{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:01:14Z","timestamp":1760144474287,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,4,25]],"date-time":"2024-04-25T00:00:00Z","timestamp":1714003200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this work, we lay the foundations for computing the behavior of a quantum heat engine whose working medium consists of an ensemble of non-harmonic quantum oscillators. In order to enable this analysis, we develop a method based on the Schr\u00f6dinger picture. We investigate different possible choices on the basis of expanding the density operator, as it is crucial to select a basis that will expedite the numerical integration of the time-evolution equation without compromising the accuracy of the computed results. For this purpose, we developed an estimation technique that allows us to quantify the error that is unavoidably introduced when time-evolving the density matrix expansion over a finite-dimensional basis. Using this and other ways of evaluating a specific choice of basis, we arrive at the conclusion that the basis of eigenstates of a harmonic Hamiltonian leads to the best computational performance. Additionally, we present a method to quantify and reduce the error that is introduced when extracting relevant physical information about the ensemble of oscillators. The techniques presented here are specific to quantum heat cycles; the coexistence within a cycle of time-dependent Hamiltonian and coupling with a thermal reservoir are particularly complex to handle for the non-harmonic case. The present investigation is paving the way for numerical analysis of non-harmonic quantum heat machines.<\/jats:p>","DOI":"10.3390\/e26050359","type":"journal-article","created":{"date-parts":[[2024,4,25]],"date-time":"2024-04-25T05:26:13Z","timestamp":1714022773000},"page":"359","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Computational Issues of Quantum Heat Engines with Non-Harmonic Working Medium"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6981-2635","authenticated-orcid":false,"given":"Andrea R.","family":"Insinga","sequence":"first","affiliation":[{"name":"Department of Energy Conversion and Storage, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2657-3424","authenticated-orcid":false,"given":"Bjarne","family":"Andresen","sequence":"additional","affiliation":[{"name":"Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6274-406X","authenticated-orcid":false,"given":"Peter","family":"Salamon","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182-7720, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1038\/s41566-022-01076-x","article-title":"A supercharged photonic quantum heat engine","volume":"16","author":"Kim","year":"2022","journal-title":"Nat. Photonics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"090602","DOI":"10.1103\/PhysRevLett.128.090602","article-title":"Spin Quantum Heat Engine Quantified by Quantum Steering","volume":"128","author":"Ji","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"135502","DOI":"10.1088\/1361-6455\/ac7097","article-title":"A nuclear quadrupolar spin quantum heat engine","volume":"55","author":"Altintas","year":"2022","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"033030","DOI":"10.1088\/1367-2630\/acc04e","article-title":"Quantum heat engine with long-range advantages","volume":"25","author":"Solfanelli","year":"2023","journal-title":"New J. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s10773-023-05498-3","article-title":"Quantum Heat Engine with Level Degeneracy for Oscillator-shaped Potential Well","volume":"62","author":"Evkaya","year":"2023","journal-title":"Int. J. Theor. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Breuer, H.P., and Petruccione, F. (2002). The Theory of Open Quantum Systems, Oxford University Press.","DOI":"10.1007\/3-540-44874-8_4"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"011134","DOI":"10.1103\/PhysRevE.82.011134","article-title":"Optimal performance of reciprocating demagnetization quantum refrigerators","volume":"82","author":"Kosloff","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"051114","DOI":"10.1103\/PhysRevE.85.051114","article-title":"Short time cycles of purely quantum refrigerators","volume":"85","author":"Feldmann","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"055102","DOI":"10.1103\/PhysRevE.65.055102","article-title":"Discrete four-stroke quantum heat engine exploring the origin of friction","volume":"65","author":"Kosloff","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"025107","DOI":"10.1103\/PhysRevE.73.025107","article-title":"Quantum lubrication: Suppression of friction in a first-principles four-stroke heat engine","volume":"73","author":"Feldmann","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"240601","DOI":"10.1103\/PhysRevLett.123.240601","article-title":"Experimental Characterization of a Spin Quantum Heat Engine","volume":"123","author":"Peterson","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"013055","DOI":"10.1088\/1367-2630\/ab6876","article-title":"Quantum signatures in the quantum Carnot cycle","volume":"22","author":"Dann","year":"2020","journal-title":"New J. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Deffner, S. (2018). Efficiency of harmonic quantum Otto engines at maximal power. Entropy, 20.","DOI":"10.3390\/e20110875"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"124140","DOI":"10.1016\/j.physa.2020.124140","article-title":"Power and efficiency optimization of an irreversible quantum Carnot heat engine working with harmonic oscillators","volume":"550","author":"Chen","year":"2020","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kosloff, R., and Rezek, Y. (2017). The Quantum Harmonic Otto Cycle. Entropy, 19.","DOI":"10.3390\/e19040136"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1088\/1367-2630\/8\/5\/083","article-title":"Irreversible performance of a quantum harmonic heat engine","volume":"8","author":"Rezek","year":"2006","journal-title":"New J. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"30008","DOI":"10.1209\/0295-5075\/85\/30008","article-title":"The Quantum Refrigerator: The quest for absolute zero","volume":"85","author":"Rezek","year":"2009","journal-title":"Europhys. Lett."},{"key":"ref_18","unstructured":"Rezek, Y. (2004). The Quantum Harmonic Oscillator as a Thermodynamic Engine. [Ph.D. Thesis, Hebrew University of Jerusalem]."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1885","DOI":"10.3390\/e12081885","article-title":"Reflections on Friction in Quantum Mechanics","volume":"12","author":"Rezek","year":"2010","journal-title":"Entropy"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"60002","DOI":"10.1209\/0295-5075\/113\/60002","article-title":"Optimal performance of a quantum Otto refrigerator","volume":"113","author":"Abah","year":"2016","journal-title":"Epl"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2237","DOI":"10.1103\/PhysRevLett.56.2237","article-title":"Dissipative quantum and classical liouville mechanics of the anharmonic-oscillator","volume":"56","author":"Milburn","year":"1986","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"053834","DOI":"10.1103\/PhysRevA.98.053834","article-title":"Dissipation and thermal noise in hybrid quantum systems in the ultrastrong-coupling regime","volume":"98","author":"Settineri","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"125002","DOI":"10.1103\/PhysRevD.101.125002","article-title":"Nonequilibrium nonlinear open quantum systems: Functional perturbative analysis of a weakly anharmonic oscillator","volume":"101","author":"Hsiang","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"023252","DOI":"10.1103\/PhysRevResearch.3.023252","article-title":"Response theory for nonequilibrium steady states of open quantum systems","volume":"3","author":"Levy","year":"2021","journal-title":"Phys. Rev. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF01609396","article-title":"Completely positive maps and entropy inequalities","volume":"40","author":"Lindblad","year":"1975","journal-title":"Commun. Math. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/S0301-0104(01)00317-2","article-title":"Dissipation in anharmonic molecular systems: Beyond the linear coupling limit","volume":"268","author":"Nest","year":"2001","journal-title":"Chem. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"052117","DOI":"10.1103\/PhysRevA.65.052117","article-title":"Model for dissipative quantum dynamics and nonlinear coupling: Lennard-Jones potential","volume":"65","author":"Nest","year":"2002","journal-title":"Phys. Rev."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"012119","DOI":"10.1103\/PhysRevE.94.012119","article-title":"Thermodynamical analysis of a quantum heat engine based on harmonic oscillators","volume":"94","author":"Insinga","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_29","unstructured":"Leonhardt, K. (2011). Die Beschreibung Dissipativer Quantensysteme: Lindblad-Dynamik. [Diploma Thesis, Technische Universit\u00e4t Chemnitz-Institut f\u00fcr Physik]."},{"key":"ref_30","unstructured":"(2024, March 04). Choose an ODE Solver. Matlab Documentation. Available online: https:\/\/www.mathworks.com\/help\/matlab\/math\/choose-an-ode-solver.html."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"062153","DOI":"10.1103\/PhysRevE.97.062153","article-title":"Quantum heat engines: Limit cycles and exceptional points","volume":"97","author":"Insinga","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_32","first-page":"011101","article-title":"Critical exponents in metastable decay via quantum activation","volume":"75","author":"Dykman","year":"2008","journal-title":"Physica"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1063\/1.1576384","article-title":"Dissipative quantum dynamics of anharmonic oscillators with the multiconfiguration time-dependent Hartree method","volume":"119","author":"Nest","year":"2003","journal-title":"J. Chem. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1137\/S0895479800371529","article-title":"A Krylov-Schur Algorithm for Large Eigenproblems","volume":"23","author":"Stewart","year":"2002","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_35","unstructured":"Rezek, Y. (2011). Heat Machines and Quantum Systems: Towards the Third Heat Machines and Quantum Systems: Towards the Third Law. [Ph.D. Thesis, Hebrew University of Jerusalem]."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"204105","DOI":"10.1063\/1.5096173","article-title":"Quantum thermodynamics and open-systems modeling","volume":"150","author":"Kosloff","year":"2019","journal-title":"J. Chem. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/5\/359\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:33:52Z","timestamp":1760106832000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/5\/359"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,25]]},"references-count":36,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["e26050359"],"URL":"https:\/\/doi.org\/10.3390\/e26050359","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,4,25]]}}}