Abstract

Multicolored multipartite entanglement is of great importance in quantum communication and quantum information networks. In this paper, we calculate the quantum fluctuations of the fundamental frequency pump beam and second-harmonic beams in a two-port frequency doubling resonator, and investigate the tripartite continuous-variable entanglement generated by this device for the first time, to our knowledge. The quantum correlation among fundamental frequency pump beam and two harmonic beams is studied using a necessary and sufficient criterion for Gaussian entanglement states, the positivity under partial transposition. It is found that two-color tripartite entanglement exists in a large range of pump intensities and analysis frequencies.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
  2. Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
    [CrossRef]
  3. W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
    [CrossRef] [PubMed]
  4. O.-K. Lim and M. Saffman, “Intensity correlations and entanglement by frequency doubling in a two-port resonator,” Phys. Rev. A 74, 023816 (2006).
    [CrossRef]
  5. O.-K. Lim, B. Bol, and M. Saffman, “Observation of twin beam correlations and quadrature entanglement by frequency doubling in a two-port resonator,” EPL 78, 40004 (2007).
    [CrossRef]
  6. A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  26. R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

2010 (1)

2009 (2)

S. Zhai, R. Yang, K. Liu, H. Zhang, J. Zhang, and J. Gao, “Bright two-color tripartite entanglement with second harmonic generation,” Opt. Express 17, 9851–9857 (2009).
[CrossRef] [PubMed]

A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009).
[CrossRef] [PubMed]

2008 (2)

N. B. Grosse, S. Assad, M. Mehmet, R. Schnabel, T. Symul, and P. K. Lam, “Observation of entanglement between two light beams spanning an octave in optical frequency,” Phys. Rev. Lett. 100, 243601 (2008).
[CrossRef] [PubMed]

S. Zhai, R. Yang, D. Fan, J. Guo, K. Liu, J. Zhang, and J. Gao, “Tripartite entanglement from the cavity with second-order harmonic generation,” Phys. Rev. A 78, 014302 (2008).
[CrossRef]

2007 (1)

O.-K. Lim, B. Bol, and M. Saffman, “Observation of twin beam correlations and quadrature entanglement by frequency doubling in a two-port resonator,” EPL 78, 40004 (2007).
[CrossRef]

2006 (2)

A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006).
[CrossRef] [PubMed]

O.-K. Lim and M. Saffman, “Intensity correlations and entanglement by frequency doubling in a two-port resonator,” Phys. Rev. A 74, 023816 (2006).
[CrossRef]

2005 (1)

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[CrossRef]

2004 (1)

M. K. Olsen, “Continuous-variable Einstein–Podolsky–Rosen paradox with traveling-wave second-harmonic generation,” Phys. Rev. A 70, 035801 (2004).
[CrossRef]

2003 (4)

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]

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
[CrossRef] [PubMed]

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl, and H. J. Kimble, “Quantum teleportation of light beams,” Phys. Rev. A 67, 033802 (2003).
[CrossRef]

2002 (3)

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein–Podolsky–Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[CrossRef] [PubMed]

Ch. Silberhorn, N. Korolkova, and G. Leuchs, “Quantum key distribution with bright entangled beams,” Phys. Rev. Lett. 88, 167902 (2002).
[CrossRef] [PubMed]

J. Zhang, C. Xie, and K. Peng, “Controlled dense coding for continuous variables using three-particle entangled states,” Phys. Rev. A 66, 032318 (2002).
[CrossRef]

2001 (3)

P. van Loock and S. L. Braunstein, “Telecloning of continuous quantum variables,” Phys. Rev. Lett. 87, 247901 (2001).
[CrossRef] [PubMed]

G. M. D’Ariano, P. Lo Presti, and M. G. A. Paris, “Using entanglement improves the precision of quantum measurements,” Phys. Rev. Lett. 87, 270404 (2001).
[CrossRef]

R. F. Werner and M. M. Wolf, “Bound entangled Gaussian states,” Phys. Rev. Lett. 86, 3658–3661 (2001).
[CrossRef] [PubMed]

2000 (2)

R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[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]

1999 (1)

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[CrossRef]

1998 (1)

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

1994 (2)

Z. Y. Ou, “Propagation of quantum fluctuations in single-pass second-harmonic generation for arbitrary interaction length,” Phys. Rev. A 49, 2106–2116 (1994).
[CrossRef] [PubMed]

R. D. Li and P. Kumar, “Quantum-noise reduction in traveling-wave second-harmonic generation,” Phys. Rev. A 49, 2157–2166 (1994).
[CrossRef] [PubMed]

1992 (2)

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef]

Assad, S.

