doi/​10.1103/​PhysRevE.87.022143.

[23] M. Campisi, S. Denisov, and P. Hänggi. Geometric magnetism in open quantum systems. Phys. Rev. A, 86: 032114, Sep 2012. 10.1103/​PhysRevA.86.032114. URL https:/​/​​doi/​10.1103/​PhysRevA.86.032114.

[24] T. V. Acconcia, M. V. S. Bonança, and S. Deffner. Shortcuts to adiabaticity from linear response theory. Physical Review E, 92 (4): 042148, 2015. URL https:/​/​​pre/​abstract/​10.1103/​PhysRevE.92.042148. 10.1103/​PhysRevE.92.042148.

[25] M. F. Ludovico, F. Battista, F. von Oppen, and L. Arrachea. Adiabatic response and quantum thermoelectrics for ac-driven quantum systems. Phys. Rev. B, 93: 075136, Feb 2016. 10.1103/​PhysRevB.93.075136. URL https:/​/​​doi/​10.1103/​PhysRevB.93.075136.

[26] M. V. S. Bonança and S. Deffner. Minimal dissipation in processes far from equilibrium. Phys. Rev. E, 98: 042103, Oct 2018. 10.1103/​PhysRevE.98.042103. URL https:/​/​​doi/​10.1103/​PhysRevE.98.042103.

[27] Patrick R. Zulkowski and Michael R. DeWeese. Optimal protocols for slowly driven quantum systems. Phys. Rev. E, 92: 032113, Sep 2015b. 10.1103/​PhysRevE.92.032113. URL https:/​/​​doi/​10.1103/​PhysRevE.92.032113.

[28] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan. Completely positive dynamical semigroups of n-level systems. Journal of Mathematical Physics, 17 (5): 821–825, 1976. 10.1063/​1.522979. URL https:/​/​​doi/​abs/​10.1063/​1.522979.

[29] Tameem Albash, Sergio Boixo, Daniel A Lidar, and Paolo Zanardi. Quantum adiabatic markovian master equations. New Journal of Physics, 14 (12): 123016, December 2012. 10.1088/​1367-2630/​14/​12/​123016. URL https:/​/​​10.1088/​1367-2630/​14/​12/​123016.

[30] Makoto Yamaguchi, Tatsuro Yuge, and Tetsuo Ogawa. Markovian quantum master equation beyond adiabatic regime. Phys. Rev. E, 95: 012136, Jan 2017. 10.1103/​PhysRevE.95.012136. URL https:/​/​​doi/​10.1103/​PhysRevE.95.012136.

[31] Roie Dann, Amikam Levy, and Ronnie Kosloff. Time-dependent markovian quantum master equation. Phys. Rev. A, 98: 052129, Nov 2018. 10.1103/​PhysRevA.98.052129. URL https:/​/​​doi/​10.1103/​PhysRevA.98.052129.

[32] V. Cavina, A. Mari, and V. Giovannetti. Slow dynamics and thermodynamics of open quantum systems. Phys. Rev. Lett., 119: 050601, Aug 2017. 10.1103/​PhysRevLett.119.050601. URL https:/​/​​doi/​10.1103/​PhysRevLett.119.050601.

[33] D. Mandal and C. Jarzynski. Analysis of slow transitions between nonequilibrium steady states. Journal of Statistical Mechanics: Theory and Experiment, 2016 (6): 063204, jun 2016. 10.1088/​1742-5468/​2016/​06/​063204. URL https:/​/​​10.1088/​1742-5468/​2016/​06/​063204.

[34] Gavin E Crooks. On the Drazin inverse of the rate matrix. 2018.

[35] F. Hiai and D. Petz. Introduction to Matrix Analysis and Applications. Springer International Publishing, Cham, 2014. ISBN 978-3-319-04150-6. 10.1007/​978-3-319-04150-6_3. URL https:/​/​​10.1007/​978-3-319-04150-6_3.

[36] T. L. Boullion and P. L. Odell. Generalised inverse matrices. Wiley-Interscience, New York, 1971.

[37] A. Müller-Hermes and D. Reeb. Monotonicity of the quantum relative entropy under positive maps. Annales Henri Poincaré, 18 (5): 1777–1788, jan 2017. 10.1007/​s00023-017-0550-9. URL https:/​/​​10.1007/​s00023-017-0550-9.

[38] B. O'Neill. Semi-Riemannian geometry with applications to relativity. Pure and Applied Mathematics. Elsevier Science, 1983. ISBN 9780080570570.

