Quantum behavior of a flux qubit coupled to a resonator

Физика Низких Температур: Том 36, Выпуск 10-11 (Октябрь 2010), c. 1117-1127 ( к оглавлению , назад )

Quantum behavior of a flux qubit coupled to a resonator

A.N. Omelyanchouk¹, S.N. Shevchenko^1,2, Ya.S. Greenberg^3,2, O. Astafiev⁴, and E. Il’ichev²

¹B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Ave., 61103 Kharkov, Ukraine
E-mail: sshevchenko@ilt.kharkov.ua

²Institute of Photonic Technology, P.O. Box 100239, D-07702 Jena, Germany

³Novosibirsk State Technical University, 20 Karl Marx Ave., Novosibirsk, 630092, Russia

⁴NEC Nano Electronics Research Laboratories, Tsukuba, Ibaraki, 305-8501, Japan

Received March 29, 2010

Abstract

The detailed theory for the system of a superconducting qubit coupled to the transmission line resonator is presented. We describe the system by solving analytically and numerically the master equation for the density matrix, which includes dissipative Lindblad term. We calculate the transmission coefficient, which provides the way to probe the dressed states of the qubit. The theoretical results are related to the experiment with the intermediate coupling between the qubit and the resonator, when the coupling energy is of the same order as the qubit relaxation rate.

PACS: 85.25.Am Superconducting device characterization, design, and modeling;
PACS: 85.25.Cp Josephson devices;
PACS: 84.37.+q Measurements in electric variables.

Key words: superconducting qubit, transmission line, resonator.

References

1. E. Ilâichev, A. Yu. Smirnov, M. Grajcar, A. Izmalkov, D. Born, N. Oukhanski, Th. Wagner, W. Krech, H.-G. Meyer, and A. Zagoskin, Fiz. Nizk. Temp. 30, 823 (2004) E. Ilâichev, A. Yu. Smirnov, M. Grajcar, A. Izmalkov, D. Born, N. Oukhanski, Th. Wagner, W. Krech, H.-G. Meyer, and A. Zagoskin, [Low Temp. Phys. 30, 620 (2004)]. CrossRef Google Scholar

2. J. Q. You and F. Nori, Phys. Today 58(11), 42 (2005). CrossRef Google Scholar

3. G. Wendin and V. S. Shumeiko, arXiv:cond-mat/0508729; G. Wendin and V. S. Shumeiko, Fiz. Nizk. Temp. 33, 957 (2007) G. Wendin and V. S. Shumeiko, [Low Temp. Phys. 33, 724 (2007)]. CrossRef Google Scholar

4. A. Zagoskin and A. Blais, Phys. Canada 63, 215 (2007). Google Scholar

5. Yu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin, and J. S. Tsai, Nature 421, 823 (2003). CrossRef Google Scholar

6. A. Izmalkov, M. Grajcar, E. Ilâichev, Th. Wagner, H.-G. Meyer, A. Yu. Smirnov, M. H. S. Amin, Alec Maassen van den Brink, and A. M. Zagoskin, Phys. Rev. Lett. 93, 037003 (2004). CrossRef Google Scholar

7. D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and M. H. Devoret, Science 296, 886 (2002). CrossRef Google Scholar

8. I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, Science 299, 1869 (2003). CrossRef Google Scholar

9. I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, Nature 431, 160 (2004). CrossRef Google Scholar

10. E. Ilâichev, N. Oukhanski, A. Izmalkov, Th. Wagner, M. Grajcar, H.-G. Meyer, A. Yu. Smirnov, Alec Maassen van den Brink, M. H. S. Amin, and A. M. Zagoskin, Phys. Rev. Lett. 91, 097906 (2003). CrossRef Google Scholar

11. J. Johansson, S. Saito, T. Meno, H. Nakano, M. Ueda, K. Semba, and H. Takayanagi, Phys. Rev. Lett. 96, 127006 (2006). CrossRef Google Scholar

12. A. N. Omelyanchouk, S. Savelâev, A. M. Zagoskin, E. Ilâichev, and F. Nori, Phys. Rev. B 80, 212503 (2009). CrossRef Google Scholar

13. Y. Nakamura, Yu. A. Pashkin, T. Yamamoto, and J. S. Tsai, Phys. Rev. Lett. 88, 047901 (2002). CrossRef Google Scholar

14. D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and M. H. Devoret, Fortschr. Phys. 51, 462 (2003). CrossRef Google Scholar

15. A. Izmalkov, M. Grajcar, E. Ilâichev, N. Oukhanski, Th. Wagner, H.-G. Meyer, W. Krech, M. H. S. Amin, Alec Maassen van den Brink, and A. M. Zagoskin, Europhys. Lett. 65, 844 (2004). CrossRef Google Scholar

