Abstract

As a highly entangled quantum network, the cluster state has the potential for greater information capacity and use in measurement-based quantum computation. Here, we report generating a continuous-variable quadripartite “square” cluster state of multiplexing orthogonal spatial modes in a single optical parametric amplifier (OPA), and further improve the quality of entanglement by optimizing the pump profile. We produce multimode entanglement of two first-order Hermite–Gauss modes within one beam in a single multimode OPA and transform it into a cluster state by phase correction. Furthermore, the pump-profile dependence of the entanglement of this state is explored. Compared with fundamental mode pumping, an enhancement of approximately 33% is achieved using the suitable pump-profile mode. Our approach is potentially scalable to multimode entanglement in the spatial domain. Such spatial cluster states may contribute to future schemes in spatial quantum information processing.

© 2018 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  27. K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016).
    [Crossref]
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    [Crossref]
  29. M. Lassen, V. Delaubert, 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 Publ. 1, 06003 (2006).
    [Crossref]
  30. S. L. W. Midgley, M. K. Olsen, A. S. Bradley, and O. Pfister, “Analysis of a continuous-variable quadripartite cluster state from a single optical parametric oscillator,” Phys. Rev. A 82, 053826 (2010).
    [Crossref]
  31. L. J. Pereira, W. T. Buono, D. S. Tasca, K. Dechoum, and A. Z. Khoury, “Orbital-angular-momentum mixing in type-II second-harmonic generation,” Phys. Rev. A 96, 053856 (2017).
    [Crossref]
  32. M. Martinelli, J. A. O. Huguenin, P. Nussenzveig, and A. Z. Khoury, “Orbital angular momentum exchange in an optical parametric oscillator,” Phys. Rev. A 70, 013812 (2004).
    [Crossref]
  33. M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [Crossref]
  34. G. Ferrini, J. P. Gazeau, T. Coudreau, C. Fabre, and N. Treps, “Compact Gaussian quantum computation by multi-pixel homodyne detection,” New J. Phys. 15, 093015 (2013).
    [Crossref]
  35. J. Zhang, J. J. Wang, R. G. Yang, K. Liu, and J. R. Gao, “Large-scale continuous-variable dual-rail cluster entangled state based on spatial mode comb,” Opt. Express 25, 27172–27181 (2017).
    [Crossref]
  36. R. C. Pooser and J. Jing, “Continuous variable cluster state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
    [Crossref]

2017 (3)

2016 (4)

K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016).
[Crossref]

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,” APL Photon. 1, 060801 (2016).
[Crossref]

R. N. Alexander, P. Wang, N. Sridhar, M. Chen, O. Pfister, and N. C. Menicucci, “One-way quantum computing with arbitrarily large time-frequency continuous-variable cluster states from a single optical parametric oscillator,” Phys. Rev. A 94, 032327 (2016).
[Crossref]

W. McCutcheon, A. Pappa, B. A. Bell, A. McMillan, A. Chailloux, T. Lawson, M. Mafu, D. Markham, E. Diamanti, I. Kerenidis, J. G. Rarity, and M. S. Tame, “Experimental verification of multipartite entanglement in quantum networks,” Nat. Commun. 7, 13251 (2016).
[Crossref]

2015 (1)

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref]

2014 (3)

J. Roslund, R. Medeiros de Arajo, S. Jiang, C. Fabre, and N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs,” Nat. Photonics 8, 109–112 (2014).
[Crossref]

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref]

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

2013 (2)

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]

G. Ferrini, J. P. Gazeau, T. Coudreau, C. Fabre, and N. Treps, “Compact Gaussian quantum computation by multi-pixel homodyne detection,” New J. Phys. 15, 093015 (2013).
[Crossref]

2012 (2)

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun. 3, 1026 (2012).
[Crossref]

X. Su, Y. Zhao, S. Hao, X. Jia, C. Xie, and K. Peng, “Experimental preparation of eight-partite cluster state forphotonic qumodes,” Opt. Lett. 37, 5178–5180 (2012).
[Crossref]

2011 (3)

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]

R. Ukai, N. Iwata, Y. Shimokawa, S. Armstrong, A. Politi, J. Yoshikawa, P. V. Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
[Crossref]

N. C. Menicucci, “Temporal-mode continuous-variable cluster states using linear optics,” Phys. Rev. A 83, 062314 (2011).
[Crossref]

2010 (3)

N. C. Menicucci, X. Ma, and T. C. Ralph, “Arbitrarily large continuous-variable cluster states from a single quantum nondemolition gate,” Phys. Rev. Lett. 104, 250503 (2010).
[Crossref]