[39] V. I. Arnold. Lagrangian mechanics on manifolds, pages 75–97. Springer New York, New York, NY, 1989. ISBN 978-1-4757-2063-1. 10.1007/​978-1-4757-2063-1_4. URL https:/​/​​10.1007/​978-1-4757-2063-1_4.

[40] F. J. Dyson, E. H. Lieb, and B. Simon. Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. Journal of Statistical Physics, 18 (4): 335–383, Apr 1978. ISSN 1572-9613. 10.1007/​BF01106729. URL https:/​/​​10.1007/​BF01106729.

[41] G. Roepstorff. Correlation inequalities in quantum statistical mechanics and their application in the Kondo problem. Comm. Math. Phys., 46 (3): 253–262, 1976. URL https:/​/​​euclid.cmp/​1103899639.

[42] G. Guarnieri, G. T. Landi, S. R. Clark, and J. Goold. Thermodynamics of precision in quantum non equilibrium steady states. arXiv preprint arXiv:1901.10428, 2019. 10.1103/​PhysRevResearch.1.033021.

[43] S. Sachdev. Quantum phase transitions. Handbook of Magnetism and Advanced Magnetic Materials, 2007.

[44] H. P. Breuer and F. Petruccione. The theory of open quantum systems. Oxford University Press, Great Clarendon Street, 2002.

[45] A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz. Experimental verification of Landauer's principle linking information and thermodynamics. Nature, 483 (7388): 187, 2012. URL https:/​/​​articles/​nature10872. 10.1038/​nature10872.

[46] Y. Jun, M. Gavrilov, and J. Bechhoefer. High-precision test of Landauer's principle in a feedback trap. Phys. Rev. Lett., 113: 190601, Nov 2014. 10.1103/​PhysRevLett.113.190601. URL https:/​/​​doi/​10.1103/​PhysRevLett.113.190601.

[47] J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin. Experimental realization of a Szilard engine with a single electron. Proceedings of the National Academy of Sciences, 111 (38): 13786–13789, 2014. URL http:/​/​​content/​111/​38/​13786. newblock 10.1073/​pnas.1406966111.

[48] R. Gaudenzi, E. Burzurí, S. Maegawa, H. S. J. Zant, and F. Luis. Quantum Landauer erasure with a molecular nanomagnet. Nature Physics, 14 (6): 565, 2018. URL https:/​/​​articles/​s41567-018-0070-7. 10.1038/​s41567-018-0070-7.

[49] T. Schmiedl and U. Seifert. Optimal finite-time processes in stochastic thermodynamics. Phys. Rev. Lett., 98: 108301, Mar 2007. 10.1103/​PhysRevLett.98.108301. URL https:/​/​​doi/​10.1103/​PhysRevLett.98.108301.

[50] M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck. Finite-time thermodynamics for a single-level quantum dot. EPL (Europhysics Letters), 89 (2): 20003, 2010. URL http:/​/​​article/​10.1209/​0295-5075/​89/​20003. 10.1209/​0295-5075/​89/​20003.

[51] V. Cavina, A. Mari, A. Carlini, and V. Giovannetti. Optimal thermodynamic control in open quantum systems. Phys. Rev. A, 98: 012139, Jul 2018a. 10.1103/​PhysRevA.98.012139. URL https:/​/​​doi/​10.1103/​PhysRevA.98.012139.

[52] Paul Menczel, Tuomas Pyhäranta, Christian Flindt, and Kay Brandner. Two-stroke optimization scheme for mesoscopic refrigerators. Phys. Rev. B, 99: 224306, Jun 2019. 10.1103/​PhysRevB.99.224306. URL https:/​/​​doi/​10.1103/​PhysRevB.99.224306.

[53] S. Deffner. Optimal control of a qubit in an optical cavity. Journal of Physics B: Atomic, Molecular and Optical Physics, 47 (14): 145502, 2014. URL http:/​/​​article/​10.1088/​0953-4075/​47/​14/​145502/​meta. 10.1088/​0953-4075/​47/​14/​145502.

[54] Vasco Cavina, Andrea Mari, Alberto Carlini, and Vittorio Giovannetti. Variational approach to the optimal control of coherently driven, open quantum system dynamics. Phys. Rev. A, 98: 052125, Nov 2018b. 10.1103/​PhysRevA.98.052125. URL https:/​/​​doi/​10.1103/​PhysRevA.98.052125.