16. W. D. Oliver, Ya. Yu, J. C. Lee, K. K. Berggren, L. S. Levitov, and T. P. Orlando, Science 310, 1653 (2005). CrossRef Google Scholar

17. M. SillanpÃ¤Ã¤, T. Lehtinen, A. Paila, Yu. Makhlin, and P. Hakonen, Phys. Rev. Lett. 96, 187002 (2006). CrossRef Google Scholar

18. S. N. Shevchenko, S. Ashhab, and F. Nori, Phys. Rep. 492, 1 (2010). CrossRef Google Scholar

19. G. Sun, X. Wen, B. Mao, J. Chen, Y. Yu, P. Wu, and S. Han, Nat. Commun. 1, 51 (2010). Google Scholar

20. R. J. Schoelkopf and S. M. Girvin, Nature 451, 664 (2008). CrossRef Google Scholar

21. A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004). CrossRef Google Scholar

22. Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A 74, 052321 (2006). CrossRef Google Scholar

23. Y.-L. Chen, Y.-F. Xiao, X. Zhou, Xu-Bo Zou, Z.-W. Zhou, and G.-C. Guo, J. Phys. B 41, 175503 (2008). CrossRef Google Scholar

24. J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, Jr., O. Astafiev, Y. Nakamura, and A. Blais, Phys. Rev. A 80, 032109 (2009). CrossRef Google Scholar

25. S. Ashhab and F. Nori, Phys. Rev. A 81, 042311 (2010). CrossRef Google Scholar

26. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004). CrossRef Google Scholar

27. D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 94, 123602 (2005). CrossRef Google Scholar

28. D. I. Schuster, Circuit Quantum Electrodynamics, PhD thesis, Yale University (2007). Google Scholar

29. T. LindstrÃ¶m, C. H. Webster, J. E. Healey, M. S. Colclough, C. M. Muirhead, and A. Y. Tzalenchuk, Supercond. Sci. Technol. 20, 814 (2007). CrossRef Google Scholar

30. A. A. Abdumalikov, Jr., O. Astafiev, Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, Phys. Rev. B 78, 180502(R) (2008). CrossRef Google Scholar

31. G. Oelsner, S. H. W. van der Ploeg, P. Macha, U. HÃ¼bner, D. Born, S. Anders, E. Ilâichev, H.-G. Meyer, M. Grajcar, S. WÃ¼nsch, M. Siegel, A. N. Omelyanchouk, and O. Astafiev, Phys. Rev. B 81, 172505 (2010). CrossRef Google Scholar

32. T. P. Orlando, J. E. Mooij, L. Tian, C. H. van der Wal, L. S. Levitov, S. Lloyd, and J. J. Mazo, Phys. Rev. B 60, 15399 (1999). CrossRef Google Scholar

33. O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Jr., Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, Science 327, 840 (2010). CrossRef Google Scholar

34. L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008). CrossRef Google Scholar

35. S. N. Shevchenko, S. H. W. van der Ploeg, M. Grajcar, E. Ilâichev, A. N. Omelyanchouk, and H.-G. Meyer, Phys. Rev. B 78, 174527 (2008). CrossRef Google Scholar

36. Ya. S. Greenberg, Phys. Rev. B 76, 104520 (2007). CrossRef Google Scholar

37. J. M. Fink, M. GÃ¶ppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, Nature 454, 315 (2008). CrossRef Google Scholar

38. W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, Berlin (2001). Google Scholar

39. M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge, Cambridge University Press (1997). Google Scholar

40. L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, Nat. Phys. 5, 105 (2009). CrossRef Google Scholar

41. D. M. Pozar, Microwave Engineering, Wiley, New York, 3rd ed. (1990). Google Scholar

42. P. Macha, S. H. W. van der Ploeg, G. Oelsner, E. Ilâichev, H.-G. Meyer, S. WÃ¼nsch, and M. Siegel, Appl. Phys. Lett. 96, 062503 (2010). CrossRef Google Scholar

43. Ya. S. Greenberg, A. Izmalkov, M. Grajcar, E. Ilâichev, W. Krech, H.-G. Meyer, M. H. S. Amin, and A. Maassen van den Brink, Phys. Rev. B 66, 214525 (2002). CrossRef Google Scholar

44. M. Grajcar, A. Izmalkov, E. Ilâichev, Th. Wagner, N. Oukhanski, U. HÃ¼bner, T. May, I. Zhilyaev, H. E. Hoenig, Ya. S. Greenberg, V. I. Shnyrkov, D. Born, W. Krech, H.-G. Meyer, Alec Maassen van den Brink, and M. H. S. Amin, Phys. Rev. B 69, 060501(R) (2004). CrossRef Google Scholar