Skip to main content
Log in

Quantum Heat Engine with Level Degeneracy for Oscillator-shaped Potential Well

  • Research
  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

In this paper, we consider positive oscillator-shaped well potential and set a Szilard-like quantum heat engine based on energy level degeneracy. By using position-dependent energy eigenvalues of the oscillator-shaped well, we compute extracted work and efficiency based on Stirling-like thermodynamical cycle. We obtain numerical results for physical quantities and discuss work and efficiency dependence of angular frequency, well width, and temperature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data Availability

The datasets used and analyzed during the current study are available from the corresponding author upon reasonable request.

References

  1. Maxwell, J.C.: Sketch of thermodynamics. In: Knott, C.G. (ed.) Life and Scientific Work of Peter Guthrie Tait, pp. 213–214. Cambridge University Press, Cambridge (1911)

    Google Scholar 

  2. Szilard, L.: über die entropieverminderung in einem thermodynamischen system bei eingriffen intelligenter wesen. Z. Physik 53, 840–856 (1929). https://doi.org/10.1007/BF01341281

    Article  ADS  MATH  Google Scholar 

  3. Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 183–191 (1961). https://doi.org/10.1147/rd.53.0183

    Article  MathSciNet  MATH  Google Scholar 

  4. Bennett, C.H.: The thermodynamics of computation–a review. Int. J. Theor. Phys. 5(3), 905–940 (1982). https://doi.org/10.1007/BF02084158

    Article  MathSciNet  Google Scholar 

  5. Leff, H., Rex, A.: Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing. Institute of Physics, London (2003)

    Google Scholar 

  6. Maruyama, K., Nori, F., Vedral, V.: Colloquium: the physics of maxwell’s demon and information. Rev. Mod. Phys. 81, 1–23 (2009). https://doi.org/10.1103/RevModPhys.81.1

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Bérut, A., et al.: Experimental verification of landauer’s principle linking information and thermodynamics. Nature 483, 187 (2012). https://doi.org/10.1038/nature10872

    Article  ADS  Google Scholar 

  8. Koski, J.V., et al.: Experimental realization of a szilard engine with a single electron. Proc. Natl. Acad. Sci. U.S.A. 111, 13786–13789 (2014). https://doi.org/10.1073/pnas.1406966111

    Article  ADS  Google Scholar 

  9. Serreli, V., Lee, C.-F., Kay, E.R., Leigh, D.A.: A molecular information ratchet. Nature 445, 523–527 (2007). https://doi.org/10.1038/nature05452

    Article  ADS  Google Scholar 

  10. Raizen, M.G.: Comprehensive control of atomic motion. Science 324, 1403–1406 (2009). https://doi.org/10.1126/science.1171506

    Article  ADS  Google Scholar 

  11. Bannerman, S.T., Price, G.N., Viering, K., Raizen, M.G.: Single-photon cooling at the limit of trap dynamics: Maxwell’s demon near maximum efficiency. New J. Phys. 11(6), 063044 (2009). https://doi.org/10.1088/1367-2630/11/6/063044

    Article  Google Scholar 

  12. Koski, J.V., Maisi, V.F., Sagawa, T., Pekola, J.P.: Experimental observation of the role of mutual information in the nonequilibrium dynamics of a maxwell demon. Phys. Rev. Lett. 113(5), 030601 (2014). https://doi.org/10.1103/PhysRevLett.113.030601

    Article  ADS  Google Scholar 

  13. Koski, J.V., Kutvonen, A., Khaymovich, I.M., Ala-Nissila, T., Pekola, J.P.: On-chip maxwell’s demon as an information-powered refrigerator. Phys. Rev. Lett. 115(5), 260602 (2015). https://doi.org/10.1103/PhysRevLett.115.260602

    Article  ADS  Google Scholar 

  14. Vinjanampathy, S., Anders, J.: Quantum thermodynamics. Contemp. Phys. 57(4), 545–579 (2016). https://doi.org/10.1080/00107514.2016.1201896

    Article  ADS  Google Scholar 

  15. Bhattacharjee, S., Dutta, A.: Quantum thermal machines and batteries. Eur. Phys. J. B 94, 239 (2021). https://doi.org/10.1140/epjb/s10051-021-00235-3

