Abstract

A new scheme is presented to directly produce fundamental, second-, and third-harmonic three-color continuous-variable (CV) entangled beams by cascaded quasi-phase-matched third-harmonic generation (THG) in an optical cavity. THG can be achieved with high efficiency through a coupled sum-frequency process between the second-harmonic and the fundamental fields. It is demonstrated that the three beams (fundamental, second-, and third-harmonic fields) are entangled with each other according to the CV entanglement criterion. In this scheme, only one crystal and one pump field can generate three-color CV entangled beams separated by an octave in frequency through quasi-phase-matched cascaded nonlinear process, which may be very useful for the applications in quantum communication and computation networks.

© 2011 OSA

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    [CrossRef] [PubMed]
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2010 (1)

2009 (2)

S. Q. Zhai, R. G. Yang, K. Liu, H. L. Zhang, J. X. Zhang, and J. R. 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 (3)

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[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]

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

N. B. Grosse, W. P. Bowen, K. McKenzie, and P. K. Lam, “Harmonic entanglement with second-order nonlinearity,” Phys. Rev. Lett. 96, 063601 (2006).
[CrossRef] [PubMed]

2005 (3)

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

A. Serafini, G. Adesso, and F. Illuminati, “Unitarily localizable entanglement of Gaussian states,” Phys. Rev. A 71, 032349 (2005).
[CrossRef]

A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
[CrossRef] [PubMed]

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 (1)

P. Lodahl, “Einstein-Podolsky-Rosen correlations in second-harmonic generation,” Phys. Rev. A 68, 023806 (2003).
[CrossRef]

2001 (2)

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

G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, “Separability properties of three-mode Gaussian states,” Phys. Rev. A 64, 052303 (2001).
[CrossRef]

2000 (1)

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

1997 (2)

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

1996 (2)

A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413–1415 (1996).
[CrossRef] [PubMed]

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1–8 (1996).
[CrossRef]

1994 (1)

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

1992 (1)

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]

1991 (1)

C. W. Gardiner, Quantum Noise (Springer, 1991).

1988 (1)

M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
[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]

1962 (1)

J. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Adesso, G.

A. Serafini, G. Adesso, and F. Illuminati, “Unitarily localizable entanglement of Gaussian states,” Phys. Rev. A 71, 032349 (2005).
[CrossRef]

Armstrong, J.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[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]

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]

Bloembergen, N.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bowen, W. P.

N. B. Grosse, W. P. Bowen, K. McKenzie, and P. K. Lam, “Harmonic entanglement with second-order nonlinearity,” Phys. Rev. Lett. 96, 063601 (2006).
[CrossRef] [PubMed]

Braunstein, S. L.

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

A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
[CrossRef] [PubMed]

Cirac, J. I.

G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, “Separability properties of three-mode Gaussian states,” Phys. Rev. A 64, 052303 (2001).
[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]

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]

Cruz, L. S.

A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
[CrossRef] [PubMed]

Drummond, P. D.

M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
[CrossRef] [PubMed]

Ducuing, J.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

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. H.

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

Gao, J. R.

Gardiner, C. W.

C. W. Gardiner, Quantum Noise (Springer, 1991).

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]

Ge, C. Z.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Giedke, G.

G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, “Separability properties of three-mode Gaussian states,” Phys. Rev. A 64, 052303 (2001).
[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]

N. B. Grosse, W. P. Bowen, K. McKenzie, and P. K. Lam, “Harmonic entanglement with second-order nonlinearity,” Phys. Rev. Lett. 96, 063601 (2006).
[CrossRef] [PubMed]

Guo, J.

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

Horodecki, M.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1–8 (1996).
[CrossRef]

Horodecki, P.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1–8 (1996).
[CrossRef]

Horodecki, R.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1–8 (1996).
[CrossRef]

Illuminati, F.

A. Serafini, G. Adesso, and F. Illuminati, “Unitarily localizable entanglement of Gaussian states,” Phys. Rev. A 71, 032349 (2005).
[CrossRef]

Kimble, H. J.

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[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]

Kraus, B.

G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, “Separability properties of three-mode Gaussian states,” Phys. Rev. A 64, 052303 (2001).
[CrossRef]

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]

N. B. Grosse, W. P. Bowen, K. McKenzie, and P. K. Lam, “Harmonic entanglement with second-order nonlinearity,” Phys. Rev. Lett. 96, 063601 (2006).
[CrossRef] [PubMed]

Lewenstein, M.

G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, “Separability properties of three-mode Gaussian states,” Phys. Rev. A 64, 052303 (2001).
[CrossRef]

Liu, K.

Lodahl, P.

