- Open Access
Quantum Otto cycle in the Anderson impurity model
Phys. Rev. Research 8, 033233 – Published 25 August, 2026
DOI: https://doi.org/10.1103/t45v-bg7h
Abstract
We study the thermodynamic performance of a periodic quantum Otto cycle operating on the single-impurity Anderson model. Using a decomposition of the time-evolution generator based on the principle of minimal dissipation, combined with the numerically exact hierarchical equations of motion method, we analyze the operating regimes of the quantum thermal machine and investigate effects of Coulomb interactions, strong system-reservoir coupling, and energy level alignments. Our results show that Coulomb interaction can change the operating regimes and may lead to an enhancement of the efficiency.
Physics Subject Headings (PhySH)
Article Text
References (82)
- J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics (Springer, Berlin, 2004).
- G. Schaller, Open Quantum Systems Far from Equilibrium (Springer, Cham, Switzerland, 2014).
- F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, Thermodynamics in the Quantum Regime (Springer, Cham, Switzerland, 2018).
- S. Deffner and S. Campbell, Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information (Morgan & Claypool, San Rafael, CA, 2019), pp. 2053–2571.
- G. T. Landi and M. Paternostro, Irreversible entropy production: From classical to quantum, Rev. Mod. Phys. 93, 035008 (2021).
- O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, and E. Lutz, Single-ion heat engine at maximum power, Phys. Rev. Lett. 109, 203006 (2012).
- J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, A single-atom heat engine, Science 352, 325 (2016).
- R. J. de Assis, T. M. de Mendonça, C. J. Villas-Boas, A. M. de Souza, R. S. Sarthour, I. S. Oliveira, and N. G. de Almeida, Efficiency of a quantum Otto heat engine operating under a reservoir at effective negative temperatures, Phys. Rev. Lett. 122, 240602 (2019).
- D. von Lindenfels, O. Gräb, C. T. Schmiegelow, V. Kaushal, J. Schulz, M. T. Mitchison, J. Goold, F. Schmidt-Kaler, and U. G. Poschinger, Spin heat engine coupled to a harmonic-oscillator flywheel, Phys. Rev. Lett. 123, 080602 (2019).
- J. P. S. Peterson, T. B. Batalhão, M. Herrera, A. M. Souza, R. S. Sarthour, I. S. Oliveira, and R. M. Serra, Experimental characterization of a spin quantum heat engine, Phys. Rev. Lett. 123, 240601 (2019).
- N. M. Myers, O. Abah, and S. Deffner, Quantum thermodynamic devices: From theoretical proposals to experimental reality, AVS Quantum Sci. 4, 027101 (2022).
- R. Uzdin, A. Levy, and R. Kosloff, Equivalence of quantum heat machines, and quantum-thermodynamic signatures, Phys. Rev. X 5, 031044 (2015).
- R. Alicki and D. Gelbwaser-Klimovsky, Non-equilibrium quantum heat machines, New J. Phys. 17, 115012 (2015).
- C. L. Latune, G. Pleasance, and F. Petruccione, Cyclic quantum engines enhanced by strong bath coupling, Phys. Rev. Appl. 20, 024038 (2023).
- L. M. Cangemi, C. Bhadra, and A. Levy, Quantum engines and refrigerators, Phys. Rep. 1087, 1 (2024).
- S. Koyanagi and Y. Tanimura, Thermodynamic quantum Fokker–Planck equations and their application to thermostatic Stirling engine, J. Chem. Phys. 161, 112501 (2024).
- M. Esposito, K. Lindenberg, and C. Van den Broeck, Thermoelectric efficiency at maximum power in a quantum dot, Europhys. Lett. 85, 60010 (2009).
- D. M. Kennes, D. Schuricht, and V. Meden, Efficiency and power of a thermoelectric quantum dot device, Europhys. Lett. 102, 57003 (2013).
- M. Josefsson, A. Svilans, H. Linke, and M. Leijnse, Optimal power and efficiency of single quantum dot heat engines: Theory and experiment, Phys. Rev. B 99, 235432 (2019).
- E. Pyurbeeva and R. Kosloff, Quantum dot thermal machines—A guide to engineering, Entropy 28, 2 (2026).
- S. Volosheniuk, R. Conte, E. Pyurbeeva, T. Baum, M. Vilas-Varela, S. Fernández, D. Peña, H. S. J. van der Zant, and P. Gehring, A single-molecule quantum heat engine, Nano Lett. 26, 984 (2026).
- H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Non-Markovian dynamics in open quantum systems, Rev. Mod. Phys. 88, 021002 (2016).
- I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017).
- M. Popovic, B. Vacchini, and S. Campbell, Entropy production and correlations in a controlled non-Markovian setting, Phys. Rev. A 98, 012130 (2018).
- P. Strasberg and M. Esposito, Non-Markovianity and negative entropy production rates, Phys. Rev. E 99, 012120 (2019).
- A. Rivas, Strong coupling thermodynamics of open quantum systems, Phys. Rev. Lett. 124, 160601 (2020).