S. L. W. Midgley, A. S. Bradley, O. Pfister, and M. K. Olsen, “Quadripartite continuous-variable entanglement via quadruply concurrent down-conversion,” Phys. Rev. A 81, 063834 (2010).
[Crossref]

S. L. W. Midgley, M. K. Olsen, A. S. Bradley, and O. Pfister, “Analysis of a continuous-variable quadripartite cluster state from a single optical parametric oscillator,” Phys. Rev. A 82, 053826 (2010).
[Crossref]

2009 (3)

B. dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

J. Janousek, K. Wagner, J. F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagating modes,” Nat. Photonics 3, 399–402 (2009).
[Crossref]

T. Aoki, G. Takahashi, T. Kajiya, J. Yoshikawa, S. L. Braunstein, P. van Loock, and A. Furusawa, “Quantum error correction beyond qubits,” Nat. Phys. 5, 541–546 (2009).
[Crossref]

2008 (3)

Y. Tokunaga, S. Kuwashiro, T. Yamamoto, M. Koashi, and N. Imoto, “Generation of high-fidelity four-photon cluster state and quantum-domain demonstration of one-way quantum computing,” Phys. Rev. Lett. 100, 210501 (2008).
[Crossref]

M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A 78, 012301 (2008).
[Crossref]

N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008).
[Crossref]

2007 (1)

X. Su, A. Tan, X. Jia, J. Zhang, C. Xie, and K. Peng, “Experimental preparation of quadripartite cluster and Greenberger-Horne-Zeilinger entangled states for continuous variables,” Phys. Rev. Lett. 98, 070502 (2007).
[Crossref]

2006 (3)

J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A 73, 032318 (2006).
[Crossref]

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[Crossref]

M. Lassen, V. Delaubert, 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 Publ. 1, 06003 (2006).
[Crossref]

2005 (1)

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref]

2004 (1)

M. Martinelli, J. A. O. Huguenin, P. Nussenzveig, and A. Z. Khoury, “Orbital angular momentum exchange in an optical parametric oscillator,” Phys. Rev. A 70, 013812 (2004).
[Crossref]

2001 (2)

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188–5191 (2001).
[Crossref]

1993 (1)

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Alexander, R. N.

R. N. Alexander, P. Wang, N. Sridhar, M. Chen, O. Pfister, and N. C. Menicucci, “One-way quantum computing with arbitrarily large time-frequency continuous-variable cluster states from a single optical parametric oscillator,” Phys. Rev. A 94, 032327 (2016).
[Crossref]

Allen, L.

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Aoki, T.

T. Aoki, G. Takahashi, T. Kajiya, J. Yoshikawa, S. L. Braunstein, P. van Loock, and A. Furusawa, “Quantum error correction beyond qubits,” Nat. Phys. 5, 541–546 (2009).
[Crossref]

Armstrong, S.

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun. 3, 1026 (2012).
[Crossref]

R. Ukai, N. Iwata, Y. Shimokawa, S. Armstrong, A. Politi, J. Yoshikawa, P. V. Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
[Crossref]

Armstrong, S. C.

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]

Aspelmeyer, M.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref]

Bachor, H. A.

J. Janousek, K. Wagner, J. F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagating modes,” Nat. Photonics 3, 399–402 (2009).
[Crossref]

M. Lassen, V. Delaubert, 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 Publ. 1, 06003 (2006).
[Crossref]

Bachor, H.-A.

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun. 3, 1026 (2012).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Bell, B. A.

W. McCutcheon, A. Pappa, B. A. Bell, A. McMillan, A. Chailloux, T. Lawson, M. Mafu, D. Markham, E. Diamanti, I. Kerenidis, J. G. Rarity, and M. S. Tame, “Experimental verification of multipartite entanglement in quantum networks,” Nat. Commun. 7, 13251 (2016).
[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]

Bradley, A. S.

S. L. W. Midgley, A. S. Bradley, O. Pfister, and M. K. Olsen, “Quadripartite continuous-variable entanglement via quadruply concurrent down-conversion,” Phys. Rev. A 81, 063834 (2010).
[Crossref]

S. L. W. Midgley, M. K. Olsen, A. S. Bradley, and O. Pfister, “Analysis of a continuous-variable quadripartite cluster state from a single optical parametric oscillator,” Phys. Rev. A 82, 053826 (2010).
[Crossref]

Braunstein, S. L.

T. Aoki, G. Takahashi, T. Kajiya, J. Yoshikawa, S. L. Braunstein, P. van Loock, and A. Furusawa, “Quantum error correction beyond qubits,” Nat. Phys. 5, 541–546 (2009).
[Crossref]

J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A 73, 032318 (2006).
[Crossref]

Briegel, H. J.

