Abstract

High-dimensional quantum system provides a higher capacity of quantum channel, which exhibits potential applications in quantum information processing. However, high-dimensional universal quantum logic gates is difficult to achieve directly with only high-dimensional interaction between two quantum systems and requires a large number of two-dimensional gates to build even a small high-dimensional quantum circuits. In this paper, we propose a scheme to implement a general controlled-flip (CF) gate where the high-dimensional single photon serve as the target qudit and stationary qubits work as the control logic qudit, by employing a three-level Λ-type system coupled with a whispering-gallery-mode microresonator. In our scheme, the required number of interaction times between the photon and solid state system reduce greatly compared with the traditional method which decomposes the high-dimensional Hilbert space into 2-dimensional quantum space, and it is on a shorter temporal scale for the experimental realization. Moreover, we discuss the performance and feasibility of our hybrid CF gate, concluding that it can be easily extended to a 2n-dimensional case and it is feasible with current technology.

© 2016 Optical Society of America

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  68. The relevant application of our scheme proposed here is going to be discussed in our following work.

2015 (6)

D. Y. Cao, B. H. Liu, Z. Wang, Y. F. Huang, C. F. Li, and G. C. Guo, “Multiuser-to-multiuser entanglement distribution based on 1550 nm polarization-entangled photons,” Sci. Bull. 60, 1128 (2015).
[Crossref]

B. E. Anderson, H. Sosa-Martinez, C.A. Riofrío, Ivan H. Deutsch, and Poul S. Jessen, “Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System,” Phys. Rev. Lett. 114, 240401 (2015).
[Crossref] [PubMed]

Ehsan Zahedinejad, Joydip Ghosh, and Barry C. Sanders, “High-Fidelity Single-Shot Toffoli Gate via Quantum Control,” Phys. Rev. Lett. 114, 200502 (2015).
[Crossref] [PubMed]

D. S. Ding, W. Zhang, Z. Y. Zhou, S. Shi, G. Y. Xiang, X. S. Wang, Y. K. Jiang, B. S. Shi, and G. C. Guo, “Quantum Storage of Orbital Angular Momentum Entanglement in an Atomic Ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref] [PubMed]

J. S. Xu and C. F. Li, “Quantum integrated circuit: classical characterization,” Sci. Bull. 60, 141 (2015).
[Crossref]

L. Zhou and Y. B. Sheng, “Complete logic Bell-state analysis assisted with photonic Faraday rotation,” Phys. Rev. A 92, 042314 (2015).
[Crossref]

2014 (8)

W. H. Zhang, Q. Q. Qi, J. Zhou, and L. X. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112, 153601 (2014).
[Crossref] [PubMed]

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilingera, “Generation and confirmation of a (100 100)-dimensional entangled quantum system,” PNAS 111, 6243 (2014).
[Crossref]

T. J. Wang and C. Wang, “Universal hybrid three-qubit quantum gates assisted by a nitrogen-vacancy center coupled with a whispering-gallery-mode microresonator,” Phys. Rev. A 90, 052310 (2014).
[Crossref]

L. Zhou and Y. B. Sheng, “Detection of nonlocal atomic entanglement assisted by single photon,” Phys. Rev. A 90, 024301(2014).
[Crossref]

C. Wang, T. J. Wang, Y. Zhang, R. Jiao, and G. Jin, “Concentration of entangled nitrogen-vacancy centers in decoherence free subspace,” Opt. Express 22, 1551 (2014).
[Crossref] [PubMed]

V. D’Ambrosio, F. Bisesto, F. Sciarrino, J. F. Barra, G. Lima, and A. Cabello, “Device-Independent Certification of High-Dimensional Quantum Systems,” Phys. Rev. Lett. 112, 140503 (2014).
[Crossref]

R. Fickler, R. Lapkiewicz, M. Huber, M. P. J. Lavery, M. J. Padgett, and A. Zeilinger, “Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information,” Nat. Commun. 5, 4502 (2014).
[Crossref] [PubMed]

C. Zheng and G.F. Long, “Quantum secure direct dialogue using Einstein-Podolsky-Rosen pairs,” Sci. China Phys. Mech. Astron. 57, 1238 (2014).
[Crossref]

2013 (7)

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

S. P. Liu, J. H. Li, R. Yu, and Y. Wu, “Achieving maximum entanglement between two nitrogen-vacancy centers coupling to a whispering-gallery-mode microresonator,” Opt. Express 21, 3501(2013).
[Crossref] [PubMed]

