Abstract

In recent continuous-variable (CV) multipartite entanglement researches, the number of fully inseparable light modes has been increased dramatically by the introduction of a multiplexing scheme in either the time domain or the frequency domain. In this paper, we propose a scheme that a large-scale (≥ 20) CV dual-rail cluster entangled state is established based on a spatial mode comb in a self-imaging optical parametric oscillator, which is pumped by two spatial Laguerre-Gaussian modes with different polarization and identical frequency. A sufficient condition of full inseparability for a CV dual-rail cluster entangled state is used to evaluate the degree of quantum entanglement. It is shown that entanglement exists over a wide range of analyzing frequency and pump parameter. We have found a new scheme that uses the optical parametric cavity to generate a large-scale entanglement based on optical spatial mode comb. The presented system will be hopefully as a practical entangled source for quantum information.

© 2017 Optical Society of America

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References

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  1. M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
    [Crossref]
  2. J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
    [Crossref] [PubMed]
  3. T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
    [Crossref]
  4. O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
    [Crossref]
  5. R. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 19, 2635–2637 (2005).
    [Crossref]
  6. H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles,” Phys. Rev. Lett. 86, 910–913 (2001).
    [Crossref] [PubMed]
  7. P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
    [Crossref]
  8. R. Shahrokhshahi and O. Pfister, “Large-scale multipartite entanglement in the quantum optical frequency comb of a depleted-pump optical parametric oscillator,” Quantum Inf. Comput. 12, 953–969 (2012).
  9. R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
    [Crossref]
  10. M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
    [Crossref] [PubMed]
  11. M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
    [Crossref] [PubMed]
  12. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]
  13. J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).
  14. R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
    [Crossref]
  15. R. G. Yang, J. Zhang, Z. H. Zhai, S. Q. Zhai, K. Liu, and J. R. Gao, “Scheme for efficient extraction of low-frequency signal beyond the quantum limit by frequency-shift detection,” Opt. Express 23, 21323–21333 (2015).
    [Crossref] [PubMed]
  16. R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
    [Crossref]
  17. R. G. Yang, J. J. Wang, J. Zhang, K. Liu, and J. R. Gao, “Generation of continuous-variable spatial cluster entangled states in optical mode comb,” J. Opt. Soc. Am. B 33(12), 2424–2429 (2016).
    [Crossref]
  18. K. Liu, J. Guo, C. X. Cai, J. X. Zhang, and J. R. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178 (2016).
    [Crossref] [PubMed]
  19. P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
    [Crossref]
  20. M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
    [Crossref] [PubMed]
  21. L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
    [Crossref]
  22. J. A. Arnaud, “Degenerate Optical Cavities,” Appl. Opt. 8, 189–196 (1969).
    [Crossref] [PubMed]
  23. M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
    [Crossref]
  24. M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
    [Crossref]
  25. G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
    [Crossref]
  26. N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014).
    [Crossref] [PubMed]

2016 (3)

2015 (1)

2014 (3)

R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
[Crossref]

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014).
[Crossref] [PubMed]

2013 (2)

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
[Crossref]

2012 (1)

R. Shahrokhshahi and O. Pfister, “Large-scale multipartite entanglement in the quantum optical frequency comb of a depleted-pump optical parametric oscillator,” Quantum Inf. Comput. 12, 953–969 (2012).

2011 (1)

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

2009 (4)

G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
[Crossref]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

2006 (2)

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

2005 (1)

R. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 19, 2635–2637 (2005).
[Crossref]

2004 (1)

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

2003 (3)

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[Crossref]

2002 (1)

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[Crossref]

2001 (1)

H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

1984 (1)

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

1969 (1)

Andersen, U. L.

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

Armstrong, S. C.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

Arnaud, J. A.

Bachor, H-A.

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

Bartlett, S. D.

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[Crossref]

Bloomer, R.

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

Briegel, H. J.

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

Browne, D. E.

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

Cai, C. X.

Cao, Z. L.

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

Chalopin, B.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

Chen, M.

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

Collett, M. J.

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

Delaubert, V.

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

Dong, P.

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

Dong, R. F.

G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
[Crossref]

Fabre, C.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

Feng, S.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

Furusawa, A.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[Crossref]

Gao, J. R.

Gardiner, C. W.