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188–5191 (2001).
[Crossref]

Buono, W. T.

L. J. Pereira, W. T. Buono, D. S. Tasca, K. Dechoum, and A. Z. Khoury, “Orbital-angular-momentum mixing in type-II second-harmonic generation,” Phys. Rev. A 96, 053856 (2017).
[Crossref]

Cai, C.

Cai, X. D.

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref]

Chailloux, A.

W. McCutcheon, A. Pappa, B. A. Bell, A. McMillan, A. Chailloux, T. Lawson, M. Mafu, D. Markham, E. Diamanti, I. Kerenidis, J. G. Rarity, and M. S. Tame, “Experimental verification of multipartite entanglement in quantum networks,” Nat. Commun. 7, 13251 (2016).
[Crossref]

Chen, M.

R. N. Alexander, P. Wang, N. Sridhar, M. Chen, O. Pfister, and N. C. Menicucci, “One-way quantum computing with arbitrarily large time-frequency continuous-variable cluster states from a single optical parametric oscillator,” Phys. Rev. A 94, 032327 (2016).
[Crossref]

Chen, M. C.

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref]

Coudreau, T.

G. Ferrini, J. P. Gazeau, T. Coudreau, C. Fabre, and N. Treps, “Compact Gaussian quantum computation by multi-pixel homodyne detection,” New J. Phys. 15, 093015 (2013).
[Crossref]

Dechoum, K.

L. J. Pereira, W. T. Buono, D. S. Tasca, K. Dechoum, and A. Z. Khoury, “Orbital-angular-momentum mixing in type-II second-harmonic generation,” Phys. Rev. A 96, 053856 (2017).
[Crossref]

B. dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

Delaubert, V.

M. Lassen, V. Delaubert, 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 Publ. 1, 06003 (2006).
[Crossref]

Diamanti, E.

W. McCutcheon, A. Pappa, B. A. Bell, A. McMillan, A. Chailloux, T. Lawson, M. Mafu, D. Markham, E. Diamanti, I. Kerenidis, J. G. Rarity, and M. S. Tame, “Experimental verification of multipartite entanglement in quantum networks,” Nat. Commun. 7, 13251 (2016).
[Crossref]

dos Santos, B.

B. dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

Fabre, C.

J. Roslund, R. Medeiros de Arajo, S. Jiang, C. Fabre, and N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs,” Nat. Photonics 8, 109–112 (2014).
[Crossref]

G. Ferrini, J. P. Gazeau, T. Coudreau, C. Fabre, and N. Treps, “Compact Gaussian quantum computation by multi-pixel homodyne detection,” New J. Phys. 15, 093015 (2013).
[Crossref]

Ferrini, G.

G. Ferrini, J. P. Gazeau, T. Coudreau, C. Fabre, and N. Treps, “Compact Gaussian quantum computation by multi-pixel homodyne detection,” New J. Phys. 15, 093015 (2013).
[Crossref]

Flammia, S. T.

N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008).
[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,” APL Photon. 1, 060801 (2016).
[Crossref]

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S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun. 3, 1026 (2012).
[Crossref]

J. Janousek, K. Wagner, J. F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagating modes,” Nat. Photonics 3, 399–402 (2009).
[Crossref]

M. Lassen, V. Delaubert, 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 Publ. 1, 06003 (2006).
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Ukai, R.

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]

R. Ukai, N. Iwata, Y. Shimokawa, S. Armstrong, A. Politi, J. Yoshikawa, P. V. Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
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M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A 78, 012301 (2008).
[Crossref]

van der Veen, H.

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

van Loock, P.

T. Aoki, G. Takahashi, T. Kajiya, J. Yoshikawa, S. L. Braunstein, P. van Loock, and A. Furusawa, “Quantum error correction beyond qubits,” Nat. Phys. 5, 541–546 (2009).
[Crossref]

M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A 78, 012301 (2008).
[Crossref]

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[Crossref]

Vedral, V.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref]

Wagner, K.

J. Janousek, K. Wagner, J. F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagating modes,” Nat. Photonics 3, 399–402 (2009).
[Crossref]

Walther, P.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref]

Wang, J. J.

Wang, P.

R. N. Alexander, P. Wang, N. Sridhar, M. Chen, O. Pfister, and N. C. Menicucci, “One-way quantum computing with arbitrarily large time-frequency continuous-variable cluster states from a single optical parametric oscillator,” Phys. Rev. A 94, 032327 (2016).
[Crossref]

Wang, X. L.

