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基于超导传输线和超导量子比特相互耦合的电路量子电动力学(quantum Electrodynamics, QED)系统, 是目前固态量子信息领域的一个倍受关注的物理系统, 也是研究量子测量和量子控制的理想实验平台. 由于其中涉及的驱动场和超导传输线谐振腔支持的光子频率都在微波区, 在量子测量和量子控制研究中往往遇到 大量光子数引起的状态空间维数过大带来的数值模拟方面的困难. 为了避免这个困难, 往往采取"消除"光子自由度的办法, 建立一个只保留量子比特状态自由度的有效描述方案. 本文通过对单比特的量子测量动力学的数值模拟, 检验了 "绝热消除"和"极化子变换"两种方案的适用条件. 结果表明, 在量子非破坏(quantum non-demolition, QND) 测量情况下, 极化子变换精确适用于 任意驱动强度和任意(光子)泄漏速率微腔; 但在非QND测量情况下, 极化子变换相对通常的绝热消除方案, 并无优势. 在强泄漏微腔和弱耦合情况下, 两种消除光子自由度的方法都可以较好地描述 测量动力学; 但如果微腔光子泄漏速率不是很大或量子比特与微腔耦合较强, 则需要纳入光子自由度做完整模拟, 此时的量子测量属性是一个尚待研究的课题.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
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[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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