Abstract

Quantum entanglement of two photons created by spontaneous parametric downconversion (SPDC) can be used to probe quantum optical phenomena during a single cycle of light. Harris [Opt. Express 98, 063602 (2007)] suggested using ultrabroad parametric fluorescenc generated from a quasi-phase-matched (QPM) device whose poling period is chirped. In the Harris’s original proposal, it is assumed that the photons are collinearly generated and then spatially separated by frequency filtering Here, we alternatively propose using noncollinearly generated SPDC. In our numerical calculation, to achieve 1.2 cycle temporal correlation for a 532 nm pump laser, only 10% -chirped device is sufficien when noncollinear condition is applied, while a largely chirped (50%) device is required in collinear condition. We also experimentally demonstrate an octave-spanning (790–1610 nm) noncollinear parametric fluorescenc from a 10% chirped MgSLT crystal using both a superconducting nanowire single-photon detector and photomultiplier tube as photon detectors. The observed SPDC bandwidth is 194 THz, which is the largest width achieved to date for a chirped QPM device. From this experimental result, our numerical analysis predicts that the bi-photon can be compressed to 1.2 cycles with appropriate phase compensation.

© 2012 OSA

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  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
  2. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316, 726–729 (2007).
    [CrossRef] [PubMed]
  3. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59, 2044–2046 (1987).
    [CrossRef] [PubMed]
  4. 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]
  5. V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett.87, 117902 (2001).
    [CrossRef] [PubMed]
  6. A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A65, 053817 (2002).
    [CrossRef]
  7. N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
    [CrossRef]
  8. J. Javanainen and P. L. Gould, “Linear intensity sependence of a two-photon transition rate,” Phys. Rev. A41, 5088–5091 (1990).
    [CrossRef] [PubMed]
  9. J. Gea-Banacloche, “Two-photon absorption of nonclassical light,” Phys. Rev. Lett.62, 1603 (1989).
    [CrossRef] [PubMed]
  10. O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett.101, 153602 (2008).
    [CrossRef] [PubMed]
  11. K. G. Katamadze and S. P. Kulik, “Control of the spectrum of the biphoton field” Journal of Experimental and Theoretical Physics112, 20 (2011).
    [CrossRef]
  12. M. Okano, R. Okamoto, A. Tanaka, S. Subashchandran, and S. Takeuchi, “Generation of broadband spontaneous parametric fluorescenc using multiple bulk nonlinear crystals,” Opt. Express20 (13), 13977–13987 (2012).
    [CrossRef] [PubMed]
  13. S. E. Harris, “ Chirp and compress: toward single-cycle biphotons,” Phys. Rev. Lett.98, 063602 (2007).
    [CrossRef] [PubMed]
  14. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers” J. Opt. Soc. Am. B25, 1402–1413 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  16. S. Sensarn, G. Y. Yin, and S. E. Harris, “Generation and compression of chirped biphotons,” Phys. Rev. Lett.104, 253602 (2010).
    [CrossRef] [PubMed]
  17. S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).
  18. M. Niigaki, T. Hirohata, T. Suzuki, H. Kan, and T. Hiruma, “Field-assisted photoemission from InP/InGaAsP photocathode with p/n junction,” Appl. Phys. Lett.71, 2493–2495 (1997).
    [CrossRef]
  19. A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett.94, 073601 (2005).
    [CrossRef]
  20. S. Du, “Atomic-resonance-enhanced nonlinear optical frequency conversion with entangled photon pairs,” Phys. Rev. A83, 033807 (2011).
    [CrossRef]
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    [CrossRef]
  22. S.-Y. Baek and Y.-H. Kim, “Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion,” Phys. Rev. A77, 043807 (2008).
    [CrossRef]
  23. A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A68, 053807 (2003).
    [CrossRef]
  24. T. Katagai, S. Jianhong, I. Shoji, M. Nakamura, and S. Kurimura, “Refractive index dispersion of Mg-doped stoichiometric LiTaO3,” in Meeting Abstracts of the Japan Society of Applied Physics, (Academic, Toyama, Japan, 2009), 3, 11a-P8-33, 1107.
  25. N. E. Yu, S. Kurimura, Y. Nomura, and K. Kitamura, “Stable high-power green light generation with a periodically poled stoichiometric lithium tantalate,” Mat. Sci. Eng. B-Solid.120, 146–149 (2005).
    [CrossRef]
  26. H. Lim, T. Katagai, S. Kurimura, T. Shimizu, K. Noguchi, N. Ohmae, N. Mio, and I. Shoji, “Thermal performance in high power SHG characterized by phase-matched calorimetry,” Opt. Express19, 22588–22593 (2011).
    [CrossRef] [PubMed]
  27. W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A56, 1627 (1997).
    [CrossRef]
  28. S. Akturk, X. Gu, M. Kimmel, and R. Trebino, “Extremely simple single-prism ultrashort-pulse compressor,” Opt. Express14, 10101–10108 (2006).
    [CrossRef] [PubMed]
  29. Note that the spectral phase of biphotons is mostly an even function of ωp/2; because a Hong-Ou-Mandel interferometer cancels even order dispersion, the HOM dip that appears in reference [15] using a chirped PPSLT crystal leads to a FWHM of 4 fs, which is nearly Fourier-transform limited.

