Abstract

The principle of superposition is a key ingredient for quantum mechanics. A recent work [Phys. Rev. Lett. 116, 110403 (2016) [CrossRef]  ] has shown that a quantum adder that deterministically generates a superposition of two unknown states is forbidden. Here we consider the implementation of the probabilistic quantum adder in the 3D cavity-transmon system. Our implementation is based on a three-level superconducting transmon qubit dispersively coupled to two cavities. Numerical simulations show that high-fidelity generation of the superposition of two coherent states is feasible with current circuit QED technology. Our method also works for other physical systems such as two optical cavities coupled to a three-level atom or two nitrogen-vacancy center ensembles interacted with one three-level superconducting flux qubit.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (4)

H. Zhang, A. Alsaedi, T. Hayat, and F. G. Deng, “Entanglement concentration and purification of two-mode squeezed microwave photons in circuit QED,” Ann. Phys. 391, 112–119 (2018).
[Crossref]

S. Rosenblum, Y. Y. Gao, P. Reinhold, C. Wang, C. J. Axline, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A CNOT gate between multiphoton qubits encoded in two cavities,” Nature Comm. 9, 652 (2018).
[Crossref]

K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Deterministic teleportation of a quantum gate between two logical qubits,” Nature 561, 368–373 (2018).
[Crossref] [PubMed]

T. Liu, B. Q. Guo, C. S. Yu, and W. N. Zhang, “One-step implementation of a hybrid Fredkin gate with quantum memories and single superconducting qubit in circuit QED and its applications,” Opt. Express 26, 4498–4511 (2018).
[Crossref] [PubMed]

2017 (9)

Q. P. Su, H. H. Zhu, L. Yu, Y. Zhang, S. J. Xiong, J. M. Liu, and C. P. Yang, “Generating double NOON states of photons in circuit QED,” Phys. Rev. A 95, 022339 (2017).
[Crossref]

M. Doosti, F. Kianvash, and V. Karimipour, “Universal superposition of orthogonal states,” Phys. Rev. A 96, 052318 (2017).
[Crossref]

G. Gatti, D. Barberena, M. Sanz, and E. Solano, “Protected state transfer via an approximate quantum adder,” Sci. Rep. 7, 6964 (2017).
[Crossref] [PubMed]

K. Li, G. Long, H. Katiyar, T. Xin, G. Feng, D. Lu, and R. Laflamme, “Experimentally superposing two pure states with partial prior knowledge,” Phys. Rev. A 95, 022334 (2017).
[Crossref]

X. Gu, A. F. Kockum, A. Miranowicz, Y. X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718, 1–102 (2017).
[Crossref]

C. S. Yu, “Quantum coherence via skew information and its polygamy,” Phys. Rev. A 95, 042337 (2017).
[Crossref]

F. Souris, H. Christiani, and J. P. Davis, “Tuning a 3D microwave cavity via superfluid helium at millikelvin temperatures,” Appl. Phys. Lett. 111, 172601 (2017).
[Crossref]

T. Liu, Y. Zhang, B. Q. Guo, C. S. Yu, and W. N. Zhang, “Circuit QED: cross-Kerr effect induced by a superconducting qutrit without classical pulses,” Quantum Inf. Process. 16, 209 (2017).
[Crossref]

C. P. Yang, Q. P. Su, S. B. Zheng, F. Nori, and S. Han, “Entangling two oscillators with arbitrary asymmetric initial states,” Phys. Rev. A 95, 052341 (2017).
[Crossref]

2016 (11)

T. Liu, Q. P. Su, S. J. Xiong, J. M. Liu, C. P. Yang, and F. Nori, “Generation of a macroscopic entangled coherent state using quantum memories in circuit QED,” Sci. Rep. 6, 32004 (2016).
[Crossref] [PubMed]

C. P. Yang, Q. P. Su, S. B. Zheng, and F. Nori, “Entangling superconducting qubits in a multi-cavity system,” New J. Phys. 18, 013025 (2016).
[Crossref]

