Quantum Information Processing

, Volume 5, Issue 6, pp 503–536 | Cite as

Dephasing of Quantum Bits by a Quasi-Static Mesoscopic Environment

Article

We examine coherent processes in a two-state quantum system that is strongly coupled to a mesoscopic spin bath and weakly coupled to other environmental degrees of freedom. Our analysis is specifically aimed at understanding the quantum dynamics of solid-state quantum bits such as electron spins in semiconductor structures and superconducting islands. The role of mesoscopic degrees of freedom with long correlation times (local degrees of freedom such as nuclear spins and charge traps) in qubit-related dephasing is discussed in terms of a quasi-static bath. A mathematical framework simultaneously describing coupling to the slow mesoscopic bath and a Markovian environment is developed and the dephasing and decoherence properties of the total system are investigated. The model is applied to several specific examples with direct relevance to current experiments. Comparisons to experiments suggests that such quasi-static degrees of freedom play an important role in current qubit implementations. Several methods of mitigating the bath-induced error are considered.

Keywords

Spin bath decoherence dephasing quantum bit quantum dot superconducting qubit 

PACS

03.65.Yz 03.67.-a 73.21.La 74.78.Na 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Loss D., DiVincenzo D. (1998). Phys. Rev. A 57:120CrossRefADSGoogle Scholar
  2. 2.
    Imamoglu A., Awschalom D.D., Burkard G., DiVincenzo D.P., Loss D., Sherwin M., Small A. (1999). Phys. Rev. Lett. 83:4204CrossRefADSGoogle Scholar
  3. 3.
    Merkulov I.A., Efros A.L., Rosen M. (2002). Phys. Rev. B 65:205309CrossRefADSGoogle Scholar
  4. 4.
    Khaetskii A.V., Loss D., Glazman L. (2002). Phys. Rev. Lett. 88:186802, URL http://publish.aps.org/abstract/prl/v88/p186802.CrossRefGoogle Scholar
  5. 5.
    Taylor J.M., Marcus C.M., Lukin M.D. (2003). Phys. Rev. Lett. 90:206803CrossRefADSGoogle Scholar
  6. 6.
    Johnson A.C., Petta J., Taylor J.M., Lukin M.D., Marcus C.M., Hanson M.P., Gossard A.C. (2005). Nature 435:925CrossRefADSGoogle Scholar
  7. 7.
    F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Hanson, L. H. Willems van , I. T. Vink, H.-P. Tranitz, W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Vandersypen, Science p. 1113719 (2005), URL http://www.sciencemag.org/cgi/content/abstract/1113719v2.Google Scholar
  8. 8.
    Petta J.R., Johnson A.C., Taylor J.M., Laird E., Yacoby A., Lukin M.D., Marcus C.M. (2005). Science 309:2180CrossRefADSGoogle Scholar
  9. 9.
    de Sousa R., Das Sarma S. (2003). Phys. Rev. B 67:033301CrossRefADSGoogle Scholar
  10. 10.
    Coish W.A., Loss D. (2004). Phys. Rev. B 70:195340CrossRefADSGoogle Scholar
  11. 11.
    Elzerman J.M., Hanson R., van Beveren L.H.W., Witkamp B., Vandersypen L.M.K., Kouwenhoven L.P. (2004). Nature 430:431CrossRefADSGoogle Scholar
  12. 12.
    Fujisawa T., Tokura Y., Hirayama Y. (2001). Phys. Rev. B. (Rapid Comm.) 63:081304ADSGoogle Scholar
  13. 13.
