Abstract

Spectral correlation of photon pairs generated in dispersion shifted fiber by a pulsed pump is theoretically analyzed and experimentally investigated. We first calculate the spectral function of photon pairs according to the deduced two-photon state generated by spontaneous four wave mixing under the assumptions close to the real experimental conditions. We then experimentally study the spectral property of the signal and idler photon pairs generated in optical fiber by photon correlation measurements, and the experimental results agree with the calculation. The investigation is useful for developing fiber-based sources of entangled photon pairs and for studying multi-photon quantum interference with multiple photon pairs.

©2008 Optical Society of America

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  1. D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
    [Crossref]
  2. J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
    [Crossref]
  3. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
    [Crossref] [PubMed]
  4. Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel, and J. W. Pan, “Experimental demonstration of five-photon entanglement and open-destination quantum teleportation,” Nature 430, 54–58 (2004).
    [Crossref] [PubMed]
  5. B. Yurke and D. Stoler, “Einstein-Podolsky-Rosen effects from independent particle sources,” Phys. Rev. Lett. 68, 1251–1254 (1992).
    [Crossref] [PubMed]
  6. M. Zukowski, A. Zeilinger, and H. Weinfurter, “Entangling Photons Radiated by Independent Pulsed Sources,” Ann. (N. Y.) Acad. Sci. 755, 91–102 (1995).
    [Crossref]
  7. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
    [Crossref] [PubMed]
  8. Z. Y. Ou and Y. J. Lu, “Cavity Enhanced Spontaneous Parametric Down-Conversion for the Prolongation of Correlation Time between Conjugate Photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
    [Crossref]
  9. M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nature Physics 3, 692–695 (2007).
    [Crossref]
  10. Z. Y. Ou, “Parametric Down-Conversion with Coherent Pulse Pumping and Quantum Interference between Independent Fields,” Quantum Semiclass Opt. 9, 599–614 (1997).
    [Crossref]
  11. W. P. Grice, A. B. U’ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multi-Photon States,” Phys. Rev. A 64, 063,815 (2001).
    [Crossref]
  12. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” Photon. Technol. Lett. 14, 983–985 (2002).
    [Crossref]
  13. X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053,601 (2005).
    [Crossref]
  14. H. Takesue and K. Inoue, “Generation of 1.5-um band time-bin entanglement using spontaneous fiber four-wave mixing and planar lightwave circuit interferometers,” Phys. Rev. A 72, 041,804 (2005).
    [Crossref]
  15. J. Fan, A. Dogariu, and L. J. Wang, “Generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. 30, 1530–1532 (2005).
    [Crossref] [PubMed]
  16. J. Fulconis, O. Alibart, W. J. Wadsworth, P. S. Russell, and J. G. Rarity, “High brightness single mode source of correlated photon pairs using a photonic crystal fiber,” Opt. Express 13, 7572–7582 (2005).
    [Crossref] [PubMed]
  17. K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31, 1905–1907 (2006).
    [Crossref] [PubMed]
  18. X. Li, J. Chen, P. L. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12, 3737–3744 (2004).
    [Crossref] [PubMed]
  19. K. Inoue and K. Shimizu, “Generation of Quantum-Correlated Photon Pairs in Optical Fiber: Influence of Spontaneous Raman Scattering,” Jpn. J. Appl. Phys 43, 8048–8052 (2004).
    [Crossref]
  20. H. Takesue and K. Inoue, “1.5 um band quantum-correlated photon pair generation in dispersion-shifted fiber: suppression of noise photons by cooling fiber,” Opt. Express 13, 7832–7839 (2005).
    [Crossref] [PubMed]
  21. J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534–544 (2005).
    [Crossref] [PubMed]
  22. J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033,801 (2005).
    [Crossref]
  23. O. Alibart, J. Fulconis, G. K. L. Wong, S. G. Murdoch, W. J. Wadsworth, and J. G. Rarity, “Photon pair generation using four-wave mixing in a microstructured fibre: theory versus experiment,” New J. of Phys. 8, 67 (2006).
    [Crossref]
  24. K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
    [Crossref]
  25. G. P. Agrawal, Nonlinear fiber optics (Elsevier Pte Ltd., Singapore, 2005).
  26. X. Li, J. Chen, K. F. Lee, P. L. Voss, and P. Kumar, “All-fiber photon-pair source for quantum communication: Influence of spectra,” Proceeding of Quantum Communication and Measurement QCMC’06, 31–34 (2006).
  27. Z. Y. Ou, J. K. Rhee, and L. J. Wang, “Photon Bunching and Multiphoton Interference in Parametric Down-Conversion,” Phys. Rev. A 60, 593–604 (1999).
    [Crossref]

2007 (2)

M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nature Physics 3, 692–695 (2007).
[Crossref]

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

2006 (3)

X. Li, J. Chen, K. F. Lee, P. L. Voss, and P. Kumar, “All-fiber photon-pair source for quantum communication: Influence of spectra,” Proceeding of Quantum Communication and Measurement QCMC’06, 31–34 (2006).