L. Y. Cheng, H. F. Wang, S. Zhang, and K. H. Yeon, “Quantum state engineering with nitrogen-vacancy centers coupled to low-Q microresonator,” Opt. Express 21, 5988 (2013).
[Crossref] [PubMed]

H. R. Wei and F.G. Deng, “Compact quantum gates on electron-spin qubits assisted by diamond nitrogen-vacancy centers inside cavities,” Phys. Rev. A 88, 042323 (2013);;H. R. Wei and G. L. Long, “Universal photonic quantum gates assisted by ancilla diamond nitrogen-vacancy centers coupled to resonators,” Phys. Rev. A 91, 032324 (2015).
[Crossref]

D. O’Shea, C. Junge, J. Volz, and A. Rauschenbeutel, “Fiber-Optical Switch Controlled by a Single Atom,” Phys. Rev. Lett. 111, 193601 (2013).
[Crossref]

F. Monifi, S. K. Ozdemir, and L. Yang, “Tunable add-drop filter using an active whispering gallery mode micro-cavity,” Appl. Phys. Lett 103, 181103 (2013).
[Crossref]

F. Monifi, S. K. Özdemir, and L. Yang, “Tunable add-drop filter using an active whispering gallery mode microcavity,” Appl. Phys. Lett. 103, 181103 (2013).
[Crossref]

2012 (5)

F. Monifi, J. Friedlein, S. K. Ozdemir, and L. Yang, “A robust and tunable addCdrop filter using whispering gallery mode microtoroid resonator,” J. Lightwave Technol. 30, 3306 (2012).
[Crossref]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref] [PubMed]

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

D. Richart, Y. Fischer, and H. Weinfurter, “Experimental implementation of higher dimensional timeCenergy entanglement,” Appl. Phys. B 106(3), 543 (2012).
[Crossref]

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett 108, 143603 (2012).
[Crossref] [PubMed]

2011 (6)

A. Dada, J. Leach, G. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011).
[Crossref]

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

P. E. Barclay, K.-M. C. Fu, C. Santori, A. Faraon, and R. G. Beausoleil, “Hybrid Nanocavity Resonant Enhancement of Color Center Emission in Diamond,” Phys. Rev. X 1, 011007 (2011).

A. Faraon, P. E. Barclay, C. Santori, K. M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301 (2011).
[Crossref]

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond,” Nat. Phys. 7, 789 (2011).
[Crossref]

Q. Chen, W. L. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
[Crossref]

2010 (4)

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
[Crossref]

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730(2010).
[Crossref] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[Crossref] [PubMed]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angleCorbital angular momentum variables,” Science 329, 662–665 (2010).
[Crossref] [PubMed]

2009 (8)

R. Ionicioiu, T. P. Spiller, and W. J. Munro, “Generalized Toffoli gates using qudit catalysis,” Phys. Rev. A 80, 012312 (2009).
[Crossref]

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nature Physics 5, 134–140 (2009).
[Crossref]

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2005 (4)

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2004 (2)

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2002 (2)

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2000 (2)

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
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D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418 (2000).
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S. Schietinger, M. Barth, T. Aichele, and O. Benson, “Plasmon-Enhanced Single Photon Emission from a Nanoassembled Metal-Diamond Hybrid Structure at Room Temperature,” Nano. Lett. 9, 1694 (2009).
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D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
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A. Faraon, P. E. Barclay, C. Santori, K. M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301 (2011).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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A. Faraon, P. E. Barclay, C. Santori, K. M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301 (2011).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
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Benson, O.

S. Schietinger, M. Barth, T. Aichele, and O. Benson, “Plasmon-Enhanced Single Photon Emission from a Nanoassembled Metal-Diamond Hybrid Structure at Room Temperature,” Nano. Lett. 9, 1694 (2009).
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V. D’Ambrosio, F. Bisesto, F. Sciarrino, J. F. Barra, G. Lima, and A. Cabello, “Device-Independent Certification of High-Dimensional Quantum Systems,” Phys. Rev. Lett. 112, 140503 (2014).
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M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
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Malcolm N. O’Sullivan-Hale, Irfan Ali Khan, Robert W. Boyd, and John C. Howell, “Pixel entanglement: experimental realization of optically entangled d=3 and d=6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
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M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
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A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
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G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond,” Nat. Phys. 7, 789 (2011).
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Cabello, A.

