臺灣博碩士論文加值系統

現今已有多種量子系統被提出及實踐作為量子位元並應用於量子資訊領域。人們期望著量子位元能與外加的控制場擁有較強的耦合強度又同時能保有較長的同調(相干)時間： 前者可使得量子操作更快速；後者則可使量子位元保有較好的量子態。然而，此兩者通常無法同時兼顧， 容易與其他量子系統耦合的系統亦容易受到環境的影響而失去其同調性，反之亦然。利用耦合能力較佳的系統進行量子操作，在其閒置時將量子態移轉到同調時間較長的量子記憶體中儲存以保持其同調性，此種混合量子系統便能解決上述之兩難。
在此論文中，我們提出了一種鑽石中單一氮-空缺中心之混合量子系統，用以儲存磁通量超導量子位元之量子態。藉由釔鐵石榴石(YIG)強化兩系統間之耦合強度，我們將量子態轉存到鑽石中單一氮-空缺中心而實現量子記憶體，且在增強耦合強度的同時， 釔鐵石榴石不會像使用群體氮-空缺中心一樣引入其他導致同調性流失的效應。我們推導了磁通量超導量子位元與單一氮-空缺中心是如何藉由釔鐵石榴石而達到耦合，接著詳述量子記憶體中量子態轉換和儲存的過程，最後考慮磁通量超導量子位元與單一氮-空缺中心的能量與相位耗散效應， 我們以量子主方程式(Lindblad 形式)來模擬整個過程。 利用真實的實驗參數估算，我們所提出的量子記憶體在完成量子態移轉、儲存(達 10 毫秒)及回傳等所有步驟後，仍可保持超過 90%的保真度。
此混合量子系統不只能作為可靠的量子記憶體，亦可作為往後欲利用以集體激發之磁振子來增強各個不同量子系統間之耦合強度的範例與參考。

There are many kinds of physical quantum systems that have been proposed and realized as qubits to implement quantum computation and information processing. One may wish to have both the strong coupling strength between the qubit and an external control field and long coherence times for qubits: the former leads to fast and easy qubit operations; the latter maintains the coherence of the quantum state of the qubit. However, it is hard to have a qubit with both advantages. The systems, which can couple to other system strongly , are normally also easily influenced by the environment resulting in decoherence, and those with good coherence property due to the isolation from their environment cannot interact with other system well. So the idea of hybrid quantum system taking advantages of their constituents’ strengths has been proposed. Using the qubit with excellent coupling
ability in the operating stage, and assisted by a quantum memory, which can transfer the quantum state between the operating qubit and storage qubit, one can avoid the decoherece in the idle time of the whole quantum processes.
Here we propose a quantum memory scheme to transfer and store the quantum state of a superconducting flux qubit (FQ), as an operating unit, into the electron spin of a single nitrogen-vacancy (NV) center in diamond, as a storage unit, via a ferromagnet transducer, yttrium iron garnet (YIG). Unlike an ensemble of NV centers, the YIG moderator can enhance the effective FQ-NV-center coupling strength without introducing additional appreciable decoherence. We derive the effective interaction between the FQ and the NV center by tracing out the degrees of freedom of the collective mode of the YIG spins. We demonstrate the transfer, storage, and retrieval procedures, taking into account the effects of spontaneous decay and pure dephasing by a master equation in Lindblad form. Using realistic experimental parameters for the FQ, NV center and YIG, we find that a combined transfer, storage, and retrieval fidelity higher than 0.9, with a long storage time of 10 ms, can be achieved.
This hybrid system not only acts as a promising quantum memory, but also provides an example of enhanced coupling between various systems through collective degrees of freedom.

