Abstract

Bright three-color continuous-variable (CV) entanglement generated by a cascaded sum-frequency process in an optical cavity is investigated. The second- and fourth-harmonic beams can be generated by two quasi-phase-matched cascaded sum-frequency processes in a quasi-periodic optical superlattice. Quantum correlations among the fundamental and second- and fourth-harmonic beams are calculated and discussed by applying a sufficient and necessary inseparability criterion for CV three-mode entanglement. Strong three-color CV entanglement beams with a double frequency interval can be produced in this simple system. It is experimentally feasible and may be very useful for applications in quantum communication and computation networks.

© 2011 Optical Society of America

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    [CrossRef]

2011 (1)

Y. B. Yu, J. T. Sheng, and M. Xiao, “Generation of bright quadricolor continuous-variable entanglement by four-wave-mixing process,” Phys. Rev. A 83, 012321 (2011).
[CrossRef]

2009 (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]

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]

2008 (4)

C. J. McKinstrie, S. J. van Enk, M. G. Raymer, and S. Radic, “Multicolor multipartite entanglement produced by vector four-wave mixing in a fiber,” Opt. Express 16, 2720–2739(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]

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

2007 (2)

C. Pennarun, A. S. Bradley, and M. K. Olsen, “Tripartite entanglement and threshold properties of coupled intracavity down-conversion and sum-frequency generation,” Phys. Rev. A 76, 063812 (2007).
[CrossRef]

K. N. Cassemiro, A. S. Villar, P. Valente, M. Martinelli, and P. Nussenzveig, “Experimental observation of three-color optical quantum correlations,” Opt. Lett. 32, 695–697 (2007).
[CrossRef] [PubMed]

2006 (3)

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]

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]

Y. B. Yu, Z. D. Xie, X. Q. Yu, H. X. Li, P. Xu, H. M. Yao, and S. N. Zhu, “Generation of three-mode continuous-variable entanglement by cascaded nonlinear interactions in a quasiperiodic superlattice,” Phys. Rev. A 74, 042332 (2006).
[CrossRef]

2005 (5)

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

J. Zhang, C. D. Xie, and K. C. Peng, “Continuous-variable quantum state transfer with partially disembodied transport,” Phys. Rev. Lett. 95, 170501 (2005).
[CrossRef] [PubMed]

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]

2004 (3)

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

X. J. Jia, X. L. Su, Q. Pan, J. R. Gao, C. D. Xie, and K. C. Peng, “Experimental demonstration of unconditional entanglement swapping for continuous variables,” Phys. Rev. Lett. 93, 250503(2004).
[CrossRef]

A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni, “Three-mode entanglement by interlinked nonlinear interactions in optical x(2) media,” J. Opt. Soc. Am. B 21, 1241–1249 (2004).
[CrossRef]

2003 (1)

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

2002 (1)

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

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 (3)

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

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (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 (2)

R. E. S. Polkinghorne and T. C. Ralph, “Continuous variable entanglement swapping,” Phys. Rev. Lett. 83, 2095–2099(1999).
[CrossRef]

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787(1999).
[CrossRef]

1998 (5)

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]

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[CrossRef]

S. L. Braunstein, “Quantum error correction for communication with linear optics,” Nature 394, 47–49 (1998).
[CrossRef]

S. Lloyd and J. E. Slotine, “Analog quantum error correction,” Phys. Rev. Lett. 80, 4088–4091 (1998).
[CrossRef]

Y. Q. Qin, Y. Y. Zhu, S. N. Zhu, and N. B. Ming, “Quasi-phase-matched harmonic generation through coupled parametric processes in a quasi-periodic optical superlattice,” J. Appl. Phys. 84, 6911–6916 (1998).
[CrossRef]

1997 (2)

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]

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]

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]

1990 (1)

P. D. Drummond and M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
[CrossRef] [PubMed]

1989 (1)

M. D. Reid and P. D. Drummond, “Correlations in nondegenerate parametric oscillation: squeezing in the presence of phase diffusion,” Phys. Rev. A 40, 4493–4506 (1989).
[CrossRef] [PubMed]

1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[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]

Allevi, A.

Andreoni, A.

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]

Bondani, M.

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]

Bradley, A. S.

C. Pennarun, A. S. Bradley, and M. K. Olsen, “Tripartite entanglement and threshold properties of coupled intracavity down-conversion and sum-frequency generation,” Phys. Rev. A 76, 063812 (2007).
[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]

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787(1999).
[CrossRef]

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[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]

S. L. Braunstein, “Quantum error correction for communication with linear optics,” Nature 394, 47–49 (1998).
[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]

K. N. Cassemiro, A. S. Villar, P. Valente, M. Martinelli, and P. Nussenzveig, “Experimental observation of three-color optical quantum correlations,” Opt. Lett. 32, 695–697 (2007).
[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]

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef] [PubMed]

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.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef] [PubMed]

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.