K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31, 1905–1907 (2006).
[Crossref] [PubMed]

O. Alibart, J. Fulconis, G. K. L. Wong, S. G. Murdoch, W. J. Wadsworth, and J. G. Rarity, “Photon pair generation using four-wave mixing in a microstructured fibre: theory versus experiment,” New J. of Phys. 8, 67 (2006).
[Crossref]

2005 (7)

2004 (3)

X. Li, J. Chen, P. L. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12, 3737–3744 (2004).
[Crossref] [PubMed]

K. Inoue and K. Shimizu, “Generation of Quantum-Correlated Photon Pairs in Optical Fiber: Influence of Spontaneous Raman Scattering,” Jpn. J. Appl. Phys 43, 8048–8052 (2004).
[Crossref]

Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel, and J. W. Pan, “Experimental demonstration of five-photon entanglement and open-destination quantum teleportation,” Nature 430, 54–58 (2004).
[Crossref] [PubMed]

2002 (1)

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

2001 (2)

W. P. Grice, A. B. U’ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multi-Photon States,” Phys. Rev. A 64, 063,815 (2001).
[Crossref]

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

1999 (2)

Z. Y. Ou and Y. J. Lu, “Cavity Enhanced Spontaneous Parametric Down-Conversion for the Prolongation of Correlation Time between Conjugate Photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
[Crossref]

Z. Y. Ou, J. K. Rhee, and L. J. Wang, “Photon Bunching and Multiphoton Interference in Parametric Down-Conversion,” Phys. Rev. A 60, 593–604 (1999).
[Crossref]

1998 (1)

J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

1997 (2)

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

Z. Y. Ou, “Parametric Down-Conversion with Coherent Pulse Pumping and Quantum Interference between Independent Fields,” Quantum Semiclass Opt. 9, 599–614 (1997).
[Crossref]

1995 (2)

M. Zukowski, A. Zeilinger, and H. Weinfurter, “Entangling Photons Radiated by Independent Pulsed Sources,” Ann. (N. Y.) Acad. Sci. 755, 91–102 (1995).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

1992 (1)

B. Yurke and D. Stoler, “Einstein-Podolsky-Rosen effects from independent particle sources,” Phys. Rev. Lett. 68, 1251–1254 (1992).
[Crossref] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Nonlinear fiber optics (Elsevier Pte Ltd., Singapore, 2005).

Alibart, O.

O. Alibart, J. Fulconis, G. K. L. Wong, S. G. Murdoch, W. J. Wadsworth, and J. G. Rarity, “Photon pair generation using four-wave mixing in a microstructured fibre: theory versus experiment,” New J. of Phys. 8, 67 (2006).
[Crossref]

J. Fulconis, O. Alibart, W. J. Wadsworth, P. S. Russell, and J. G. Rarity, “High brightness single mode source of correlated photon pairs using a photonic crystal fiber,” Opt. Express 13, 7572–7582 (2005).
[Crossref] [PubMed]

Beveratos, A.

M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nature Physics 3, 692–695 (2007).
[Crossref]

Bouwmeester, D.

J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

Briegel, H. J.

Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel, and J. W. Pan, “Experimental demonstration of five-photon entanglement and open-destination quantum teleportation,” Nature 430, 54–58 (2004).
[Crossref] [PubMed]

Chen, J.

K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31, 1905–1907 (2006).
[Crossref] [PubMed]

X. Li, J. Chen, K. F. Lee, P. L. Voss, and P. Kumar, “All-fiber photon-pair source for quantum communication: Influence of spectra,” Proceeding of Quantum Communication and Measurement QCMC’06, 31–34 (2006).

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033,801 (2005).
[Crossref]

X. Li, J. Chen, P. L. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12, 3737–3744 (2004).
[Crossref] [PubMed]

Chen, Y. A.

Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel, and J. W. Pan, “Experimental demonstration of five-photon entanglement and open-destination quantum teleportation,” Nature 430, 54–58 (2004).
[Crossref] [PubMed]

Cohen, O.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

Dogariu, A.

Duligall, J.

Eibl, M.

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

Fan, J.

Fiorentino, M.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

Fulconis, J.

Garay-Palmett, K.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

Gisin, N.

M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nature Physics 3, 692–695 (2007).
[Crossref]

Grice, W. P.

W. P. Grice, A. B. U’ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multi-Photon States,” Phys. Rev. A 64, 063,815 (2001).
[Crossref]

Halder, M.

M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nature Physics 3, 692–695 (2007).
[Crossref]

Inoue, K.

H. Takesue and K. Inoue, “Generation of 1.5-um band time-bin entanglement using spontaneous fiber four-wave mixing and planar lightwave circuit interferometers,” Phys. Rev. A 72, 041,804 (2005).
[Crossref]

H. Takesue and K. Inoue, “1.5 um band quantum-correlated photon pair generation in dispersion-shifted fiber: suppression of noise photons by cooling fiber,” Opt. Express 13, 7832–7839 (2005).
[Crossref] [PubMed]

K. Inoue and K. Shimizu, “Generation of Quantum-Correlated Photon Pairs in Optical Fiber: Influence of Spontaneous Raman Scattering,” Jpn. J. Appl. Phys 43, 8048–8052 (2004).
[Crossref]

Knill, E.

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

Kumar, P.

X. Li, J. Chen, K. F. Lee, P. L. Voss, and P. Kumar, “All-fiber photon-pair source for quantum communication: Influence of spectra,” Proceeding of Quantum Communication and Measurement QCMC’06, 31–34 (2006).

K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31, 1905–1907 (2006).
[Crossref] [PubMed]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053,601 (2005).
[Crossref]

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033,801 (2005).
[Crossref]

X. Li, J. Chen, P. L. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12, 3737–3744 (2004).
[Crossref] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

Kwiat, P. G.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Laflamme, R.

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

Lee, K. F.

X. Li, J. Chen, K. F. Lee, P. L. Voss, and P. Kumar, “All-fiber photon-pair source for quantum communication: Influence of spectra,” Proceeding of Quantum Communication and Measurement QCMC’06, 31–34 (2006).

K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31, 1905–1907 (2006).
[Crossref] [PubMed]

Li, X.

K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31, 1905–1907 (2006).
[Crossref] [PubMed]

X. Li, J. Chen, K. F. Lee, P. L. Voss, and P. Kumar, “All-fiber photon-pair source for quantum communication: Influence of spectra,” Proceeding of Quantum Communication and Measurement QCMC’06, 31–34 (2006).

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033,801 (2005).
[Crossref]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053,601 (2005).
[Crossref]

X. Li, J. Chen, P. L. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12, 3737–3744 (2004).
[Crossref] [PubMed]

Liang, C.

Lu, Y. J.

Z. Y. Ou and Y. J. Lu, “Cavity Enhanced Spontaneous Parametric Down-Conversion for the Prolongation of Correlation Time between Conjugate Photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
[Crossref]

Lundeen, J. S.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

Mattle, K.

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

McGuinness, H. J.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

McKinstrie, C. J.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

Milburn, G. J.

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

Murdoch, S. G.

O. Alibart, J. Fulconis, G. K. L. Wong, S. G. Murdoch, W. J. Wadsworth, and J. G. Rarity, “Photon pair generation using four-wave mixing in a microstructured fibre: theory versus experiment,” New J. of Phys. 8, 67 (2006).
[Crossref]

Ou, Z. Y.

Z. Y. Ou and Y. J. Lu, “Cavity Enhanced Spontaneous Parametric Down-Conversion for the Prolongation of Correlation Time between Conjugate Photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
[Crossref]

Z. Y. Ou, J. K. Rhee, and L. J. Wang, “Photon Bunching and Multiphoton Interference in Parametric Down-Conversion,” Phys. Rev. A 60, 593–604 (1999).
[Crossref]

Z. Y. Ou, “Parametric Down-Conversion with Coherent Pulse Pumping and Quantum Interference between Independent Fields,” Quantum Semiclass Opt. 9, 599–614 (1997).
[Crossref]

Pan, J. W.

Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel, and J. W. Pan, “Experimental demonstration of five-photon entanglement and open-destination quantum teleportation,” Nature 430, 54–58 (2004).
[Crossref] [PubMed]

J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

Radic, S.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

Rangel-Rojo, R.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

Rarity, J. G.

Raymer, M. G.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

Rhee, J. K.

Z. Y. Ou, J. K. Rhee, and L. J. Wang, “Photon Bunching and Multiphoton Interference in Parametric Down-Conversion,” Phys. Rev. A 60, 593–604 (1999).
[Crossref]

Russell, P. S.

Russell, P. S. J.

Scarani, V.

M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nature Physics 3, 692–695 (2007).
[Crossref]

Sergienko, A. V.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Sharping, J.

Sharping, J. E.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053,601 (2005).
[Crossref]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

Shih, Y. H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Shimizu, K.

K. Inoue and K. Shimizu, “Generation of Quantum-Correlated Photon Pairs in Optical Fiber: Influence of Spontaneous Raman Scattering,” Jpn. J. Appl. Phys 43, 8048–8052 (2004).
[Crossref]

Simon, C.

M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nature Physics 3, 692–695 (2007).
[Crossref]

Stoler, D.

B. Yurke and D. Stoler, “Einstein-Podolsky-Rosen effects from independent particle sources,” Phys. Rev. Lett. 68, 1251–1254 (1992).
[Crossref] [PubMed]

Takesue, H.

H. Takesue and K. Inoue, “1.5 um band quantum-correlated photon pair generation in dispersion-shifted fiber: suppression of noise photons by cooling fiber,” Opt. Express 13, 7832–7839 (2005).
[Crossref] [PubMed]

H. Takesue and K. Inoue, “Generation of 1.5-um band time-bin entanglement using spontaneous fiber four-wave mixing and planar lightwave circuit interferometers,” Phys. Rev. A 72, 041,804 (2005).
[Crossref]

U’Ren, A. B.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

W. P. Grice, A. B. U’ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multi-Photon States,” Phys. Rev. A 64, 063,815 (2001).
[Crossref]

Voss, P. L.

X. Li, J. Chen, K. F. Lee, P. L. Voss, and P. Kumar, “All-fiber photon-pair source for quantum communication: Influence of spectra,” Proceeding of Quantum Communication and Measurement QCMC’06, 31–34 (2006).

K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31, 1905–1907 (2006).
[Crossref] [PubMed]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053,601 (2005).
[Crossref]

X. Li, J. Chen, P. L. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12, 3737–3744 (2004).
[Crossref] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

Wadsworth, W. J.

Walmsley, I. A.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14,870–14,886 (2007).
[Crossref]

W. P. Grice, A. B. U’ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multi-Photon States,” Phys. Rev. A 64, 063,815 (2001).
[Crossref]

Wang, L. J.

J. Fan, A. Dogariu, and L. J. Wang, “Generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. 30, 1530–1532 (2005).
[Crossref] [PubMed]

Z. Y. Ou, J. K. Rhee, and L. J. Wang, “Photon Bunching and Multiphoton Interference in Parametric Down-Conversion,” Phys. Rev. A 60, 593–604 (1999).
[Crossref]

Weinfurter, H.

J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

M. Zukowski, A. Zeilinger, and H. Weinfurter, “Entangling Photons Radiated by Independent Pulsed Sources,” Ann. (N. Y.) Acad. Sci. 755, 91–102 (1995).
[Crossref]

Wong, G. K. L.

O. Alibart, J. Fulconis, G. K. L. Wong, S. G. Murdoch, W. J. Wadsworth, and J. G. Rarity, “Photon pair generation using four-wave mixing in a microstructured fibre: theory versus experiment,” New J. of Phys. 8, 67 (2006).
[Crossref]

Yang, T.

Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel, and J. W. Pan, “Experimental demonstration of five-photon entanglement and open-destination quantum teleportation,” Nature 430, 54–58 (2004).
[Crossref] [PubMed]

Yurke, B.

B. Yurke and D. Stoler, “Einstein-Podolsky-Rosen effects from independent particle sources,” Phys. Rev. Lett. 68, 1251–1254 (1992).
[Crossref] [PubMed]

Zbinden, H.

M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nature Physics 3, 692–695 (2007).
[Crossref]

Zeilinger, A.