Acknowledgements I
Chinese abstract II
Abstract III
1. Introduction 1
2. Open quantum system 5
2.1. Dynamics of closed and open quantum system 6
2.2. Quantum master equation 8
2.2.1. Born and Markov Approximations 9
2.2.2. Lindblad equation and dissipator 10
3. The nitrogen-vacancy center in diamond 12
3.1. The states of the NV 13
3.2. Decoherence of the NV 16
3.3. The applications of the NV 19
4. Hybrid quantum system 22
4.1. Superconducting qubits 23
4.1.1. Superconductivity 23
4.1.2. Josephson junction 23
4.1.3. The classification of the superconducting qubit 25
4.1.3.1. The charge qubits 25
4.1.3.2. The flux qubits 28
4.1.3.3. The phase qubits 29
4.2. Yttrium iron garnet (YIG) 31
4.3. The effective coupling between a FQ and a NV-center spin via YIG 33
5. The hybrid quantum memory 39
5.1. The protocol 39
5.2. The system parameters 41
5.3. The numerical results and discussion 42
6. Conclusion 48
A. : Spin wave, magnon, and the Holstein-Primakoff transformation 49
B. : Coupling with magnons 51
C. : Derivation of the effective Hamiltonian by the Schriffer-Wolff transformation 54
Bibliography 57

[1] R. Shankar, Principles of Quantum Mechanics. Springer US, 2012.
[2] J. Sakurai and J. Napolitano, Modern Quantum Mechanics. Cambridge University Press, 2017.
[3] G. E. Moore, “Cramming more components onto integrated circuits, reprinted from electronics, volume 38, number 8, april 19, 1965, pp.114 ff.,” IEEE Solid-State Circuits Society Newsletter, vol. 11, no. 3, pp. 33–35, Sept.
[4] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010.
[5] P. W. Shor, “Algorithms for quantum computation: discrete logarithms and factoring,” in Proceedings 35th Annual Symposium on Foundations of Computer Science,
pp. 124–134, 20-2.
[6] L. K. Grover, “A fast quantum mechanical algorithm for database search,” in Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing,
STOC ’96, (New York, NY, USA), pp. 212–219, ACM, 1996.
[7] H. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations. Physics and Astronomy Online Library, Springer, 1999.
[8] R. P. Feynman, “Simulating physics with computers,” International Journal of Theoretical Physics, vol. 21, pp. 467–488, June 1982.
[9] J. Q. You, X. Hu, S. Ashhab, and F. Nori, “Low-decoherence flux qubit,” Phys. Rev. B, vol. 75, p. 140515, Apr. 2007.
[10] J. Clarke and F. K. Wilhelm, “Superconducting quantum bits,” Nature, vol. 453, pp. 1031–1042, June 2008.
[11] J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature, vol. 474, p. 589, June 2011.
[12] X. Gu, A. F. Kockum, A. Miranowicz, Y.-x. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Microwave photonics with superconducting
quantum circuits, vol. 718-719, pp. 1–102, Nov. 2017.
[13] M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Physica Scripta, vol. 2009, no. T137, p. 014001, 2009.
[14] I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Reports on Progress in Physics, vol. 74, no. 10, p. 104401, 2011.
[15] Z.-L. Xiang, X.-Y. LÃŒ, T.-F. Li, J. Q. You, and F. Nori, “Hybrid quantum circuit consisting of a superconducting flux qubit coupled to a spin ensemble and a transmission-line resonator,” Phys. Rev. B, vol. 87, p. 144516, Apr. 2013.
[16] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys., vol. 85, pp. 623–653, Apr 2013.
[17] G. Kurizki, P. Bertet, Y. Kubo, K. MÞlmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc Natl Acad Sci USA,
vol. 112, p. 3866, Mar. 2015.
[18] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, “The nitrogen-vacancy colour centre in diamond,” Physics Reports, vol. 528,
no. 1, pp. 1 – 45, 2013.
[19] J. Twamley and S. D. Barrett, “Superconducting cavity bus for single nitrogenvacancy defect centers in diamond,” Phys. Rev. B, vol. 81, p. 241202, Jun 2010.
[20] Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve,
“Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett., vol. 105, p. 140502, Sep 2010.
[21] Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves,
D. Vion, D. Esteve, and P. Bertet, “Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble,” Phys. Rev. Lett., vol. 107, p. 220501, Nov 2011.
[22] X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S.-i. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nature, vol. 478, p. 221, Oct. 2011.
[23] X.-Y. Lü, Z.-L. Xiang, W. Cui, J. Q. You, and F. Nori, “Quantum memory using a hybrid circuit with flux qubits and nitrogen-vacancy centers,” Phys. Rev. A, vol. 88, p. 012329, Jul 2013.
[24] V. Cherepanov, I. Kolokolov, and V. L’vov, “The saga of yig: Spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet,” Physics Reports, vol. 229, no. 3, pp. 81 – 144, 1993.