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref]

Weedbrook, C.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[Crossref]

Weinfurter, H.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Wu, D.

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
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Xie, C.

X. Su, Y. Zhao, S. Hao, X. Jia, C. Xie, and K. Peng, “Experimental preparation of eight-partite cluster state forphotonic qumodes,” Opt. Lett. 37, 5178–5180 (2012).
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X. Su, A. Tan, X. Jia, J. Zhang, C. Xie, and K. Peng, “Experimental preparation of quadripartite cluster and Greenberger-Horne-Zeilinger entangled states for continuous variables,” Phys. Rev. Lett. 98, 070502 (2007).
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Y. Tokunaga, S. Kuwashiro, T. Yamamoto, M. Koashi, and N. Imoto, “Generation of high-fidelity four-photon cluster state and quantum-domain demonstration of one-way quantum computing,” Phys. Rev. Lett. 100, 210501 (2008).
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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,” APL Photon. 1, 060801 (2016).
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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]

Yonezawa, H.

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]

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,” APL Photon. 1, 060801 (2016).
[Crossref]

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]

R. Ukai, N. Iwata, Y. Shimokawa, S. Armstrong, A. Politi, J. Yoshikawa, P. V. Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
[Crossref]

T. Aoki, G. Takahashi, T. Kajiya, J. Yoshikawa, S. L. Braunstein, P. van Loock, and A. Furusawa, “Quantum error correction beyond qubits,” Nat. Phys. 5, 541–546 (2009).
[Crossref]

Yukawa, M.

M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A 78, 012301 (2008).
[Crossref]

Zeilinger, A.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref]

Zhang, J.

J. Zhang, J. J. Wang, R. G. Yang, K. Liu, and J. R. Gao, “Large-scale continuous-variable dual-rail cluster entangled state based on spatial mode comb,” Opt. Express 25, 27172–27181 (2017).
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K. Liu, J. Guo, C. Cai, J. Zhang, and J. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178–5181 (2016).
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X. Su, A. Tan, X. Jia, J. Zhang, C. Xie, and K. Peng, “Experimental preparation of quadripartite cluster and Greenberger-Horne-Zeilinger entangled states for continuous variables,” Phys. Rev. Lett. 98, 070502 (2007).
[Crossref]

J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A 73, 032318 (2006).
[Crossref]

Zhao, Y.

APL Photon. (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,” APL Photon. 1, 060801 (2016).
[Crossref]

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

M. Lassen, V. Delaubert, 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 Publ. 1, 06003 (2006).
[Crossref]

Nat. Commun. (2)

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun. 3, 1026 (2012).
[Crossref]

W. McCutcheon, A. Pappa, B. A. Bell, A. McMillan, A. Chailloux, T. Lawson, M. Mafu, D. Markham, E. Diamanti, I. Kerenidis, J. G. Rarity, and M. S. Tame, “Experimental verification of multipartite entanglement in quantum networks,” Nat. Commun. 7, 13251 (2016).
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Nat. Photonics (3)

J. Roslund, R. Medeiros de Arajo, S. Jiang, C. Fabre, and N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs,” Nat. Photonics 8, 109–112 (2014).
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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]

J. Janousek, K. Wagner, J. F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagating modes,” Nat. Photonics 3, 399–402 (2009).
[Crossref]

Nat. Phys. (1)

T. Aoki, G. Takahashi, T. Kajiya, J. Yoshikawa, S. L. Braunstein, P. van Loock, and A. Furusawa, “Quantum error correction beyond qubits,” Nat. Phys. 5, 541–546 (2009).
[Crossref]

Nature (2)

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref]

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
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New J. Phys. (1)

G. Ferrini, J. P. Gazeau, T. Coudreau, C. Fabre, and N. Treps, “Compact Gaussian quantum computation by multi-pixel homodyne detection,” New J. Phys. 15, 093015 (2013).
[Crossref]

Opt. Commun. (1)

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (9)

R. N. Alexander, P. Wang, N. Sridhar, M. Chen, O. Pfister, and N. C. Menicucci, “One-way quantum computing with arbitrarily large time-frequency continuous-variable cluster states from a single optical parametric oscillator,” Phys. Rev. A 94, 032327 (2016).
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J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A 73, 032318 (2006).
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M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A 78, 012301 (2008).
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N. C. Menicucci, “Temporal-mode continuous-variable cluster states using linear optics,” Phys. Rev. A 83, 062314 (2011).
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K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
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N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
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N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008).
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Y. Tokunaga, S. Kuwashiro, T. Yamamoto, M. Koashi, and N. Imoto, “Generation of high-fidelity four-photon cluster state and quantum-domain demonstration of one-way quantum computing,” Phys. Rev. Lett. 100, 210501 (2008).
[Crossref]

