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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 15 — Jul. 18, 2011
  • pp: 13949–13956
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Directly produced three-color entanglement by quasi-phase-matched third-harmonic generation

Y. B. Yu, H. J. Wang, M. Xiao, and S. N. Zhu  »View Author Affiliations


Optics Express, Vol. 19, Issue 15, pp. 13949-13956 (2011)
http://dx.doi.org/10.1364/OE.19.013949


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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

1. Introduction

Quantum entanglement attracts many interests in recent years since it is the central resource in the applications such as quantum communication and computation. Multipartite continuous-variable (CV) entangled beams with different frequencies are necessary to connect different physical systems at the nodes of quantum networks [1

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

], which can facilitate many quantum information protocols of interspecies quantum teleportation. Two-color CV entanglement produced by nondegenerate optical parametric oscillator (OPO) was predicted [2

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

], and experimentally demonstrated both below [3

3. 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]

] and above [4

4. 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]

] the oscillation threshold. Then, it was predicated that three-color CV entanglement among pump, signal and idler beams can be obtained using an OPO operating above the threshold [5

5. 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]

], which also has been realized by the recent experiment [6

6. 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]

]. Besides the above schemes, up-conversion, such as second-harmonic generation (SHG), has been suggested as a source of CV entanglement both in an optical cavity [7

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

] and without optical cavity [8

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

]. It was predicted that the perfect entanglement could be produced between the fundamental and the second-harmonic fields by the second-order nonlinear interaction [9

9. 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]

], which has been experimentally verified recently [10

10. 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]

]. Other suggestions were proposed to generate two-color tripartite entanglement among the fundamental and the second-harmonic fields through the type-II SHG system operating below [11

11. 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]

] and above [12

12. 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]

] the threshold. In addition, multicolored tripartite CV entanglement generated in a two-port resonator of SHG process was also investigated by applying a necessary and sufficient entanglement criterion [13

13. R. G. Yang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generation of multicolored tripartite entanglement by frequency doubling in a two-port resonator,” J. Opt. Soc. Am. B 27, 2721–2726 (2010). [CrossRef]

].

Fig. 1 (a)Sketch of the optical cavity. (b)The sketched momentum geometry for coupled quasi-phase-matching processes.

2. The stationary solutions and output fields

The interaction Hamiltonian for this quasi-phase-matched THG can be written as
HI=ih¯κ0a02a1+ih¯κ1a0a1a2+h.c.,
(1)
where κ 0 and κ 1 are the dimensionless nonlinear coupling coefficients, which are taken to be real for simplicity.

The cavity pumping is
Hpump=ih¯(ɛa0ɛ*a0),
(2)
where ɛ is the classical pumping laser amplitude and taken as a real field here for ɛ = ɛ* = E 0.

The losses of the three modes are given by
Λiρ=γi(2aiρaiaiaiρρaiai),
(3)
where γi (i = 0, 1, 2) stand for the damping rates of the three cavity fields.

We proceed by mapping the master equation onto a set of stochastic differential equations by the Fokker-Planck equation in the positive-P representation [17

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

]. We find the Fokker-Planck equation for the P function of this system as
dpdt={α0[ɛ2κ0α0*α1κ1α1*α2γ0α0]α0*[ɛ*2κ0α0α1*κ1α1α2*γ0α0*]α1[κ0α02κ1α0*α2γ1α1]α1*[κ0α0*2κ1α0α2*γ1α1*]α2[κ1α0α1γ2α2]α2*[κ1α0*α1*γ2α2*]+122α0[2κ0α1]+122α0*[2κ0α1*]+12α0α1[2κ1α2]+12α0*α1*[2κ1α2*]}P,
(4)
where αi (i = 0, 1, 2) correspond to ai in the P representation, respectively. The above Fokker-Planck equation does not possess a positive-definite diffusion matrix. So we must double the phase space and use the positive-P representation to find the appropriate stochastic differential equations. α* will be substituted by α in the stochastic differential equations. However, it should be noted that α and α are independent complex variables in the positive-P representation [17

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

]. Following the standard procedure [17

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

], the stochastic differential equations for the cavity fields can be written as
α0/t=ɛ2κ0α0α1κ1α1α2γ0α0+κ0α1η1+κ1α2η2α0/t=ɛ*2κ0α0α1κ1α1α2γ0α0+κ0α1η1+κ1α2η3α1/t=κ0α02κ1α0α2γ1α1+κ1α2η2α1/t=κ0α02κ1α0α2γ1α1+κ1α2η3α2/t=κ1α0α1γ2α2,α2/t=κ1α0α1γ2α2,
(5)
ηi (i = 1, 2, 3) are the complex noise terms, which satisfy the relations 〈ηi(t)〉 = 0 and ηi(t)ηj(t)=ηi(t)ηj(t)=δijδ(tt).