[25] A. A. Serga, A. V. Chumak, and B. Hillebrands, “Yig magnonics,” Journal of Physics D: Applied Physics, vol. 43, no. 26, p. 264002, 2010.
[26] L. Trifunovic, F. L. Pedrocchi, S. Hoffman, P. Maletinsky, A. Yacoby, and D. Loss, “High-efficiency resonant amplification of weak magnetic fields for single spin magnetometry at room temperature,” Nat Nano, vol. 10, pp. 541–546, June 2015.
[27] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett., vol. 113, p. 083603, Aug 2014.
[28] D. Zhang, X.-M. Wang, T.-F. Li, X.-Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet
sphere,” Npj Quantum Information, vol. 1, p. 15014, Nov. 2015.
[29] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, “Magnonics,” Journal of Physics D: Applied Physics, vol. 43, no. 26, p. 264001, 2010.
[30] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Coherent coupling between a ferromagnetic magnon and a superconducting
qubit,” Science, vol. 349, no. 6246, pp. 405–408, 2015.
[31] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, “Bidirectional conversion between microwave and light via
ferromagnetic magnons,” Phys. Rev. B, vol. 93, p. 174427, May 2016.
[32] L. Trifunovic, F. L. Pedrocchi, and D. Loss, “Long-distance entanglement of spin qubits via ferromagnet,” Phys. Rev. X, vol. 3, p. 041023, Dec 2013.
[33] T. Douce, M. Stern, N. Zagury, P. Bertet, and P. Milman, “Coupling a single nitrogenvacancy center to a superconducting flux qubit in the far-off-resonance regime,” Phys. Rev. A, vol. 92, p. 052335, Nov 2015.
[34] H. Breuer, F. Petruccione, P. Breuer, and S. Petruccione, The Theory of Open Quantum Systems. Oxford University Press, 2002.
[35] H. Wiseman and G. Milburn, Quantum Measurement and Control. Cambridge University Press, 2010.
[36] W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys., vol. 75, pp. 715–775, May 2003.
[37] M. Schlosshauer, “Decoherence, the measurement problem, and interpretations of quantum mechanics,” Rev. Mod. Phys., vol. 76, pp. 1267–1305, Feb. 2005.
[38] C.-C. Chen and H.-S. Goan, “Effects of initial system-environment correlations on open-quantum-system dynamics and state preparation,” Phys. Rev. A, vol. 93,
p. 032113, Mar. 2016.
[39] G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics, vol. 48, pp. 119–130, June 1976.
[40] P. L. Stanwix, L. M. Pham, J. R. Maze, D. Le Sage, T. K. Yeung, P. Cappellaro, P. R. Hemmer, A. Yacoby, M. D. Lukin, and R. L. Walsworth, “Coherence of nitrogenvacancy electronic spin ensembles in diamond,” Phys. Rev. B, vol. 82, p. 201201, Nov 2010.
[41] N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nature Communications,
vol. 4, p. 1743, Apr. 2013.
[42] G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat Mater, vol. 8, pp. 383–387, May 2009.
[43] J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, “High-sensitivity diamond magnetometer with nanoscale resolution,” Nature Physics, vol. 4, p. 810, Sept. 2008.
[44] J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M. Taylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E. Togan, A. S. Zibrov, A. Yacoby, R. L. Walsworth, and M. D. Lukin, “Nanoscale magnetic sensing with an individual electronic spin in diamond,” Nature, vol. 455, p. 644, Oct. 2008.
[45] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hemmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Bratschitsch, F. Jelezko, and J. Wrachtrup, “Nanoscale imaging magnetometry with diamond spins under ambient conditions,” Nature, vol. 455, p. 648, Oct. 2008.
[46] S. Kolkowitz, Q. P. Unterreithmeier, S. D. Bennett, and M. D. Lukin, “Sensing distant nuclear spins with a single electron spin,” Phys. Rev. Lett., vol. 109, p. 137601, Sept. 2012.
[47] T. H. Taminiau, J. J. T. Wagenaar, T. van der Sar, F. Jelezko, V. V. Dobrovitski, and R. Hanson, “Detection and control of individual nuclear spins using a weakly
coupled electron spin,” Phys. Rev. Lett., vol. 109, p. 137602, Sept. 2012.