N. B. Grosse, S. Assad, M. Mehmet, R. Schnabel, T. Symul, and P. K. Lam, “Observation of entanglement between two light beams spanning an octave in optical frequency,” Phys. Rev. Lett. 100, 243601 (2008).
[CrossRef] [PubMed]

Bachor, H. -A.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

Barbosa, F. A. S.

A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009).
[CrossRef] [PubMed]

Bernard, F.

Bol, B.

O.-K. Lim, B. Bol, and M. Saffman, “Observation of twin beam correlations and quadrature entanglement by frequency doubling in a two-port resonator,” EPL 78, 40004 (2007).
[CrossRef]

Bowen, W. P.

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
[CrossRef] [PubMed]

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

Braunstein, S. L.

P. van Loock and S. L. Braunstein, “Telecloning of continuous quantum variables,” Phys. Rev. Lett. 87, 247901 (2001).
[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]

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Buchler, B. C.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

Cassemiro, K. N.

A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009).
[CrossRef] [PubMed]

Cerf, N. J.

Chou, C. W.

T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl, and H. J. Kimble, “Quantum teleportation of light beams,” Phys. Rev. A 67, 033802 (2003).
[CrossRef]

Coelho, A. S.

A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009).
[CrossRef] [PubMed]

D’Ariano, G. M.

G. M. D’Ariano, P. Lo Presti, and M. G. A. Paris, “Using entanglement improves the precision of quantum measurements,” Phys. Rev. Lett. 87, 270404 (2001).
[CrossRef]

Daems, D.

Fabre, C.

A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006).
[CrossRef] [PubMed]

Fan, D.

S. Zhai, R. Yang, D. Fan, J. Guo, K. Liu, J. Zhang, and J. Gao, “Tripartite entanglement from the cavity with second-order harmonic generation,” Phys. Rev. A 78, 014302 (2008).
[CrossRef]

Fuchs, C. A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Furusawa, A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Gao, J.

S. Zhai, R. Yang, K. Liu, H. Zhang, J. Zhang, and J. Gao, “Bright two-color tripartite entanglement with second harmonic generation,” Opt. Express 17, 9851–9857 (2009).
[CrossRef] [PubMed]

S. Zhai, R. Yang, D. Fan, J. Guo, K. Liu, J. Zhang, and J. Gao, “Tripartite entanglement from the cavity with second-order harmonic generation,” Phys. Rev. A 78, 014302 (2008).
[CrossRef]

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[CrossRef]

Goh, K. W.

T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl, and H. J. Kimble, “Quantum teleportation of light beams,” Phys. Rev. A 67, 033802 (2003).
[CrossRef]

Grosse, N. B.

N. B. Grosse, S. Assad, M. Mehmet, R. Schnabel, T. Symul, and P. K. Lam, “Observation of entanglement between two light beams spanning an octave in optical frequency,” Phys. Rev. Lett. 100, 243601 (2008).
[CrossRef] [PubMed]

Guo, J.

S. Zhai, R. Yang, D. Fan, J. Guo, K. Liu, J. Zhang, and J. Gao, “Tripartite entanglement from the cavity with second-order harmonic generation,” Phys. Rev. A 78, 014302 (2008).
[CrossRef]

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[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]

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein–Podolsky–Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[CrossRef] [PubMed]

Jonathan, D.

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[CrossRef]

Kimble, H. J.

T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl, and H. J. Kimble, “Quantum teleportation of light beams,” Phys. Rev. A 67, 033802 (2003).
[CrossRef]

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef]

Kolobov, M. I.

Korolkova, N.

Ch. Silberhorn, N. Korolkova, and G. Leuchs, “Quantum key distribution with bright entangled beams,” Phys. Rev. Lett. 88, 167902 (2002).
[CrossRef] [PubMed]

Kumar, P.

R. D. Li and P. Kumar, “Quantum-noise reduction in traveling-wave second-harmonic generation,” Phys. Rev. A 49, 2157–2166 (1994).
[CrossRef] [PubMed]

Lam, P. K.