T. Liu, X. Z. Cao, Q. P. Su, S. J. Xiong, and C. P. Yang, “Multi-target-qubit unconventional geometric phase gate in a multi-cavity system,” Sci. Rep. 6, 21562 (2016).
[Crossref] [PubMed]

N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Extending the lifetime of a quantum bit with error correction in superconducting circuits,” Nature 536, 441–445 (2016).
[Crossref] [PubMed]

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

C. P. Yang, Q. P. Su, S. B. Zheng, and F. Nori, “Crosstalk-insensitive method for simultaneously coupling multiple pairs of resonators,” Phys. Rev. A 93, 042307 (2016).
[Crossref]

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 Commun. 7, 12964 (2016).
[Crossref]

M. Oszmaniec, A. Grudka, M. Horodecki, and A. Wójcik, “Creating a superposition of unknown quantum states,” Phys. Rev. Lett. 116, 110403 (2016).
[Crossref] [PubMed]

R. Sharma and F. W. Strauch, “Quantum state synthesis of superconducting resonators,” Phys. Rev. A 93, 012342 (2016).
[Crossref]

Y. J. Zhao, C. Q. Wang, X. B. Zhu, and Y. X. Liu, “Engineering entangled microwave photon states through multiphoton interactions between two cavity fields and a superconducting qubit,” Sci. Rep. 6, 23646 (2016).
[Crossref] [PubMed]

X. M. Hu, M. J. Hu, J. S. Chen, B. H. Liu, Y. F. Huang, C. F. Li, G. C. Guo, and Y. S. Zhang, “Experimental creation of superposition of unknown photonic quantum states,” Phys. Rev. A 94, 033844 (2016).
[Crossref]

2015 (3)

Y. Alvarez-Rodriguez, M. Sanz, L. Lamata, and E. Solano, “The forbidden quantum adder,” Sci. Rep. 5, 11983 (2015).
[Crossref] [PubMed]

S. J. Xiong, Z. Sun, J. M. Liu, T. Liu, and C. P. Yang, “Efficient scheme for generation of photonic NOON states in circuit QED,” Opt. Lett. 40, 2221–2224 (2015).
[Crossref] [PubMed]

M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gudmundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gustavsson, “Coherence and decay of higher energy levels of a superconducting transmon qubit,” Phys. Rev. Lett. 114, 010501 (2015).
[Crossref] [PubMed]

2014 (1)

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[Crossref] [PubMed]

2013 (6)

M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013).
[Crossref] [PubMed]

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref] [PubMed]

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623 (2013).
[Crossref]

R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
[Crossref] [PubMed]

J. D. Strand, M. Ware, F. Beaudoin, T. A. Ohki, B. R. Johnson, A. Blais, and B. L. T. Plourde, “First-order sideband transitions with flux-driven asymmetric transmon qubits,” Phys. Rev. B 87, 220505 (2013).
[Crossref]

Z. L. Wang, Y. P. Zhong, L. J. He, H. Wang, J. M. Martinis, A. N. Cleland, and Q. W. Xie, “Quantum state characterization of a fast tunable superconducting resonator,” Appl. Phys. Lett. 102, 163503 (2013).
[Crossref]

2012 (2)

C. P. Yang, Q. P. Su, and S. Han, “Generation of Greenberger-Horne-Zeilinger entangled states of photons in multiple cavities via a superconducting qutrit or an atom through resonant interaction,” Phys. Rev. A 86, 022329 (2012).
[Crossref]

C. Rigetti, J. M. Gambetta, S. Poletto, B. L. T. Plourde, J. M. Chow, A. D. Córcoles, J. A. Smolin, S. T. Merkel, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, “Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms,” Phys. Rev. B 86, 100506 (2012).
[Crossref]

2011 (5)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature 474, 589–597 (2011).
[Crossref] [PubMed]