- S. Marcantoni, S. Alipour, F. Benatti, R. Floreanini, and A. T. Rezakhani, Entropy production and non-Markovian dynamical maps, Sci. Rep. 7, 12447 (2017).
- I. A. Picatoste, A. Colla, and H.-P. Breuer, Local and global approaches to the thermodynamics of pure decoherence processes in open quantum systems, Phys. Rev. A 112, 022210 (2025).
- A. Colla, B. Vacchini, and A. Smirne, Local energy assignment for two interacting quantum thermal reservoirs, New J. Phys. 27, 124510 (2025).
- H. Weimer, M. J. Henrich, F. Rempp, H. Schröder, and G. Mahler, Local effective dynamics of quantum systems: A generalized approach to work and heat, Europhys. Lett. 83, 30008 (2008).
- M. Esposito, K. Lindenberg, and C. V. den Broeck, Entropy production as correlation between system and reservoir, New J. Phys. 12, 013013 (2010).
- J. Teifel and G. Mahler, Autonomous modular quantum systems: Contextual Jarzynski relations, Phys. Rev. E 83, 041131 (2011).
- S. Alipour, F. Benatti, F. Bakhshinezhad, M. Afsary, S. Marcantoni, and A. T. Rezakhani, Correlations in quantum thermodynamics: Heat, work, and entropy production, Sci. Rep. 6, 35568 (2016).
- U. Seifert, First and second law of thermodynamics at strong coupling, Phys. Rev. Lett. 116, 020601 (2016).
- P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, Quantum and information thermodynamics: A unifying framework based on repeated interactions, Phys. Rev. X 7, 021003 (2017).
- S. Alipour, A. T. Rezakhani, A. Chenu, A. del Campo, and T. Ala-Nissila, Entropy-based formulation of thermodynamics in arbitrary quantum evolution, Phys. Rev. A 105, L040201 (2022).
- J. Liu, K. A. Jung, and D. Segal, Periodically driven quantum thermal machines from warming up to limit cycle, Phys. Rev. Lett. 127, 200602 (2021).
- X. Y. Zhang, X. L. Huang, and X. X. Yi, Quantum Otto heat engine with a non-Markovian reservoir, J. Phys. A: Math. Theor. 47, 455002 (2014).
- A. Pozas-Kerstjens, E. G. Brown, and K. V. Hovhannisyan, A quantum Otto engine with finite heat baths: Energy, correlations, and degradation, New J. Phys. 20, 043034 (2018).
- G. Thomas, N. Siddharth, S. Banerjee, and S. Ghosh, Thermodynamics of non-Markovian reservoirs and heat engines, Phys. Rev. E 97, 062108 (2018).
- M. Pezzutto, M. Paternostro, and Y. Omar, An out-of-equilibrium non-Markovian quantum heat engine, Quantum Sci. Technol. 4, 025002 (2019).
- V. Mukherjee, A. G. Kofman, and G. Kurizki, Anti-Zeno quantum advantage in fast-driven heat machines, Commun. Phys. 3, 8 (2020).
- M. Wiedmann, J. T. Stockburger, and J. Ankerhold, Non-Markovian dynamics of a quantum heat engine: Out-of-equilibrium operation and thermal coupling control, New J. Phys. 22, 033007 (2020).
- M. Wiedmann, J. T. Stockburger, and J. Ankerhold, Non-Markovian quantum Otto refrigerator, Eur. Phys. J. Spec. Top. 230, 851 (2021).
- S. Chakraborty, A. Das, and D. Chruściński, Strongly coupled quantum Otto cycle with single qubit bath, Phys. Rev. E 106, 064133 (2022).
- M. Kaneyasu and Y. Hasegawa, Quantum Otto cycle under strong coupling, Phys. Rev. E 107, 044127 (2023).
- M. Ishizaki, N. Hatano, and H. Tajima, Switching the function of the quantum Otto cycle in non-Markovian dynamics: Heat engine, heater, and heat pump, Phys. Rev. Res. 5, 023066 (2023).
- D. Gelbwaser-Klimovsky and A. Aspuru-Guzik, Strongly coupled quantum heat machines, J. Phys. Chem. Lett. 6, 3477 (2015).
- A. Colla and H.-P. Breuer, Open-system approach to nonequilibrium quantum thermodynamics at arbitrary coupling, Phys. Rev. A 105, 052216 (2022).
- I. A. Picatoste, A. Colla, and H.-P. Breuer, Dynamically emergent quantum thermodynamics: Non-Markovian Otto cycle, Phys. Rev. Res. 6, 013258 (2024).
- S. Gatto, A. Colla, H.-P. Breuer, and M. Thoss, Quantum thermodynamics of the spin-boson model using the principle of minimal dissipation, Phys. Rev. A 110, 032210 (2024).
- Y. Tanimura and R. Kubo, Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath, J. Phys. Soc. Jpn. 58, 101 (1989).
- J. Bätge, Y. Ke, C. Kaspar, and M. Thoss, Nonequilibrium open quantum systems with multiple bosonic and fermionic environments: A hierarchical equations of motion approach, Phys. Rev. B 103, 235413 (2021).