    Golovach V.N., Khaetskii A., Loss D. (2004). Phys. Rev. Lett. 93:016601CrossRefADSGoogle Scholar
  14. 14.
    Hanson R., Witkamp B., Vandersypen L.M.K., van Beveren L.H.W., Elzerman J.M., Kouwenhoven L.P. (2003). Phys. Rev. Lett. 91:196802CrossRefADSGoogle Scholar
  15. 15.
    Vion D., Aassime A., Cottet A., Joyez P., Pothier H., Urbina C., Esteve D., Devorett M. (2002). Science 296:886CrossRefADSGoogle Scholar
  16. 16.
    Pashkin A., Yamamoto T., Astafiev O., Nakamura Y., Averin D., Tsai J. (2003). Nature 421:823CrossRefADSGoogle Scholar
  17. 17.
    Chiorescu I., Nakamura Y., Harmans C., Mooij J. (2003). Science 299:1869CrossRefADSGoogle Scholar
  18. 18.
    Martinis J., Nam S., Aumentado J., Urbina C. (2002). Phys. Rev. Lett. 89:117901CrossRefADSGoogle Scholar
  19. 19.
    Simmonds R., Lang K.M., Hite D.A., Nam S., Pappas D.P., Martinis J.M. (2004). Phys. Rev. Lett. 93:077003CrossRefADSGoogle Scholar
  20. 20.
    Makhlin Y., Shnirman A. (2004). Phys. Rev. Lett. 92:178301CrossRefADSGoogle Scholar
  21. 21.
    Falci G., D’Arrigo A., Mastellone A., Paladino E. (2005). Phys. Rev. Lett. 94:167002CrossRefADSGoogle Scholar
  22. 22.
    Stamp P. (2003). The Physics of Communication. World Scientific, New Jersey, chap. 3, pp. 39–82.Google Scholar
  23. 23.
    Taylor J.M., Imamoglu A., Lukin M.D. (2003). Phys. Rev. Lett. 91:246802CrossRefADSGoogle Scholar
  24. 24.
    Weissman M.B. (1988). Rev. Mod. Phys. 60:537CrossRefADSGoogle Scholar
  25. 25.
    D. Klauser, W. A. Coish, and D. Loss, e-print: cond-mat/0510177 (2005).Google Scholar
  26. 26.
    Coish W.A., Loss D. (2005). Phys. Rev. B 72:125337CrossRefADSGoogle Scholar
  27. 27.
    X. Hu and S. D. Sarma, e-print: cond-mat/0507725 (2005).Google Scholar
  28. 28.
    Zurek W.H. (1981). Phys. Rev. D 24:1516MathSciNetCrossRefADSGoogle Scholar
  29. 29.
    Prokof’ev N.V., Stamp P.C.E. (2000). Reports Prog Phys 63:669CrossRefADSGoogle Scholar
  30. 30.
    Rose G., Smirnov A.Y. (2001). J. Phys.: Cond. Mat. 13:11027CrossRefADSGoogle Scholar
  31. 31.
    Zanardi P., Rasetti M. (1997). Phys. Rev. Lett. 79:3306CrossRefADSGoogle Scholar
  32. 32.
    Viola L., Lloyd S. (1998). Phys. Rev. A 58:2733MathSciNetCrossRefADSGoogle Scholar
  33. 33.
    The breakdown of the two-level approximation in superconductor-based qubit designs has already been explored in great detail (Burkard et al. Phys. Rev B. 69, 064503 (2004)) and we instead focus on other sources of error due to local spins, charge traps, etc.Google Scholar
  34. 34.
    Feynman R.P., Vernon F.L. (1963). Ann. Phys. 24:118MathSciNetCrossRefADSGoogle Scholar
  35. 35.
    Magnus W. (1954). Commun. Pure Appl. Math 7:649MATHMathSciNetGoogle Scholar
  36. 36.
    Cottet A., et al. (2001). Macroscopic Quantum Coherence and Quantum Computing. Kluwer/Plenum, New York, p. 111Google Scholar
  37. 37.
    G. Giedke, J. M. Taylor, D. D’Alessandro, M. D. Lukin, and A. Imamoglu, e-print: quantph/ 0508144 (2005).Google Scholar
  38. 38.
    J. M. Taylor, J. Petta, A. C. Johnson, A. Yacoby, C. M. Marcus, and M. D. Lukin, (in preparation) (2006).Google Scholar