[55] M. V. S. Bonança and S. Deffner. Optimal driving of isothermal processes close to equilibrium. The Journal of chemical physics, 140 (24): 244119, 2014. URL https:/​/​​doi/​abs/​10.1063/​1.4885277?journalCode=jcp. 10.1063/​1.4885277.

[56] G. M. Rotskoff and G. E. Crooks. Optimal control in nonequilibrium systems: dynamic Riemannian geometry of the Ising model. Phys. Rev. E, 92: 060102, Dec 2015. 10.1103/​PhysRevE.92.060102. URL https:/​/​​doi/​10.1103/​PhysRevE.92.060102.

[57] T. R. Gingrich, G. M. Rotskoff, G. E. Crooks, and P. L. Geissler. Near-optimal protocols in complex nonequilibrium transformations. Proceedings of the National Academy of Sciences, 113 (37): 10263–10268, aug 2016. 10.1073/​pnas.1606273113. URL https:/​/​​10.1073/​pnas.1606273113.

[58] G. M. Rotskoff, G. E. Crooks, and E. Vanden-Eijnden. Geometric approach to optimal nonequilibrium control: minimizing dissipation in nanomagnetic spin systems. Phys. Rev. E, 95: 012148, Jan 2017. 10.1103/​PhysRevE.95.012148. URL https:/​/​​doi/​10.1103/​PhysRevE.95.012148.

[59] Harry JD Miller, Matteo Scandi, Janet Anders, and Martí Perarnau-Llobet. Work fluctuations in slow processes: quantum signatures and optimal control. arXiv preprint arXiv:1905.07328, 2019.

[60] Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk. Thermodynamics of quantum systems with multiple conserved quantities. Nature communications, 7: ncomms12049, 2016. URL https:/​/​​articles/​ncomms12049/​. 10.1038/​ncomms12049.

[61] M. Lostaglio, D. Jennings, and T. Rudolph. Thermodynamic resource theories, non-commutativity and maximum entropy principles. New Journal of Physics, 19 (4): 043008, 2017. URL http:/​/​​article/​10.1088/​1367-2630/​aa617f/​meta. 10.1088/​1367-2630/​aa617f.

[62] N. Y. Halpern, P. Faist, J. Oppenheim, and A. Winter. Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges. Nature communications, 7: 12051, 2016. URL https:/​/​​articles/​ncomms12051. 10.1038/​ncomms12051.

[63] M. Perarnau-Llobet, A. Riera, R. Gallego, H. Wilming, and J. Eisert. Work and entropy production in generalised Gibbs ensembles. New Journal of Physics, 18 (12): 123035, 2016. URL http:/​/​​article/​10.1088/​1367-2630/​aa4fa6/​meta. 10.1088/​1367-2630/​aa4fa6.

[64] André M. Timpanaro, Giacomo Guarnieri, John Goold, and Gabriel T. Landi. Thermodynamic uncertainty relations from exchange fluctuation theorems. Phys. Rev. Lett., 123: 090604, Aug 2019. 10.1103/​PhysRevLett.123.090604. URL https:/​/​​doi/​10.1103/​PhysRevLett.123.090604.

[65] M. F. Gelin and M. Thoss. Thermodynamics of a subensemble of a canonical ensemble. Phys. Rev. E, 79: 051121, May 2009. 10.1103/​PhysRevE.79.051121. URL https:/​/​​doi/​10.1103/​PhysRevE.79.051121.

[66] M. Campisi, P. Talkner, and P. Hänggi. Fluctuation theorem for arbitrary open quantum systems. Phys. Rev. Lett., 102: 210401, May 2009. 10.1103/​PhysRevLett.102.210401. URL https:/​/​​doi/​10.1103/​PhysRevLett.102.210401.

[67] S. Hilt, B. Thomas, and E. Lutz. Hamiltonian of mean force for damped quantum systems. Phys. Rev. E, 84: 031110, Sep 2011. 10.1103/​PhysRevE.84.031110. URL https:/​/​​doi/​10.1103/​PhysRevE.84.031110.

[68] R Gallego, A Riera, and J Eisert. Thermal machines beyond the weak coupling regime. New Journal of Physics, 16 (12): 125009, 2014. URL http:/​/​​1367-2630/​16/​i=12/​a=125009. newblock 10.1088/​1367-2630/​16/​12/​125009.

