Bright two-color tripartite entanglement with second harmonic generation

The bright two-color tripartite entanglement is investigated in the process of type II second harmonic generation (SHG) operating above threshold. The two pump fields and the second harmonic field are proved to be entangled, and the dependence of the entanglement degree on pump parameterσ and normalized frequency Ω is also analyzed. © 2009 Optical Society of America OCIS codes: (270.0270) Quantum optics; (270.6570) Squeezed states; (190.2620) Harmonic generition References and links 1. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991). 2. 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(16), 167903 (2003). 3. S. L. Braunstein, and P. van Loock, “Multipartite entanglement for continuous variables: A quantum teleportation network,” Phys. Rev. Lett. 84(15), 3482–3485 (2000). 4. D. Deutsch, and R. Jozsa, “Rapid solution of problems by quantum computation,” Proc. R. Soc. Lond. A 439(1907), 553–558 (1992). 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(14), 140504 (2006). 6. K. N. Cassemiro, A. S. Villar, P. Valente, M. Martinelli, and P. Nussenzveig, “Experimental observation of three-color optical quantum correlations,” Opt. Lett. 32(6), 695–697 (2007). 7. K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, “The quest for three-color entanglement: experimental investigation of new multipartite quantum correlations,” Opt. Express 15(26), 18236–18246 (2007). 8. N. B. Grosse, W. P. Bowen, K. McKenzie, and P. K. Lam, “Harmonic entanglement with second-order nonlinearity,” Phys. Rev. Lett. 96(6), 063601 (2006). 9. 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(24), 243601 (2008). 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(1), 014302 (2008). 11. L. Longchambon, N. Treps, T. Coudreau, J. Laurat, and C. Fabre, “Experimental evidence of spontaneous symmetry breaking in intracavity type II second-harmonic generation with triple resonance,” Opt. Lett. 30(3), 284–286 (2005). 12. M. J. Collett, and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30(3), 1386–1391 (1984). 13. P. van Loock, and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67(5), 052315 (2003). 14. K. Schneider, and S. Schiller, “Multiple conversion and optical limiting in a subharmonicpumped parametric oscillator,” Opt. Lett. 22(6), 363-365 (1997). Entanglement between more than two parts plays a central role in the process of quantum information, such as quantum cryptography [1], controlled dense coding [2], teleportation [3], and quantum computation [4]. And the multipartite entanglement states with different frequencies will be more important since it would facilitate many quantum information protocols of interspecies quantum teleportation, disparate nodes in a quantum information network. The generation of continuous variable (CV) multicolored entangled beams can be realized with nonlinear process. It was reported that the tripartite entanglement among signal, idler and #110003 $15.00 USD Received 13 Apr 2009; revised 24 May 2009; accepted 25 May 2009; published 27 May 2009 (C) 2009 OSA 8 June 2009 / Vol. 17, No. 12 / OPTICS EXPRESS 9851 pump modes was existent when a nondegenerate optical parametric oscillator (OPO) was operated above threshold [5]. And subsequently the quantum correlation experiments based on this scheme were demonstrated [6,7]. Apart from the parametric down-conversion, the second harmonic generation (SHG) can also be used to yield the multicolored entanglement. The bipartite entanglement with different wavelength between the fundamental and second-harmonic fields was theoretically investigated and experimentally demonstrated [8,9]. Thereafter, we suggested the generation of the tripartite entanglement of the fundamental and second-harmonic fields with two frequencies through the type-II SHG below threshold [10]. In this paper, we discuss the quantum