2012 (1)

2011 (3)

H. Lim, T. Katagai, S. Kurimura, T. Shimizu, K. Noguchi, N. Ohmae, N. Mio, and I. Shoji, “Thermal performance in high power SHG characterized by phase-matched calorimetry,” Opt. Express19, 22588–22593 (2011).
[CrossRef] [PubMed]

K. G. Katamadze and S. P. Kulik, “Control of the spectrum of the biphoton field” Journal of Experimental and Theoretical Physics112, 20 (2011).
[CrossRef]

S. Du, “Atomic-resonance-enhanced nonlinear optical frequency conversion with entangled photon pairs,” Phys. Rev. A83, 033807 (2011).
[CrossRef]

2010 (1)

S. Sensarn, G. Y. Yin, and S. E. Harris, “Generation and compression of chirped biphotons,” Phys. Rev. Lett.104, 253602 (2010).
[CrossRef] [PubMed]

2009 (1)

N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
[CrossRef]

2008 (5)

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett.101, 153602 (2008).
[CrossRef] [PubMed]

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100, 183601 (2008).
[CrossRef] [PubMed]

S.-Y. Baek and Y.-H. Kim, “Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion,” Phys. Rev. A77, 043807 (2008).
[CrossRef]

M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifier using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B25, 463–480 (2008).
[CrossRef]

M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers” J. Opt. Soc. Am. B25, 1402–1413 (2008).
[CrossRef]

2007 (2)

S. E. Harris, “ Chirp and compress: toward single-cycle biphotons,” Phys. Rev. Lett.98, 063602 (2007).
[CrossRef] [PubMed]

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316, 726–729 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett.94, 073601 (2005).
[CrossRef]

2003 (1)

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A68, 053807 (2003).
[CrossRef]

2002 (1)

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A65, 053817 (2002).
[CrossRef]

2001 (1)

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett.87, 117902 (2001).
[CrossRef] [PubMed]

1997 (2)

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A56, 1627 (1997).
[CrossRef]

M. Niigaki, T. Hirohata, T. Suzuki, H. Kan, and T. Hiruma, “Field-assisted photoemission from InP/InGaAsP photocathode with p/n junction,” Appl. Phys. Lett.71, 2493–2495 (1997).
[CrossRef]

1995 (1)

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]

1990 (1)

J. Javanainen and P. L. Gould, “Linear intensity sependence of a two-photon transition rate,” Phys. Rev. A41, 5088–5091 (1990).
[CrossRef] [PubMed]

1989 (1)

J. Gea-Banacloche, “Two-photon absorption of nonclassical light,” Phys. Rev. Lett.62, 1603 (1989).
[CrossRef] [PubMed]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59, 2044–2046 (1987).
[CrossRef] [PubMed]

Abouraddy, A. F.

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A65, 053817 (2002).
[CrossRef]

Afeyan, B.

Akturk, S.