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

Gu, M.

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

Guo, J.

Harb, C. C.

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

ichi Yoshikawa, J.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

Jennings, G.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

Jing, J.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Jing, J. T.

R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
[Crossref]

Johnson, T. J.

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[Crossref]

Kaji, T.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

Lam, P. K.

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

Lassen, M.

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

Leuchs, G.

G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
[Crossref]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

Liu, K.

Lopez, L.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

Maître, A.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

Makino, K.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

Menicucci, N. C.

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014).
[Crossref] [PubMed]

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

Miwa, Y.

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

Peng, K.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Pfister, O.

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

R. Shahrokhshahi and O. Pfister, “Large-scale multipartite entanglement in the quantum optical frequency comb of a depleted-pump optical parametric oscillator,” Quantum Inf. Comput. 12, 953–969 (2012).

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

R. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 19, 2635–2637 (2005).
[Crossref]

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

Pooser, R.

R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
[Crossref]

R. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 19, 2635–2637 (2005).
[Crossref]

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

Pysher, M.

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

Ralph, T. C.

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

Raussendorf, R.

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

Riviére de la Souchére, A.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

Sanders, B. C.

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[Crossref]

Shahrokhshahi, R.

R. Shahrokhshahi and O. Pfister, “Large-scale multipartite entanglement in the quantum optical frequency comb of a depleted-pump optical parametric oscillator,” Quantum Inf. Comput. 12, 953–969 (2012).

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

Shiozawa, Y.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

Sornphiphatphong, C.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

Suzuki, S.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

Sych, D.

G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
[Crossref]

Treps, N.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

Ukai, R.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

van Loock, P.

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[Crossref]

Wang, J. J.

Weedbrook, C.

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

Xie, C.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Xie, D.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

Xue, Z. Y.

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

Yan, Y.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Yang, M.

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

Yang, R. G.

Yokoyama, S.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

Yonezawa, H.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

Yoshikawa, J.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

Zhai, S. Q.

R. G. Yang, J. Zhang, Z. H. Zhai, S. Q. Zhai, K. Liu, and J. R. Gao, “Scheme for efficient extraction of low-frequency signal beyond the quantum limit by frequency-shift detection,” Opt. Express 23, 21323–21333 (2015).
[Crossref] [PubMed]

R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
[Crossref]

Zhai, Z. H.

Zhang, J.

R. G. Yang, J. J. Wang, J. Zhang, K. Liu, and J. R. Gao, “Generation of continuous-variable spatial cluster entangled states in optical mode comb,” J. Opt. Soc. Am. B 33(12), 2424–2429 (2016).
[Crossref]

R. G. Yang, J. Zhang, Z. H. Zhai, S. Q. Zhai, K. Liu, and J. R. Gao, “Scheme for efficient extraction of low-frequency signal beyond the quantum limit by frequency-shift detection,” Opt. Express 23, 21323–21333 (2015).
[Crossref] [PubMed]

R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
[Crossref]

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Zhang, J. X.

K. Liu, J. Guo, C. X. Cai, J. X. Zhang, and J. R. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178 (2016).
[Crossref] [PubMed]

R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
[Crossref]

Zhao, F.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Photonics (1)

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

J. Eur. Opt. Soc.-Rapid (1)

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Opt. Soc.Am. B (1)

R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
[Crossref]

Nat. Photonics (1)

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi 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]

New J. Phys. (1)

G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (9)

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[Crossref]

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[Crossref]

R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
[Crossref]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

Phys. Rev. Lett. (6)

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014).
[Crossref] [PubMed]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

Quantum Inf. Comput. (1)

R. Shahrokhshahi and O. Pfister, “Large-scale multipartite entanglement in the quantum optical frequency comb of a depleted-pump optical parametric oscillator,” Quantum Inf. Comput. 12, 953–969 (2012).