characteristics of the two reflected fundamental modes and the generated second harmonic mode in the triple resonant type-II SHG system operating above threshold. In this case, second-harmonic generation and parametric downconversion occur simultaneously. The tripartite quadrature entanglement between three modes are analyzed in terms of experiment-related parameters such as normalized pump parametric, cavity losses and normalized frequency. Under appropriate conditions, triply resonant type-IISHG system can be also exploited as a bright continuous variable tripartite entanglement resource, but the quantum correlation is different between above and below threshold. Fig. 1. The sketch of SHG The system under analysis is shown in Fig. 1. It consists of a type II phase-matched crystal ( 2) χ placed inside a one-side cavity. Two sub-harmonic pump modes 1 α and 2 α are incident upon the optical medium. Their frequencies are degenerate 1 2 ( ) ω ω = , and the polarizations are orthogonal for the phase-matching condition. Then the harmonic mode 0 α is generated when the energy-matching condition 1 2 0 ω ω ω + = is satisfied. All of the sub-harmonic modes and harmonic mode resonate in the cavity simultaneously. Assuming perfect phase matching, the semi-classical dynamic Eqs. can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 1 2 0 0 0 0 ˆ ˆ ˆ ˆ ˆ ˆ 2 2 i in b c a t a t a t a t a t e c t φ τ γ χ γ γ = − − + + ɺ #110003 $15.00 USD Received 13 Apr 2009; revised 24 May 2009; accepted 25 May 2009; published 27 May 2009 (C) 2009 OSA 8 June 2009 / Vol. 17, No. 12 / OPTICS EXPRESS 9852 ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 2 0 1 1 1 1 ˆ ˆ ˆ ˆ ˆ ˆ 2 2 i in b c a t a t a t a t a t e c t φ τ γ χ γ γ + = − + + + ɺ (1) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 0 2 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ 2 2 i in b c a t a t a t a t a t e c t φ τ γ χ γ γ + = − + + + ɺ Here 0 â , 1 â , and 2 â are the annihilation operators of harmonic and two sub-harmonic pump fields inside the cavity, respectively. ˆ ( 0,1, 2) in i a i = denotes the input amplitude operators of the three fields. The roundtrip time τ in the cavity is assumed to be same for all the three fields. χ is the effective nonlinear coupling parameter, which is proportional to the second order susceptibility of the crystal. The total loss parameter for each mode is ( ) , 0,1, 2 i bi ci i γ γ γ = + = , where bi γ is related to the amplitude reflection coefficients i r and amplitude transmission coefficients i t of the coupling mirror of the optical cavity by the formula: 1 i bi r γ = − , 2 i bi t γ = , and ci γ represents extra intracavity loss parameter. ( ) ˆ i c t is the vacuum noise term corresponding to intracavity loss. In general case, assuming that the two pump modes and harmonic mode have zero initial phase 0 1 2 1 i i i e e e φ φ φ = = = , the two pumping modes have the same positive real amplitude β , the cavity transmission factor and the extra-losses for the two pump fields are assumed to be the same 1 2 γ γ γ = = , 1 2 b b b γ γ γ = = , 1 2 c c c γ γ γ = = . Stationary mean field solutions 0 α , 1 α , and 2 α can be obtained by setting 0 â , 1 â and 2 â to be zero: 0 0 1 2 0 γ α χα α − − = (2a) * 1 2 0 2 0 b γα χα α γ β − + + = (2b) * 2 1 0 2 0 b γα χα α γ β − + + = (2c) Equations (2b) and (2c) show that both 1 α and 2 α are real numbers. The pumping threshold th β and pump parameterσ can be expressed as

Entanglement between more than two parts plays a central role in the process of quantum information, such as quantum cryptography [1], controlled dense coding [2], teleportation [3], and quantum computation [4]. And the multipartite entanglement states with different frequencies will be more important since it would facilitate many quantum information protocols of interspecies quantum teleportation, disparate nodes in a quantum information network.