In the following, one can decompose the system variables into their steady-state values and small fluctuations around the steady-state values as αi = Ai + δαi and use this linearization procedure to rewrite the Eq. (5) as [17

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

, 18

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

]
dδα˜=Aδα˜dt+BdW,
(8)
where δα˜=[δα0,δα0,δα1,δα1,δα2,δα2]T; A is the drift matrix with the steady-state values; B contains the steady-state coefficients of the noise terms of Eq. (5); dW is a vector of Wiener increment [17

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

].

Then, one can obtain the intracavity spectral matrix in the frequency domain as [17

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

, 18

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

]
S(ω)=(A+iωI)1BBT(ATiωI)1,
(9)
where I is the identity matrix and ω is the analysis frequency. The output fields can be obtained by applying the well-known input-output relations [19

19. 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]

].

However, the condition for the validity of the linearization process is that the eigenvalues of the drift matrix A have no negative real parts [18

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

]. In Fig. 2 we plot the 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, where γ = γ 1 = γ 2 for simplicity;σ = E 0/ɛth and ɛth = γ 0 γ 1/κ 0 is the well-known threshold of OPO which used here as a terminology to investigate entanglement for different values of E 0.

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.

3. Discussions on the entanglement characteristic among three beams

There are two nonlinear sum-frequency processes in quasi-phase-matched THG. In the first sum-frequency process, when two fundamental photons are destroyed, one second-harmonic photon is created. So the intensity of the fundamental field is anticorrelated to that of the second-harmonic field. Similarly, in the second cascaded sum-frequency process, when one second-harmonic photon and one fundamental photon are destroyed, one third-harmonic photon is created. So the intensity of the third-harmonic field is anticorrelated to that of the second-harmonic field, as well as that of the fundamental field. Finally, the three fields are entangled with each other and bright three-color entanglement can be obtained in this scheme.

A necessary and sufficient criterion for bipartite Gaussian states CV entanglement was proposed [20

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

] based on the positivity of the partial transpose (PPT) [21

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

, 22

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

]. Then, necessary and sufficient criteria for 1×N bipartite multi-mode Gaussian states entanglement [23

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

] and tripartite three-mode Gaussian states [24

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

] were also developed,respectively. According to these criteria, the correlation matrix (CM) can be defined as [25

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

] σkk =< μ k μ k + μ k μk > /2, where μ = (X 0, Y 0, X 1, Y 1, X 2, Y 2) and X^i=(αiout+αiout) (i=0, 1, 2) represent the amplitude quadratures of the fields; Y^i=i(αioutαiout) stand for their phase quadratures. The Heisenberg uncertainty principle is [20

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

]
σ+iΩ0,
(10)
where
Ω(J000J000J),J=(0110).
(11)

Above uncertainty relation (10) can be replaced by [26

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

]
νi1,
(12)
where νi is the symplectic eigenvalues. We denote σ˜A as the partial transpose on the CM σ for subsystem A and EA is the symplectic eigenvalues of the matrix |iΩσ˜|. Then the subsystems A and B will be inseparable (i. e. entangled with each other) when [26

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

]
EA<1.
(13)

According to the criterion for tripartite entanglement [24

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

], the three fields are entangled when all conditions for two parties entanglement are satisfied. In present three-mode system, we denote Ei (i = 0, 1, 2) as the symplectric eigenvalues of the bipartite system between the field ai and the subsystem including the remaining fields (1×2 bipartition). When Ei < 1 are all satisfied simultaneously, the three fields are entangled with each other. In the following, we will employ this necessary and sufficient CV entanglement criterion to discuss the quantum entanglement features among the three fields. Smaller of the symplectric eigenvalue is, larger of the degree of entanglement will be. Figure 3(a) depicts the symplectic eigenvalues Ei versus analysis frequency ω with γ 0 = 0.01, γ = 1.5γ 0, κ 0 = 0.1, κ 1 = 1.5κ 0, and σ = 2. One can see in Fig. 3(a) that the three values of Ei are all below 1 simultaneously in a wide range of the analysis frequency which clearly indicates that the fundamental, second-, and third-harmonic fields are CV entangled with each other.

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.