[48] R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen, “Nitrogen-vacancy centers in diamond: Nanoscale sensors for physics and biology,” Annu. Rev. Phys. Chem., vol. 65, pp. 83–105, Apr. 2014.
[49] I. I. Vlasov, V. G. Ralchenko, A. V. Khomich, S. V. Nistor, D. Shoemaker, and R. A. Khmelnitskii, “Relative abundance of single and vacancyâbonded substitutional
nitrogen in cvd diamond,” phys. stat. sol. (a), vol. 181, pp. 83–90, Sept. 2000.
[50] J. Meijer, B. Burchard, M. Domhan, C. Wittmann, T. Gaebel, I. Popa, F. Jelezko, and J. Wrachtrup, “Generation of single color centers by focused nitrogen implantation,” Appl. Phys. Lett., vol. 87, p. 261909, Dec. 2005.
[51] J. Martin, R. Wannemacher, J. Teichert, L. Bischoff, and B. KÃ{hler, “Generation and detection of fluorescent color centers in diamond with submicron resolution,”
Appl. Phys. Lett., vol. 75, pp. 3096–3098, Nov. 1999.
[52] G. Davies, “Dynamic jahn-teller distortions at trigonal optical centres in diamond,” Journal of Physics C: Solid State Physics, vol. 12, no. 13, p. 2551, 1979.
[53] T. Gaebel, M. Domhan, C. Wittmann, I. Popa, F. Jelezko, J. Rabeau, A. Greentree, S. Prawer, E. Trajkov, P. R. Hemmer, and J. Wrachtrup, “Photochromism in single
nitrogen-vacancy defect in diamond,” Applied Physics B, vol. 82, pp. 243–246, Feb. 2006.
[54] A. Gruber, A. DrÃbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. v. Borczyskowski, “Scanning confocal optical microscopy and magnetic resonance on single
defect centers,” Science, vol. 276, p. 2012, June 1997.
[55] A. Lenef and S. C. Rand, “Electronic structure of the n-v center in diamond: Theory,” Phys. Rev. B, vol. 53, pp. 13441–13455, May 1996.
[56] J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, “Properties of nitrogen-vacancy centers in diamond: the group theoretic approach,”
New Journal of Physics, vol. 13, no. 2, p. 025025, 2011.
[57] L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent dynamics of coupled electron and nuclear spin qubits in diamond,” Science, vol. 314, no. 5797, pp. 281–285, 2006.
[58] C. M. Breeding and J. E. Shigley, “The "type" classification system of diamonds and its importance in gemology,” GEMS & GEMOLOGY, vol. 45, no. 2, p. 96, 2009.
[59] W. Yang, Z.-Y. Wang, and R.-B. Liu, “Preserving qubit coherence by dynamical decoupling,” Frontiers of Physics in China, vol. 6, pp. 2–14, Mar. 2011.
[60] Z.-H. Wang, G. de Lange, D. Ristè, R. Hanson, and V. V. Dobrovitski, “Comparison of dynamical decoupling protocols for a nitrogen-vacancy center in diamond,” Phys.
Rev. B, vol. 85, p. 155204, Apr 2012.
[61] Y. Chou, S.-Y. Huang, and H.-S. Goan, “Optimal control of fast and high-fidelity quantum gates with electron and nuclear spins of a nitrogen-vacancy center in diamond,” Phys. Rev. A, vol. 91, p. 052315, May 2015.
[62] E. L. Hahn, “Spin echoes,” Phys. Rev., vol. 80, pp. 580–594, Nov 1950.
[63] H. Y. Carr and E. M. Purcell, “Effects of diffusion on free precession in nuclear magnetic resonance experiments,” Phys. Rev., vol. 94, pp. 630–638, May 1954.
[64] G. S. Uhrig, “Exact results on dynamical decoupling by Ï pulses in quantum information processes,” New Journal of Physics, vol. 10, no. 8, p. 083024, 2008.
[65] W. M. Witzel and S. D. Sarma, “Multiple-pulse coherence enhancement of solid state spin qubits,” Phys. Rev. Lett., vol. 98, p. 077601, Feb. 2007.
