## Directly produced three-color entanglement by quasi-phase-matched third-harmonic generation |

Optics Express, Vol. 19, Issue 15, pp. 13949-13956 (2011)

http://dx.doi.org/10.1364/OE.19.013949

Acrobat PDF (982 KB)

### 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

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]

_{3}by using QPM method [15

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 LiTaO_{3},” 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]

_{3}placed inside an optical cavity, as shown in Fig. 1(a). The fundamental wave with the frequency of

*ω*

_{0}is incident upon the QPOS in the cavity. The second-harmonic field with the frequency of

*ω*

_{1}is generated by the first sum-frequency process. The third-harmonic field with the frequency of

*ω*

_{2}is generated by the second sum-frequency process between the fundamental and the second-harmonic fields. The phase mismatches in the two nonlinear processes are compensated by two different reciprocal vectors

**G**

_{1}and

**G**

_{2}of the QPOS, respectively. The QPM conditions can be written as

**k**

_{1}= 2

**k**

_{0}+

**G**

_{1}for the SHG and

**k**

_{2}=

**k**

_{0}+

**k**

_{1}+

**G**

_{2}for the THG, respectively, which are depicted in Fig. 1(b).

**k**

_{0},

**k**

_{1}, and

**k**

_{2}are the wave vectors of the fundamental, second-, and third-harmonic fields, respectively.

## 2. The stationary solutions and output fields

*κ*

_{0}and

*κ*

_{1}are the dimensionless nonlinear coupling coefficients, which are taken to be real for simplicity.

*ɛ*is the classical pumping laser amplitude and taken as a real field here for

*ɛ*=

*ɛ** =

*E*

_{0}.

*γ*(

_{i}*i*= 0, 1, 2) stand for the damping rates of the three cavity fields.

*P*representation [17]. We find the Fokker-Planck equation for the P function of this system as

*α*(

_{i}*i*= 0, 1, 2) correspond to

*a*in the

_{i}*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]. Following the standard procedure [17], the stochastic differential equations for the cavity fields can be written as

*η*(

_{i}*i*= 1, 2, 3) are the complex noise terms, which satisfy the relations 〈

*η*(

_{i}*t*)〉 = 0 and

*α*=

_{i}*A*+

_{i}*δα*and use this linearization procedure to rewrite the Eq. (5) as [17, 18] where

_{i}**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].

**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]

**A**have no negative real parts [18]. 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}/

*ɛ*and

_{th}*ɛ*=

_{th}*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}.

## 3. Discussions on the entanglement characteristic among three beams

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]

*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]

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]

_{kk′}=<

*μ*

_{k}

*μ*

_{k′}+

*μ*

_{k′}

*μ*> /2, where

_{k}*μ*= (

*X*

_{0},

*Y*

_{0},

*X*

_{1},

*Y*

_{1},

*X*

_{2},

*Y*

_{2}) and

*i*=0, 1, 2) represent the amplitude quadratures of the fields;

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

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

*is the symplectic eigenvalues. We denote*

_{i}*as the partial transpose on the CM*σ ˜

_{A}*σ*for subsystem

*A*and

*E*is the symplectic eigenvalues of the matrix |

_{A}*i*Ω

*|. Then the subsystems*σ ˜

*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]

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]

*E*(

_{i}*i*= 0, 1, 2) as the symplectric eigenvalues of the bipartite system between the field

*a*and the subsystem including the remaining fields (1×2 bipartition). When

_{i}*E*< 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

_{i}*E*versus analysis frequency

_{i}*ω*with

*γ*

_{0}= 0.01,

*γ*= 1.5

*γ*

_{0},

*κ*

_{0}= 0.1,

*κ*

_{1}= 1.5

*κ*

_{0}, and

*σ*= 2. One can see in Fig. 3(a) that the three values of

*E*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.

_{i}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]

*E*

_{0}. In Fig. 3(b) the symplectic eigenvalues

*E*are presented versus

_{i}*σ*with

*γ*

_{0}= 0.01,

*γ*= 1.5

*γ*

_{0},

*κ*

_{0}= 0.1,

*κ*

_{1}= 1.5

*κ*

_{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.

*γ*/

*γ*

_{0}with

*γ*

_{0}= 0.01,

*κ*

_{0}= 0.1,

*κ*

_{1}= 1.5

*κ*

_{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].

*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}= 1.5

*γ*

_{0},

*κ*

_{0}= 0.1,

*κ*

_{1}= 1.5

*κ*

_{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}= 1.6

*γ*

_{1}.

## 4. Conclusions

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]

## Acknowledgments

## References and links

1. | H. J. Kimble, “The quantum internet,” Nature |

2. | M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. |

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. |

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. |

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. |

6. | A. S. Coelho, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “Three-color entanglement,” Science |

7. | P. Lodahl, “Einstein-Podolsky-Rosen correlations in second-harmonic generation,” Phys. Rev. A |

8. | M. K. Olsen, “Continuous-variable Einstein-Podolsky-Rosen paradox with traveling-wave second-harmonic generation,” Phys. Rev. A |

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

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. |

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 |

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 |

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 |

14. | J. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. |

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 LiTaO |

16. | S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science |

17. | C. W. Gardiner, |

18. | D. F. Walls and G. J. Milburn, |

19. | M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A |

20. | R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. |

21. | A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. |

22. | M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A |

23. | R. F. Werner and M. M. Wolf, “Bound entangled Gaussian states,” Phys. Rev. Lett. |

24. | G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, “Separability properties of three-mode Gaussian states,” Phys. Rev. A |

25. | S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. |

26. | A. Serafini, G. Adesso, and F. Illuminati, “Unitarily localizable entanglement of Gaussian states,” Phys. Rev. A |

**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

- H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008). [CrossRef] [PubMed]
- M. D. Reid and P. D. Drummond, “Quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988). [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]
- 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]
- 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. 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]
- P. Lodahl, “Einstein-Podolsky-Rosen correlations in second-harmonic generation,” Phys. Rev. A 68, 023806 (2003). [CrossRef]
- M. K. Olsen, “Continuous-variable Einstein-Podolsky-Rosen paradox with traveling-wave second-harmonic generation,” Phys. Rev. A 70, 035801 (2004). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- C. W. Gardiner, Quantum Noise (Springer, 1991).
- D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).
- 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]
- R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000). [CrossRef] [PubMed]
- 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]
- 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]
- 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]

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