    Article  ADS  Google Scholar 

  16. Scovil, H.E.D., Schulz-DuBois, E.O.: Three-level masers as heat engines. Phys. Rev. Lett., 2 2, 262–263 (1959). https://doi.org/10.1103/PhysRevLett.2.262

    Article  Google Scholar 

  17. Quan, H.T., Liu, Y.-X., Sun, C.P., Nori, F.: Quantum thermodynamic cycles and quantum heat engines. Phys. Rev. E 76(18), 031105 (2007). https://doi.org/10.1103/PhysRevE.76.031105

    Article  ADS  MathSciNet  Google Scholar 

  18. Mahler, G.: Quantum Thermodynamic Processes: Energy and Information Flow at the Nanoscale. Pan Stanford Publ., Singapore (2015)

    MATH  Google Scholar 

  19. Quan, H.T.: Quantum thermodynamic cycles and quantum heat engines. ii. Phys. Rev. E 79(10), 041129 (2009). https://doi.org/10.1103/PhysRevE.79.041129

    Article  ADS  MathSciNet  Google Scholar 

  20. Chen, L., Liu, X., Wu, F., Xia, S., Feng, H.: Exergy-based ecological optimization of an irreversible quantum carnot heat pump with harmonic oscillators. Phys. A 537, 122597 (2020). https://doi.org/10.1016/j.physa.2019.122597

    Article  MATH  Google Scholar 

  21. Tajima, H., Hayashi, M.: Finite-size effect on optimal efficiency of heat engines. Phys. Rev. E 96(38), 012128 (2017). https://doi.org/10.1103/PhysRevE.96.012128

    Article  ADS  Google Scholar 

  22. Wang, H., Liu, S., He, J.: Performance analysis and parametric optimum criteria of a quantum otto heat engine with heat transfer effects. Appl. Therm. Eng. 29(4), 706–711 (2009). https://doi.org/10.1016/j.applthermaleng.2008.03.042

    Article  ADS  Google Scholar 

  23. Stefanatos, D.: Optimal efficiency of a noisy quantum heat engine. Phys. Rev. E 90(5), 012119 (2014). https://doi.org/10.1103/PhysRevE.90.012119

    Article  ADS  MathSciNet  Google Scholar 

  24. Wu, F., Chen, L., Sun, F., Wu, C., Li, Q.: Generalized model and optimum performance of an irreversible quantum brayton engine with spin systems. Phys. Rev. E 73(7), 016103 (2006). https://doi.org/10.1103/PhysRevE.73.016103

    Article  ADS  Google Scholar 

  25. Dong, C.D., Lefkidis, G., Hubner, W.: Magnetic quantum diesel engine in ni\({}_{2}\). Phys. Rev. B 88(11), 214421 (2013). https://doi.org/10.1103/PhysRevB.88.214421

    Article  ADS  Google Scholar 

  26. Dinis, L., et al.: Thermodynamics at the microscale: from effective heating to the brownian carnot engine. J. Stat. Mech., 054003 (2016). https://doi.org/10.1088/1742-5468/2016/05/054003

  27. Agarwal, G.S., Chaturvedi, S.: Quantum dynamical framework for brownian heat engines. Phys. Rev. E 88(12), 012130 (2013). https://doi.org/10.1103/PhysRevE.88.012130

    Article  ADS  Google Scholar 

  28. Aydiner, E., Han, S.D.: Quantum heat engine model of mixed triangular spin system as a working substance. Phys. A 509, 766–776 (2018). https://doi.org/10.1016/j.physa.2018.06.018

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang, X.Y., Huang, X.L., Yi, X.X.: Quantum otto heat engine with a non-markovian reservoir. J. Phys. A: Math. Theor. 47(45), 455002 (2014). https://doi.org/10.1088/1751-8113/47/45/455002

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  31. Quan, H.T., Zhang, P., Sun, C.P.: Quantum-classical transition of photon-carnot engine induced by quantum decoherence. Phys. Rev. E 73(6), 036122 (2006). https://doi.org/10.1103/PhysRevE.73.036122

    Article  ADS  Google Scholar 

  32. Huang, X.L., Wang, T., Yi, X.X.: Effects of reservoir squeezing on quantum systems and work extraction. Phys. Rev. E 86(6), 051105 (2012). https://doi.org/10.1103/PhysRevE.86.051105