R. Ukai, N. Iwata, Y. Shimokawa, S. Armstrong, A. Politi, J. Yoshikawa, P. V. Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
[Crossref]

X. Su, A. Tan, X. Jia, J. Zhang, C. Xie, and K. Peng, “Experimental preparation of quadripartite cluster and Greenberger-Horne-Zeilinger entangled states for continuous variables,” Phys. Rev. Lett. 98, 070502 (2007).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Schematic of the underlying principle involved in generating the CV quadripartite spatial-mode Gaussian cluster state. (a) Pump laser drives the multimode OPA to produce entanglement of two spatial modes, HG01 and HG10, within one beam. By performing the transformation U, the multimode entanglement is transformed into a cluster state containing four spatial orthogonal modes, HG01, HG10, HG45°, and HG135°. (b) Square representation in the graph-state picture. Each cluster node, corresponding to a spatial mode, is represented by a circle. Neighboring nodes are connected by lines and represent the bipartite entanglement between two spatial modes.
Fig. 2.
Fig. 2. Experimental layout for the generation and characterization of the spatial-mode cluster state. Entanglement is generated using a multimode OPA and measured by BHD detection with spatial tailored local oscillators. LG10 pump reconstruction is shown in the right panel. NOPA: nondegenerate optical parametric amplifier; LO: local oscillator; DBS: dichroic beam splitter; PBS: polarization beam splitter; HWP: half-wavelength plate; QWP: quarter-wave plate; BHD: balanced homodyne detection; DP: Dove prism; SA: spectrum analyzer; FQ-PM: four-quadrant phase mask; MC: HG11LG10 mode converter.
Fig. 3.
Fig. 3. Measured quantum correlations of the cluster state in the mode basis (a) HG01, (b) HG10, (c) HG45°, and (d) HG135° using the fundamental mode pump. Trace (2) is the shot noise limit (SNL), and traces (1) and (3) are the correlation variance and anti-correlation variance, respectively, normalized to SNL. Measurement settings: resolution bandwidth 300 kHz, video bandwidth 390 Hz.
Fig. 4.
Fig. 4. Enhanced entanglement for the (a) HG01, (b) HG10, (c) HG45°, and (d) HG135° modes with LG10 pumping. Trace (2) is SNL; traces (1) and (3) are the correlation variances normalized to SNL. Measurement settings: resolution bandwidth 300 kHz, video bandwidth 390 Hz.
Fig. 5.
Fig. 5. Experimental measurement of inseparability for the (a) HG01, (b) HG10, (c) HG45°, and (d) HG135° modes with LG10 pumping. Values below unity indicate entanglement.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

Hint=iχk(a^pa^+1ia^1s+a^pa^1ia^+1s+h.c),
Hint=iχk(a^pa^10ia^10s+a^pa^01ia^01s+h.c),
V(X^01i+X^01s)<1,
V(Y^01iY^01s)<1,
V(X^10i+X^10s)<1,
V(Y^10iY^10s)<1,
X^01s=(Y^135°s+Y^45°s)/2,
Y^01s=(X^135°s+X^45°s)/2,
X^10s=(Y^135°sY^45°s)/2,
Y^10s=(X^135°sX^45°s)/2.
V(Y^01i+12X^135°s+12X^45°s)<1,
V(Y^10i+12X^135°s12X^45°s)<1,
V(Y^45°s+12X^01i12X^10i)<1,
V(Y^135°s+12X^01i+12X^10i)<1,
Y^AX^0,
V(Y^01iY^01s)<1,
V(Y^10iY^10s)<1,
V(Y^45°iY^45°s)<1,
V(Y^135°iY^135°s)<1.
Γ=+νp(r)μs(r)μi(r)αdr,
Γ00=+ν00(r)μ01/102(r)+μ01/104(r)drdr=13,
Γ02/20=+ν02/20(r)μ01/102(r)+μ01/104(r)drdr=23,
V(Y^01iY^01s)=0.62±0.02<1,
V(Y^10iY^10s)=0.59±0.02<1,
V(Y^45°iY^45°s)=0.64±0.02<1,
V(Y^135°iY^135°s)=0.61±0.02<1.
V(Y^01iY^01s)=0.48±0.02<1,
V(Y^10iY^10s)=0.43±0.02<1,
V(Y^45°iY^45°s)=0.48±0.02<1,
V(Y^135°iY^135°s)=0.47±0.02<1.

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