[66] S. K. Saikin, W. Yao, and L. J. Sham, “Single-electron spin decoherence by nuclear spin bath: Linked-cluster expansion approach,” Phys. Rev. B, vol. 75, p. 125314, Mar. 2007.
[67] W. Yao, R.-B. Liu, and L. J. Sham, “Restoring coherence lost to a slow interacting mesoscopic spin bath,” Phys. Rev. Lett., vol. 98, p. 077602, Feb 2007.
[68] R.-B. Liu, W. Yao, and L. J. Sham, “Control of electron spin decoherence caused by electron-nuclear spin dynamics in a quantum dot,” New Journal of Physics, vol. 9, no. 7, p. 226, 2007.
[69] J. R. Maze, J. M. Taylor, and M. D. Lukin, “Electron spin decoherence of single nitrogen-vacancy defects in diamond,” Phys. Rev. B, vol. 78, p. 094303, Sept. 2008.
[70] W. Yang and R.-B. Liu, “Quantum many-body theory of qubit decoherence in a finite-size spin bath,” Phys. Rev. B, vol. 78, p. 085315, Aug. 2008.
[71] W. Yang and R.-B. Liu, “Quantum many-body theory of qubit decoherence in a finite-size spin bath. ii. ensemble dynamics,” Phys. Rev. B, vol. 79, p. 115320, Mar. 2009.
[72] W. Yang, W.-L. Ma, and R.-B. Liu, “Quantum many-body theory for electron spin decoherence in nanoscale nuclear spin baths,” Reports on Progress in Physics, vol. 80,
no. 1, p. 016001, 2017.
[73] F. Reinhard, F. Shi, N. Zhao, F. Rempp, B. Naydenov, J. Meijer, L. T. Hall, L. Hollenberg, J. Du, R.-B. Liu, and J. Wrachtrup, “Tuning a spin bath through the quantum-classical transition,” Phys. Rev. Lett., vol. 108, p. 200402, May 2012.
[74] N. F. Ramsey, “A molecular beam resonance method with separated oscillating fields,” Phys. Rev., vol. 78, pp. 695–699, June 1950.
[75] P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nature Physics, vol. 6, p. 602, May 2010.
[76] L.-g. Zhou, L. F. Wei, M. Gao, and X.-b. Wang, “Strong coupling between two distant electronic spins via a nanomechanical resonator,” Phys. Rev. A, vol. 81, p. 042323, Apr. 2010.
[77] Y. Makhlin, G. SchÃ{n, and A. Shnirman, “Quantum-state engineering with josephson-junction devices,” Rev. Mod. Phys., vol. 73, pp. 357–400, May 2001.
[78] V. Bouchiat, D. Vion, P. Joyez, D. Esteve, and M. H. Devoret, “Quantum coherence with a single cooper pair,” Physica Scripta, vol. 1998, no. T76, p. 165, 1998.
[79] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-cooper-pair box,” Nature, vol. 398, p. 786, Apr. 1999.
[80] Y. A. Pashkin, O. Astafiev, T. Yamamoto, Y. Nakamura, and J. S. Tsai, “Josephson charge qubits: a brief review,” Quantum Information Processing, vol. 8, pp. 55–80, June 2009.
[81] J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal, and S. Lloyd, “Josephson persistent-current qubit,” Science, vol. 285, p. 1036, Aug. 1999.
[82] T. P. Orlando, J. E. Mooij, L. Tian, C. H. van der Wal, L. S. Levitov, S. Lloyd, and J. J. Mazo, “Superconducting persistent-current qubit,” Phys. Rev. B, vol. 60, pp. 15398–15413, Dec. 1999.
[83] M. H. Devoret, J. M. Martinis, and J. Clarke, “Measurements of macroscopic quantum tunneling out of the zero-voltage state of a current-biased josephson junction,” Phys. Rev. Lett., vol. 55, pp. 1908–1911, Oct. 1985.
[84] J. M. Martinis, M. H. Devoret, and J. Clarke, Energy-level quantization in the zero-voltage state of a current-biased josephson junction,” Phys. Rev. Lett., vol. 55, pp. 1543–1546, Oct. 1985.