    Article  ADS  Google Scholar 

  33. Roßnagel, J., Abah, O., Schmidt-Kaler, F., Singer, K., Lutz, E.: Nanoscale heat engine beyond the carnot limit. Phys. Rev. Lett. 112(5), 030602 (2014). https://doi.org/10.1103/PhysRevLett.112.030602

    Article  ADS  Google Scholar 

  34. Niedenzu, W., Mukherjee, V., Ghosh, A., Kofman, A.G., G., K.: Quantum engine efficiency bound beyond the second law of thermodynamics. Nat. Commun. 9, 165 (2018). https://doi.org/10.1038/s41467-017-01991-6

  35. Wang, J., He, J., Ma, Y.: Finite-time performance of a quantum heat engine with a squeezed thermal bath. Phys. Rev. E 100(8), 052126 (2019). https://doi.org/10.1103/PhysRevE.100.052126

    Article  ADS  Google Scholar 

  36. Dillenschneider, R., Lutz, E.: Energetics of quantum correlations. EPL 88(5), 50003 (2009). https://doi.org/10.1209/0295-5075/88/50003

    Article  ADS  Google Scholar 

  37. Geva, E., Kosloff, R.: A quantum-mechanical heat engine operating in finite time. a model consisting of spin-1/2 systems as the working fluid. J. Chem. Phys. 96, 3054 (1992). https://doi.org/10.1063/1.461951

    Article  ADS  Google Scholar 

  38. Harbola, U., Rahav, S., Mukamel, S.: Quantum heat engines: a thermodynamic analysis of power and efficiency. EPL 99, 50005 (2012). https://doi.org/10.1209/0295-5075/99/50005

    Article  ADS  Google Scholar 

  39. Rahav, S., Harbola, U., Mukamel, S.: Heat fluctuations and coherences in a quantum heat engine. Phys. Rev. A 86(8), 043843 (2012). https://doi.org/10.1103/PhysRevA.86.043843

    Article  ADS  Google Scholar 

  40. Su, S., Zhang, Y., Su, G., Chen, J.: The carnot efficiency enabled by complete degeneracies. Phys. Lett. A 382(32), 2108–2112 (2018). https://doi.org/10.1016/j.physleta.2018.05.042

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Lin, B., Chen, J.: Performance analysis of an irreversible quantum heat engine working with harmonic oscillators. Phys. Rev. E 67(8), 046105 (2003). https://doi.org/10.1103/PhysRevE.67.046105

    Article  ADS  Google Scholar 

  42. Lin, B., Chen, J.: Optimization on the performance of a harmonic quantum brayton heat engine. J. Appl. Phys. 94, 6185–6191 (2003). https://doi.org/10.1063/1.1616983

    Article  ADS  Google Scholar 

  43. Insinga, A., Andresen, B., Salamon, P.: Thermodynamical analysis of a quantum heat engine based on harmonic oscillators. Phys. Rev. E 94(10), 012119 (2016). https://doi.org/10.1103/PhysRevE.94.012119

    Article  ADS  Google Scholar 

  44. Kosloff, R., Rezek, Y.: The quantum harmonic otto cycle. Entropy 19, 136 (2017). https://doi.org/10.3390/e19040136

    Article  ADS  Google Scholar 

  45. Kim, S.V., et al.: Quantum szilard engine. Phys. Rev. Lett. 106(4), 070401 (2011). https://doi.org/10.1103/PhysRevLett.106.070401

    Article  ADS  Google Scholar 

  46. Li, H., Zou, J., Li, J.-G., Shao, B., Wu, L.-A.: Quantum isothermal reversible process of particles in a box with a delta potential. J. Korean Phys. Soc. 66(6), 739–743 (2015). https://doi.org/10.3938/jkps.66.739

    Article  Google Scholar 

  47. Cai, C.Y., Dong, H., Sun, C.P.: Multiparticle quantum szilard engine with optimal cycles assisted by a maxwell’s demon. Phys. Rev. E 85(12), 031114 (2012). https://doi.org/10.1103/PhysRevE.85.031114