N. B. Grosse, S. Assad, M. Mehmet, R. Schnabel, T. Symul, and P. K. Lam, “Observation of entanglement between two light beams spanning an octave in optical frequency,” Phys. Rev. Lett. 100, 243601 (2008).
[CrossRef] [PubMed]

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
[CrossRef] [PubMed]

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

Leuchs, G.

Ch. Silberhorn, N. Korolkova, and G. Leuchs, “Quantum key distribution with bright entangled beams,” Phys. Rev. Lett. 88, 167902 (2002).
[CrossRef] [PubMed]

Li, R. D.

R. D. Li and P. Kumar, “Quantum-noise reduction in traveling-wave second-harmonic generation,” Phys. Rev. A 49, 2157–2166 (1994).
[CrossRef] [PubMed]

Li, X.

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein–Podolsky–Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[CrossRef] [PubMed]

Lim, O. -K.

O.-K. Lim, B. Bol, and M. Saffman, “Observation of twin beam correlations and quadrature entanglement by frequency doubling in a two-port resonator,” EPL 78, 40004 (2007).
[CrossRef]

O.-K. Lim and M. Saffman, “Intensity correlations and entanglement by frequency doubling in a two-port resonator,” Phys. Rev. A 74, 023816 (2006).
[CrossRef]

Liu, K.

S. Zhai, R. Yang, K. Liu, H. Zhang, J. Zhang, and J. Gao, “Bright two-color tripartite entanglement with second harmonic generation,” Opt. Express 17, 9851–9857 (2009).
[CrossRef] [PubMed]

S. Zhai, R. Yang, D. Fan, J. Guo, K. Liu, J. Zhang, and J. Gao, “Tripartite entanglement from the cavity with second-order harmonic generation,” Phys. Rev. A 78, 014302 (2008).
[CrossRef]

Lo Presti, P.

G. M. D’Ariano, P. Lo Presti, and M. G. A. Paris, “Using entanglement improves the precision of quantum measurements,” Phys. Rev. Lett. 87, 270404 (2001).
[CrossRef]

Lodahl, P.

T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl, and H. J. Kimble, “Quantum teleportation of light beams,” Phys. Rev. A 67, 033802 (2003).
[CrossRef]

Martinelli, M.

A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009).
[CrossRef] [PubMed]

A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006).
[CrossRef] [PubMed]

Mehmet, M.

N. B. Grosse, S. Assad, M. Mehmet, R. Schnabel, T. Symul, and P. K. Lam, “Observation of entanglement between two light beams spanning an octave in optical frequency,” Phys. Rev. Lett. 100, 243601 (2008).
[CrossRef] [PubMed]

Murao, M.

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[CrossRef]

Nussenzveig, P.

A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009).
[CrossRef] [PubMed]

A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006).
[CrossRef] [PubMed]

Olsen, M. K.

M. K. Olsen, “Continuous-variable Einstein–Podolsky–Rosen paradox with traveling-wave second-harmonic generation,” Phys. Rev. A 70, 035801 (2004).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou, “Propagation of quantum fluctuations in single-pass second-harmonic generation for arbitrary interaction length,” Phys. Rev. A 49, 2106–2116 (1994).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef]

Pan, Q.

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein–Podolsky–Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[CrossRef] [PubMed]

Paris, M. G. A.

G. M. D’Ariano, P. Lo Presti, and M. G. A. Paris, “Using entanglement improves the precision of quantum measurements,” Phys. Rev. Lett. 87, 270404 (2001).
[CrossRef]

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]

J. Zhang, C. Xie, and K. Peng, “Controlled dense coding for continuous variables using three-particle entangled states,” Phys. Rev. A 66, 032318 (2002).
[CrossRef]

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein–Podolsky–Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[CrossRef] [PubMed]

Peng, K. C.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

Pereira, S. F.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef]

Plenio, M. B.

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[CrossRef]

Polzik, E. S.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Ralph, T. C.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
[CrossRef] [PubMed]

Saffman, M.

O.-K. Lim, B. Bol, and M. Saffman, “Observation of twin beam correlations and quadrature entanglement by frequency doubling in a two-port resonator,” EPL 78, 40004 (2007).
[CrossRef]

O.-K. Lim and M. Saffman, “Intensity correlations and entanglement by frequency doubling in a two-port resonator,” Phys. Rev. A 74, 023816 (2006).
[CrossRef]

Schnabel, R.