The conversion efficiency of sum-frequency process is much higher than that of the parametric process and the sum-frequency generation does not have threshold. In the single pass experiment in Ref. [16

16. 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]

], the conversion efficiency of THG is about 23%. If we put this setup into a cavity, the efficiency will be farther increased. So the threshold for the cavity is very smaller than that of OPO. Bright three-color entanglement can be easy obtained in this scheme. Equation (6) is a fifth-order equation, it is well known, which has no analytic solution. Therefore, we can not obtain the threshold for the cavity. However, we will continue to use the threshold of OPO as a terminology to investigate entanglement among the three fields for different values of E 0. In Fig. 3(b) the symplectic eigenvalues Ei are presented versus σ with γ 0 = 0.01, γ = 1.5γ 0, κ 0 = 0.1, κ 1 = 1.5κ 0, and ω = γ 0. Form Fig. 3(b) one can see that the three eigen-values are below 1 which indicates the three fields are entangled with each other below and above the threshold. From Fig. 2 one can see that the linearization is valid and the three fields are entangled with each other in this range.

Figure 3(d) depicts the symplectic eigenvalues versus the damping rates γ/γ 0 with γ 0 = 0.01, κ 0 = 0.1, κ 1 = 1.5κ 0, σ = 2, and ω = γ 0. One can see that the three fields are entangled with each other and better three-color entanglement can be obtained at about γ/γ 0 = 3.1. Actually, better squeezed state of the fundamental field can be obtained when the damping rate γ 0 is smaller [18

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

].

In the above calculations, we set γ = γ 1 = γ 2 for simplicity. In Fig. 3(e) we plot the symplectic eigenvalues versus different values of γ 1 and γ 2 with γ 0 = 0.01, γ 1 = 1.5γ 0, κ 0 = 0.1, κ 1 = 1.5κ 0, σ = 2, and ω = γ 0. One can see that the three fields are also entangled for different values of γ 1 and γ 2 and better three-color entanglement can be obtained at about γ 2 = 1.6γ 1.

4. Conclusions

We have presented a new scheme to directly produce fundamental, second-, and third-harmonic entangled beams by quasi-phase-matched THG. Three-color CV entanglement among the fundamental, second-, and third-harmonic beams are demonstrated by applying a necessary and sufficient tripartite CV entanglement criterion. In this scheme, only one crystal and one pump field can generate bright three-color CV entangled beams separated by an octave in frequency, which is very different from the scheme reported in Ref. [6

6. 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]

] and may be very useful for the applications in quantum communication and computation networks.

Acknowledgments

This work is supported by the National Natural Science Foundations of China (No. 10804059), the Sate Key Program for Basic Research of China (Nos. 2006CB921804 and 2011CBA00205), the Zhejiang Provincial Natural Science Foundation (No. Y6090488), the Ningbo Natural Science Foundation (Nos. 2008A610004 and 2008A610006), the K. C. Wong Education Foundation, Hong Kong and the National Science Foundation (USA).

References and links

1.

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

2.

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

3.

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]

4.

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]

5.

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]

6.

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]

7.

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

8.

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

9.

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]

10.

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]

11.

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]

12.

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]

13.

R. G. Yang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generation of multicolored tripartite entanglement by frequency doubling in a two-port resonator,” J. Opt. Soc. Am. B 27, 2721–2726 (2010). [CrossRef]

14.

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]

15.

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]

16.

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]

17.

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

18.

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

19.

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]

20.

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

21.

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

22.

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

23.

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

24.

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

25.

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

26.

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

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(270.0270) Quantum optics : Quantum optics
(270.6570) Quantum optics : Squeezed states

ToC Category:
Nonlinear Optics

History
Original Manuscript: February 7, 2011
Revised Manuscript: March 18, 2011
Manuscript Accepted: June 17, 2011
Published: July 7, 2011

Citation
Y. B. Yu, H. J. Wang, M. Xiao, and S. N. Zhu, "Directly produced three-color entanglement by quasi-phase-matched third-harmonic generation," Opt. Express 19, 13949-13956 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-13949


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References

  1. H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008). [CrossRef] [PubMed]
  2. M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988). [CrossRef] [PubMed]
  3. 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]
  4. 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]
  5. 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]
  6. 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]
  7. P. Lodahl, “Einstein-Podolsky-Rosen correlations in second-harmonic generation,” Phys. Rev. A 68, 023806 (2003). [CrossRef]
  8. M. K. Olsen, “Continuous-variable Einstein-Podolsky-Rosen paradox with traveling-wave second-harmonic generation,” Phys. Rev. A 70, 035801 (2004). [CrossRef]
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. R. G. Yang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generation of multicolored tripartite entanglement by frequency doubling in a two-port resonator,” J. Opt. Soc. Am. B 27, 2721–2726 (2010). [CrossRef]
  14. 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]
  15. 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]
  16. 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]
  17. C. W. Gardiner, Quantum Noise (Springer, 1991).
  18. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).
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