The generation of continuous variable (CV) multicolored entangled beams can be realized with nonlinear process. It was reported that the tripartite entanglement among signal, idler and pump modes was existent when a nondegenerate optical parametric oscillator (OPO) was operated above threshold [5]. And subsequently the quantum correlation experiments based on this scheme were demonstrated [6,7]. Apart from the parametric down-conversion, the second harmonic generation (SHG) can also be used to yield the multicolored entanglement. The bipartite entanglement with different wavelength between the fundamental and second-harmonic fields was theoretically investigated and experimentally demonstrated [8,9]. Thereafter, we suggested the generation of the tripartite entanglement of the fundamental and second-harmonic fields with two frequencies through the type-II SHG below threshold [10]. In this paper, we discuss the quantum characteristics of the two reflected fundamental modes and the generated second harmonic mode in the triple resonant type-II SHG system operating above threshold. In this case, second-harmonic generation and parametric downconversion occur simultaneously. The tripartite quadrature entanglement between three modes are analyzed in terms of experiment-related parameters such as normalized pump parametric, cavity losses and normalized frequency. Under appropriate conditions, triply resonant type-IISHG system can be also exploited as a bright continuous variable tripartite entanglement resource, but the quantum correlation is different between above and below threshold. The system under analysis is shown in Fig. 1. It consists of a type II phase-matched crystal ( 2 ) χ placed inside a one-side cavity. Two sub-harmonic pump modes 1 α and 2 α are incident ω ω ω + = is satisfied. All of the sub-harmonic modes and harmonic mode resonate in the cavity simultaneously. Assuming perfect phase matching, the semi-classical dynamic Eqs. can be expressed as Stationary mean field solutions 0 α , 1 α , and 2 α can be obtained by setting 0 a ɺ , 1 a ɺ and 2 a ɺ to be zero: Equations (2b) and (2c) show that both 1 α and 2 α are real numbers. The pumping threshold th β and pump parameter σ can be expressed as Using the input-output relations 2 can be written as respectively. We also show the case below threshold ( ) Fig. 2 [10]. It can be seen obviously that when the cavity is operating below threshold, the two output fundamental modes are equal, but when it is above threshold, the two modes are different from each other because of spontaneous symmetry-breaking phenomenon [11], which will affect the entanglement in the system. The solid line in Fig. 2 is for the case of existing an internal cavity loss ( In the following section, entanglement between two pumping sub-harmonic modes and the harmonic mode of triple resonant SHG operating above threshold is considered.
The dynamics of the quantum fluctuations can be described by linearizing the classical Eqs. of motion around the stationary state. Setting , the Fourier transformation of the output quadrature components of harmonic field and two pump fields above threshold can be solved with the Eqs. (1)(2)(3)(4)(5).
In order to discuss the bright tripartite entanglement among the two pumping sub-harmonic modes and harmonic mode of triple resonant SHG operating above threshold, we introduce the sufficient inseparability criterion for tripartite CV entanglement proposed by van The satisfaction of any pair of the inequalities is sufficient for full inseparability of three-party entanglement. The smaller the values of the inequalities are, the larger the correlation degree will be obtained. The correlation spectra of these three quantities with optimized choice of the three parameters i g are plotted in Figs. 3 and 4., the dot lines are corresponding to standard quantum limit (SQL) 4. Figure 3 shows the minimum correlation spectra S (dash-dotted curve)) are satisfied when the normalized frequency is below 1.5. It satisfy sufficient inseparability criterion for tripartite CV entanglement below normalized analysis frequency 1.5. The correlation noise spectrum 3 out S is lower than others, it means the quantum correlation between one of the fundamental modes and harmonic mode is stronger than others. Compared to the result below threshold, which quantum correlation between two fundamental modes is strong, the fundamental modes and harmonic mode presents strong coupling above threshold.
In summary, by means of semi-classical method, we analyzed the continuous-variable tripartite entanglement characteristic of the reflected fundamental fields and the second harmonic field in a triple resonant type-II SHG operating above threshold. The full inseparability of the three output modes as a function of normalized frequency and pump parameter are calculated. Under certain conditions, type-II SHG system operating above threshold can be also exploited as a bright two color tripartite entanglement resource. But the quantum correlation characters are very different because of spontaneous symmetry-breaking. With the advantage of simplicity of SHG process, the calculated results may be a useful reference for the generation and application of entanglement in quantum communication networks. Furthermore, with multiple conversion in a subharmonic-pumped parametric oscillator operating above threshold [14], it is possible to develop multicolor multiparty (more than 3) entanglement with some special condition. It will be more interesting.