N. B. Grosse, S. Assad, M. Mehmet, R. Schnabel, T. Symul, and P. K. Lam, “Observation of entanglement between two light beams spanning an octave in optical frequency,” Phys. Rev. Lett. 100, 243601 (2008).
[CrossRef] [PubMed]

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
[CrossRef] [PubMed]

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

Silberhorn, Ch.

Ch. Silberhorn, N. Korolkova, and G. Leuchs, “Quantum key distribution with bright entangled beams,” Phys. Rev. Lett. 88, 167902 (2002).
[CrossRef] [PubMed]

Simon, R.

R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef] [PubMed]

Sørensen, J. L.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Symul, T.

N. B. Grosse, S. Assad, M. Mehmet, R. Schnabel, T. Symul, and P. K. Lam, “Observation of entanglement between two light beams spanning an octave in optical frequency,” Phys. Rev. Lett. 100, 243601 (2008).
[CrossRef] [PubMed]

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

Treps, N.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

van Loock, P.

P. van Loock and S. L. Braunstein, “Telecloning of continuous quantum variables,” Phys. Rev. Lett. 87, 247901 (2001).
[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]

Vedral, V.

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[CrossRef]

Villar, A. S.

A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009).
[CrossRef] [PubMed]

A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006).
[CrossRef] [PubMed]

Werner, R. F.

R. F. Werner and M. M. Wolf, “Bound entangled Gaussian states,” Phys. Rev. Lett. 86, 3658–3661 (2001).
[CrossRef] [PubMed]

Wolf, M. M.

R. F. Werner and M. M. Wolf, “Bound entangled Gaussian states,” Phys. Rev. Lett. 86, 3658–3661 (2001).
[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]

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein–Podolsky–Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[CrossRef] [PubMed]

J. Zhang, C. Xie, and K. Peng, “Controlled dense coding for continuous variables using three-particle entangled states,” Phys. Rev. A 66, 032318 (2002).
[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, R.

S. Zhai, R. Yang, K. Liu, H. Zhang, J. Zhang, and J. Gao, “Bright two-color tripartite entanglement with second harmonic generation,” Opt. Express 17, 9851–9857 (2009).
[CrossRef] [PubMed]

S. Zhai, R. Yang, D. Fan, J. Guo, K. Liu, J. Zhang, and J. Gao, “Tripartite entanglement from the cavity with second-order harmonic generation,” Phys. Rev. A 78, 014302 (2008).
[CrossRef]

Zhai, S.

S. Zhai, R. Yang, K. Liu, H. Zhang, J. Zhang, and J. Gao, “Bright two-color tripartite entanglement with second harmonic generation,” Opt. Express 17, 9851–9857 (2009).
[CrossRef] [PubMed]

S. Zhai, R. Yang, D. Fan, J. Guo, K. Liu, J. Zhang, and J. Gao, “Tripartite entanglement from the cavity with second-order harmonic generation,” Phys. Rev. A 78, 014302 (2008).
[CrossRef]

Zhai, Z.

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[CrossRef]

Zhang, H.

Zhang, J.

S. Zhai, R. Yang, K. Liu, H. Zhang, J. Zhang, and J. Gao, “Bright two-color tripartite entanglement with second harmonic generation,” Opt. Express 17, 9851–9857 (2009).
[CrossRef] [PubMed]

S. Zhai, R. Yang, D. Fan, J. Guo, K. Liu, J. Zhang, and J. Gao, “Tripartite entanglement from the cavity with second-order harmonic generation,” Phys. Rev. A 78, 014302 (2008).
[CrossRef]

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[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]

J. Zhang, C. Xie, and K. Peng, “Controlled dense coding for continuous variables using three-particle entangled states,” Phys. Rev. A 66, 032318 (2002).
[CrossRef]

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein–Podolsky–Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[CrossRef] [PubMed]

Zhang, T. C.

T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl, and H. J. Kimble, “Quantum teleportation of light beams,” Phys. Rev. A 67, 033802 (2003).
[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]

Zou, H.