H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107, 240501 (2011).
[Crossref]

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. 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 478, 221–224 (2011).
[Crossref] [PubMed]

2010 (4)

M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, “High-fidelity readout in circuit quantum electrodynamics using the Jaynes-Cummings nonlinearity,” Phys. Rev. Lett. 105, 173601 (2010).
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S. B. Zheng, C. P. Yang, and F. Nori, “Arbitrary control of coherent dynamics for distant qubits in a quantum network,” Phys. Rev. A 82, 042327 (2010).
[Crossref]

S. T. Merkel and F. K. Wilhelm, “Generation and detection of NOON states in superconducting circuits,” New J. Phys. 12, 093036 (2010).
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F. W. Strauch, K. Jacobs, and R. W. Simmonds, “Arbitrary control of entanglement between two superconducting resonators,” Phys. Rev. Lett. 105, 050501 (2010).
[Crossref] [PubMed]

2009 (2)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
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P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
[Crossref]

2008 (3)

M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Physics 4, 523–526 (2008).
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M. Sandberg, C. M. Wilson, F. Persson, T. Bauch, G. Johansson, V. Shumeiko, T. Duty, and P. Delsing, “Tuning the field in a microwave resonator faster than the photon lifetime,” Appl. Phys. Lett. 92, 203501 (2008).
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M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wilhelm, and E. Solano, “Two-resonator circuit quantum electrodynamics: A superconducting quantum switch,” Phys. Rev. B 78, 104508 (2008).
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2007 (3)

J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitive qubit design derived from the Cooper pair box,” Phys. Rev. A 76, 042319 (2007).
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D. F. James and J. Jerke, “Effective Hamiltonian theory and its applications in quantum information,” Can. J. Phys. 85, 625–632 (2007).
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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
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2005 (1)

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
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2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 745 (2002).
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1999 (1)

P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM Rev. 41, 303–332 (1999).
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Alsaedi, A.

H. Zhang, A. Alsaedi, T. Hayat, and F. G. Deng, “Entanglement concentration and purification of two-mode squeezed microwave photons in circuit QED,” Ann. Phys. 391, 112–119 (2018).
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Alvarez-Rodriguez, Y.

Y. Alvarez-Rodriguez, M. Sanz, L. Lamata, and E. Solano, “The forbidden quantum adder,” Sci. Rep. 5, 11983 (2015).
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Ansmann, M.

M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Physics 4, 523–526 (2008).
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Ashhab, S.

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623 (2013).
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Atatüre, M.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
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Axline, C.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
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Axline, C. J.

S. Rosenblum, Y. Y. Gao, P. Reinhold, C. Wang, C. J. Axline, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A CNOT gate between multiphoton qubits encoded in two cavities,” Nature Comm. 9, 652 (2018).
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K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Deterministic teleportation of a quantum gate between two logical qubits,” Nature 561, 368–373 (2018).
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Bader, S. J.

M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gudmundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gustavsson, “Coherence and decay of higher energy levels of a superconducting transmon qubit,” Phys. Rev. Lett. 114, 010501 (2015).
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Badolato, A.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
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Barberena, D.

G. Gatti, D. Barberena, M. Sanz, and E. Solano, “Protected state transfer via an approximate quantum adder,” Sci. Rep. 7, 6964 (2017).
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Barends, R.

R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
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Bauch, T.

M. Sandberg, C. M. Wilson, F. Persson, T. Bauch, G. Johansson, V. Shumeiko, T. Duty, and P. Delsing, “Tuning the field in a microwave resonator faster than the photon lifetime,” Appl. Phys. Lett. 92, 203501 (2008).
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Baumgratz, T.

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
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Baur, M.

P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
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Beaudoin, F.

J. D. Strand, M. Ware, F. Beaudoin, T. A. Ohki, B. R. Johnson, A. Blais, and B. L. T. Plourde, “First-order sideband transitions with flux-driven asymmetric transmon qubits,” Phys. Rev. B 87, 220505 (2013).
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H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
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M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Physics 4, 523–526 (2008).
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Bianchetti, R.

