Abstract

An analytical calculation of the interaction geometry of two interlinked second-order nonlinear processes fulfilling phase-matching conditions is presented. The method is developed for type-I uniaxial crystals and gives the positions on a screen beyond the crystal of the entangled triplets generated by the interactions. The analytical results are compared to experiments realized in the macroscopic regime. Preliminary tests to identify the triplets are also performed based on intensity correlations.

© 2006 Optical Society of America

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  1. A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni, "Three-mode entanglement by interlinked nonlinear interactions in optical |(2) media," J. Opt. Soc. Am. B 21, 1241-1249 (2004).
    [CrossRef]
  2. A. V. Rodionov and A. S. Chirkin, "Entangled photon states in consecutive nonlinear optical interactions," JETP Lett. 79, 253-256 (2004).
    [CrossRef]
  3. A. S. Bradley, M. K. Olsen, O. Pfister, and R. C. Pooser, "Bright tripartite entanglement in triply concurrent parametric oscillation," Phys. Rev. A 72, 053805 (2005).
    [CrossRef]
  4. P. van Loock and S. L. Braunstein, "Multipartite entanglement for continuous variables: a quantum teleportation network," Phys. Rev. Lett. 84, 3482-3485 (2000).
    [CrossRef] [PubMed]
  5. 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]
  6. T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
    [CrossRef] [PubMed]
  7. O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
    [CrossRef]
  8. S. L. Braunstein and A. K. Pati, Quantum Information with Continuous Variables (Kluwer Academic, Dordrecht, 2003).
  9. S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005).
    [CrossRef]
  10. Chao-Kuei Lee, Jing-Yuan Zhang, J. Huang, and Ci-Ling Pan, "Generation of femtosecond laser pulses tunable from 380 nm to 465 nm via cascaded nonlinear optical mixing in a noncollinear optical parametric amplifier with a type-I phase matched BBO crystal," Opt. Express 11, 1702-1708 (2003).
    [CrossRef] [PubMed]
  11. M. Bondani, A. Allevi, E. Puddu, A. Andreoni, A. Ferraro, and M. G. A. Paris, "Properties of two interlinked chi(2) interactions in noncollinear phase matching," Opt. Lett. 29, 180-182 (2004), and erratum, Opt. Lett. 29, 1417-1417 (2004).
    [CrossRef]
  12. A. Agliati, M. Bondani, A. Andreoni, G. De Cillis, and M. G. A. Paris, "Quantum and classical correlations of intense beams of light via joint photodetection," J. Opt. B - Quantum Semiclassal Opt. 7, S652-S663 (2005).
    [CrossRef]
  13. M. G. A. Paris, A. Allevi, M. Bondani, and A. Andreoni are preparing a manuscript to be called "Quantum and classical correlations in tripartite states of light".

2005

A. S. Bradley, M. K. Olsen, O. Pfister, and R. C. Pooser, "Bright tripartite entanglement in triply concurrent parametric oscillation," Phys. Rev. A 72, 053805 (2005).
[CrossRef]

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005).
[CrossRef]

2004

2003

Chao-Kuei Lee, Jing-Yuan Zhang, J. Huang, and Ci-Ling Pan, "Generation of femtosecond laser pulses tunable from 380 nm to 465 nm via cascaded nonlinear optical mixing in a noncollinear optical parametric amplifier with a type-I phase matched BBO crystal," Opt. Express 11, 1702-1708 (2003).
[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]

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef] [PubMed]

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

2000

P. van Loock and S. L. Braunstein, "Multipartite entanglement for continuous variables: a quantum teleportation network," Phys. Rev. Lett. 84, 3482-3485 (2000).
[CrossRef] [PubMed]

Allevi, A.

Andreoni, A.

Aoki, T.

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef] [PubMed]

Bondani, M.

Bradley, A. S.

A. S. Bradley, M. K. Olsen, O. Pfister, and R. C. Pooser, "Bright tripartite entanglement in triply concurrent parametric oscillation," Phys. Rev. A 72, 053805 (2005).
[CrossRef]

Braunstein, S. L.