    Article  ADS  Google Scholar 

  48. Zhuang, Z., Liang, S.-D.: Quantum szilard engines with arbitrary spin. Phys. Rev. E 90(11), 052117 (2014). https://doi.org/10.1103/PhysRevE.90.052117

    Article  ADS  Google Scholar 

  49. Bengtsson, J., et al.: Quantum szilard engine with attractively interacting bosons. Phys. Rev. Lett. 120(5), 100601 (2018). https://doi.org/10.1103/PhysRevLett.120.100601

    Article  ADS  Google Scholar 

  50. Park, J.J., et al.: Heat engine driven by purely quantum information. Phys. Rev. Lett. 111(5), 230402 (2013). https://doi.org/10.1103/PhysRevLett.111.230402

    Article  ADS  Google Scholar 

  51. Mehta, V., Johal, R.S.: Quantum otto engine with exchange coupling in the presence of level degeneracy. Phys. Rev. Lett. 96(7), 032110 (2017). https://doi.org/10.1103/PhysRevE.96.032110

    Article  ADS  Google Scholar 

  52. Thomas, G., Das, D., Ghosh, S.: Quantum heat engine based on level degeneracy. Phys. Rev. E 100(7), 012123 (2019). https://doi.org/10.1103/PhysRevE.100.012123

    Article  ADS  Google Scholar 

  53. Chatterjee, S., Koner, A., Chatterjee, S., Kumar, C.: Temperature-dependent maximization of work and efficiency in a degeneracy-assisted quantum stirling heat engine. Phys. Rev. E 103(12), 062109 (2021). https://doi.org/10.1103/PhysRevE.103.062109

    Article  ADS  MathSciNet  Google Scholar 

  54. Davies, P., Thomas, L., Zahariade, G.: The harmonic quantum szilárd engine. Am. J. Phys. 89(12), 1123–1131 (2019). https://doi.org/10.1119/10.0005946

    Article  ADS  Google Scholar 

  55. Aydiner, E.: Quantum szilard engine for the fractional power-law potentials. Sci. Rep. 11, 1576 (2021). https://doi.org/10.1038/s41598-020-80639-w

    Article  ADS  Google Scholar 

  56. Aydiner, E.: Space-fractional quantum heat engine based on level degeneracy. Sci. Rep. 11, 17901 (2021). https://doi.org/10.1038/s41598-021-97304-5

    Article  ADS  Google Scholar 

  57. Zhang, H.W., Huang, X.L., Wu, S.L.: Quantum heat engine with identical particles and level degeneracy. Int. J. Mod. Phys. B, 2450109. https://doi.org/10.1142/S0217979224501091

  58. Jafarov, E.I., Nagiyev, S.M.: Exact solution of the position-dependent mass schrödinger equation with the completely positive oscillator-shaped quantum well potential. https://doi.org/10.48550/arXiv.2212.13062

  59. Mathews, P.M., Lakshmanan, M.: A quantum-mechanically solvable nonpolynomial lagrangian with velocity-dependent interaction. Nuovo Cim. 26, 299–316 (1975). https://doi.org/10.1007/BF02769015

    Article  ADS  MathSciNet  Google Scholar 

  60. Schmidt, A.G.M.: Time evolution for harmonic oscillators with position-dependent mass. Phys. Scr. 75(4), 480 (2007). https://doi.org/10.1088/0031-8949/75/4/019

    Article  ADS  MathSciNet  Google Scholar 

  61. Amir, N., Iqbal, S.: Exact solutions of schrödinger equation for the position-dependent effective mass harmonic oscillator. Commun. Theor. Phys. 62(6), 790 (2014). https://doi.org/10.1088/0253-6102/62/6/03

    Article  MathSciNet  MATH  Google Scholar 

  62. Quesne, C.: Generalized nonlinear oscillators with quasi-harmonic behaviour: classical solutions. J. Math. Phys. 56, 012903 (2015). https://doi.org/10.1063/1.4906113

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. Karthiga, S., et al.: Quantum solvability of a general ordered position dependent mass system: mathews-lakshmanan oscillator. J. Math. Phys. 58, 102110 (2017). https://doi.org/10.1063/1.5008993