P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
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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 Commun. 7, 12964 (2016).
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Birnbaum, K. M.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
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Bishop, L. S.

H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107, 240501 (2011).
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Blais, A.

J. D. Strand, M. Ware, F. Beaudoin, T. A. Ohki, B. R. Johnson, A. Blais, and B. L. T. Plourde, “First-order sideband transitions with flux-driven asymmetric transmon qubits,” Phys. Rev. B 87, 220505 (2013).
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J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitive qubit design derived from the Cooper pair box,” Phys. Rev. A 76, 042319 (2007).
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Blumoff, J.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
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Blumoff, J. Z.

K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Deterministic teleportation of a quantum gate between two logical qubits,” Nature 561, 368–373 (2018).
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Boca, A.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
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Boozer, A. D.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
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Cao, X. Z.

T. Liu, X. Z. Cao, Q. P. Su, S. J. Xiong, and C. P. Yang, “Multi-target-qubit unconventional geometric phase gate in a multi-cavity system,” Sci. Rep. 6, 21562 (2016).
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Catelani, G.

H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107, 240501 (2011).
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Chen, J. S.

X. M. Hu, M. J. Hu, J. S. Chen, B. H. Liu, Y. F. Huang, C. F. Li, G. C. Guo, and Y. S. Zhang, “Experimental creation of superposition of unknown photonic quantum states,” Phys. Rev. A 94, 033844 (2016).
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Chen, Y.

R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
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Chiaro, B.

R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
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Chou, K.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
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Chou, K. S.

K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Deterministic teleportation of a quantum gate between two logical qubits,” Nature 561, 368–373 (2018).
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Chow, J. M.

C. Rigetti, J. M. Gambetta, S. Poletto, B. L. T. Plourde, J. M. Chow, A. D. Córcoles, J. A. Smolin, S. T. Merkel, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, “Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms,” Phys. Rev. B 86, 100506 (2012).
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Christiani, H.

F. Souris, H. Christiani, and J. P. Davis, “Tuning a 3D microwave cavity via superfluid helium at millikelvin temperatures,” Appl. Phys. Lett. 111, 172601 (2017).
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Clarke, J.

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 Commun. 7, 12964 (2016).
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Cleland, A. N.

R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
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Z. L. Wang, Y. P. Zhong, L. J. He, H. Wang, J. M. Martinis, A. N. Cleland, and Q. W. Xie, “Quantum state characterization of a fast tunable superconducting resonator,” Appl. Phys. Lett. 102, 163503 (2013).
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H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
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M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Physics 4, 523–526 (2008).
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Córcoles, A. D.

C. Rigetti, J. M. Gambetta, S. Poletto, B. L. T. Plourde, J. M. Chow, A. D. Córcoles, J. A. Smolin, S. T. Merkel, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, “Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms,” Phys. Rev. B 86, 100506 (2012).
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Cramer, M.

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
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Davis, J. P.

F. Souris, H. Christiani, and J. P. Davis, “Tuning a 3D microwave cavity via superfluid helium at millikelvin temperatures,” Appl. Phys. Lett. 111, 172601 (2017).
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Delsing, P.

M. Sandberg, C. M. Wilson, F. Persson, T. Bauch, G. Johansson, V. Shumeiko, T. Duty, and P. Delsing, “Tuning the field in a microwave resonator faster than the photon lifetime,” Appl. Phys. Lett. 92, 203501 (2008).
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Deng, F. G.

H. Zhang, A. Alsaedi, T. Hayat, and F. G. Deng, “Entanglement concentration and purification of two-mode squeezed microwave photons in circuit QED,” Ann. Phys. 391, 112–119 (2018).
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Deppe, F.

M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wilhelm, and E. Solano, “Two-resonator circuit quantum electrodynamics: A superconducting quantum switch,” Phys. Rev. B 78, 104508 (2008).
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Devoret, M. H.