- F. Shibata, Y. Takahashi, and N. Hashitsume, A generalized stochastic Liouville equation. Non-Markovian versus memoryless master equations, J. Stat. Phys. 17, 171 (1977).
- S. Chaturvedi and F. Shibata, Time-convolutionless projection operator formalism for elimination of fast variables. Application to Brownian motion, Z. Phys. B 35, 297 (1979).
- H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
- J. Sorce and P. M. Hayden, A canonical Hamiltonian for open quantum systems, J. Phys. A: Math. Theor. 55, 225302 (2022).
- H. Spohn, Entropy production for quantum dynamical semigroups, J. Math. Phys. 19, 1227 (1978).
- H. Spohn and J. L. Lebowitz, Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs, in Advances in Chemical Physics, edited by S. A. Rice (John Wiley & Sons, New York, 1978), pp. 109–142.
- R. Alicki, The quantum open system as a model of the heat engine, J. Phys. A: Math. Gen. 12, L103 (1979).
- R. Kosloff, A quantum mechanical open system as a model of a heat engine, J. Chem. Phys. 80, 1625 (1984).
- R. Kosloff, Quantum thermodynamics: A dynamical viewpoint, Entropy 15, 2100 (2013).
- S. Deffner and E. Lutz, Nonequilibrium work distribution of a quantum harmonic oscillator, Phys. Rev. E 77, 021128 (2008).
- R. Alicki and R. Kosloff, Introduction to quantum thermodynamics: History and prospects, in Thermodynamics in the Quantum Regime, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer, Cham, 2018), pp. 1–33.
- P. W. Anderson, Localized magnetic states in metals, Phys. Rev. 124, 41 (1961).
- C. Schinabeck, A. Erpenbeck, R. Härtle, and M. Thoss, Hierarchical quantum master equation approach to electronic-vibrational coupling in nonequilibrium transport through nanosystems, Phys. Rev. B 94, 201407(R) (2016).
- R. P. Feynman, An operator calculus having applications in quantum electrodynamics, Phys. Rev. 84, 108 (1951).
- R. Feynman and F. Vernon, The theory of a general quantum system interacting with a linear dissipative system, Ann. Phys. (NY) 24, 118 (1963).
- J. Jin, X. Zheng, and Y. Yan, Exact dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach, J. Chem. Phys. 128, 234703 (2008).
- J. Hu, R.-X. Xu, and Y. Yan, Communication: Padé spectrum decomposition of Fermi function and Bose function, J. Chem. Phys. 133, 101106 (2010).
- J. Hu, M. Luo, F. Jiang, R.-X. Xu, and Y. Yan, Padé spectrum decompositions of quantum distribution functions and optimal hierarchical equations of motion construction for quantum open systems, J. Chem. Phys. 134, 244106 (2011).
- A. Kato and Y. Tanimura, Quantum heat transport of a two-qubit system: Interplay between system-bath coherence and qubit-qubit coherence, J. Chem. Phys. 143, 064107 (2015).
- A. Erpenbeck and M. Thoss, Hierarchical quantum master equation approach to vibronic reaction dynamics at metal surfaces, J. Chem. Phys. 151, 191101 (2019).
- C. Schinabeck and M. Thoss, Hierarchical quantum master equation approach to current fluctuations in nonequilibrium charge transport through nanosystems, Phys. Rev. B 101, 075422 (2020).
- C. Schinabeck, R. Härtle, and M. Thoss, Hierarchical quantum master equation approach to electronic-vibrational coupling in nonequilibrium transport through nanosystems: Reservoir formulation and application to vibrational instabilities, Phys. Rev. B 97, 235429 (2018).
- R. Härtle, G. Cohen, D. R. Reichman, and A. J. Millis, Decoherence and lead-induced interdot coupling in nonequilibrium electron transport through interacting quantum dots: A hierarchical quantum master equation approach, Phys. Rev. B 88, 235426 (2013).
- A. Kato and Y. Tanimura, Quantum heat current under non-perturbative and non-Markovian conditions: Applications to heat machines, J. Chem. Phys. 145, 224105 (2016).
- A. Colla, H.-P. Breuer, and G. Gasbarri, Unveiling coherent dynamics in non-Markovian open quantum systems: Exact expression and recursive perturbation expansion, Phys. Rev. A 112, L050203 (2025).
- R. Kosloff and Y. Rezek, The quantum harmonic Otto cycle, Entropy 19, 136 (2017).
- A. Colla, F. Hasse, D. Palani, T. Schaetz, H.-P. Breuer, and U. Warring, Observing time-dependent energy level renormalisation in an ultrastrongly coupled open system, Nat. Commun. 16, 2502 (2025).
- A. Colla, H.-P. Breuer, and G. Gasbarri, Recursive perturbation approach to time-convolutionless master equations: Explicit construction of generalized Lindblad generators for arbitrary open systems, Phys. Rev. A 112, 052222 (2025).
- S. A. Gurvitz and Y. S. Prager, Microscopic derivation of rate equations for quantum transport, Phys. Rev. B 53, 15932 (1996).