Baek, S.-Y.

S.-Y. Baek and Y.-H. Kim, “Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion,” Phys. Rev. A77, 043807 (2008).
[CrossRef]

Carrasco, S.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100, 183601 (2008).
[CrossRef] [PubMed]

Charbonneau-Lefort, M.

Chen, J.

S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

Dayan, B.

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett.94, 073601 (2005).
[CrossRef]

Du, S.

S. Du, “Atomic-resonance-enhanced nonlinear optical frequency conversion with entangled photon pairs,” Phys. Rev. A83, 033807 (2011).
[CrossRef]

Fejer, M. M.

Friesem, A. A.

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett.94, 073601 (2005).
[CrossRef]

Gatti, A.

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A68, 053807 (2003).
[CrossRef]

Gea-Banacloche, J.

J. Gea-Banacloche, “Two-photon absorption of nonclassical light,” Phys. Rev. Lett.62, 1603 (1989).
[CrossRef] [PubMed]

Giovannetti, V.

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett.87, 117902 (2001).
[CrossRef] [PubMed]

Goltsman, G. N.

N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
[CrossRef]

Gould, P. L.

J. Javanainen and P. L. Gould, “Linear intensity sependence of a two-photon transition rate,” Phys. Rev. A41, 5088–5091 (1990).
[CrossRef] [PubMed]

Grice, W. P.

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A56, 1627 (1997).
[CrossRef]

Gu, X.

Harris, S. E.

S. Sensarn, G. Y. Yin, and S. E. Harris, “Generation and compression of chirped biphotons,” Phys. Rev. Lett.104, 253602 (2010).
[CrossRef] [PubMed]

S. E. Harris, “ Chirp and compress: toward single-cycle biphotons,” Phys. Rev. Lett.98, 063602 (2007).
[CrossRef] [PubMed]

Hirohata, T.

M. Niigaki, T. Hirohata, T. Suzuki, H. Kan, and T. Hiruma, “Field-assisted photoemission from InP/InGaAsP photocathode with p/n junction,” Appl. Phys. Lett.71, 2493–2495 (1997).
[CrossRef]

Hiruma, T.

M. Niigaki, T. Hirohata, T. Suzuki, H. Kan, and T. Hiruma, “Field-assisted photoemission from InP/InGaAsP photocathode with p/n junction,” Appl. Phys. Lett.71, 2493–2495 (1997).
[CrossRef]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59, 2044–2046 (1987).
[CrossRef] [PubMed]

Hum, D. S.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100, 183601 (2008).
[CrossRef] [PubMed]

Javanainen, J.

J. Javanainen and P. L. Gould, “Linear intensity sependence of a two-photon transition rate,” Phys. Rev. A41, 5088–5091 (1990).
[CrossRef] [PubMed]

Jianhong, S.

T. Katagai, S. Jianhong, I. Shoji, M. Nakamura, and S. Kurimura, “Refractive index dispersion of Mg-doped stoichiometric LiTaO3,” in Meeting Abstracts of the Japan Society of Applied Physics, (Academic, Toyama, Japan, 2009), 3, 11a-P8-33, 1107.

Kan, H.

M. Niigaki, T. Hirohata, T. Suzuki, H. Kan, and T. Hiruma, “Field-assisted photoemission from InP/InGaAsP photocathode with p/n junction,” Appl. Phys. Lett.71, 2493–2495 (1997).
[CrossRef]

Kang, L.

S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).

Katagai, T.

H. Lim, T. Katagai, S. Kurimura, T. Shimizu, K. Noguchi, N. Ohmae, N. Mio, and I. Shoji, “Thermal performance in high power SHG characterized by phase-matched calorimetry,” Opt. Express19, 22588–22593 (2011).
[CrossRef] [PubMed]

T. Katagai, S. Jianhong, I. Shoji, M. Nakamura, and S. Kurimura, “Refractive index dispersion of Mg-doped stoichiometric LiTaO3,” in Meeting Abstracts of the Japan Society of Applied Physics, (Academic, Toyama, Japan, 2009), 3, 11a-P8-33, 1107.

