International Journal of Theoretical Physics

, Volume 57, Issue 5, pp 1391–1403 | Cite as

On the Transition Probability for the Near or Exact Resonance with the RWA

  • Dafa Li
  • Meng Zhao
  • Shuwang Li


Rotating wave approximation (RWA) has been used to evaluate the transition probability and solve the Schrödinger equation approximately in quantum optics. Examples include the invalidity of the traditional adiabatic condition for the adiabaticity invoking a two-level coupled system near resonance. Here, using a two-state system driven by an oscillatory force, we derive the exact transition probability by solving the Schrödinger equation analytically for a general wave function. Our results reveal that the application of the RWA may lead to false conclusions on the transition probability for the near resonance with weak coupling, especially when the coupling strength is about a half of the transition frequency. We also investigate conditions for which RWA may work or fail.


RWA (rotating wave approximation) The Schrödinger equation Transition probability Near resonance Weak coupling 



S. Li thanks the stimulating discussion with Prof. Z. Sullivan at the Physics Department of Illinois Institute of Technology. D. Li thanks Man-Hong Yung and Yanjun Hao for their discussion. All authors thank the reviewer for his useful suggestions. This work was supported by NSFC (Grant No. 10875061) and Tsinghua National Laboratory for Information Science and Technology.


  1. 1.
    Satanin, A.M., Denisenko, M.V., Ashhab, S., Nori, F.: Phys. Rev. B 85, 184524 (2012)ADSCrossRefGoogle Scholar
  2. 2.
    Jaynes, E.T., Cummings, F.W.: Proc. IEEE 51, 89 (1963)CrossRefGoogle Scholar
  3. 3.
    Larson, J.: Phys. Rev. Lett. 108, 033601 (2012)ADSCrossRefGoogle Scholar
  4. 4.
    Greentree, A.D., Koch, J., Larson, J.: J. Phys. B 46, 220201 (2013)ADSCrossRefGoogle Scholar
  5. 5.
    Larson, J.: Phys. Scr. T153, 014040 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    Meystre, P., Sargent III, M.: Elements of Quantum Optics. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  7. 7.
    Griffiths, D.J.: Introduction to Quantum Mechanics, 2nd edn. Prentice Hall (2005)Google Scholar
  8. 8.
    Grifoni, M., Hänggi, P.: Phys. Rep. 304, 229–354 (1998)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Wang, X.G., Sun, C.P.: Acta physica Sinica 5, 881–889 (1996)ADSGoogle Scholar
  10. 10.
    Gerry, C.C., Knight, P.L.: Introductory Quantum Optics. Cambridge University Press, New York (2005)Google Scholar
  11. 11.
    Amin, M.H.S.: Phys. Rev. Lett. 102, 220401 (2009)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Sun, Z., Ma, J., Wang, X., Nori, F.: Phys. Rev. A 86, 012107 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    Silveri, M.P., Kumar, K.S., Li, J., Tuorila, J., Vepsäl äinen, A., Thuneberg, E. V., Paraoanu, G. S.: New J. Phys. 17, 043058 (2015)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Boyd, J.K.: J. Math. Phys. 41, 4330 (2000)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Xie, Q., Hai, W.: Phys. Rev. A 82, 032117 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    Ashhab, S., Johansson, J.R., Zagoskin, A.M., Nori, F.: Phys. Rev. A 75, 063414 (2007)ADSCrossRefGoogle Scholar
  17. 17.
    Angelo, R.M., Wreszinski, W.F.: Phys. Rev. A 72, 034105 (2005)ADSCrossRefGoogle Scholar
  18. 18.
    Irish, E.K., Gea-Banacloche, J., Martin, I., Schwab, K.C.: Phys. Rev. B 72, 195410 (2005)ADSCrossRefGoogle Scholar
  19. 19.
    Ford, G. W., O’Connell, R.F.: Physica A 243, 377–381 (1997)ADSCrossRefGoogle Scholar
  20. 20.
    Fujii, K.: ArXiv:quant-ph/1301.3585v3 (2014)
  21. 21.
    Ashhab, S., Johansson, J.R., Nori, F.: Phys. Rev. A 74, 052330 (2006)ADSCrossRefGoogle Scholar
  22. 22.
    Shevchenko, S.N., Ashhab, S.S., Nori, F.: Phys. Rep. 492, 1–30 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    Satanin, A.M., Denisenko, M.V., Gelman, A.I., Nori, F.: Phys. Rev. B 90, 104516 (2014)ADSCrossRefGoogle Scholar
  24. 24.
    Ashhab, S., Nori, F.: Phys. Rev. A 81, 042311 (2010)ADSCrossRefGoogle Scholar
  25. 25.
    Cao, X., You, J.Q., Zheng, H., Nori, F.: Phys. Rev. A 82, 022119 (2010)ADSCrossRefGoogle Scholar
  26. 26.
    Cao, X., You, J. Q., Zheng, H., Nori, F.: New J. Phys. 13, 073002 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    Cao, X., Ai, Q., Sun, C.P., Nori, F.: Phys. Lett. A 376, 349–357 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    Garziano, L., Stassi, R., Macrì, V., Kockum, A.F., Savasta, S., Nori, F.: Phys. Rev. A 92, 063830 (2015)ADSCrossRefGoogle Scholar
  29. 29.
    Cirio, M., Liberato, S.D., Lambert, N., Nori, F.: Phys. Rev. Lett. 116, 113601 (2016)ADSCrossRefGoogle Scholar
  30. 30.
    Garziano, L., Macrì, V., Stassi, R., Stefano, O.D., Nori, F., Savasta, S.: Phys. Rev. Lett. 117, 043601 (2016)ADSCrossRefGoogle Scholar
  31. 31.
    Stassi, R., Savasta, S., Garziano, L., Spagnolo, B., Nori, F.: New J. Phys. 18, 123005 (2016)ADSCrossRefGoogle Scholar
  32. 32.
    Stefano, O.D., Stassi, R., Garziano, L., Kockum, A.F., Savasta, S., Nori, F.: New J. Phys. 19, 053010 (2017)CrossRefGoogle Scholar
  33. 33.
    Kockum, A.F., Miranowicz, A., Macrì, V., Savasta, S., Nori, F.: Phys. Rev. A 95, 063849 (2017)ADSCrossRefGoogle Scholar
  34. 34.
    Kockum, A.F., MacRì, V., Garziano, L., Savasta, S., Nori, F: Sci. Rep. 7, 5313 (2017)ADSCrossRefGoogle Scholar
  35. 35.
    Chen, Z., Wang, Y., Li, T., Tian, L., Qiu, Y., Inomata, K., Yoshihara, F., Han, S., Nori, F., Tsai, J. S., You, J. Q.: Phys. Rev. A 96, 012325 (2017)ADSCrossRefGoogle Scholar
  36. 36.
    Cirio, M., Debnath, K., Lambert, N., Nori, F.: Phys. Rev. Lett. 119, 053601 (2017)ADSCrossRefGoogle Scholar
  37. 37.
    Stassi, R., Macrì, V., Kockum, A.F., Stefano, O.D., Miranowicz, A., Savasta, S., Nori, F.: Phys. Rev. A 96, 023818 (2017)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.Center for Quantum Information Science and TechnologyTsinghua National Laboratory for Information Science and Technology (TNList)BeijingChina
  3. 3.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA

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