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[CrossRef]

Appl. Phys. B (1)

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables in nondegenerate parametric amplification,” Appl. Phys. B 55, 265–278 (1992).
[CrossRef]

EPL (1)

O.-K. Lim, B. Bol, and M. Saffman, “Observation of twin beam correlations and quadrature entanglement by frequency doubling in a two-port resonator,” EPL 78, 40004 (2007).
[CrossRef]

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

Opt. Express (1)

Phys. Rev. A (10)

S. Zhai, R. Yang, D. Fan, J. Guo, K. Liu, J. Zhang, and J. Gao, “Tripartite entanglement from the cavity with second-order harmonic generation,” Phys. Rev. A 78, 014302 (2008).
[CrossRef]

Z. Y. Ou, “Propagation of quantum fluctuations in single-pass second-harmonic generation for arbitrary interaction length,” Phys. Rev. A 49, 2106–2116 (1994).
[CrossRef] [PubMed]

R. D. Li and P. Kumar, “Quantum-noise reduction in traveling-wave second-harmonic generation,” Phys. Rev. A 49, 2157–2166 (1994).
[CrossRef] [PubMed]

M. K. Olsen, “Continuous-variable Einstein–Podolsky–Rosen paradox with traveling-wave second-harmonic generation,” Phys. Rev. A 70, 035801 (2004).
[CrossRef]

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[CrossRef]

J. Zhang, C. Xie, and K. Peng, “Controlled dense coding for continuous variables using three-particle entangled states,” Phys. Rev. A 66, 032318 (2002).
[CrossRef]

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[CrossRef]

O.-K. Lim and M. Saffman, “Intensity correlations and entanglement by frequency doubling in a two-port resonator,” Phys. Rev. A 74, 023816 (2006).
[CrossRef]

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl, and H. J. Kimble, “Quantum teleportation of light beams,” Phys. Rev. A 67, 033802 (2003).
[CrossRef]

Phys. Rev. Lett. (12)

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein–Podolsky–Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[CrossRef] [PubMed]

Ch. Silberhorn, N. Korolkova, and G. Leuchs, “Quantum key distribution with bright entangled beams,” Phys. Rev. Lett. 88, 167902 (2002).
[CrossRef] [PubMed]

G. M. D’Ariano, P. Lo Presti, and M. G. A. Paris, “Using entanglement improves the precision of quantum measurements,” Phys. Rev. Lett. 87, 270404 (2001).
[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]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef] [PubMed]

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
[CrossRef] [PubMed]

A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006).
[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]

P. van Loock and S. L. Braunstein, “Telecloning of continuous quantum variables,” Phys. Rev. Lett. 87, 247901 (2001).
[CrossRef] [PubMed]

R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef] [PubMed]

R. F. Werner and M. M. Wolf, “Bound entangled Gaussian states,” Phys. Rev. Lett. 86, 3658–3661 (2001).
[CrossRef] [PubMed]

N. B. Grosse, S. Assad, M. Mehmet, R. Schnabel, T. Symul, and P. K. Lam, “Observation of entanglement between two light beams spanning an octave in optical frequency,” Phys. Rev. Lett. 100, 243601 (2008).
[CrossRef] [PubMed]

Science (2)

A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science 326, 823–826 (2009).
[CrossRef] [PubMed]

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Dual-port cavity of single resonant SHG: Output1, one harmonic field; Output2, the other harmonic field and the reflect pump field.

Fig. 2
Fig. 2

Lines about symplectic eigenvalues are plotted as functions of P i n when Ω = 0 . (i) Transposition of the fundamental frequency field, (ii) transposition of SH1, (iii) transposition of SH2.

Fig. 3
Fig. 3

Lines about symplectic eigenvalues are plotted as functions of Ω when P i n = 1.5   W . (i) Transposition of the fundamental frequency field, (ii) transposition of SH1, (iii) transposition of SH2.

Equations (58)