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005).
[CrossRef]

P. van Loock and S. L. Braunstein, "Multipartite entanglement for continuous variables: a quantum teleportation network," Phys. Rev. Lett. 84, 3482-3485 (2000).
[CrossRef] [PubMed]

Brownnutt, M.

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

Chirkin, A. S.

A. V. Rodionov and A. S. Chirkin, "Entangled photon states in consecutive nonlinear optical interactions," JETP Lett. 79, 253-256 (2004).
[CrossRef]

Ferraro, A.

Furusawa, A.

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef] [PubMed]

Glöckl, O.

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

Heersink, J.

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

Hiraoka, T.

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef] [PubMed]

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]

Korolkova, N.

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

Leuchs, G.

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

Lorenz, S.

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

Marquardt, C.

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

Olsen, M. K.

A. S. Bradley, M. K. Olsen, O. Pfister, and R. C. Pooser, "Bright tripartite entanglement in triply concurrent parametric oscillation," Phys. Rev. A 72, 053805 (2005).
[CrossRef]

Pan, Q.

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

Paris, M. G. A.

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.

A. S. Bradley, M. K. Olsen, O. Pfister, and R. C. Pooser, "Bright tripartite entanglement in triply concurrent parametric oscillation," Phys. Rev. A 72, 053805 (2005).
[CrossRef]

Pooser, R. C.

A. S. Bradley, M. K. Olsen, O. Pfister, and R. C. Pooser, "Bright tripartite entanglement in triply concurrent parametric oscillation," Phys. Rev. A 72, 053805 (2005).
[CrossRef]

Puddu, E.

Rodionov, A. V.

A. V. Rodionov and A. S. Chirkin, "Entangled photon states in consecutive nonlinear optical interactions," JETP Lett. 79, 253-256 (2004).
[CrossRef]

Silberhorn, C.

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

Takei, N.

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef] [PubMed]

van Loock, P.

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005).
[CrossRef]

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef] [PubMed]

P. van Loock and S. L. Braunstein, "Multipartite entanglement for continuous variables: a quantum teleportation network," Phys. Rev. Lett. 84, 3482-3485 (2000).
[CrossRef] [PubMed]

Wakui, K.

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef] [PubMed]

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]

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]

Yonezawa, H.

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef] [PubMed]

Zhang, 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]

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]

J. Opt. Soc. Am. B

JETP Lett.

A. V. Rodionov and A. S. Chirkin, "Entangled photon states in consecutive nonlinear optical interactions," JETP Lett. 79, 253-256 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

A. S. Bradley, M. K. Olsen, O. Pfister, and R. C. Pooser, "Bright tripartite entanglement in triply concurrent parametric oscillation," Phys. Rev. A 72, 053805 (2005).
[CrossRef]

O. Glöckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef]

Phys. Rev. Lett.

P. van Loock and S. L. Braunstein, "Multipartite entanglement for continuous variables: a quantum teleportation network," Phys. Rev. Lett. 84, 3482-3485 (2000).
[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]

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef] [PubMed]

Rev. Mod. Phys.

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005).
[CrossRef]

Other

A. Agliati, M. Bondani, A. Andreoni, G. De Cillis, and M. G. A. Paris, "Quantum and classical correlations of intense beams of light via joint photodetection," J. Opt. B - Quantum Semiclassal Opt. 7, S652-S663 (2005).
[CrossRef]

M. G. A. Paris, A. Allevi, M. Bondani, and A. Andreoni are preparing a manuscript to be called "Quantum and classical correlations in tripartite states of light".

S. L. Braunstein and A. K. Pati, Quantum Information with Continuous Variables (Kluwer Academic, Dordrecht, 2003).

Supplementary Material (1)

» Media 1: MPG (1271 KB)     

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Figures (3)

Fig. 1.
Fig. 1.

Scheme of the phase-matched interlinked interactions: (x,y)-plane coincides with the crystal entrance face; α, tuning angle; βj’s, angles to (y, z)-plane; ϑj’s, angles on the (y,z)-plane; φ, angle to the optical axis (OA).

