Abstract

Ultra-bright source of entangled photons is an essential component in optical quantum information processing. Here we propose a counterpropagating path-entangled photon pair sources using a quasi-periodic modulated lithium niobate crystal. The nonlinear crystal designed by a dual-grid method, simultaneously phase-matched two spontaneous parametric down-conversion processes. Signal and idler modes have opposite propagation directions in the two spontaneous parametric down-conversion processes, which is the key to generating path-entangled photon pairs. Compared to copropagating entangled sources, the counterpropagating path-entangled sources result in a much narrower spectrum. The quantum state of the path-entanglement source is not only suited for quantum coding, but also to allow the implementation of complex quantum algorithms on a photonic chip.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (5)

X. Wang, W. Liu, P. Wang, and Y. Li, “Experimental study on all-fiber-based unidimensional continuous-variable quantum key distribution,” Phys. Rev. A 95, 062330 (2017).
[Crossref]

F. -G. Deng, B. -C. Ren, and X. -H. Li, “Quantum hyperentanglement and its applications in quantum information processing,” Sci. Bull. 62, 46–68 (2017).
[Crossref]

S. Saravi, T. Pertsch, and F. Setzpfandt, “Generation of counterpropagating path-entangled photon pairs in a single periodic waveguide,” Phys. Rev. Lett. 118, 183603 (2017).
[Crossref] [PubMed]

A. S. Solntsev and A. A. Sukhorukov, “Path-entangled photon sources on nonlinear chips,” Rev. Phys. 2, 19–31 (2017).
[Crossref]

F. Laudenbach, S. Kalista, M. Hentschel, P. Walther, and H. Hübel, “A novel single-crystal & single-pass source for polarisation- and colour-entangled photon pairs,” Sci. Reports 7, 7235 (2017).
[Crossref]

2015 (2)

B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015).
[Crossref] [PubMed]

S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein, “Advances in quantum teleportation,” Nat. Photonics 9, 641–652 (2015).
[Crossref]

2014 (2)

S. Sharabi, G. Porat, and A. Arie, “Improved idler beam quality via simultaneous parametric oscillation and signal-to-idler conversion,” Opt. Lett. 39, 2152–2155 (2014).
[Crossref] [PubMed]

J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, “On-chip quantum interference between silicon photon-pair sources,” Nat. Photonics 8, 104–108 (2014).
[Crossref]

2013 (1)

2012 (6)

G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A 85, 022316 (2012)
[Crossref]

J. Yin, J. -G. Ren, H. Lu, Y. Cao, H. -L. Yong, Y. -P. Wu, C. Liu, S. -K. Liao, F. Zhou, Y. Jiang, X. -D. Cai, P. Xu, G. -S. Pan, J. -J. Jia, Y. -M. Huang, H. Yin, J. -Y. Wang, Y. -A. Chen, C. -Z. Peng, and J. -W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref] [PubMed]

H. -K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108, 130503 (2012).
[Crossref] [PubMed]

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A 86, 022338 (2012).
[Crossref]

X. -C. Yao, T. -X. Wang, P. Xu, H. Lu, G. -S. Pan, X. -H. Bao, C. -Z. Peng, C. -Y. Lu, Y. -A. Chen, and J. -W. Pan, “Observation of eight-photon entanglement,” Nat. Photonics 6, 225–228 (2012).
[Crossref]

X. Jia, Z. Yan, Z. Duan, X. Su, H. Wang, C. Xie, and K. Peng, “Experimental realization of three-color entanglement at optical fiber communication and atomic storage wavelengths,” Phys. Rev. Lett. 109, 253604 (2012).
[Crossref]

2011 (1)

G. He, J. Zhang, J. Zhu, and G. Zeng, “Continuous-variable quantum teleportation in bosonic structured environments,” Phys. Rev. A 84, 034305 (2011).
[Crossref]

2010 (3)

W. -B. Gao, P. Xu, X. -C. Yao, O. Gühne, A. Cabello, C. -Y. Lu, C. -Z. Peng, Z. -B. Chen, and J. -W. Pan, “Experimental realization of a controlled-NOT gate with four-photon six-qubit cluster states,” Phys. Rev. Lett. 104, 020501 (2010).
[Crossref] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[Crossref] [PubMed]

G. Porat, O. Gayer, and A. Arie, “Simultaneous parametric oscillation and signal-to-idler conversion for efficient downconversion,” Opt. Lett. 35, 1401–1403 (2010).
[Crossref] [PubMed]

2009 (2)

T. Suhara, G. Nakaya, J. Kawashima, and M. Fujimura, “Quasi-phase-matched waveguide devices for generation of postselection-free polarization-entangled twin photons,” IEEE Photonics Technol. Lett. 21, 1096–1098 (2009).
[Crossref]

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photoncis 3, 346–350 (2009).
[Crossref]

2008 (1)

J. -W. Pan, Z. -B. Chen, C. -Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2008).
[Crossref]

2007 (1)

2006 (1)