P. Lodahl, “Einstein-Podolsky-Rosen correlations in second-harmonic generation,” Phys. Rev. A 68, 023806 (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]

A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
[CrossRef] [PubMed]

McKenzie, K.

N. B. Grosse, W. P. Bowen, K. McKenzie, and P. K. Lam, “Harmonic entanglement with second-order nonlinearity,” Phys. Rev. Lett. 96, 063601 (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]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

Ming, N. B.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[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]

A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
[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, 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]

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]

Peres, A.

A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413–1415 (1996).
[CrossRef] [PubMed]

Pershan, P. S.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Qin, Y. Q.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Reid, M. D.

M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
[CrossRef] [PubMed]

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]

Serafini, A.

A. Serafini, G. Adesso, and F. Illuminati, “Unitarily localizable entanglement of Gaussian states,” Phys. Rev. A 71, 032349 (2005).
[CrossRef]

Simon, R.

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

van Loock, P.

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

A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
[CrossRef] [PubMed]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

Wang, H. F.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

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]

Yang, R. G.

Zhai, S. Q.

Zhang, H. L.

Zhang, J. X.

Zhu, S. N.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Zhu, Y. Y.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a)Sketch of the optical cavity. (b)The sketched momentum geometry for coupled quasi-phase-matching processes.

Fig. 2
Fig. 2

Real parts of the eigenvalues (RPEA) versus (a)σ, (b)κ 1/κ 0, (c)γ/γ 0, and (d)γ 2/γ 1, respectively.

Fig. 3
Fig. 3

The symplectic eigenvalues versus (a)ω, (b)σ, (c)κ 1/κ 0, (d)γ/γ 0, and (e)γ 2/γ 1, respectively.

Equations (13)

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H I = i h ¯ κ 0 a 0 2 a 1 + i h ¯ κ 1 a 0 a 1 a 2 + h . c . ,
H pump = i h ¯ ( ɛ a 0 ɛ * a 0 ) ,
Λ i ρ = γ i ( 2 a i ρ a i a i a i ρ ρ a i a i ) ,
d p d t = { α 0 [ ɛ 2 κ 0 α 0 * α 1 κ 1 α 1 * α 2 γ 0 α 0 ] α 0 * [ ɛ * 2 κ 0 α 0 α 1 * κ 1 α 1 α 2 * γ 0 α 0 * ] α 1 [ κ 0 α 0 2 κ 1 α 0 * α 2 γ 1 α 1 ] α 1 * [ κ 0 α 0 * 2 κ 1 α 0 α 2 * γ 1 α 1 * ] α 2 [ κ 1 α 0 α 1 γ 2 α 2 ] α 2 * [ κ 1 α 0 * α 1 * γ 2 α 2 * ] + 1 2 2 α 0 [ 2 κ 0 α 1 ] + 1 2 2 α 0 * [ 2 κ 0 α 1 * ] + 1 2 α 0 α 1 [ 2 κ 1 α 2 ] + 1 2 α 0 * α 1 * [ 2 κ 1 α 2 * ] } P ,
α 0 / t = ɛ 2 κ 0 α 0 α 1 κ 1 α 1 α 2 γ 0 α 0 + κ 0 α 1 η 1 + κ 1 α 2 η 2 α 0 / t = ɛ * 2 κ 0 α 0 α 1 κ 1 α 1 α 2 γ 0 α 0 + κ 0 α 1 η 1 + κ 1 α 2 η 3 α 1 / t = κ 0 α 0 2 κ 1 α 0 α 2 γ 1 α 1 + κ 1 α 2 η 2 α 1 / t = κ 0 α 0 2 κ 1 α 0 α 2 γ 1 α 1 + κ 1 α 2 η 3 α 2 / t = κ 1 α 0 α 1 γ 2 α 2 , α 2 / t = κ 1 α 0 α 1 γ 2 α 2 ,
Δ 0 Δ 1 2 + κ 0 2 γ 2 A 0 3 ( 3 Δ 1 γ 1 γ 2 ) = 0 ,
A 1 = A 0 [ κ 0 2 γ 2 A 0 ( 2 γ 1 γ 2 3 Δ 1 ) κ 1 2 Δ 0 Δ 1 ] / κ 0 γ 1 2 γ 2 2 , A 2 = κ 1 A 0 A 1 / γ 2 ,
d δ α ˜ = A δ α ˜ d t + B d W ,
S ( ω ) = ( A + i ω I ) 1 BB T ( A T i ω I ) 1 ,
σ + i Ω 0 ,
Ω ( J 0 0 0 J 0 0 0 J ) , J = ( 0 1 1 0 ) .
ν i 1 ,
E A < 1.

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