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a ̂ i C ( t + τ ) = r 1 t 4 L N ( ζ 2 ) r 2 t 3 L N ( ζ 1 ) a ̂ i C ( t ) + t 1 b ̂ i 1 ( t ) + r 1 t 4 L N ( ζ 2 ) t 2 b ̂ i 2 ( t ) r 1 t 4 L N ( ζ 2 ) r 2 r 3 L b ̂ i 3 ( t ) r 1 r 4 L b ̂ i 4 ( t ) .
t j = diag ( T 1 j , T 2 j , T 1 j , T 2 j ) ,
r j = diag ( 1 T 1 j , 1 T 2 j , 1 T 1 j , 1 T 2 j ) .
t j L = diag ( 1 L 1 j , 1 L 2 j , 1 L 1 j , 1 L 2 j ) ,
r j L = diag ( L 1 j , L 2 j , L 1 j , L 2 j ) .
a ̂ i C ( ω ) e i ω τ = r 1 t 4 L N ( ζ 2 ) r 2 t 3 L N ( ζ 1 ) a ̂ i C ( ω ) + t 1 b ̂ i 1 ( ω ) + r 1 t 4 L N ( ζ 2 ) t 2 b ̂ i 2 ( ω ) r 1 t 4 L N ( ζ 2 ) r 2 r 3 L b ̂ i 3 ( ω ) r 1 r 4 L b ̂ i 4 ( ω ) .
x i C ( ω ) = a i c ( ω ) + a i c ( ω ) ,     y i C ( ω ) = i [ a i c ( ω ) a i c ( ω ) ] ,
u i j ( ω ) = b i j ( ω ) + b i j ( ω ) ,     v i j ( ω ) = i [ b i j ( ω ) b i j ( ω ) ] ,
X C = D r 1 t 4 L N ( ζ 2 ) r 2 t 3 L N ( ζ 1 ) X C + D [ t 1 v 1 + r 1 t 4 L N ( ζ 2 ) t 2 v 2 r 1 t 4 L N ( ζ 2 ) r 2 r 3 L v 3 r 1 r 4 L v 4 ] ,
X j = ( X 1 j , X 2 j , Y 1 j , Y 2 j ) T .
X 1 = t 2 t 3 L N ( ζ 1 ) X C r 2 v 2 t 2 r 3 L v 3 ,
X 2 = t 1 X C / D r 1 v 1 / r 1 .
X 21 ( ω ) = f 11 u 11 ( ω ) + f 13 u 13 ( ω ) + f 14 u 14 ( ω ) + f 21 u 21 ( ω ) + f 22 u 22 ( ω ) ,
Y 21 ( ω ) = g 11 v 11 ( ω ) + g 13 v 13 ( ω ) + g 14 v 14 ( ω ) + g 21 v 21 ( ω ) + g 22 v 22 ( ω ) ,
X 22 ( ω ) = h 11 u 11 ( ω ) + h 13 u 13 ( ω ) + h 14 u 14 ( ω ) + h 21 u 21 ( ω ) + h 22 u 22 ( ω ) ,
Y 22 ( ω ) = k 11 v 11 ( ω ) + k 13 v 13 ( ω ) + k 14 v 14 ( ω ) + k 21 v 21 ( ω ) + k 22 v 22 ( ω ) .
X 12 ( ω ) = m 11 u 11 ( ω ) + m 13 u 13 ( ω ) + m 14 u 14 ( ω ) + m 21 u 21 ( ω ) + m 22 u 22 ( ω ) ,
Y 12 ( ω ) = n 11 v 11 ( ω ) + n 13 v 13 ( ω ) + n 14 v 14 ( ω ) + n 21 v 21 ( ω ) + n 22 v 22 ( ω ) .
m 11 = e i Ω / υ c 1 1 L 13 1 L 14 1 T 11 F ,
m 13 = L 13 1 L 14 N 11 ( ζ 2 ) T 11 F ,
m 14 = L 14 T 11 F ,
m 21 = e i w / v c 2 1 L 13 1 L 14 N 11 ( ζ 2 ) N 12 ( ζ 1 ) T 11 F ,
m 22 = 1 L N 12 ( ζ 2 ) T 11 F ,
n 11 = e i w / v c 1 1 L 13 1 L 14 N 33 ( ζ 2 ) N 33 ( ζ 1 ) 1 T 11 G ,
n 13 = L 13 1 L 14 N 33 ( ζ 2 ) T 11 G ,
n 14 = L 14 T 11 G ,
n 21 = e i w / v c 2 1 L 13 1 L 14 N 33 ( ζ 2 ) N 34 ( ζ 1 ) T 11 G ,
n 22 = 1 L 14 N 34 ( ζ 2 ) T 11 G .