  39. 39.
    de Sousa R., Das Sarma S. (2003). Phys. Rev. B 68:115322CrossRefADSGoogle Scholar
  40. 40.
    W. Yao, R.-B. Liu, and L. J. Sham, e-print: cond-mat/0508441 (2005).Google Scholar
  41. 41.
    Deng C., Hu X. (2005). Phys. Rev. B 72:165333CrossRefADSGoogle Scholar
  42. 42.
    Paget D. (1982). Phys. Rev. B 25:4444CrossRefADSGoogle Scholar
  43. 43.
    G. Teklemariam, E. M. Fortunato, C. C. Lopez, J. Emerson, J. P. Paz, T. F. Havel, and D. G. Cory, e-print: quant-ph/0303115 (2003).Google Scholar
  44. 44.
    For an arbitrary, quasi-static bath (i.e., not necessary a spin-bath) with a density matrix that is diagonal in the eigenbasis of \(\hat{A}_z\), \(\Phi_{\rm FID} = e^{-i \delta t} \int_{-\infty}^{\infty} d \Lambda \rho(\Lambda) e^{-i \Lambda t}\), demonstrating that ΦFID is exactly the inverse Fourier transform of the bath degree of freedom in this case.Google Scholar
  45. 45.
    By assuming the bath density matrix is diagonal in the \(\hat{A}_z\) eigenbasis, the result derived (Eqn. 40) in fact is generally true for any bath that is non-singular (ρsym(ω ≥ Ω) not singular) and satisfies u ≥ 0, not just a spin-bath. However, the spin-bath provides a natural case for \([\hat{H}_B,\hat{A}_z] \simeq 0\), as mentioned in the text.Google Scholar
  46. 46.
    Well-separated singularities in ρsym can be treated as additional stationary phase integral terms, and for each, corresponding oscillations at the resonance with different time-scales u j will emerge.Google Scholar
  47. 47.
    Gardiner C.W. (1985). Handbook of stochastic methods. Spinger, Berlin, 2nd ed.Google Scholar
  48. 48.
    Mehring M. (1976). High Resolution NMR Spectroscopy in Solids. Springer-Verlag, BerlinGoogle Scholar
  49. 49.
    Paget D., Lampel G., Sapoval B., Safarov V. (1977). Phys. Rev. B 15:5780CrossRefADSGoogle Scholar
  50. 50.
    Waugh J., Huber L., Haeberlen U. (1968). Phys. Rev. Lett. 20:180CrossRefADSGoogle Scholar
  51. 51.
    Kautz R., Martinis J. (1990). Phys. Rev. B 42:9903CrossRefADSGoogle Scholar
  52. 52.
    Galperin Y.M., Gurevich V.L. (1991). Phys. Rev. B 43:12900CrossRefADSGoogle Scholar
  53. 53.
    Caldeira A.O., Leggett A.J. (1983). Ann. Phys. 149:347Google Scholar
  54. 54.
    Chattah A.K., lvarez G.A., Levstein P.R., Cucchietti F.M., Pastawski H.M., Raya J., Hirschinger J. (2003). J. Chem. Phys. 119:7943CrossRefADSGoogle Scholar
  55. 55.
    Danieli E.P., Pastawski H.M., Álvarez G.A. (2005). Chem. Phys. Lett. 402:88CrossRefGoogle Scholar
  56. 56.
    Facchi P., Tasaki S., Pascazio S., Nakazato H., Tokuse A., Lidar D. (2005). Phys. Rev. A 71:022302CrossRefADSGoogle Scholar
  57. 57.
    Imamolgu A., Knill E., Tian L., Zoller P. (2003). Phys. Rev. Lett. 91:017402CrossRefADSGoogle Scholar
  58. 58.
    Arecchi F.T., Courtens E., Gilmore R., Thomas H. (1972). Phys. Rev. A 6:2211, URL http://80-link.aps.org.ezp1.harvard.edu/abstract/PRA/v6/p2211.CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of PhysicsHarvard UniversityCambridgeUSA

Personalised recommendations