Katamadze, K. G.

K. G. Katamadze and S. P. Kulik, “Control of the spectrum of the biphoton field” Journal of Experimental and Theoretical Physics112, 20 (2011).
[CrossRef]

Kim, Y.-H.

S.-Y. Baek and Y.-H. Kim, “Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion,” Phys. Rev. A77, 043807 (2008).
[CrossRef]

Kimmel, M.

Kitamura, K.

N. E. Yu, S. Kurimura, Y. Nomura, and K. Kitamura, “Stable high-power green light generation with a periodically poled stoichiometric lithium tantalate,” Mat. Sci. Eng. B-Solid.120, 146–149 (2005).
[CrossRef]

Kulik, S. P.

K. G. Katamadze and S. P. Kulik, “Control of the spectrum of the biphoton field” Journal of Experimental and Theoretical Physics112, 20 (2011).
[CrossRef]

Kurimura, S.

H. Lim, T. Katagai, S. Kurimura, T. Shimizu, K. Noguchi, N. Ohmae, N. Mio, and I. Shoji, “Thermal performance in high power SHG characterized by phase-matched calorimetry,” Opt. Express19, 22588–22593 (2011).
[CrossRef] [PubMed]

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett.101, 153602 (2008).
[CrossRef] [PubMed]

N. E. Yu, S. Kurimura, Y. Nomura, and K. Kitamura, “Stable high-power green light generation with a periodically poled stoichiometric lithium tantalate,” Mat. Sci. Eng. B-Solid.120, 146–149 (2005).
[CrossRef]

T. Katagai, S. Jianhong, I. Shoji, M. Nakamura, and S. Kurimura, “Refractive index dispersion of Mg-doped stoichiometric LiTaO3,” in Meeting Abstracts of the Japan Society of Applied Physics, (Academic, Toyama, Japan, 2009), 3, 11a-P8-33, 1107.

Kuzucu, O.

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett.101, 153602 (2008).
[CrossRef] [PubMed]

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]

Lim, H.

Lloyd, S.

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett.87, 117902 (2001).
[CrossRef] [PubMed]

Lugiato, L. A.

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A68, 053807 (2003).
[CrossRef]

Maccone, L.

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett.87, 117902 (2001).
[CrossRef] [PubMed]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59, 2044–2046 (1987).
[CrossRef] [PubMed]

Mattle, K.

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]

Minaeva, O.

N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
[CrossRef]

Mio, N.

Mohan, N.

N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
[CrossRef]

Nagata, T.

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316, 726–729 (2007).
[CrossRef] [PubMed]

Nakamura, M.

T. Katagai, S. Jianhong, I. Shoji, M. Nakamura, and S. Kurimura, “Refractive index dispersion of Mg-doped stoichiometric LiTaO3,” in Meeting Abstracts of the Japan Society of Applied Physics, (Academic, Toyama, Japan, 2009), 3, 11a-P8-33, 1107.

Nasr, M. B.

N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
[CrossRef]

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100, 183601 (2008).
[CrossRef] [PubMed]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A65, 053817 (2002).
[CrossRef]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

Niigaki, M.

M. Niigaki, T. Hirohata, T. Suzuki, H. Kan, and T. Hiruma, “Field-assisted photoemission from InP/InGaAsP photocathode with p/n junction,” Appl. Phys. Lett.71, 2493–2495 (1997).
[CrossRef]

Noguchi, K.

Nomura, Y.

N. E. Yu, S. Kurimura, Y. Nomura, and K. Kitamura, “Stable high-power green light generation with a periodically poled stoichiometric lithium tantalate,” Mat. Sci. Eng. B-Solid.120, 146–149 (2005).
[CrossRef]

O’Brien, J. L.

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316, 726–729 (2007).
[CrossRef] [PubMed]

Ohmae, N.

Okamoto, R.

M. Okano, R. Okamoto, A. Tanaka, S. Subashchandran, and S. Takeuchi, “Generation of broadband spontaneous parametric fluorescenc using multiple bulk nonlinear crystals,” Opt. Express20 (13), 13977–13987 (2012).
[CrossRef] [PubMed]

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316, 726–729 (2007).
[CrossRef] [PubMed]

S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).