2005 (2)

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[Crossref]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref] [PubMed]

2003 (1)

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, 167903 (2003).
[Crossref] [PubMed]

2002 (1)

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

2001 (2)

M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
[Crossref] [PubMed]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref] [PubMed]

2000 (1)

J. Zhang and K. Peng, “Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a Bell state,” Phys. Rev. A 62, 064302 (2000).
[Crossref]

1997 (1)

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A 56, 1627 (1997).
[Crossref]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

1982 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment: a new violation of Bell’s inequalities,” Phys. Rev. Lett. 49, 91 (1982).
[Crossref]

1981 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47, 460 (1981).
[Crossref]

1978 (1)

J. F. Clauser and A. Shimony, “Bell’s theorem: experimental tests and implications,” Reports on Prog. Phys. 41, 1881–1927 (1978).
[Crossref]

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Abellán, C.

B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015).
[Crossref] [PubMed]

Amaya, W.

B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015).
[Crossref] [PubMed]

Andersen, U. L.

G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A 85, 022316 (2012)
[Crossref]

Arie, A.

Aspect, A.

A. Aspect, P. Grangier, and G. Roger, “Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment: a new violation of Bell’s inequalities,” Phys. Rev. Lett. 49, 91 (1982).
[Crossref]

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47, 460 (1981).
[Crossref]

Bahabad, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref] [PubMed]

Bao, X. -H.

X. -C. Yao, T. -X. Wang, P. Xu, H. Lu, G. -S. Pan, X. -H. Bao, C. -Z. Peng, C. -Y. Lu, Y. -A. Chen, and J. -W. Pan, “Observation of eight-photon entanglement,” Nat. Photonics 6, 225–228 (2012).
[Crossref]

Barreiro, J. T.

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[Crossref]

Bernien, H.

B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015).
[Crossref] [PubMed]

Björk, G.

G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A 85, 022316 (2012)
[Crossref]

Blok, M. S.

B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015).
[Crossref] [PubMed]

Bonneau, D.

J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, “On-chip quantum interference between silicon photon-pair sources,” Nat. Photonics 8, 104–108 (2014).
[Crossref]

Braunstein, S. L.

S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein, “Advances in quantum teleportation,” Nat. Photonics 9, 641–652 (2015).
[Crossref]

Bromberg, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[Crossref] [PubMed]

Cabello, A.

W. -B. Gao, P. Xu, X. -C. Yao, O. Gühne, A. Cabello, C. -Y. Lu, C. -Z. Peng, Z. -B. Chen, and J. -W. Pan, “Experimental realization of a controlled-NOT gate with four-photon six-qubit cluster states,” Phys. Rev. Lett. 104, 020501 (2010).
[Crossref] [PubMed]

Cai, X. -D.

J. Yin, J. -G. Ren, H. Lu, Y. Cao, H. -L. Yong, Y. -P. Wu, C. Liu, S. -K. Liao, F. Zhou, Y. Jiang, X. -D. Cai, P. Xu, G. -S. Pan, J. -J. Jia, Y. -M. Huang, H. Yin, J. -Y. Wang, Y. -A. Chen, C. -Z. Peng, and J. -W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref] [PubMed]

Cao, Y.

J. Yin, J. -G. Ren, H. Lu, Y. Cao, H. -L. Yong, Y. -P. Wu, C. Liu, S. -K. Liao, F. Zhou, Y. Jiang, X. -D. Cai, P. Xu, G. -S. Pan, J. -J. Jia, Y. -M. Huang, H. Yin, J. -Y. Wang, Y. -A. Chen, C. -Z. Peng, and J. -W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref] [PubMed]

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J. Yin, J. -G. Ren, H. Lu, Y. Cao, H. -L. Yong, Y. -P. Wu, C. Liu, S. -K. Liao, F. Zhou, Y. Jiang, X. -D. Cai, P. Xu, G. -S. Pan, J. -J. Jia, Y. -M. Huang, H. Yin, J. -Y. Wang, Y. -A. Chen, C. -Z. Peng, and J. -W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
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J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, “On-chip quantum interference between silicon photon-pair sources,” Nat. Photonics 8, 104–108 (2014).
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Zeilinger, A.

J. -W. Pan, Z. -B. Chen, C. -Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2008).
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G. He, J. Zhang, J. Zhu, and G. Zeng, “Continuous-variable quantum teleportation in bosonic structured environments,” Phys. Rev. A 84, 034305 (2011).
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H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A 86, 022338 (2012).
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Zhang, J.

G. He, J. Zhang, J. Zhu, and G. Zeng, “Continuous-variable quantum teleportation in bosonic structured environments,” Phys. Rev. A 84, 034305 (2011).
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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, 167903 (2003).
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X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
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J. Zhang and K. Peng, “Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a Bell state,” Phys. Rev. A 62, 064302 (2000).
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Zhao, F.

