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Optimal quantum estimation of the coupling constant of Jaynes-Cummings interaction

  • M. G. GenoniEmail author
  • C. Invernizzi
Regular Article

Abstract

We address the estimation of the coupling constant of the Jaynes-Cummings Hamiltonian for a coupled qubit-oscillator system. We evaluate the quantum Fisher Information (QFI) for the system undergone the Jaynes-Cummings evolution, considering that the probe initial state is prepared in a Fock state for the oscillator and in a generic pure state for the qubit; we obtain that the QFI is exactly equal to the number of excitations present in the probe state. We then focus on the two subsystems, namely the qubit and the oscillator alone, deriving the two QFIs of the two reduced states, and comparing them with the previous result. Next we focus on possible measurements on the system, and we find out that if population measurement on the qubit and Fock number measurement on the oscillator are performed together, the Cramer-Rao bound is saturated, that is the corresponding Fisher Information (FI) is always equal to the QFI. We compare also the performances of these energy measurements performed alone, that is when one of the two subsystem is ignored. We show that, when the qubit is prepared in either the ground or the excited state, the local measurements are still optimal. Finally we investigate the case when the harmonic oscillator is prepared in a thermal state and observe how, particularly for small values of the coupling constant, the QFI increases with the average number of thermal photons of the initial state.

Keywords

Harmonic Oscillator European Physical Journal Special Topic Fisher Information Qubit State Quantum Fisher Information 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    E.T. Jaynes, F.W. Cummings, Proc. IEEE 51, 89 (1963)CrossRefGoogle Scholar
  2. 2.
    S. Haroche, J.M. Raimond, Exploring the quantum (Oxford University Press, 2006)Google Scholar
  3. 3.
    J.M. Raimond, M. Brune, S. Haroce, Rev. Mod. Phys. 73, 565 (2001)ADSzbMATHCrossRefGoogle Scholar
  4. 4.
    H. Walther, B.T.H. Varcoe, B.-G. Englert, T. Becker, Reports Progr. Phys. 69, 1325 (2006)ADSCrossRefGoogle Scholar
  5. 5.
    H. Mabuchi, A.C. Doherty, Science 298, 1372 (2002)ADSCrossRefGoogle Scholar
  6. 6.
    D. Leibfried, R. Blatt, C. Monroe, D. Wineland, Rev. Mod. Phys. 75, 281 (2003)ADSCrossRefGoogle Scholar
  7. 7.
    A. Blais, R.-S. Huang, A. Walraff, S.M. Girvin, R.J. Schoelkopf, Phys. Rev. A 69, 062320 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    A. Walraff, D.I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Mayer, S. Kumar, S.M. Girvin, R.J. Schoelkopf, Nature 431, 161 (2004)ADSCrossRefGoogle Scholar
  9. 9.
    R.J.Schoelkopf, S.M.Girvin, Nature 451, 664 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    H.J.Kimble, Nature 453, 1023 (2008)ADSCrossRefGoogle Scholar
  11. 11.
    M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Oxford University Press, Oxford, 2006)Google Scholar
  12. 12.
    V. Giovannetti, S. Lloyd, L. Maccone, Phys. Rev. Lett. 96, 010401 (2006)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    A. Monras, Phys. Rev. A 73, 0338821 (2006)CrossRefGoogle Scholar
  14. 14.
    M.G. Genoni, S. Olivares, M.G.A. Paris, Phys. Rev. Lett. 106, 153603 (2011)ADSCrossRefGoogle Scholar
  15. 15.
    M.G. Genoni, P. Giorda, M.G.A. Paris, Phys. Rev. A 78, 032303 (2008)ADSCrossRefGoogle Scholar
  16. 16.
    G. Brida, I. Degiovanni, A. Florio, M. Genovese, P. Giorda, A. Meda, M.G.A. Paris, A. Shurupov, Phys. Rev. Lett. 104, 100501 (2010)ADSCrossRefGoogle Scholar
  17. 17.
    M. Brunelli, S. Olivares, M.G.A. Paris, Phys. Rev. A 84, 032105 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    P. Zanardi, M.G.A. Paris, L. Campos-Venuti, Phys. Rev. A 78, 042105 (2008)ADSCrossRefGoogle Scholar
  19. 19.
    C. Invernizzi, M. Korbmann, L. Campos-Venuti, M.G.A. Paris, Phys. Rev. A 78, 042106 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    A. Monras, M.G.A. Paris, Phys. Rev. Lett. 98, 160401 (2007)ADSCrossRefGoogle Scholar
  21. 21.
    M.G. Genoni, C. Invernizzi, M.G.A. Paris, Phys. Rev. A 80, 033842 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    D. Bures, Trans. Am. Math. Soc. 135, 199 (1969)MathSciNetzbMATHGoogle Scholar
  23. 23.
    A. Uhlmann, Rep. Math. Phys. 9, 273 (1976)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    W.K. Wootters, Phys. Rev. D 23, 357 (1981)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976)Google Scholar
  26. 26.
    A.S. Holevo, Statistical Structure of Quantum Theory, Lect. Not. Phys. 61(Springer Berlin, 2001)Google Scholar
  27. 27.
    S.L. Braunstein, C.M. Caves, Phys. Rev. Lett. 72, 3439 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. 28.
    S. Braunstein, C.M. Caves, G.J. Milburn, Ann. Phys. 247, 135 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    D.C. Brody, L.P. Hughston, Proc. Roy. Soc. Lond. A 454, 2445 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 30.
    D.C. Brody, L.P. Hughston, Proc. Roy. Soc. Lond. A 455, 1683 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    M.G.A. Paris, Int. J. Quant. Inf. 7, 125 (2009)zbMATHCrossRefGoogle Scholar
  32. 32.
    A.D. O’Connell, M. Hofheinz, M. Ansmann, R.C. Bialczak, M.Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J.M. Martinis, A.N. Cleland, Nature 464, 697(2010)ADSCrossRefGoogle Scholar
  33. 33.
    D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 76, 1796 (1996)ADSCrossRefGoogle Scholar
  34. 34.
    B.T.H. Varcoe, S. Brattke, M. Weidinger, H. Walther, Nature 403, 743 (2000)ADSCrossRefGoogle Scholar
  35. 35.
    P. Bertet, S. Osnaghi, P. Milman, A. Auffeves, P. Maioli, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 88, 143601 (2002)ADSCrossRefGoogle Scholar
  36. 36.
    C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J.M. Raimond, S. Haroce, Nature 477, 73 (2011)ADSCrossRefGoogle Scholar
  37. 37.
    M. Hofheinz, E.M. Wieg, M. Ansmann, R.C. Bialczak, E. Lucero, M. Neeley, A.D. O’Connell, H. Wang, J.M. Martinis, A.N. Cleland, Nature 454, 310 (2008)ADSCrossRefGoogle Scholar
  38. 38.
    M. Hofheinz, H. Wang, M. Ansmann, R.C. Bialczak, E. Lucero, M. Neeley, A.D. O’Connell, D. Sank, J. Wenner, J.M. Martinis, A.N. Cleland, Nature 459, 546 (2009)ADSCrossRefGoogle Scholar
  39. 39.
    M. Hofheinz, H. Wang, M. Ansmann, R.C. Bialczak, E. Lucero, M. Neeley, A.D. O’Connell, D. Sank, J. Wenner, J.M. Martinis, A.N. Cleland, Nature 459, 546(2009)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2012

Authors and Affiliations

  1. 1.QOLS, Blackett Laboratory, Imperial College LondonLondonUK
  2. 2.Dipartimento di FisicaUniversità degli Studi di MilanoMilanoItalia

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