J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

M. Zukowski, A. Zeilinger, and H. Weinfurter, “Entangling Photons Radiated by Independent Pulsed Sources,” Ann. (N. Y.) Acad. Sci. 755, 91–102 (1995).
[Crossref]

Zhang, A. N.

Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel, and J. W. Pan, “Experimental demonstration of five-photon entanglement and open-destination quantum teleportation,” Nature 430, 54–58 (2004).
[Crossref] [PubMed]

Zhao, Z.

Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel, and J. W. Pan, “Experimental demonstration of five-photon entanglement and open-destination quantum teleportation,” Nature 430, 54–58 (2004).
[Crossref] [PubMed]

Zukowski, M.

M. Zukowski, A. Zeilinger, and H. Weinfurter, “Entangling Photons Radiated by Independent Pulsed Sources,” Ann. (N. Y.) Acad. Sci. 755, 91–102 (1995).
[Crossref]

Ann. (N. Y.) Acad. Sci. (1)

M. Zukowski, A. Zeilinger, and H. Weinfurter, “Entangling Photons Radiated by Independent Pulsed Sources,” Ann. (N. Y.) Acad. Sci. 755, 91–102 (1995).
[Crossref]

Jpn. J. Appl. Phys (1)

K. Inoue and K. Shimizu, “Generation of Quantum-Correlated Photon Pairs in Optical Fiber: Influence of Spontaneous Raman Scattering,” Jpn. J. Appl. Phys 43, 8048–8052 (2004).
[Crossref]

Nature (3)

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

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

Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel, and J. W. Pan, “Experimental demonstration of five-photon entanglement and open-destination quantum teleportation,” Nature 430, 54–58 (2004).
[Crossref] [PubMed]

Nature Physics (1)

M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nature Physics 3, 692–695 (2007).
[Crossref]

New J. of Phys. (1)

O. Alibart, J. Fulconis, G. K. L. Wong, S. G. Murdoch, W. J. Wadsworth, and J. G. Rarity, “Photon pair generation using four-wave mixing in a microstructured fibre: theory versus experiment,” New J. of Phys. 8, 67 (2006).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Photon. Technol. Lett. (1)

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communication,” Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

Phys. Rev. A (4)

W. P. Grice, A. B. U’ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multi-Photon States,” Phys. Rev. A 64, 063,815 (2001).
[Crossref]

H. Takesue and K. Inoue, “Generation of 1.5-um band time-bin entanglement using spontaneous fiber four-wave mixing and planar lightwave circuit interferometers,” Phys. Rev. A 72, 041,804 (2005).
[Crossref]

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033,801 (2005).
[Crossref]

Z. Y. Ou, J. K. Rhee, and L. J. Wang, “Photon Bunching and Multiphoton Interference in Parametric Down-Conversion,” Phys. Rev. A 60, 593–604 (1999).
[Crossref]

Phys. Rev. Lett. (5)

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053,601 (2005).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Z. Y. Ou and Y. J. Lu, “Cavity Enhanced Spontaneous Parametric Down-Conversion for the Prolongation of Correlation Time between Conjugate Photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
[Crossref]

B. Yurke and D. Stoler, “Einstein-Podolsky-Rosen effects from independent particle sources,” Phys. Rev. Lett. 68, 1251–1254 (1992).
[Crossref] [PubMed]

J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

Proceeding of Quantum Communication and Measurement (1)

X. Li, J. Chen, K. F. Lee, P. L. Voss, and P. Kumar, “All-fiber photon-pair source for quantum communication: Influence of spectra,” Proceeding of Quantum Communication and Measurement QCMC’06, 31–34 (2006).

Quantum Semiclass Opt. (1)

Z. Y. Ou, “Parametric Down-Conversion with Coherent Pulse Pumping and Quantum Interference between Independent Fields,” Quantum Semiclass Opt. 9, 599–614 (1997).
[Crossref]

Other (1)

G. P. Agrawal, Nonlinear fiber optics (Elsevier Pte Ltd., Singapore, 2005).

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Figures (8)

Fig. 1.
Fig. 1. The envelope of the function |α(ωs,ωi )|2 for the Gaussian shaped pump pulse, whose central wavelength and FWHM are 1541 nm and 1 nm, respectively. In this and subsequent figures, the two axes represent the signal and idler frequencies, although as an aid to the reader, the frequencies have been converted to wavelengths. The symmetry line of the function |α(ωs , ωi )|2 can be defined by ωs +ωi =2ω p0, i.e., λi =λsλ p0/(2λs -λ p0).