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. Jafarov, E.I., et al.: Exact solution of the position-dependent effective mass and angular frequency schrödinger equation: harmonic oscillator model with quantized confinement parameter. J. Phys. A: Math. Theor. 53(48), 485301 (2020). https://doi.org/10.1088/1751-8121/abbd1a

    Article  MATH  Google Scholar 

  65. Jafarov, E.I., Nagiyev, S.M., Jafarova, A.M.: Quantum-mechanical explicit solution for the confined harmonic oscillator model with the von roos kinetic energy operator. Rep. Math. Phys. 86(1), 25–37 (2020). https://doi.org/10.1016/S0034-4877(20)30055-0

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. Jafarov, E.I., Nagiyev, S.M., Seyidova, A.M.: Dynamical symmetry of a semiconfined harmonic oscillator model with a position-dependent effective mass. https://doi.org/10.48550/arXiv.2305.11702

  67. Jafarov, E.I., Nagiyev, S.M.: Exact solutions of schrödinger equation for the position-dependent effective mass harmonic oscillator. https://doi.org/10.48550/arXiv.2212.13062

  68. Nagiyev, S.M.: On two direct limits relating pseudo-jacobi polynomials to hermite polynomials and the pseudo-jacobi oscillator in a homogeneous gravitational field. Theor. Math. Phys. 210, 121–134 (2022). https://doi.org/10.1134/S0040577922010093

    Article  MathSciNet  MATH  Google Scholar 

  69. Nagiyev, S.M., et al.: Exactly solvable model of the linear harmonic oscillator with a position-dependent mass under external homogeneous gravitational field. European Phys. J. Plus 62(5), 540 (2022). https://doi.org/10.1140/epjp/s13360-022-02715-6

    Article  ADS  Google Scholar 

  70. Jafarov, E.I., Nagiyev, S.M.: On the exactly-solvable semi-infinite quantum well of the non-rectangular step-harmonic profile. Quantum Stud.: Math. Found 9, 387–404 (2022). https://doi.org/10.1007/s40509-022-00275-z

    Article  MathSciNet  Google Scholar 

  71. Ross, O.: Position-dependent effective masses in semiconductor theory. Phys. Rev. B 27, 7547–7552 (1983). https://doi.org/10.1103/PhysRevB.27.7547

    Article  ADS  Google Scholar 

  72. Smith, D.L., Mailhiot, C.: Theory of semiconductor superlattice electronic structure. Rev. Mod. Phys. 62, 173–234 (1990). https://doi.org/10.1103/RevModPhys.62.173

    Article  ADS  Google Scholar 

  73. Barranco, M., Pi, M., Gatica, S.M., Hernández, E.S., Navarro, J.: Structure and energetics of mixed \({}^{4}\)he-\({}^{3}\)he drops. Phys. Rev. B 56, 8997–9003 (1997). https://doi.org/10.1103/PhysRevB.56.8997

    Article  ADS  Google Scholar 

  74. Einevoll, G.T.: Operator ordering in effective-mass theory for heterostructures ii. strained systems. Phys. Rev. B 42, 3497–3502 (1990). https://doi.org/10.1103/PhysRevB.42.3497

    Article  ADS  Google Scholar 

  75. Morrow, R.A.: Establishment of an effective-mass hamiltonian for abrupt heterojunctions. Phys. Rev. B 35, 8074–8079 (1987). https://doi.org/10.1103/PhysRevB.35.8074

    Article  ADS  Google Scholar 

  76. BenDaniel, D.J., Duke, C.B.: Space-charge effects on electron tunneling. Phys. Rev. 152, 683–692 (1966). https://doi.org/10.1103/PhysRev.152.683

    Article  ADS  Google Scholar 

Download references

Acknowledgements

Authors thank Ramazan Sever for valuable suggestions. This work is supported by İstanbul University Post-Doctoral Research Project: MAB-2021-38032.

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed equally to this work.

Corresponding author

Correspondence to Ekrem Aydiner.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Evkaya, Y., Ökcü, Ö. & Aydiner, E. Quantum Heat Engine with Level Degeneracy for Oscillator-shaped Potential Well. Int J Theor Phys 62, 237 (2023). https://doi.org/10.1007/s10773-023-05498-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10773-023-05498-3

Keywords

Navigation