Okano, M.

M. Okano, R. Okamoto, A. Tanaka, S. Subashchandran, and S. Takeuchi, “Generation of broadband spontaneous parametric fluorescenc using multiple bulk nonlinear crystals,” Opt. Express20 (13), 13977–13987 (2012).
[CrossRef] [PubMed]

S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59, 2044–2046 (1987).
[CrossRef] [PubMed]

Pe’er, A.

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett.94, 073601 (2005).
[CrossRef]

Saleh, B. E. A.

N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
[CrossRef]

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100, 183601 (2008).
[CrossRef] [PubMed]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A65, 053817 (2002).
[CrossRef]

Saleh, M. F.

N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
[CrossRef]

San Miguel, M.

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A68, 053807 (2003).
[CrossRef]

Sasaki, K.

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316, 726–729 (2007).
[CrossRef] [PubMed]

Sensarn, S.

S. Sensarn, G. Y. Yin, and S. E. Harris, “Generation and compression of chirped biphotons,” Phys. Rev. Lett.104, 253602 (2010).
[CrossRef] [PubMed]

Sergienko, A. V.

N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
[CrossRef]

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100, 183601 (2008).
[CrossRef] [PubMed]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A65, 053817 (2002).
[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]

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, T.

Shoji, I.

H. Lim, T. Katagai, S. Kurimura, T. Shimizu, K. Noguchi, N. Ohmae, N. Mio, and I. Shoji, “Thermal performance in high power SHG characterized by phase-matched calorimetry,” Opt. Express19, 22588–22593 (2011).
[CrossRef] [PubMed]

T. Katagai, S. Jianhong, I. Shoji, M. Nakamura, and S. Kurimura, “Refractive index dispersion of Mg-doped stoichiometric LiTaO3,” in Meeting Abstracts of the Japan Society of Applied Physics, (Academic, Toyama, Japan, 2009), 3, 11a-P8-33, 1107.

Silberberg, Y.

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett.94, 073601 (2005).
[CrossRef]

Subashchandran, S.

M. Okano, R. Okamoto, A. Tanaka, S. Subashchandran, and S. Takeuchi, “Generation of broadband spontaneous parametric fluorescenc using multiple bulk nonlinear crystals,” Opt. Express20 (13), 13977–13987 (2012).
[CrossRef] [PubMed]

S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).

Suzuki, T.

M. Niigaki, T. Hirohata, T. Suzuki, H. Kan, and T. Hiruma, “Field-assisted photoemission from InP/InGaAsP photocathode with p/n junction,” Appl. Phys. Lett.71, 2493–2495 (1997).
[CrossRef]

Takeuchi, S.

M. Okano, R. Okamoto, A. Tanaka, S. Subashchandran, and S. Takeuchi, “Generation of broadband spontaneous parametric fluorescenc using multiple bulk nonlinear crystals,” Opt. Express20 (13), 13977–13987 (2012).
[CrossRef] [PubMed]

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316, 726–729 (2007).
[CrossRef] [PubMed]

S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).

Tanaka, A.

M. Okano, R. Okamoto, A. Tanaka, S. Subashchandran, and S. Takeuchi, “Generation of broadband spontaneous parametric fluorescenc using multiple bulk nonlinear crystals,” Opt. Express20 (13), 13977–13987 (2012).
[CrossRef] [PubMed]

S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).

Teich, M. C.

N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
[CrossRef]

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100, 183601 (2008).
[CrossRef] [PubMed]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A65, 053817 (2002).
[CrossRef]

Torner, L.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100, 183601 (2008).
[CrossRef] [PubMed]

Torres, J. P.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100, 183601 (2008).
[CrossRef] [PubMed]

Tovstonog, S.

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett.101, 153602 (2008).
[CrossRef] [PubMed]

Trebino, R.

Walmsley, I. A.

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A56, 1627 (1997).
[CrossRef]

Weinfurter, 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]

Wong, F. N. C.