[69] D. Gelbwaser-Klimovsky and A. Aspuru-Guzik. Strongly coupled quantum heat machines. The journal of physical chemistry letters, 6 (17): 3477–3482, 2015. URL https:/​/​​doi/​abs/​10.1021/​acs.jpclett.5b01404. 10.1021/​acs.jpclett.5b01404.

[70] P. Strasberg, G. Schaller, N. Lambert, and T. Brandes. Nonequilibrium thermodynamics in the strong coupling and non-Markovian regime based on a reaction coordinate mapping. New Journal of Physics, 18 (7): 073007, 2016. URL http:/​/​​article/​10.1088/​1367-2630/​18/​7/​073007/​meta. 10.1088/​1367-2630/​18/​7/​073007.

[71] D. Newman, F. Mintert, and A. Nazir. Performance of a quantum heat engine at strong reservoir coupling. Phys. Rev. E, 95: 032139, Mar 2017. 10.1103/​PhysRevE.95.032139. URL https:/​/​​doi/​10.1103/​PhysRevE.95.032139.

[72] M. Perarnau-Llobet, H. Wilming, A. Riera, R. Gallego, and J. Eisert. Strong coupling corrections in quantum thermodynamics. Phys. Rev. Lett., 120: 120602, Mar 2018. 10.1103/​PhysRevLett.120.120602. URL https:/​/​​doi/​10.1103/​PhysRevLett.120.120602.

[73] Elisa Bäumer, Martí Perarnau-Llobet, Philipp Kammerlander, Henrik Wilming, and Renato Renner. Imperfect Thermalizations Allow for Optimal Thermodynamic Processes. Quantum, 3: 153, June 2019. ISSN 2521-327X. 10.22331/​q-2019-06-24-153. URL https:/​/​​10.22331/​q-2019-06-24-153.

[74] Roie Dann, Ander Tobalina, and Ronnie Kosloff. Shortcut to equilibration of an open quantum system. Phys. Rev. Lett., 122: 250402, Jun 2019. 10.1103/​PhysRevLett.122.250402. URL https:/​/​​doi/​10.1103/​PhysRevLett.122.250402.

Cited by

[1] Paul M. Riechers and Mile Gu, "Initial-state dependence of thermodynamic dissipation for any quantum process", Physical Review E 103 4, 042145 (2021).

[2] Harry J. D. Miller, Giacomo Guarnieri, Mark T. Mitchison, and John Goold, "Quantum Fluctuations Hinder Finite-Time Information Erasure near the Landauer Limit", Physical Review Letters 125 16, 160602 (2020).

[3] Matteo Scandi, Harry J. D. Miller, Janet Anders, and Martí Perarnau-Llobet, "Quantum work statistics close to equilibrium", Physical Review Research 2 2, 023377 (2020).

[4] Jin-Fu Chen, Ying Li, and Hui Dong, "Simulating Finite-Time Isothermal Processes with Superconducting Quantum Circuits", Entropy 23 3, 353 (2021).

[5] Paolo Abiuso, Harry J. D. Miller, Martí Perarnau-Llobet, and Matteo Scandi, "Geometric Optimisation of Quantum Thermodynamic Processes", Entropy 22 10, 1076 (2020).

[6] Carlo Cafaro, Shannon Ray, and Paul M. Alsing, "Geometric aspects of analog quantum search evolutions", Physical Review A 102 5, 052607 (2020).

[7] Yuki Hino and Hisao Hayakawa, "Geometrical formulation of adiabatic pumping as a heat engine", Physical Review Research 3 1, 013187 (2021).

[8] Bibek Bhandari, Pablo Terrén Alonso, Fabio Taddei, Felix von Oppen, Rosario Fazio, and Liliana Arrachea, "Geometric properties of adiabatic quantum thermal machines", Physical Review B 102 15, 155407 (2020).

[9] Nicola Pancotti, Matteo Scandi, Mark T. Mitchison, and Martí Perarnau-Llobet, "Speed-Ups to Isothermality: Enhanced Quantum Thermal Machines through Control of the System-Bath Coupling", Physical Review X 10 3, 031015 (2020).

[10] Harry J. D. Miller, Matteo Scandi, Janet Anders, and Martí Perarnau-Llobet, "Work Fluctuations in Slow Processes: Quantum Signatures and Optimal Control", Physical Review Letters 123 23, 230603 (2019).

[11] Paul M. Riechers and Mile Gu, "Impossibility of achieving Landauer's bound for almost every quantum state", Physical Review A 104 1, 012214 (2021).

