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The solid-state superconducting circuit-QED (quantum electrodynamics) system is a promising candidate for quantum computing and quantum information processing, which serves also as an ideal platform for quantum measurement and quantum control studies. In this context, a large number of cavity photons may be involved in the quantum dynamics and will degrade the simulation efficiency. To avoid this difficulty, it is helpful to eliminate the degrees of freedom of the cavity photons, and obtain an effective master-equation description which contains only the qubit states. In this work, we examine two such schemes, the adiabatic elimination (AE) and the more recently proposed polaron transformation (PT) approaches, by comparing their results with exact numerical simulations. We find that in the absence of qubit-flip, which is a specific quantum nondemolition (QND) measurement, the PT scheme is superior to the AE method. Actually, in this case the PT scheme catches the measurement dynamics exactly. However, in the presence of qubit-flip such as for qubit oscillation measurement, the PT scheme is no longer better than the AE approach. We conclude that both schemes, in the weak measurement regime, can work almost equally well. This corresponds to strong cavity damping or weak coupling between the qubit and cavity photons. Out of this regime, unfortunately, one has to include the cavity photons into numerical simulations and more advanced methods/techniques are waiting for their exploration in this field.
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Keywords:
- circuit-QED /
- adiabatic elimination /
- polaron transformation /
- QND measurement
[1] Blais A, Huang R S, Wallraff A, Girvin S M, Schoelkopf R J 2004 Phys. Rev. A 69 062320
[2] Wallraff A, Schuster D I, Blais A, Frunzio L, Huang R S, Majer J, Kumar S, Girvin S M, Schoelkopf R J 2004 Nature 431 162
[3] Haroche S, Kleppner D 1989 Phys. Today 24
[4] Schuster D I, Houck A A, Schreier J A, Wallraff A, Gambetta J M, Blais A, Frunzio L, Majer J, Johnson B, Devoret M H, Girvin S M, Schoelkopf R J 2007 Nature 445 515
[5] Houck A A, Schuster D I, Gambetta J, Schreier J A, Johnson B R, Chow J M, Frunzio L, Majer J, Devoret M H, Girvin S M, Schoelkopf R J 2007 Nature 449 328
[6] Hofheinz M, Weig E M, Ansmann M, Bialczak R C, Lucero E, Neeley M, O'Connell A D, Wang H, Martinis J M, Cleland A N 2008 Nature 454 310
[7] Leek P J, Fink J M, Blais A, Bianchetti R, Gppl M, Gambetta J M, Schuster D I, Frunzio L, Schoelkopf R J, Wallraff A 2007 Science 318 1889
[8] Astafiev O, Inomata K, Niskanen A O, Yamamoto T, Pashkin Y A, Nakamura Y, Tsai J S 2007 Nature 449 588
[9] Schuster D I, Wallraff A, Blais A, Frunzio L, Huang R S, Majer J, Girvin S M, Schoelkopf R J 2005 Phys. Rev. Lett. 94 123602
[10] Gambetta J, Blais A, Schuster D I, Wallraff A, Frunzio L, Majer J, Devoret M H, Girvin S M, Schoelkopf R J 2006 Phys. Rev. A 74 042318
[11] Majer J, Chow J M, Gambetta J M, Koch J, Johnson B R, Schreier J A, Frunzio L, Schuster D I, Houck A A, Wallraff A, Blais A, Devoret M H, Girvin S M, Schoelkopf R J 2007 Nature 449 443
[12] Sarovar M, Goan H S, Spiller T P, Milburn G J 2005 Phys. Rev. A 72 062327
[13] Liu Z, Kuang L, Hu K, Xu L, Wei S, Guo L, Li X Q 2010 Phys. Rev. A 82 032335
[14] Feng W, Wang P, Ding X, Xu L, Li X Q 2011 Phys. Rev. A 83 042313