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Figures (5)

Fig. 1
Fig. 1 Schematic of experimental setup, the green lines represent pump modes lg 1 p z and lg 1 p y , and the red lines represent down-converted spatial modes with different polarizations z and y, which are generated by the pump fields lg 1 p z and lg 1 p y . Two output mode combs are shown in the dashed box, only one part of the down-converted modes are given, and intensity profile of the down-converted modes are given below.
Fig. 2
Fig. 2 (a) Structural diagram: the EPR pairs that were generated in the DOPO, the z and y modes are denoted by the red solid lines and the dashed lines, respectively. The yellow curved arrows (top) connected the zzz EPR pairs that were generated by the pump lg 1 p z and the yellow curved arrows (bottom) connected the yyy EPR pairs that were generated by the pump lg 1 p y ; the vertical arrows denote the pump modes. (b) A large-scale CV dual-rail cluster entangled state: the initial EPR pairs generated by the OPO (top) turn, after passing through a single beam splitter (gray ellipses), into a CV dual-rail cluster state (bottom). Whose ±1/2 weight edges are color coded (contrary to the qubit case, weighted cluster CV states are still stabilizer states).
Fig. 3
Fig. 3 Visualization of quantum entanglements of Eqs. (5) to (8) in the CV dual-rail cluster states shown in Fig. 2(b).
Fig. 4
Fig. 4 Quantum entanglement versus normalized frequency Ω = ωτ/γ with γp = 0.025, γ = 0.02, γb = 0.018, γc = 0.002, χ1 = χ(2), χ2 = 0.707χ(2), χ3 = 0.433χ(2), χ4 = 0.250χ(2), χ5 = 0.140χ(2) and σ = 0.8. (I): Q1,0 = Q0,−1 = P1,0 = P0,−1, blue line; (II): Q2,−1 = Q1,−2 = P2,−1 = P1,−2, red line; (III): Q3,−2 = Q2,−3 = P3,−2 = P2,−3, purple line; (IV): Q4,−3 = Q3,−4 = P4,−3 = P3,−4, green line; (V): Q5,−4 = Q4,−5 = P5,−4 = P4,−5, black line.
Fig. 5
Fig. 5 Quantum entanglement versus pump parameter σ = ε/εth with γp = 0.025, γ = 0.02, γb = 0.018, γc = 0.002, χ1 = χ(2), χ2 = 0.707χ(2), χ3 = 0.433χ(2), χ4 = 0.250χ(2), χ5 = 0.140χ(2) and Ω = 0.2. (I): Q1,0 = Q0,−1 = P1,0 = P0,−1, blue line; (II): Q2,−1 = Q1,−2 = P2,−1 = P1,−2, red line; (III): Q3,−2 = Q2,−3 = P3,−2 = P2,−3, purple line; (IV): Q4,−3 = Q3,−4 = P4,−3 = P3,−4, green line; (V): Q5,−4 = Q4,−5 = P5,−4 = P4,−5, black line.

Tables (1)

Tables Icon

Table 1 The overlap integrals and normalizations of the down-converted modes and pump modes

Equations (8)