[12] Martin Josefsson and Martin Leijnse, "Double quantum-dot engine fueled by entanglement between electron spins", Physical Review B 101 8, 081408 (2020).

[13] Paolo Abiuso and Martí Perarnau-Llobet, "Optimal Cycles for Low-Dissipation Heat Engines", Physical Review Letters 124 11, 110606 (2020).

[14] Michael P. Frank and Karpur Shukla, "Quantum Foundations of Classical Reversible Computing", Entropy 23 6, 701 (2021).

[15] Marcus V. S. Bonança, Pierre Nazé, and Sebastian Deffner, "Negative entropy production rates in Drude-Sommerfeld metals", Physical Review E 103 1, 012109 (2021).

[16] Eoin O'Connor, Giacomo Guarnieri, and Steve Campbell, "Action quantum speed limits", Physical Review A 103 2, 022210 (2021).

[17] Harry J. D. Miller, M. Hamed Mohammady, Martí Perarnau-Llobet, and Giacomo Guarnieri, "Joint statistics of work and entropy production along quantum trajectories", Physical Review E 103 5, 052138 (2021).

[18] Jin-Fu Chen, C. P. Sun, and Hui Dong, "Extrapolating the thermodynamic length with finite-time measurements", Physical Review E 104 3, 034117 (2021).

[19] Sebastian Deffner and Marcus V. S. Bonança, "Thermodynamic control —An old paradigm with new applications", EPL (Europhysics Letters) 131 2, 20001 (2020).

[20] Harry J. D. Miller and Mohammad Mehboudi, "Geometry of Work Fluctuations versus Efficiency in Microscopic Thermal Machines", Physical Review Letters 125 26, 260602 (2020).

[21] Tan Van Vu and Yoshihiko Hasegawa, "Geometrical Bounds of the Irreversibility in Markovian Systems", Physical Review Letters 126 1, 010601 (2021).

[22] Vasco Cavina, Paolo A. Erdman, Paolo Abiuso, Leonardo Tolomeo, and Vittorio Giovannetti, "Maximum-power heat engines and refrigerators in the fast-driving regime", Physical Review A 104 3, 032226 (2021).

[23] S Hamedani Raja, S Maniscalco, G S Paraoanu, J P Pekola, and N Lo Gullo, "Finite-time quantum Stirling heat engine", New Journal of Physics 23 3, 033034 (2021).

[24] Kay Brandner and Keiji Saito, "Thermodynamic Geometry of Microscopic Heat Engines", Physical Review Letters 124 4, 040602 (2020).

[25] John Goold, "Geometry and quantum thermodynamics", Quantum Views 3, 28 (2019).

[26] Harry J. D. Miller, M. Hamed Mohammady, Martí Perarnau-Llobet, and Giacomo Guarnieri, "Thermodynamic Uncertainty Relation in Slowly Driven Quantum Heat Engines", Physical Review Letters 126 21, 210603 (2021).

[27] Giacomo Guarnieri, Gabriel T. Landi, Stephen R. Clark, and John Goold, "Thermodynamics of precision in quantum nonequilibrium steady states", Physical Review Research 1, 033021 (2019).

[28] Roie Dann, Ander Tobalina, and Ronnie Kosloff, "Shortcut to Equilibration of an Open Quantum System", Physical Review Letters 122 25, 250402 (2019).

[29] Paolo Abiuso and Vittorio Giovannetti, "Non-Markov enhancement of maximum power for quantum thermal machines", Physical Review A 99 5, 052106 (2019).

[30] Pablo Terrén Alonso, Javier Romero, and Liliana Arrachea, "Work exchange, geometric magnetization, and fluctuation-dissipation relations in a quantum dot under adiabatic magnetoelectric driving", Physical Review B 99 11, 115424 (2019).

[31] Elisa Bäumer, Martí Perarnau-Llobet, Philipp Kammerlander, Henrik Wilming, and Renato Renner, "Imperfect Thermalizations Allow for Optimal Thermodynamic Processes", arXiv:1712.07128.

The above citations are from Crossref's cited-by service (last updated successfully 2021-10-27 18:36:40) and SAO/NASA ADS (last updated successfully 2021-10-27 18:36:41). The list may be incomplete as not all publishers provide suitable and complete citation data.

1 thought on “Thermodynamic length in open quantum systems

  1. Pingback: Perspective in Quantum Views by John Goold "Geometry and quantum thermodynamics"