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Fig. 2.
Fig. 2. The three top plots are the envelopes of phase-matching function |ϕ(ωs , ωi )|2, the frequency (wavelength) difference Δ/2π in plots (a), (b), and (c) is 2.17 THz (17.16 nm), 2.67 THz (21.16 nm), and 3.18 THz (25.16 nm), respectively; the three bottom plots are the normalized joint spectral intensity |F(ωs , ωi )|2 for spontaneous FWM in DSF, the frequency (wavelength) difference Δ/2π in plots (d), (e), and (f) is 2.17 THz (17.16 nm), 2.67 THz (21.16 nm), and 3.18 THz (25.16 nm), respectively. The solid line in each plot is the symmetry axis of the exponential function |α(ωs , ωi )|2; the cross-hairs in each plots corresponding to Ω i =Ω s =0. The parameters in Eq. (11) are λ 0=1540 nm, γ=2/W·km, L=0.3 km, Dslope =0.075 ps/(nm2·km), λ p0=1541 nm and γPpL=1.

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Fig. 3.
Fig. 3. Plots of phase-matching function ∣ sin ⁡ c [ k ″ L 8 ( Ω s + Δ ) 2 + k ‴ L 16 Δ 2 Ω s + γ P p L ] ∣ 2 = ∣ sin ⁡ c [ k ″ L 8 ( ω s − ω i 0 ) 2 + k ‴ L 4 ( ω i 0 − ω p 0 ) 2 ( ω s + ω i 0 − 2 ω p 0 ) + γ P p L ] ∣ 2 (a, c, e) and joint spectral intensity |F(ωs , ωi )|2 (b, d, f) under the assumption Ω i =0. The zero dispersion wavelength and dispersion slope of the DSF are λ 0=1540 nm and Dslope =0.075 ps/(nm2·km), respectively. For (a) and (b), the central wavelength of pump and SPM term are λ p0=1541 nm and γPpL=1, respectively; for (c) and (d), λ p0=1541 nm and γPpL=0.5; for (e) and (f), λ p0=1542 nm and γPpL=1. In the plots (a), (c) and (e), the solid line is the symmetry axis of the corresponding exponential function |α(ωs , ωi )|2, and in the vicinity of the cross-hairs, whose corresponding frequency difference is Δ0, the sum of second order dispersion term and SPM term in Eq. (15) (and Eq. (11)) is negligible.

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Fig. 4.
Fig. 4. A schematic of the experimental setup; scattered signal and idler photons emerging from the port labeled “Out” are detected; F, filter; G, grating; TF, tunable filter; AWG, array waveguide grating.

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Fig. 5.
Fig. 5. (a) True coincidences versus the central wavelength of filter F2 in signal band when the central wavelength of idler is fixed at 1533.3 nm, 1531.74 nm, and 1530.18 nm, respectively. In the experiment, λ p0=1540.65 nm, and the average pump power is about 0.4mW. The blue, pink and purple solid curves are fits of the Gaussian function f ( λ ) = 2.12 exp ⁡ [ − ( λ − 1547.96 1.16 ) 2 ] , f ( λ ) = 1.48 exp ⁡ [ − ( λ − 1549.69 1.26 ) 2 ] , and f ( λ ) = 1.56 exp ⁡ [ − ( λ − 1551.18 1.21 ) 2 ] , respectively. (b) Idler central wavelengths versus the central wavelengths of the corresponding fitting curves. The solid line defined by λi =λsλ p0/(2λs -λ p0) is the symmetry axis of the pump pulses.

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Fig. 6.
Fig. 6. The same as in Fig. 5, except λ p0=1541.7 nm, and the central wavelength of idler is fixed at 1534.88 nm, 1533.3 nm, and 1531.74 nm, respectively. In the experiment, the average pump power is about 0.4 mW (in (a) and (b)) and 0.2 mW (in (c) and (d)), respectively. The blue, purple and pink solid curves in (a) are fits to the Gaussian function f ( λ ) = 2.3 exp ⁡ [ − ( λ − 1548.47 1.41 ) 2 ] , f ( λ ) = 1.64 exp ⁡ [ − ( λ − 1549.97 1.3 ) 2 ] , and f ( λ ) = 2.15 exp ⁡ [ − ( λ − 1551.42 1.4 ) 2 ] , respectively; the blue, purple and pink solid curves in (c) are fits to the Gaussian function f ( λ ) = 3.92 exp ⁡ [ − ( λ − 1548.64 0.84 ) 2 ] , f ( λ ) = 3.16 exp ⁡ [ − ( λ − 1550.06 0.85 ) 2 ] , and f ( λ ) = 3.51 exp ⁡ [ − ( λ − 1551.78 0.79 ) 2 ] , respectively.