S. Rosenblum, Y. Y. Gao, P. Reinhold, C. Wang, C. J. Axline, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A CNOT gate between multiphoton qubits encoded in two cavities,” Nature Comm. 9, 652 (2018).
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K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Deterministic teleportation of a quantum gate between two logical qubits,” Nature 561, 368–373 (2018).
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C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
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N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Extending the lifetime of a quantum bit with error correction in superconducting circuits,” Nature 536, 441–445 (2016).
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B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
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M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013).
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H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107, 240501 (2011).
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J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitive qubit design derived from the Cooper pair box,” Phys. Rev. A 76, 042319 (2007).
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DiCarlo, L.

M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, “High-fidelity readout in circuit quantum electrodynamics using the Jaynes-Cummings nonlinearity,” Phys. Rev. Lett. 105, 173601 (2010).
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Doosti, M.

M. Doosti, F. Kianvash, and V. Karimipour, “Universal superposition of orthogonal states,” Phys. Rev. A 96, 052318 (2017).
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Duty, T.

M. Sandberg, C. M. Wilson, F. Persson, T. Bauch, G. Johansson, V. Shumeiko, T. Duty, and P. Delsing, “Tuning the field in a microwave resonator faster than the photon lifetime,” Appl. Phys. Lett. 92, 203501 (2008).
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Everitt, M. S.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. 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 478, 221–224 (2011).
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Fält, S.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
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Feng, G.

K. Li, G. Long, H. Katiyar, T. Xin, G. Feng, D. Lu, and R. Laflamme, “Experimentally superposing two pure states with partial prior knowledge,” Phys. Rev. A 95, 022334 (2017).
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P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
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Fink, J. M.

P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
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Frunzio, L.

K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Deterministic teleportation of a quantum gate between two logical qubits,” Nature 561, 368–373 (2018).
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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 Commun. 7, 12964 (2016).
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Figures (7)

Fig. 1
Fig. 1 A diagram of the quantum adder. Here, | ψ A and | φ B are the two arbitrary input pure states, while | ψ A + | φ B is the out superposition state with the referential state | χ .
Fig. 2
Fig. 2 (a) Schematic of a single transmon qubit dispersively coupled to two three-dimensional microwave cavities A and B. (b) Schematic diagram of transmon-cavity interaction. Cavity j is far-off resonant with the | g * * | e * * ( | e * * | f * *) transition of transmon qubit with coupling strength gj ( 2 g j) and detuning Δj (δj). Here, Δ j = ω e g ω j and δ j = ω f e ω j ( j = A , B).
Fig. 3
Fig. 3 Fidelity F versus θ and by taking the unwanted inter-cavity crosstalk into account for gAB =0, 0.01g, 0.1g. The parameters used in the numerical simulation are referred in the text.
Fig. 4
Fig. 4 Fidelity F versus c and θ by taking into account the inhomogeneity in transmon-cavity interaction. Here gA = g and gB = cg with c ∈ [0.95, 1.05].
Fig. 5
Fig. 5 Fidelity F versus d for θ = π/4 by considering the unequal cavity-transmon frequency detuning.
Fig. 6
Fig. 6 Fidelity F versus ϵ for θ = π/4 by considering the effect of the operation time errors.
Fig. 7
Fig. 7 Fidelity F versus f for θ = π/4 by considering the effect of the anharmonicity errors.