P. D. Drummond and M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
[CrossRef] [PubMed]

M. D. Reid and P. D. Drummond, “Correlations in nondegenerate parametric oscillation: squeezing in the presence of phase diffusion,” Phys. Rev. A 40, 4493–4506 (1989).
[CrossRef] [PubMed]

Duan, L. M.

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[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]

Ferraro, A.

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. R.

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]

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]

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

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Figures (6)

Fig. 1
Fig. 1

Sketch of the experiment setup. λ / 4 , quarter-wave plate; QPOS, quasi-periodic optical superlattice.

Fig. 2
Fig. 2

(a) Schematic diagram of a QPLT with a Fibonacci sequence. (b) Schematic diagram of the two QPM processes.

Fig. 3
Fig. 3

Real parts of the eigenvalues of the drift matrix A (RPEA) versus (a) ω, (b)  κ 1 / κ 0 , and (c)  γ / γ 0 .

Fig. 4
Fig. 4

Symplectric eigenvalues E i versus the analysis frequency ω with the identical parameters in Fig. 3a.

Fig. 5
Fig. 5

Symplectric eigenvalues E i versus the nonlinear coupling parameters κ 1 / κ 0 with the identical parameters in Fig. 3b.

Fig. 6
Fig. 6

Symplectric eigenvalues E i versus the damping rates γ / γ 0 with the identical parameters in Fig. 3c.

Equations (16)

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H ^ I = i κ 0 a ^ 0 2 a ^ 1 + i κ 1 a ^ 1 2 a ^ 2 + h.c. ,
H ^ P = i ( ϵ a ^ 0 ϵ * 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 ( ϵ 2 κ 0 α 0 * α 1 γ 0 α 0 ) + α 0 * ( ϵ * 2 κ 0 α 0 α 1 * γ 0 α 0 * ) + α 1 ( κ 0 α 0 2 2 κ 1 α 1 * α 2 γ 1 α 1 ) + α 1 * ( κ 0 α 0 * 2 2 κ 1 α 1 α 2 * γ 1 α 1 * ) + α 2 ( κ 1 α 1 2 γ 2 α 2 ) + α 2 * ( κ 1 α 1 * 2 γ 2 α 2 * ) ] + 1 2 [ 2 2 α 0 ( 2 κ 0 α 1 ) + 2 2 α 0 * ( 2 κ 0 α 1 * ) + 2 2 α 1 ( 2 κ 1 α 2 ) + 2 2 α 1 * ( 2 κ 1 α 2 * ) ] } P ,
d α 0 d t = ϵ 2 κ 0 α 0 α 1 γ 0 α 0 + κ 0 α 1 / 2 ( η 1 + i η 2 ) , d α 0 d t = ϵ * 2 κ 0 α 0 α 1 γ 0 α 0 + κ 0 α 1 / 2 ( η 1 i η 2 ) , d α 1 d t = κ 0 α 0 2 2 κ 1 α 1 α 2 γ 1 α 1 + κ 1 α 2 / 2 ( η 3 + i η 4 ) , d α 1 d t = κ 0 α 0 2 2 κ 1 α 1 α 2 γ 1 α 1 + κ 1 α 2 / 2 ( η 3 i η 4 ) , d α 2 d t = κ 1 α 1 2 γ 2 α 2 , d α 2 d t = κ 1 α 1 2 γ 2 α 2 ,
η j ( t ) = 0 , η j ( t ) η k ( t ) = δ j k δ ( t t ) .
Δ 0 2 Δ 1 Δ 2 = 0 ,
A 0 = 1 Δ 0 , A 2 = κ 1 γ 2 A 1 2 ,
d δ α ˜ = A δ α ˜ d t + B d W ,
A = ( γ 0 2 κ 0 A 1 2 κ 0 A 0 * 0 0 0 2 κ 0 A 1 * γ 0 0 2 κ 0 A 0 0 0 2 κ 0 A 0 0 γ 1 2 κ 1 A 2 2 κ 1 A 1 * 0 0 2 κ 0 A 0 * 2 κ 1 A 2 * γ 1 0 2 κ 1 A 1 0 0 2 κ 1 A 1 0 γ 2 0 0 0 0 2 κ 1 A 1 * 0 γ 2 ) .
S ( ω ) = ( A + i ω I ) 1 B B T ( A T i ω I ) 1 ,
S X i out ( ω ) = 1 + 2 γ i S X i ( ω ) , S X i , X j out ( ω ) = 2 γ i γ j S X i , X j ( ω ) ,
σ + i Ω 0 ,
Ω ( 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 ) .
ν i 1 ,
E A 1.

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