[15] Wiseman H M, Milburn G J 1993 Phys. Rev. A 47 642
[16] Gambetta J, Blais A, Boissonneault M, Houck A A, Schuster D I, Girvin S M 2008 Phys. Rev. A 77 012112
[17] Hutchison C L, Gambetta J M, Blais A, Wilhelm F K 2009 Can. J. Phys. 87 225
[18] Jaynes E T, Cummings F W 1963 Proc. IEEE 51 89
[19] Tavis M, Cummings F W 1968 Phys. Rev. 170 379
[20] Makhlin Y, Schön G, Shnirman A 2001 Rev. Mod. Phys. 73 357
[21] Korotkov A N, Averin D V 2001 Phys. Rev. B 64 165310
[22] Gurvitz S A, Berman G P 2005 Phys. Rev. B 72 073303
[23] Li X Q, Cui P, Yan Y J 2005 Phys. Rev. Lett. 94 066803
[24] Wiseman H M, Milburn G J 2010 Quantum Measurement and Control (Cambridge: Cambridge University Press)
[25] Ruskov R, Korotkov A N 2002 Phys. Rev. B 66 041401(R)
[26] Jin J S, Li X Q, Yan Y J 2006 Phys. Rev. B 73 233302
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[1] Blais A, Huang R S, Wallraff A, Girvin S M, Schoelkopf R J 2004 Phys. Rev. A 69 062320
[2] Wallraff A, Schuster D I, Blais A, Frunzio L, Huang R S, Majer J, Kumar S, Girvin S M, Schoelkopf R J 2004 Nature 431 162
[3] Haroche S, Kleppner D 1989 Phys. Today 24
[4] Schuster D I, Houck A A, Schreier J A, Wallraff A, Gambetta J M, Blais A, Frunzio L, Majer J, Johnson B, Devoret M H, Girvin S M, Schoelkopf R J 2007 Nature 445 515
[5] Houck A A, Schuster D I, Gambetta J, Schreier J A, Johnson B R, Chow J M, Frunzio L, Majer J, Devoret M H, Girvin S M, Schoelkopf R J 2007 Nature 449 328
[6] Hofheinz M, Weig E M, Ansmann M, Bialczak R C, Lucero E, Neeley M, O'Connell A D, Wang H, Martinis J M, Cleland A N 2008 Nature 454 310
[7] Leek P J, Fink J M, Blais A, Bianchetti R, Gppl M, Gambetta J M, Schuster D I, Frunzio L, Schoelkopf R J, Wallraff A 2007 Science 318 1889
[8] Astafiev O, Inomata K, Niskanen A O, Yamamoto T, Pashkin Y A, Nakamura Y, Tsai J S 2007 Nature 449 588
[9] Schuster D I, Wallraff A, Blais A, Frunzio L, Huang R S, Majer J, Girvin S M, Schoelkopf R J 2005 Phys. Rev. Lett. 94 123602
[10] Gambetta J, Blais A, Schuster D I, Wallraff A, Frunzio L, Majer J, Devoret M H, Girvin S M, Schoelkopf R J 2006 Phys. Rev. A 74 042318
[11] Majer J, Chow J M, Gambetta J M, Koch J, Johnson B R, Schreier J A, Frunzio L, Schuster D I, Houck A A, Wallraff A, Blais A, Devoret M H, Girvin S M, Schoelkopf R J 2007 Nature 449 443
[12] Sarovar M, Goan H S, Spiller T P, Milburn G J 2005 Phys. Rev. A 72 062327
[13] Liu Z, Kuang L, Hu K, Xu L, Wei S, Guo L, Li X Q 2010 Phys. Rev. A 82 032335
[14] Feng W, Wang P, Ding X, Xu L, Li X Q 2011 Phys. Rev. A 83 042313
[15] Wiseman H M, Milburn G J 1993 Phys. Rev. A 47 642
[16] Gambetta J, Blais A, Boissonneault M, Houck A A, Schuster D I, Girvin S M 2008 Phys. Rev. A 77 012112
[17] Hutchison C L, Gambetta J M, Blais A, Wilhelm F K 2009 Can. J. Phys. 87 225
[18] Jaynes E T, Cummings F W 1963 Proc. IEEE 51 89
[19] Tavis M, Cummings F W 1968 Phys. Rev. 170 379
[20] Makhlin Y, Schön G, Shnirman A 2001 Rev. Mod. Phys. 73 357
[21] Korotkov A N, Averin D V 2001 Phys. Rev. B 64 165310
[22] Gurvitz S A, Berman G P 2005 Phys. Rev. B 72 073303
[23] Li X Q, Cui P, Yan Y J 2005 Phys. Rev. Lett. 94 066803
[24] Wiseman H M, Milburn G J 2010 Quantum Measurement and Control (Cambridge: Cambridge University Press)
[25] Ruskov R, Korotkov A N 2002 Phys. Rev. B 66 041401(R)
[26] Jin J S, Li X Q, Yan Y J 2006 Phys. Rev. B 73 233302
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