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, 167903 (2003).
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A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
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G. He, J. Zhang, J. Zhu, and G. Zeng, “Continuous-variable quantum teleportation in bosonic structured environments,” Phys. Rev. A 84, 034305 (2011).
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Zukowski, M.

J. -W. Pan, Z. -B. Chen, C. -Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2008).
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Zwiller, V.

J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, “On-chip quantum interference between silicon photon-pair sources,” Nat. Photonics 8, 104–108 (2014).
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B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015).
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Figures (5)

Fig. 1
Fig. 1 The SPDC processes with counterpropagating photon pairs. The signal light is shown in red, the idler light is shown in blue, and the pump light is shown in yellow. The top right of the figure is the partial zoom of our nonlinear crystal. The lower part of the figure is a schematic diagram of the crystal design. The figure on the lower left is the one-dimensional quasi-periodic lattice composed of two basic lattice tiles. The green is marked as TA and the blue is marked as TB. Their lengths are la = 82.4 nm and lb = 89.29 nm respectively. The lower right figure is the final nonlinear crystal. The black blocks indicate that the nonlinear coefficient is positive and the white blocks are opposite.
Fig. 2
Fig. 2 The sketch of showing the momentum conversation. ΔkA and ΔkB are the phase mismatches of processes ��A and ��B. The left half of the picture is the phase matching condition of process A. The right half of the picture is the phase matching condition of process B.
Fig. 3
Fig. 3 The Fourier transform of our quasi-periodic lattice. The main reciprocal lattice vectors of the nonlinear photonic crystal are GA and GB and other vectors can be ignored.
Fig. 4
Fig. 4 The spectrum distribution of simultaneous spontaneous parametric processes A and B. (a), (c) phase-matching functions of process A and process B, respectively. (b), (d) joint-spectral amplitude distribution of process A and process B, respectively. The joint-spectral amplitude of process A is determined by the continuous-wave laser source and its phase matching function ΓA in the first row. The process B is the same in the second row.
Fig. 5
Fig. 5 The nonlinear efficiencies of processes A and B as the function of crystal length. (a) process A. (b) process B. From the inserted figures, the nonlinear efficiency fluctuates with distance but the overall trend is increased. The effective difference between the nonlinear efficiencies of processes A and B is only one percent.

Equations (10)

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H ^ I ( t ) = ε 0 L 2 L 2 d z χ ( 2 ) ( z ) E ^ p ( z , t ) E ^ s ( z , t ) E ^ i ( z , t ) ,
E ^ m ( z , t ) = E ^ m ( + ) ( z , t ) + E ^ m ( ) ( z , t ) , E ^ m ( + ) ( z , t ) = E ^ m ( ) ( z , t ) = d ω m A m ( ω m ) e j [ k m ( ω m ) z ω m t ] a ^ m ( ω m ) .
H ^ f ( t ) ε 0 L 2 L 2 d z χ ( 2 ) ( z ) E p ( + ) ( z , t ) E ^ s ( ) ( z , t ) E ^ i ( ) ( z , t ) + H . c .
E ^ s ( ) ( z , t ) = d ω s A s * ( ω s ) [ e j [ k s ( ω s ) z ω s t ] a ^ F , s ( ω s ) + e j [ k s ( ω s ) z ω s t ] a ^ B , s ( ω s ) ] ,
E ^ i ( ) ( z , t ) = d ω i A i * ( ω i ) [ e j [ k i ( ω i ) z ω i t ] a ^ F , i ( ω i ) + e j [ k i ( ω i ) z ω i t ] a ^ B , i ( ω i ) ] .
E p ( + ) ( z , t ) = E p ( ) * ( z , t ) = A p d ω p α ( ω p ) e i ( k p z ω p t ) ,
H ^ f ( t ) L 2 L 2 d z d ω p d ω s d ω i χ ( 2 ) ( ω p , ω s , ω i , z ) A s A i A p α ( ω p ) × { e j { [ k p ( ω p ) k s ( ω s ) + k i ( ω i ) ] z [ ω p ω s ω i ] t } a ^ F , s ( ω s ) a ^ B , i ( ω i ) + e j { [ k p ( ω p ) + k s ( ω s ) k i ( ω i ) ] z [ ω p ω s ω i ] t } a ^ F , i ( ω i ) a ^ B , s ( ω s ) } .
| ψ ( t + ) | ψ ( t ) i + H ^ f ( t ) | ψ ( t ) d t .
| ψ 2 = T d ω s d ω i α ( ω s + ω i ) [ Γ A ( ω s , ω i ) | s , F , ω s | i , B , ω i + Γ B ( ω s , ω i ) | s , B , ω s | i , F , ω i ] ,
Γ A ( ω s , ω i ) = L 2 L 2 χ ( 2 ) e j [ k p ( ω s + ω i ) k s ( ω s ) + k i ( ω i ) ] z d z Γ B ( ω s , ω i ) = L 2 L 2 χ ( 2 ) e j [ k p ( ω s + ω i ) + k s ( ω s ) k i ( ω i ) ] z d z

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