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Fig. 7.
Fig. 7. Plot of joint spectral function |F(ωs , ωi )|2 with A/B=10 for (a) σp =3.3/A, (b) σp =0.37/A, and (c) σp =1.1/A.

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Fig. 8.
Fig. 8. Plot of joint spectral function |F(ωs , ωi )|2 with σp =3.3/A for (a) A=B and (b) A=-B.

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Equations (17)

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∣ Ψ ( ω s , ω i ) 〉 = A ∫ dt ∫ − L 0 d z E s ( − ) ( t , z ) E i ( − ) ( t , z ) E p ( + ) ( t , z ) E p ( + ) ( t , z ) ∣ 0 〉 ,
E s ( − ) ( t , z ) = ∫ d ω s a s + e − i ( k s z − ω s t )
E i ( − ) ( t , z ) = ∫ d ω i a i + e − i ( k i z − ω i t )
E p ( + ) ( t , z ) = e − i ω p 0 t e − i γ P p z ∫ d Ω p e − Ω p 2 2 σ p 2 e i ( k p z − Ω p t ) ,
∣ Ψ ( ω s , ω i ) 〉 = g ∫ d ω s ∫ d ω i F ( ω s , ω i ) ∣ ω s 〉 ∣ ω i 〉 ,
g ∝ π 2 χ ( 3 ) P p σ p ,
F ( ω s , ω i ) = ∫ − L 0 dz exp ⁡ { − i Δ kz − 2 i γ P p z } 1 − i k ″ σ p 2 z − i 2 k ‴ ( Ω s + Ω i ) z σ p 2 exp ⁡ { − ( Ω s + Ω i ) 2 4 σ p 2 } ,
Δ k = k ″ 4 ( Ω s − Ω i + Δ ) 2 + k ‴ 8 Δ 2 ( Ω s + Ω i ) ,
F ( ω s , ω i ) = exp ⁡ [ iL ( Δ k 2 + γ P p ) ] α ( ω s , ω i ) ϕ ( ω s , ω i ) ,
α ( ω s , ω i ) = exp ⁡ [ − ( Ω s + Ω i ) 2 4 σ p 2 ]
ϕ ( ω s , ω i ) = sin ⁡ c [ k ″ L 8 ( Ω s − Ω i + Δ ) 2 + k ‴ L 16 Δ 2 ( Ω s + Ω i ) + γ P p L ]
S s ( i ) ( Ω ) = ∫ d Ω i ( s ) e − Ω i ( s ) 2 σ 0 2 ∣ F ( ω s , ω i ) ∣ 2 = ∫ d Ω i ( s ) e − Ω i ( s ) 2 σ 0 2 ∣ α ( ω s , ω i ) ϕ ( ω s , ω i ) ∣ 2
S s ( i ) ( Ω ) = g ′ exp ⁡ ( − Ω 2 [ 1 + Γ σ p 2 ( k ‴ L Δ 2 4 ) 2 ] 2 σ p 2 + σ 0 2 + Γ σ p 2 σ 0 2 ( k ‴ L Δ 2 4 ) 2 )
S s ( Ω ) = S i ( Ω ) ≈ 2 π σ 0 σ p 2 σ p 2 + σ 0 2 exp ⁡ [ − Ω 2 2 σ p 2 + σ 0 2 ]
S s ( Ω s ) ∝ ∣ F ( ω s , ω i ) ∣ 2 = exp ⁡ ( − Ω s 2 2 σ p 2 ) ∣ sin ⁡ c [ k ″ L 8 ( Ω s + Δ ) 2 + k ‴ L 16 Δ 2 Ω s + γ P p L ] ∣ 2
ϕ ( ω s , ω i ) = sin ⁡ c [ k ″ L 8 Δ 2 + γ P p L + k ″ L 4 Δ ( Ω s − Ω i ) + k ‴ L 16 Δ 2 ( Ω s + Ω i ) ] ,
ϕ ( ω s , ω i ) = sin ⁡ c [ A ( Ω s − Ω i ) + B ( Ω s + Ω i ) ] ,

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