F = 1 e i Ω / υ c 1 1 T 11 1 L 13 1 L 14 N 11 ( ζ 1 ) N 11 ( ζ 2 ) ,
G = 1 e i Ω / υ c 1 1 T 11 1 L 13 1 L 14 N 33 ( ζ 1 ) N 33 ( ζ 2 ) .
σ = ( c 1212 x 0 c 1221 x 0 c 1222 x 0 0 c 1212 y 0 c 1221 y 0 c 1222 y c 2112 x 0 c 2121 x 0 c 2122 x 0 0 c 2112 y 0 c 2121 y 0 c 2122 y c 2212 x 0 c 2221 x 0 c 2222 x 0 0 c 2212 y 0 c 2221 y 0 c 2222 y ) .
S = I 1 ( 1 2 0 1 2 0 0 1 2 0 1 2 1 2 0 1 2 0 0 1 2 0 1 2 ) ,
E = min { 1 2 ( c 2121 x + c 2222 x ± ( c 2121 x ) 2 + 4 c 2122 x c 2221 x 2 c 2121 x c 2222 x + ( c 2222 x ) 2 ) , 1 2 ( c 2121 y + c 2222 y ± ( c 2121 y ) 2 + 4 c 2122 y c 2221 y 2 c 2121 y c 2222 y + ( c 2222 y ) 2 ) } .
N 11 ( ζ ) = 1 ζ   tanh   ζ cosh   ζ ,     N 12 ( ζ ) = 2   tanh   ζ cosh   ζ ,
N 21 ( ζ ) = 1 2 ( tanh   ζ + ζ sech 2 ζ ) ,     N 22 ( ζ ) = sech 2 ζ ,
N 33 ( ζ ) = sech   ζ ,     N 34 ( ζ ) = 1 2 ( sinh   ζ + ζ   sech   ζ ) ,
N 43 ( ζ ) = 2   tanh   ζ ,     N 44 ( ζ ) = 1 ζ   tanh   ζ ,
ζ 1 = n 1 n 2 ε 1 E NL 1 p i n ,     ζ 2 = n 1 n 2 ε 2 E NL 2 p i n .
f 11 = e i Ω / ν c 1 T 11 N 21 ( ζ 1 ) F ,
f 13 = e i Ω / ν c 1 1 T 11 L 13 1 L 14 N 21 ( ζ 1 ) N 11 ( ζ 2 ) F ,
f 14 = e i Ω / ν c 1 1 T 11 L 14 N 21 ( ζ 1 ) F ,
f 21 = N 22 ( ζ 1 ) e i Ω / ν c 1 1 T 11 1 L 13 1 L 14 N 11 c 2 F ,
f 22 = e i Ω / ν c 1 1 T 11 1 L 14 N 21 ( ζ 1 ) N 12 ( ζ 2 ) F ,
g 11 = e i Ω / ν c 1 T 11 N 43 ( ζ 1 ) G ,
g 13 = e i Ω / ν c 1 1 T 11 L 13 1 L 14 N 43 ( ζ 1 ) N 33 ( ζ 2 ) G ,
g 14 = e i Ω / ν c 1 1 T 11 L 14 N 43 ( ζ 1 ) G ,
g 21 = N 44 ( ζ 1 ) e i Ω / ν c 1 1 T 11 1 L 13 1 L 14 N 33 c 2 G ,
g 22 = e i Ω / ν c 1 1 T 11 1 L 14 N 43 ( ζ 1 ) N 34 ( ζ 2 ) G ,
h 11 = e i Ω / ν c 1 T 11 1 L 13 N 11 ( ζ 1 ) N 21 ( ζ 2 ) F ,
h 13 = L 13 N 21 ( ζ 2 ) F ,
h 14 = e i Ω / ν c 1 1 T 11 1 L 13 L 14 N 11 ( ζ 1 ) N 21 ( ζ 2 ) F ,
h 21 = 1 L 13 N 12 ( ζ 1 ) N 21 ( ζ 2 ) F ,
h 22 = N 22 ( ζ 2 ) e i Ω / ν c 1 1 T 11 1 L 13 1 L 14 N 11 c 1 F ,
k 11 = e i Ω / ν c 1 T 11 1 L 13 N 33 ( ζ 1 ) N 43 ( ζ 2 ) G ,
k 13 = L 11 N 43 ( ζ 2 ) G ,
k 14 = e i Ω / ν c 1 1 T 11 1 L 13 L 14 N 33 ( ζ 1 ) N 43 ( ζ 2 ) G ,
k 21 = 1 L 13 N 34 ( ζ 1 ) N 43 ( ζ 2 ) G ,
k 22 = N 44 ( ζ 2 ) e i Ω / ν c 1 1 T 11 1 L 13 1 L 14 N 33 c 1 G ,

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