V. D’Ambrosio, F. Bisesto, F. Sciarrino, J. F. Barra, G. Lima, and A. Cabello, “Device-Independent Certification of High-Dimensional Quantum Systems,” Phys. Rev. Lett. 112, 140503 (2014).
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Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075(2006).
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V. D’Ambrosio, F. Bisesto, F. Sciarrino, J. F. Barra, G. Lima, and A. Cabello, “Device-Independent Certification of High-Dimensional Quantum Systems,” Phys. Rev. Lett. 112, 140503 (2014).
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A. Dada, J. Leach, G. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011).
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A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
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J. S. Kim, A. Das, and B. C. Sanders, “Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity,” Phys. Rev. A 79, 012329 (2009).
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B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. I. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062 (2008).
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Q. Chen, W. L. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
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L. M. Duan and H. J. Kimble, “Scalable Photonic Quantum Computation through Cavity-Assisted Interactions,” Phys. Rev. Lett. 92, 127902 (2004).
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P. E. Barclay, K.-M. C. Fu, C. Santori, A. Faraon, and R. G. Beausoleil, “Hybrid Nanocavity Resonant Enhancement of Color Center Emission in Diamond,” Phys. Rev. X 1, 011007 (2011).

A. Faraon, P. E. Barclay, C. Santori, K. M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301 (2011).
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Q. Chen, W. L. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
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A. Faraon, P. E. Barclay, C. Santori, K. M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301 (2011).
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P. E. Barclay, K.-M. C. Fu, C. Santori, A. Faraon, and R. G. Beausoleil, “Hybrid Nanocavity Resonant Enhancement of Color Center Emission in Diamond,” Phys. Rev. X 1, 011007 (2011).

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Other (3)

The relevant application of our scheme proposed here is going to be discussed in our following work.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).
[Crossref]

A d-dimensional controlled-flip gate is used to perform a flip operation on a target qudit or not, depending on the states of the control qudit. That is, if the state of the control qudit is |i〉, after the gating operation the the state of the target qudit is changed as |j〉 → |j + i〉, where i + j in the |j + i〉 means modding d.

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Figures (5)

Fig. 1
Fig. 1 The Λ-type atom is fixed on the exterior surface of the WGM microresonator, which is coupled with four tapered fibers (add-drop structure). The waveguides with the ports a′0a′3 are the bus waveguides and that with ports a0a3 are the drop waveguides, respectively. The effective energy-level diagram of an atom is shown in the dashed frame. The possible cavity-mode-induced transitions are |+ 1〉 → |A2〉 driven resonantly by absorbing a σ-circular polarized photon and |− 1〉 → |A2〉 by absorbing a σ+-circular polarized photon.
Fig. 2
Fig. 2 Schematic diagram for a 4-dimensional controlled-flip gate (a 4-dimensional CF gate) operating on the 2 atomic qubits and a 4-dimensional single-photon system. The HWP represents a half-wave plate which serves as a π-phase-add operation on the |σ〉 photon. c1, c2, c3, and c4 are four circulators, and the transmission direction of the c1 and c4 is clockwise while the transmission direction of the c2 and c3 is counter clockwise.
Fig. 3
Fig. 3 Schematic diagram for a 8-dimensional controlled-flip gate (a 8-d CF gate) operating on the 3-qubit atoms and a 8-dimensional single-photon system. The 50:50 beam splitter (BS) works as the Hadamard gate on the 2-dimensional spatial-mode photonic state.
Fig. 4
Fig. 4 The required interaction times between the photon and atoms of the present schemes for constructing a d-dimensional CF gate. The black curve S1 stands for the traditional method in which a d-dimensional quantum gates can be achieved by decomposing into sequentially implemented single- and two-qubit gates, whereas the line S2 represents that of the our scheme.
Fig. 5
Fig. 5 The fidelity and efficiency of our scheme in 4-dimensional case as a function of the parameter g / κ γ. The blue solid curve represents the fidelity of the 4-dimensional CF gate while the red dashed line corresponds to the efficiency. Here, g / κ γ 0.5.