[85] J. O. H. N. CLARKE, A. N. D. R. E. W. N. CLELAND, M. I. C. H. E. L. H. DEVORET, D. A. N. I. E. L. ESTEVE, and J. O. H. N. M. MARTINIS, “Quantum mechanics of a macroscopic variable: The phase difference of a josephson junction,” Science, vol. 239, p. 992, Feb. 1988.
[86] R. C. Ramos, M. A. Gubrud, A. J. Berkley, J. R. Anderson, C. J. Lobb, and F. C. Wellstood, “Design for effective thermalization of junctions for quantum coherence,” IEEE Transactions on Applied Superconductivity, vol. 11, no. 1, pp. 998–1001, 2001.
[87] J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina, “Rabi oscillations in a large josephson-junction qubit,” Phys. Rev. Lett., vol. 89, p. 117901, Aug. 2002.
[88] A. Shnirman, G. Schön, and Z. Hermon, “Quantum manipulations of small josephson junctions,” Phys. Rev. Lett., vol. 79, pp. 2371–2374, Sept. 1997.
[89] D. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.
[90] T. A. Fulton and L. N. Dunkleberger, “Lifetime of the zero-voltage state in josephson tunnel junctions,” Phys. Rev. B, vol. 9, pp. 4760–4768, Jun 1974.
[91] C. M. Srivastava and R. Aiyar, “Spin wave stiffness constants in some ferrimagnetics,” Journal of Physics C: Solid State Physics, vol. 20, no. 8, p. 1119, 1987.
[92] T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev., vol. 58, pp. 1098–1113, Dec 1940.
[93] J. Norpoth, S. Dreyer, and C. Jooss, “Straightforward field calculations for uniaxial hardmagnetic prisms: stray field distributions and dipolar coupling in regular arrays,” Journal of Physics D: Applied Physics, vol. 41, no. 2, p. 025001, 2008.
[94] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Quantum magnonics: The magnon meets the superconducting qubit,” Comptes Rendus Physique, vol. 17, no. 7, pp. 729 – 739, 2016.
[95] M. Stern, G. Catelani, Y. Kubo, C. Grezes, A. Bienfait, D. Vion, D. Esteve, and P. Bertet, “Flux qubits with long coherence times for hybrid quantum circuits,” Phys. Rev. Lett., vol. 113, p. 123601, Sep 2014.
[96] T. Ishikawa, K.-M. C. Fu, C. Santori, V. M. Acosta, R. G. Beausoleil, H. Watanabe, S. Shikata, and K. M. Itoh, “Optical and spin coherence properties of nitrogenvacancy centers placed in a 100 nm thick isotopically purified diamond layer,” Nano Lett., vol. 12, pp. 2083–2087, Apr. 2012.
[97] C. Hahn, V. V. Naletov, G. de Loubens, O. Klein, O. d’Allivy Kelly, A. Anane, R. Bernard, E. Jacquet, P. Bortolotti, V. Cros, J. L. Prieto, and M. Muñoz, “Measurement of the intrinsic damping constant in individual nanodisks of y3fe5o12 and y3fe5o12|pt,” Appl. Phys. Lett., vol. 104, p. 152410, Apr. 2014.
[98] O. Klein. (private communication), 2018.
[99] F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, G. Samach, S. Weber, J. L. Yoder, T. P. Orlando, J. Clarke,
A. J. Kerman, and W. D. Oliver, “The flux qubit revisited to enhance coherence and reproducibility,” Nature Communications, vol. 7, p. 12964, Nov. 2016.
[100] C. Kittel, Introduction to Solid State Physics. Wiley, 2004.
[101] T. A. Salaoru and J. R. Woodward, “Rapid rise time pulsed magnetic field circuit for pump-probe field effect studies,” Review of Scientific Instruments, vol. 78, no. 3, p. 036104, 2007.
[102] J. R. Schrieffer and P. A. Wolff, “Relation between the anderson and kondo hamiltonians,” Phys. Rev., vol. 149, pp. 491–492, Sep 1966.
[103] M. M. Salomaa, “Schrieffer-wolff transformation for the anderson hamiltonian in a superconductor,” Phys. Rev. B, vol. 37, pp. 9312–9317, Jun 1988.
[104] S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schriefferâwolff transformation for quantum many-body systems,” Annals of Physics, vol. 326, no. 10, pp. 2793 – 2826, 2011