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H ^ = i [ a 1 i χ m b ^ 1 p z a ^ s a ^ i + s 1 i χ m b ^ 1 p y a ^ s a ^ i ] + H . C .
τ b ^ ˙ 1 ( z ) ( t ) = γ p ( z ) b ^ 1 ( z ) ( t ) + ε 1 χ 1 a ^ 0 ( z ) ( t ) a ^ 1 ( z ) ( t ) χ 2 a ^ 1 ( z ) ( t ) a ^ 2 ( z ) ( t ) χ 3 a ^ 2 ( z ) ( t ) a ^ 3 ( z ) ( t ) + 2 γ p b ( z ) b ^ 1 i n ( z ) ( t ) + 2 γ p c ( z ) c ^ b 1 ( z ) ( t ) , τ b ^ ˙ 1 ( y ) ( t ) = γ p ( y ) b ^ 1 ( y ) ( t ) + ε 1 χ 1 a ^ 0 ( y ) ( t ) a ^ 1 ( y ) ( t ) χ 2 a ^ 1 ( y ) ( t ) a ^ 2 ( y ) ( t ) χ 3 a ^ 2 ( y ) ( t ) a ^ 3 ( y ) ( t ) + 2 γ p b ( y ) b ^ 1 i n ( y ) ( t ) + 2 γ p c ( y ) c ^ b 1 ( y ) ( t ) , τ a ^ ˙ 0 ( z ) ( t ) = γ 0 ( z ) a ^ 0 ( z ) ( t ) + χ 1 b ^ 1 ( z ) ( t ) a ^ 1 ( z ) ( t ) 2 γ b 0 ( z ) a ^ 0 i n ( z ) ( t ) + 2 γ c 0 ( z ) c ^ 0 ( z ) ( t ) , τ a ^ ˙ 0 ( y ) ( t ) = γ 0 ( y ) a ^ 0 ( y ) ( t ) + χ 1 b ^ 1 ( y ) ( t ) a ^ 1 ( y ) ( t ) 2 γ b 0 ( y ) a ^ 0 i n ( y ) ( t ) + 2 γ c 0 ( y ) c ^ 0 ( y ) ( t ) , τ a ^ ˙ 1 ( z ) ( t ) = γ 1 ( z ) a ^ 1 ( z ) ( t ) + χ 1 b ^ 1 ( z ) ( t ) a ^ 0 ( z ) ( t ) 2 γ b 1 ( z ) a ^ 1 i n ( z ) ( t ) + 2 γ c 1 ( z ) c ^ 1 ( z ) ( t ) , τ a ^ ˙ 1 ( y ) ( t ) = γ 1 ( y ) a ^ 1 ( y ) ( t ) + χ 2 b 1 ( y ) ( t ) a ^ 2 ( y ) ( t ) 2 γ b 1 ( z ) a ^ 1 i n ( y ) ( t ) + 2 γ c 1 ( y ) c ^ 1 ( y ) ( t ) , τ a ^ ˙ 1 ( z ) ( t ) = γ 1 ( z ) a ^ 1 ( z ) ( t ) + χ 2 b 1 ( z ) ( t ) a ^ 2 ( z ) ( t ) 2 γ b 1 ( z ) a ^ 1 i n ( z ) ( t ) + 2 γ c 1 ( z ) c ^ 1 ( z ) ( t ) , τ a ^ ˙ 1 ( z ) ( t ) = γ 1 ( y ) a ^ 1 ( y ) ( t ) + χ 1 b 1 ( y ) ( t ) a ^ 0 ( y ) ( t ) 2 γ b 1 ( y ) a ^ 1 i n ( y ) ( t ) + 2 γ c 1 ( y ) c ^ 1 ( y ) ( t ) , τ a ^ ˙ 1 ( y ) ( t ) = γ 1 ( y ) a ^ 1 ( y ) ( t ) + χ 1 b 1 ( y ) ( t ) a ^ 0 ( y ) ( t ) 2 γ b 1 ( y ) a ^ 1 i n ( y ) ( t ) + 2 γ c 1 ( y ) c ^ 1 ( y ) ( t ) , τ a ^ ˙ 2 ( z ) ( t ) = γ 2 ( z ) a ^ 2 ( z ) ( t ) + χ 2 b 1 ( z ) ( t ) a ^ 1 ( z ) ( t ) 2 γ b 2 ( z ) a ^ 2 i n ( z ) ( t ) + 2 γ c 2 ( z ) c ^ 2 ( z ) ( t ) , τ a ^ ˙ 2 ( y ) ( t ) = γ 2 ( y ) a ^ 2 ( y ) ( t ) + χ 3 b 1 ( y ) ( t ) a ^ 3 ( z ) ( t ) 2 γ b 2 ( y ) a ^ 2 i n ( z ) ( t ) + 2 γ c 2 ( y ) c ^ 2 ( y ) ( t ) , τ a ^ ˙ 2 ( z ) ( t ) = γ 2 ( z ) a ^ 2 ( z ) ( t ) + χ 3 b 1 ( z ) ( t ) a ^ 3 ( z ) ( t ) 2 γ b 2 ( z ) a ^ 2 i n ( z ) ( t ) + 2 γ c 2 ( z ) c ^ 2 ( z ) ( t ) , τ a ^ ˙ 2 ( y ) ( t ) = γ 2 ( y ) a ^ 2 ( y ) ( t ) + χ 2 b 1 ( y ) ( t ) a ^ 1 ( z ) ( t ) 2 γ b 2 ( y ) a ^ 2 i n ( z ) ( t ) + 2 γ c 2 ( y ) c ^ 2 ( y ) ( t ) , τ a ^ ˙ 3 ( z ) ( t ) = γ 3 ( z ) a ^ 3 ( z ) ( t ) + χ 3 b 1 ( z ) ( t ) a ^ 2 ( z ) ( t ) 2 γ b 3 ( z ) a ^ 3 i n ( z ) ( t ) + 2 γ c 3 ( z ) c ^ 3 ( z ) ( t ) , τ a ^ ˙ 3 ( y ) ( t ) = γ 3 ( y ) a ^ 3 ( y ) ( t ) + χ 3 b 1 ( y ) ( t ) a ^ 2 ( y ) ( t ) + 2 γ b 3 ( y ) a ^ 3 i n ( y ) ( t ) + 2 γ c 3 ( y ) c ^ 3 ( z ) ( t ) .
M x ( δ Q ^ 0 ( z ) ( ω ) δ Q ^ 1 ( z ) ( ω ) δ Q ^ 1 ( z ) ( ω ) δ Q ^ 2 ( z ) ( ω ) δ Q ^ 2 ( z ) ( ω ) δ Q ^ 3 ( z ) ( ω ) ) = 2 γ b ( δ Q ^ 0 i n ( z ) ( ω ) δ Q ^ 1 i n ( z ) ( ω ) δ Q ^ 1 i n ( z ) ( ω ) δ Q ^ 2 i n ( z ) ( ω ) δ Q ^ 2 i n ( z ) ( ω ) δ Q ^ 3 i n ( z ) ( ω ) ) + 2 γ c ( δ Q ^ c 0 ( z ) ( ω ) δ Q ^ c 1 ( z ) ( ω ) δ Q ^ c 1 ( z ) ( ω ) δ Q ^ c 2 ( z ) ( ω ) δ Q ^ c 2 ( y ) ( ω ) δ Q ^ c 3 ( y ) ( ω ) ) , M y ( δ Q ^ 0 ( y ) ( ω ) δ Q ^ 1 ( y ) ( ω ) δ Q ^ 1 ( y ) ( ω ) δ Q ^ 2 ( y ) ( ω ) δ Q ^ 2 ( y ) ( ω ) δ Q ^ 3 ( y ) ( ω ) ) = 2 γ b ( δ Q ^ 0 i n ( y ) ( ω ) δ Q ^ 1 i n ( y ) ( ω ) δ Q ^ 1 i n ( y ) ( ω ) δ Q ^ 2 i n ( y ) ( ω ) δ Q ^ 2 i n ( y ) ( ω ) δ Q ^ 3 i n ( y ) ( ω ) ) + 2 γ c ( δ Q ^ c 0 ( y ) ( ω ) δ Q ^ c 1 ( y ) ( ω ) δ Q ^ c 1 ( y ) ( ω ) δ Q ^ c 2 ( y ) ( ω ) δ Q ^ c 2 ( y ) ( ω ) δ Q ^ c 3 ( y ) ( ω ) ) .
M x = ( i ω τ + γ χ 1 β 1 ( z ) 0 0 0 0 χ 1 β 1 ( z ) i ω τ + γ 0 0 0 0 0 0 i ω τ + γ χ 2 β 2 ( z ) 0 0 0 χ 2 β 2 ( z ) i ω τ + γ 0 0 0 0 0 0 i ω τ + γ χ 3 β 3 ( z ) 0 0 0 0 χ 3 β 3 ( z ) i ω τ + γ ) , M y = ( i ω τ + γ 0 χ 1 β 1 ( y ) 0 0 0 0 i ω τ + γ 0 0 χ 2 β 2 ( y ) 0 χ 1 β 1 ( y ) 0 i ω τ + γ 0 0 0 0 0 0 i ω τ + γ 0 χ 3 β 3 ( y ) 0 χ 2 β 2 ( y ) 0 0 i ω τ + γ 0 0 0 0 χ 3 β 3 ( y ) 0 i ω τ + γ ) .
Q s . i = ( ( Q ^ s z + Q ^ s y ) ( Q ^ i z + Q ^ i y ) ) 2 < 1 / 2 ,
P s . i = ( ( P ^ s z + P ^ s y ) ( P ^ i z + P ^ i y ) ) 2 < 1 / 2 ,
Q s . i = ( ( Q ^ i z Q ^ i y ) + ( Q ^ s z Q ^ s y ) ) 2 < 1 / 2 ,
P s . i = ( ( P ^ i z + P ^ i y ) ( P ^ s z + P ^ s y ) ) 2 < 1 / 2 .

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