Equations (28)

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α T | ψ A | φ B | 0 T + β T | φ A | ψ B | 1 T ,
| Ψ A = 1 N ( γ | ψ ± η | φ ) ,
H 0 = ω e g | e e | + ( ω e g + ω f e ) | f f | + ω A a a + ω B b b ,
H I = g A ( a σ e g + + a σ e g ) + g B ( b σ e g + + b σ e g ) + 2 g A ( a σ f e + + a σ f e ) + 2 g B ( b σ f e + + b σ f e ) ,
H I = g A ( a σ e g + e i Δ A t + a σ e g e i Δ A t ) + g B ( b σ e g + e i Δ B t + b σ e g e i Δ B t ) + 2 g A ( a σ f e + e i δ A t + a σ f e e i δ A t ) + 2 g B ( b σ f e + e i δ B t + b σ f e e i δ B t ) ,
H e = ( 2 g A 2 δ A a a + 2 g B 2 δ B b b ) | f f | + ( g A 2 Δ A a a + g B 2 Δ B b b 2 g A 2 δ A a a 2 g B 2 δ B b b ) | e e | ( g A 2 Δ A a a + g B 2 Δ B b b ) | g g | + g A g B 2 ( 1 Δ A + 1 Δ B ) ( a b e i Δ A B t + a b e i Δ A B t ) σ z e g + g A g B ( 1 δ A + 1 δ B ) ( a b e i δ A B t + a b e i δ A B t ) σ z f e ,
H e = λ ( a a + b b + 2 ) | f f | + [ Λ ( a a + b b ) + 2 χ ] | e e | χ ( a a + b b ) | g g | χ ( a b + a b ) | g g | + λ ( a b + a b ) | f f | + Λ ( a b + a b ) | e e | ,
| ϕ T = sin  θ | g + cos  θ | e .
H e = [ Λ ( a a + b b ) + 2 χ ] | e e | + Λ ( a b + a b ) | e e | χ ( a a + b b ) | g g | χ ( a b + a b ) | g g | .
e i H 0 g t e i H I g t sin  θ | g | ψ A | φ B + e i H 0 e t e i H I e t cos  θ | e | ψ A | φ B
a ( t ) = cos   ( χ t ) a + i sin   ( χ t ) b , b ( t ) = cos   ( χ t ) b + i sin   ( χ t ) a ,
a ( t ) = cos   ( Λ t ) a i sin   ( Λ t ) b , b ( t ) = cos   ( Λ t ) b i sin   ( Λ t ) a ,
Δ = 4 k 1 + 4 k 2 + 5 4 k 2 4 k 1 + 3 α
sin  θ | g | φ A | ψ B cos  θ | e | ψ A | φ B
sin  θ | φ A | ψ B cos  θ | ψ A | φ B .
| Ψ A = 1 N ( γ | φ η | ψ ) ,
Δ = 4 k 1 + 4 k 2 + 5 4 k 2 4 k 1 3 α
sin  θ | g | ψ A | φ B + cos  θ | e | φ A | ψ B
cos  θ | ψ A | φ B ± sin  θ | φ A | ψ B .
| Ψ A = 1 N ( γ | ψ ± η | φ ) ,
d ρ d t = i [ H I , ρ ] + κ A D [ a ] + κ B D [ b ] + γ e g D [ σ e g ] + γ f e D [ σ f e ] + γ f g D [ σ f g ] + γ φ e D [ σ e e ] + γ φ f D [ σ f f ] ,
H I g = χ ( a b + a b ) ,    H I e = Λ ( a b + a b ) ,
a ( t ) = cos   ( χ t ) a + i sin   ( χ t ) b , b ( t ) = cos   ( χ t ) b + i sin   ( χ t ) a ,
a ( t ) = cos   ( Λ t ) a i sin   ( Λ t ) b , b ( t ) = cos   ( Λ t ) b i sin   ( Λ t ) a ,
sin θ exp ( i H 0 g t ) exp ( i H I g t ) | g | ψ A | φ B + cos θ exp ( i H 0 e t ) exp ( i H I e t ) | e | ψ A | φ B