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett.101, 153602 (2008).
[CrossRef] [PubMed]

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett.87, 117902 (2001).
[CrossRef] [PubMed]

Wu, P.

S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).

Yin, G. Y.

S. Sensarn, G. Y. Yin, and S. E. Harris, “Generation and compression of chirped biphotons,” Phys. Rev. Lett.104, 253602 (2010).
[CrossRef] [PubMed]

Yu, N. E.

N. E. Yu, S. Kurimura, Y. Nomura, and K. Kitamura, “Stable high-power green light generation with a periodically poled stoichiometric lithium tantalate,” Mat. Sci. Eng. B-Solid.120, 146–149 (2005).
[CrossRef]

Zambrini, R.

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A68, 053807 (2003).
[CrossRef]

Zeilinger, A.

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]

Zhang, L.

S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).

Appl. Optics (1)

N. Mohan, O. Minaeva, G. N. Goltsman, M. F. Saleh, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ultrabroadband coherence-domain imaging using parametric downconversion and superconducting single-photon detectors at 1064 nm,” Appl. Optics48, 4009–4017 (2009).
[CrossRef]

Appl. Phys. Lett. (1)

M. Niigaki, T. Hirohata, T. Suzuki, H. Kan, and T. Hiruma, “Field-assisted photoemission from InP/InGaAsP photocathode with p/n junction,” Appl. Phys. Lett.71, 2493–2495 (1997).
[CrossRef]

J. Opt. Soc. Am. B (2)

Journal of Experimental and Theoretical Physics (1)

K. G. Katamadze and S. P. Kulik, “Control of the spectrum of the biphoton field” Journal of Experimental and Theoretical Physics112, 20 (2011).
[CrossRef]

Opt. Express (3)

Phys. Rev. A (6)

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A56, 1627 (1997).
[CrossRef]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A65, 053817 (2002).
[CrossRef]

S. Du, “Atomic-resonance-enhanced nonlinear optical frequency conversion with entangled photon pairs,” Phys. Rev. A83, 033807 (2011).
[CrossRef]

S.-Y. Baek and Y.-H. Kim, “Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion,” Phys. Rev. A77, 043807 (2008).
[CrossRef]

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A68, 053807 (2003).
[CrossRef]

J. Javanainen and P. L. Gould, “Linear intensity sependence of a two-photon transition rate,” Phys. Rev. A41, 5088–5091 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (9)

J. Gea-Banacloche, “Two-photon absorption of nonclassical light,” Phys. Rev. Lett.62, 1603 (1989).
[CrossRef] [PubMed]

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett.101, 153602 (2008).
[CrossRef] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59, 2044–2046 (1987).
[CrossRef] [PubMed]

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]

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett.87, 117902 (2001).
[CrossRef] [PubMed]

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett.100, 183601 (2008).
[CrossRef] [PubMed]

S. Sensarn, G. Y. Yin, and S. E. Harris, “Generation and compression of chirped biphotons,” Phys. Rev. Lett.104, 253602 (2010).
[CrossRef] [PubMed]

S. E. Harris, “ Chirp and compress: toward single-cycle biphotons,” Phys. Rev. Lett.98, 063602 (2007).
[CrossRef] [PubMed]

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett.94, 073601 (2005).
[CrossRef]

Science (1)

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316, 726–729 (2007).
[CrossRef] [PubMed]

Other (5)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

S. Subashchandran, R. Okamoto, A. Tanaka, M. Okano, L. Zhang, L. Kang, J. Chen, P. Wu, and S. Takeuchi, “Spectral dependence of ultra-low dark count superconducting single photon detector for the evaluation of broadband parametric fluorescence” Proc. SPIE8268, 82681V (2011).

T. Katagai, S. Jianhong, I. Shoji, M. Nakamura, and S. Kurimura, “Refractive index dispersion of Mg-doped stoichiometric LiTaO3,” in Meeting Abstracts of the Japan Society of Applied Physics, (Academic, Toyama, Japan, 2009), 3, 11a-P8-33, 1107.