Equations (25)

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U A ˜ B ˜ = C B 1 A 1 C B 2 A 2 C B n A n T B 1 A 2 B 2 T B 2 A 3 B 3 T B n 1 A n B n T B 1 A 3 B 3 B 2 T B 2 A 4 B 4 B 3 T B n 2 A n B n B n 1 T B 1 A n 1 B n 1 B n 2 B 2 T B 2 A n B n B n 1 B 3 T B 1 A n B n B n 1 B 2 ,
d a d t = [ i ( ω c ω p ) + κ 1 2 + κ 2 2 ] a ( t ) g σ ( t ) + κ 1 a in , d σ d t = [ i ( ω 0 ω p ) + γ 2 ] σ ( t ) g σ z ( t ) a ( t ) .
t ( ω p ) = i ( ω c ω p ) [ i ( ω 0 ω p ) + γ 2 ] + g 2 [ i ( ω c ω p ) + κ ] [ i ( ω 0 ω p ) + γ 2 ] + g 2 , r ( ω p ) = κ [ i ( ω 0 ω p ) + γ / 2 ] [ i ( ω 0 ω p ) + γ / 2 ] [ i ( ω c ω p ) + κ ] + g 2 .
t ( ω p ) = 2 g 2 γ κ + 2 g 2 , r ( ω p ) = κ γ γ κ + 2 g 2
| a j σ , + 1 | a j σ , + 1 , | a j σ , 1 | a j + 1 σ , 1 ,
| 0 ˜ = | + 1 , + 1 Λ 2 Λ 1 , | 1 ˜ = | + 1 , 1 Λ 2 Λ 1 , | 2 ˜ = | 1 + 1 Λ 2 Λ 1 , | 3 ˜ = | 1 , 1 Λ 2 Λ 1 .
X = [ 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 ]
CX = [ I 4 0 0 X 4 ] , CX 2 = [ I 4 0 0 X 4 2 ]
CF 4 = CX CX 2 = [ I 4 0 0 0 0 X 0 0 0 0 X 2 0 0 0 0 X 3 ]
| φ A 0 | φ e 0 | σ A [ α ˜ | 0 ˜ Λ 2 Λ 1 ( α | a 0 + β | a 1 + γ | a 2 + ξ | a 3 ) A + β ˜ | 1 ˜ Λ 2 Λ 1 ( α | a 1 + β | a 2 + γ | a 3 + ξ | a 0 ) A + γ ˜ | 2 ˜ Λ 2 Λ 1 ( α | a 0 + β | a 1 + γ | a 2 + ξ | a 3 ) A + ξ ˜ | 3 ˜ Λ 2 Λ 1 ( α | a 1 + β | a 2 + γ | a 3 + ξ | a 0 ) A ] .
| a j σ , 1 | a j + 2 σ , 1 , | a j σ , + 1 | a j σ + , + 1 ,
| σ A [ α ˜ | 0 ˜ Λ 2 Λ 1 ( α | a 0 + β | a 1 + γ | a 2 + ξ | a 3 ) A + β ˜ | 1 ˜ Λ 2 Λ 1 ( α | a 1 + β | a 2 + γ | a 3 + ξ | a 0 ) A + γ ˜ | 2 ˜ Λ 2 Λ 1 ( α | a 2 + β | a 3 + γ | a 0 + ξ | a 1 ) A + ξ ˜ | 3 ˜ Λ 2 Λ 1 ( α | a 3 + β | a 0 + γ | a 1 + ξ | a 2 ) A ] .
| 0 ˜ = | + 1 , + 1 , + 1 Λ 3 Λ 2 Λ 1 , | 1 ˜ = | + 1 , + 1 , 1 Λ 3 Λ 2 Λ 1 , | 2 ˜ = | + 1 , 1 , + 1 Λ 3 Λ 2 Λ 1 , | 3 ˜ = | + 1 , 1 , 1 Λ 3 Λ 2 Λ 1 , | 4 ˜ = | 1 , + 1 , + 1 Λ 3 Λ 2 Λ 1 , | 5 ˜ = | 1 , + 1 , 1 Λ 3 Λ 2 Λ 1 , | 6 ˜ = | 1 , 1 , + 1 Λ 3 Λ 2 Λ 1 , | 7 ˜ = | 1 , 1 , 1 Λ 3 Λ 2 Λ 1 .
X 8 = [ 0 I 7 1 0 ]
CF 8 = CX 8 CX 8 2 CX 8 4 = [ I 8 0 0 X 8 ] [ I 8 0 0 X 8 2 ] [ I 8 0 0 X 8 4 ] = [ I 8 0 0 0 0 0 0 0 0 X 8 0 0 0 0 0 0 0 0 X 8 2 0 0 0 0 0 0 0 0 X 8 3 0 0 0 0 0 0 0 0 X 8 4 0 0 0 0 0 0 0 0 X 8 5 0 0 0 0 0 0 0 0 X 8 6 0 0 0 0 0 0 0 0 X 8 7 ]