sin   θ exp   ( i H 0 g t ) exp   ( i H I g t ) | g | ψ A | φ B + cos   θ exp   ( i H 0 e t ) exp   ( i H I e t ) | e | ψ A | φ B = sin   θ exp   ( i H 0 g t ) exp   ( i H I g t ) | g n = 0 c n n ! ( a ) n | 0 A m = 0 d m m ! ( b ) m | 0 B + cos   θ exp   ( i H 0 e t ) exp   ( i H I e t ) | e n = 0 c n n ! ( a ) n | 0 A m = 0 d m m ! ( b ) m | | 0 B = sin   θ exp   ( i H 0 g t ) | g n = 0 c n n ! ( i b ) n | 0 B m = 0 d m m ! ( i a ) m | 0 | A + cos   θ exp   ( i H 0 e t ) | e n = 0 c n n ! ( a ) n | 0 A m = 0 d m m ! ( b ) m | 0 B = sin   θ exp   ( i H 0 g t ) | g n = 0 ( i ) n c n | n B m = 0 ( i ) m d m | m A + cos   θ exp   ( i H 0 e t ) | e n = 0 c n | n A m = 0 d m | m B = sin   θ exp   ( i χ m t ) exp   ( i χ n t ) exp   ( i π 2 m ) exp   ( i π 2 n ) | g | ψ B | φ A + cos   θ exp   ( i Λ n t ) exp   ( i Λ m t ) exp   ( i 2 χ t ) | e | ψ A | φ B = sin   θ exp   ( i m π [ ± ( 1 2 + 2 k 1 ) + 1 2 ] ) exp   ( i n π [ ± ( 1 2 + 2 k 1 ) + 1 2 ] ) | g | φ A | ψ B + cos   θ exp   [ ± i 2 n π ( 1 + k 2 ) ] exp   [ ± i 2 m π ( 1 + k 2 ) ] exp   ( i 2 π ( 1 2 + 2 k 1 ) ) | e | ψ A | φ B = sin   θ | g | φ A | ψ B cos θ | e | ψ A | φ B ,
sin θ exp ( i H 0 g t ) exp ( i H I g t ) | g | ψ A | φ B + cos θ exp ( i H 0 e t ) exp ( i H I e t ) | e | ψ A | φ B
sin θ exp ( i H 0 g t ) exp ( i H I g t ) | g | ψ A | φ B + cos θ exp ( i H 0 e t ) exp ( i H I e t ) | e | ψ A | φ B = sin θ exp ( i H 0 g t ) exp ( i H I g t ) | g n = 0 c n n ! ( a ) n | 0 A m = 0 d m m ! ( b ) m | 0 B + cos θ exp ( i H 0 e t ) exp ( i H I e t ) | e n = 0 c n n ! ( a ) n | 0 A m = 0 d m m ! ( b ) m | 0 B = sin θ exp ( i H 0 g t ) | g n = 0 c n n ! ( a ) n | 0 A m = 0 d m m ! ( b ) m | 0 B + cos θ exp ( i H 0 e t ) | e n = 0 c n n ! ( i b ) n | 0 B m = 0 d m m ! ( i a ) m | 0 A = sin θ exp ( i H 0 g t ) | g n = 0 c n | n A m = 0 d m | m B + cos θ exp ( i H 0 e t ) | e n = 0 ( i ) n c n | n B m = 0 ( i ) m d m | m A = sin θ exp ( i χ m t ) exp ( i χ n t ) | g | ψ A | φ B + cos θ exp ( i Λ n t ) exp ( i Λ m t ) exp ( i 2 χ t ) exp ( i π 2 m ) exp ( i π 2 n ) | e | ψ B | φ A = sin θ exp [ ± i 2 n π ( 1 + k 2 ) ] exp [ ± i 2 m π ( 1 + k 2 ) ] | g | ψ A | φ B + cos θ exp ( i m π [ ± ( 1 2 + 2 k 1 ) 1 2 ] ) exp ( i n π [ ± ( 1 2 + 2 k 1 ) 1 2 ] ) × exp [ i 4 π ( 1 + k 2 ) ] | e | ψ A | φ B = sin θ | g | ψ A | ψ B + cos θ | e | φ A | ψ B ,

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