N. E. Yu, S. Kurimura, Y. Nomura, and K. Kitamura, “Stable high-power green light generation with a periodically poled stoichiometric lithium tantalate,” Mat. Sci. Eng. B-Solid.120, 146–149 (2005).
[CrossRef]

Note that the spectral phase of biphotons is mostly an even function of ωp/2; because a Hong-Ou-Mandel interferometer cancels even order dispersion, the HOM dip that appears in reference [15] using a chirped PPSLT crystal leads to a FWHM of 4 fs, which is nearly Fourier-transform limited.

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic of the chirped quasi-phase-matched device.

Fig. 2
Fig. 2

Simulations of (a) collinear (47% chirp) and (b) noncollinear(10% chirp) parametric f uorescence spectra to achieve monocycle temporal correlation. Inset: calculation of the expected SFG signals of R S F G col ( τ ) and R S F G noncol ( τ ) for a two-photon state with a FWHM of 4.4 fs based on spectra in (a) and (b).

Fig. 3
Fig. 3

Calculated tuning curve for Type-0 SPDC from the chirped-QPM device. The parametric fluorescenc in the experiment lies within the two pink boxes centered at angles of ±0.25 deg and spanning a width of 0.28 deg.

Fig. 4
Fig. 4

Experimental setup for the noncollinear geometry. MMF: multi-mode fibe , DSF: dispersion-shifted fibe , SNSPD: superconducting single photon detector, PMT: photomultiplier tube. Inset: photograph of the chirped-QPM device on a peltier unit.

Fig. 5
Fig. 5

(a) Measured parametric fluorescenc spectra after calibration for noncollinear propagation detected by an SNSPD, together with a theoretical curve including the wavelength resolution of the tunable bandpass filter. We inserted a long wavelength pass filter for the wavelength longer than 1400 nm. (b) The raw data of the fluorescence spectra at longer wavelength region (accumulation time: 5sec) after inserting the same long wavelength pass filter. Pale shadow: typical dark counts.

Fig.6
Fig.6

Logarithmic plots of the parametric fluorescenc spectra for noncollinear propagation detected by the PMT after intensity calibration. The black line is a theoretical curve accounting for the wavelength resolution of the tunable bandpass filter.

Fig. 7
Fig. 7

(a) Schematic of the prism compressor with a separation of l. (b) Simulated spectral phase of the two-photon wavefunction before (blue) and after (red) insertion of the prism compressor. SFG signals (c) before and (d) after insertion of the compressor, having a FWHM of 3600 fs (1000 cycles) and 26.6 fs (7.5 cycles). (e) SFG signal after perfect compression with a FWHM of 4.4 fs, corresponding to 1.2 cycles.

Equations (9)

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K ( z ) 2 π Λ ( z ) = 2 π Λ 0 η z
ω p = ω s + ω i
Δ k ( ω , z ; φ ) = k ( ω p ) k ( ω ) 1 ( sin φ n e ( ω ) ) 2 k ( ω p ω ) 1 ( ω ω p ω ) 2 ( sin φ n e ( ω p ω ) ) 2 K ( z ) = 0
ψ ( ω , L ; φ ) = i κ 2 π 2 η e i { k ( ω ) + k ( ω p ω ) } L e i Δ k ( ω , 0 ; φ ) 2 / 2 η × [ f ( 1 + i 2 η Δ k ( ω , 0 ; φ ) ) f ( 1 + i 2 η ( Δ k ( ω , 0 ; φ ) + η L ) ) ]
| Φ col = d ω ψ ( ω , L ; φ = 0 ) | ω , ω p ω col
| Ψ noncol = d ω ψ ( ω , L ; φ ) | ω s | ω p ω i
R S F G col ( τ ) μ | 1 2 π ω p / 2 H ( ω ) G ( ω p ω ; τ ) ψ ( ω , L ; φ = 0 ) e i ω τ d ω | 2
R S F G noncol ( τ ) μ | 1 2 π H ( ω ) G ( ω p ω ; τ ) ψ ( ω , L ; φ ) e i ω τ d ω | 2 .
ϕ spec ( ω ) k ( ω ) L + k ( ω p ω ) L [ Δ k ( ω , 0 ) ] 2 / 2 η .

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