| φ A 8 | φ e 3 [ ( β ˜ 0 | 0 ˜ + β ˜ 2 | 2 ˜ + β ˜ 4 | 4 ˜ + β ˜ 6 | 6 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 | a 0 + α 1 | a 1 + α 2 | a 2 + α 3 | a 3 + α 4 | a 4 + α 5 | a 5 + α 6 | a 6 + α 7 | a 7 ) + ( β ˜ 1 | 1 ˜ + β ˜ 3 | 3 ˜ + β ˜ 5 | 5 ˜ + β ˜ 7 | 7 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 | a 1 + α 1 | a 2 + α 2 | a 3 + α 3 | a 0 + α 4 | a 5 + α 5 | a 6 + α 6 | a 7 + α 7 | a 4 ) ] | σ A .
| a 0 B S 1 2 ( | a 0 1 + | a 0 2 ) , | a 4 B S 1 2 ( | a 4 1 + | a 4 2 ) .
| σ A [ ( β ˜ 0 | 0 ˜ + β ˜ 2 | 2 ˜ + β ˜ 4 | 4 ˜ + β ˜ 6 | 6 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 2 | a 0 1 + α 0 2 | a 0 2 + α 1 | a 1 + α 2 | a 2 + α 3 | a 3 + α 4 2 | a 4 1 + α 4 2 | a 4 2 + α 5 | a 5 + α 6 | a 6 + α 7 | a 7 ) A + ( β ˜ 1 | 1 ˜ + β ˜ 3 | 3 ˜ + β ˜ 5 | 5 ˜ + β ˜ 7 | 7 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 | a 1 + α 1 | a 2 + α 2 | a 3 + α 3 2 | a 4 1 + α 3 2 | a 4 2 + α 4 | a 5 + α 5 | a 6 + α 6 | a 7 + α 7 2 | a 0 1 + α 7 2 | a 0 2 ) A ] .
[ ( β ˜ 0 | 0 ˜ + β ˜ 2 | 2 ˜ + β ˜ 4 | 4 ˜ + β ˜ 6 | 6 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 | a 0 + α 1 | a 1 + α 2 | a 2 + α 3 | a 3 + α 4 | a 4 + α 5 | a 5 + α 6 | a 6 + α 7 | a 7 ) + ( β ˜ 1 | 1 ˜ + β ˜ 3 | 3 ˜ + β ˜ 5 | 5 ˜ + β ˜ 7 | 7 ˜ ) Λ 3 Λ 2 Λ 1 ( α 0 | a 1 + α 1 | a 2 + α 2 | a 3 + α 3 | a 4 + α 4 | a 5 + α 5 | a 6 + α 6 | a 7 + α 7 | a 0 ) ] | σ A .
| a 0 | a 2 , | a 2 | a 4 , | a 4 | a 6 , | a 6 | a 0 , | a 1 | a 3 , | a 3 | a 5 , | a 5 | a 7 , | a 7 | a 1 ,
[ ( β ˜ 0 | 0 ˜ + β ˜ 4 | 4 ˜ ) Λ 3 Λ 2 Λ 1 j = 0 7 α j | a j A + ( β ˜ 2 | 2 ˜ + β ˜ 6 | 6 ˜ ) Λ 3 Λ 2 Λ 1 j = 0 7 α j | a j + 2 A + ( β ˜ 1 | 1 ˜ + β ˜ 5 | 5 ˜ ) Λ 3 Λ 2 Λ 1 j = 0 7 α j | a j + 1 A + ( β ˜ 3 | 3 ˜ + β ˜ 7 | 7 ˜ ) Λ 3 Λ 2 Λ 1 j = 0 7 α j | a j + 3 A ] .
| a 0 | a o 4 , | a 1 | a o 5 , | a 2 | a o 6 , | a 3 | a o 7 , | a 4 | a o 0 , | a 5 | a o 1 , | a 6 | a o 2 , | a 7 | a o 3 .
| σ A i = 0 7 β ˜ i | i ˜ Λ 3 Λ 2 Λ 1 j = 0 7 α j | a o j + i A .
CF 2 n = CX 2 n CX 2 n 2 CX 2 n 4 CX 2 n 2 n 1 ,
F = ( 1 + 2 t + t 2 ) 2 4 [ 1 + 2 t 2 + ( 1 + t 2 ) ( r 2 + r 4 ) + t 4 + r 6 ] ,

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