Fig. 2.
Fig. 2.

(1.271 MB) Movie showing the variation of the measured and calculated outputs of the crystal as a function of the tuning angle α for fixed external angle between the pumps. Left panel: picture of the output of the crystal taken with a commercial digital photocamera (no color correction applied). LHS: a portion of the downconversion cones. RHS: output of the upconversion process. Right panel: calculated output of the upconversion process.

Fig. 3.
Fig. 3.

Upper panel: scheme of the experimental setup. BBO, nonlinear crystal; F1–3, filters; P1–3, pin-holes; f1–5,s, lenses; D1–3, p-i-n photodiodes; SGI, synchronous gated-integrator; MCA+PC, computer integrated multichannel analyzer. Lower panel: picture of the visible portion of the output states on a screen located beyond the nonlinear crystal.

Equations (19)

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H int = γ 1 a 1 a 3 + γ 2 a 2 a 3 + h . c . ,
N 1 ( t ) = N 2 ( t ) + N 3 ( t ) ,
k 1 sin β 1 + k 3 sin β 3 = 0
k 1 cos β 1 sin ϑ 1 + k 3 cos β 3 sin ϑ 3 = 0
k 1 cos β 1 cos ϑ 1 + k 3 cos β 3 cos ϑ 3 = k 4
k 2 sin β 2 = k 3 sin β 3
k 2 cos β 2 sin ϑ 2 = k 3 cos β 3 sin ϑ 3 + k 5 sin ϑ 5
k 2 cos β 2 cos ϑ 2 = k 3 cos β 3 cos ϑ 3 + k 5 cos ϑ 5
n j ( ω j ) = n o ( ω j ) j = 1 , 3 , 5
n 2 ( ω 2 , φ ) = [ cos 2 φ n o 2 ( ω 2 ) + sin 2 φ n e 2 ( ω 2 ) ] 1 2 ; n 4 ( ω 4 , α ) = [ cos 2 α n o 2 ( ω 4 ) + sin 2 α n e 2 ( ω 4 ) ] 1 2 ,
Γ ( k ) = 1 K i = 1 K [ m 1 ( i ) m 1 ] [ m 2 ( i + k ) + m 3 ( i + k ) m 2 + m 3 ] σ ( m 1 ) σ ( m 2 + m 3 ) ,
k 2 = [ G 2 cos 2 ( ϑ 2 α ) cos 2 β 2 + L 2 ] 1 2 ,
cos β 3 = A cos ϑ 3 ,
k 2 2 = k 3 2 + k 5 2 + 2 k 3 k 5 cos β 3 ( sin β 3 sin ϑ 5 + cos β 3 cos ϑ 5 )
k 2 2 cos 2 β 2 = k 3 2 cos 2 β 3 + k 5 2 + 2 k 3 k 5 cos β 3 ( sin β 3 sin ϑ 5 + cos β 3 cos ϑ 5 ) .
cos 2 β 2 = k 3 2 A 2 tan 2 ϑ 3 + k 5 2 + 2 k 3 k 5 A ( tan ϑ 3 sin ϑ 5 + cos ϑ 5 ) k 3 2 + k 5 2 + 2 k 3 k 5 A ( tan ϑ 3 sin ϑ 5 + cos ϑ 5 ) .
tan ϑ 2 = k 3 A tan ϑ 3 + k 5 sin ϑ 5 k 3 A + k 5 cos ϑ 5 .
cos 2 ( ϑ 2 α ) = 1 L 2 [ k 3 2 + k 5 2 + 2 k 3 k 5 cos β 3 ( sin β 3 sin ϑ 5 + cos β 3 cos ϑ 5 ) ] G 2 [ k 3 2 cos 2 β 3 + k 5 2 + 2 k 3 k 5 cos β 3 ( sin β 3 sin ϑ 5 + cos β 3 cos ϑ 5 ) ] .
a tan 2 ϑ 3 + b tan ϑ 3 + c = 0 ,

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