Abstract

Quantum beats can be produced in fourth-order interference such as in a Hong–Ou–Mandel (HOM) interferometer by using photons with different frequencies. Here we present theoretically the appearance of interference of quantum beats when the HOM interferometer is combined with a Franson-type interferometer. This combination can make the interference effect of photons with different colors take place not only within the coherence time of downconverted fields but also in the region beyond that. We expect that it can provide a new method in quantum metrology, as it can realize the measurement of time intervals in three scales.

© 2015 Chinese Laser Press

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  5. K. Cho and J. Noh, “Temporal ghost imaging of a time object, dispersion cancelation, and nonlocal time lens with bi-photon state,” Opt. Commun. 285, 1275–1282 (2012).
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    [Crossref]
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    [Crossref]

2015 (1)

J. Qiu, Y. S. Zhang, G. Y. Xiang, S. S. Han, and Y. Z. Gui, “Unified view of the second-order and the fourth-order interferences in a single interferometer,” Opt. Commun. 336, 9–13 (2015).
[Crossref]

2013 (6)

A. Zhang, M. Li, and Y. H. Feng, “Experimentally achieve two photon entanglement on various emitting angle,” Chin. Opt. Lett. 11, 092701 (2013).

P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X. M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computing in a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013).
[Crossref]

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

J. Zhang, D. L. Matthias, and M. C. Carlton, “Mixing nonclassical pure states in a linear-optical network almost always generates modal entanglement,” Phys. Rev. A 88, 044301 (2013).
[Crossref]

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, and W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[Crossref]

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor quantum metrology with entangled photons,” Phys. Rev. Lett. 111, 093603 (2013).
[Crossref]

2012 (4)

C. Liu, J. F. Chen, S. C. Zhang, S. Y. Zhou, Y. H. Kim, M. M. T. Loy, G. K. L. Wong, and S. W. Du, “Two-photon interferences with degenerate and nondegenerate paired photons,” Phys. Rev. A 85, 021803 (2012).
[Crossref]

E. B. Flagg, S. V. Polyakov, T. Thomay, and G. S. Solomon, “Dynamics of nonclassical light from a single solid-state quantum emitter,” Phys. Rev. Lett. 109, 163601 (2012).
[Crossref]

K. Cho and J. Noh, “Temporal ghost imaging of a time object, dispersion cancelation, and nonlocal time lens with bi-photon state,” Opt. Commun. 285, 1275–1282 (2012).
[Crossref]

H. Y. Zhang, J. F. Li, X. Y. Liang, H. Lin, L. H. Zheng, L. B. Su, and J. Xu, “High-power and wavelength tunable diode-pumped continuous wave Yb:SSO laser,” Chin. Opt. Lett. 10, 111404 (2012).

2011 (3)

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “Quantum imaging and spatio-temporal correlations,” Opt. Spectrosc. 111, 505–509 (2011).
[Crossref]

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

D. Cavalcanti, N. Brunner, P. Skrzypczyk, A. Salles, and V. Scarani, “Large violation of Bell inequalities using both particle and wave measurements,” Phys. Rev. A 84, 022105 (2011).
[Crossref]

2009 (2)

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[Crossref]

S. Y. Baek, Y. W. Cho, and Y. H. Kim, “Nonlocal dispersion cancellation using entangled photons,” Opt. Express 17, 19241 (2009).
[Crossref]

2006 (1)

S. Mori, J. S. derholm, N. Namekata, and S. Inoue, “On the distribution of 1550-nm photon pairs efficiently generated using a periodically poled lithium niobate waveguide,” Opt. Commun. 264, 156–162 (2006).
[Crossref]

2005 (1)

Y. Li and T. Kobayashi, “Multi-photon entangled states from two-crystal geometry parametric down-conversion and their application in quantum teleportation,” Opt. Commun. 244, 285–289 (2005).
[Crossref]

2004 (2)

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pdua, “Time control of two-photon interference,” Phys. Rev. A 69, 053817 (2004).
[Crossref]

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004).
[Crossref]

2001 (2)

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref]

Y. H. Kim, S. P. Kulik, and Y. H. Shih, “Bell-state preparation using pulsed nondegenerate two-photon entanglement,” Phys. Rev. A 63, 060301(R) (2001).
[Crossref]

2000 (1)

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Quantum communication with correlated nonclassical states,” Phys. Rev. A 62, 042311 (2000).
[Crossref]

1998 (1)

D. V. Strekalov, T. B. Pittman, and Y. H. Shih, “What we can learn about single photons in a two-photon interference experiment,” Phys. Rev. A 57, 567–570 (1998).
[Crossref]

1994 (2)

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[Crossref]

Y. H. Shih and A. V. Sergienko, “Observation of quantum beating in a simple beam-splitting experiment: two-particle entanglement in spin and space-time,” Phys. Rev. A 50, 2564–2568 (1994).
[Crossref]

1992 (2)

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer,” Phys. Rev. A 45, 6659–6665 (1992).
[Crossref]

J. D. Franson, “Nonlocal cancellation of dispersion,” Phys. Rev. A 45, 3126–3132 (1992).
[Crossref]

1988 (2)

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref]

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54–57 (1988).
[Crossref]

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]

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref]

Andersson, J.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Baek, S. Y.

Barbieri, M.

P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X. M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computing in a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013).
[Crossref]

Bell, B.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor quantum metrology with entangled photons,” Phys. Rev. Lett. 111, 093603 (2013).
[Crossref]

Bonato, C.

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[Crossref]

Brambilla, E.

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “Quantum imaging and spatio-temporal correlations,” Opt. Spectrosc. 111, 505–509 (2011).
[Crossref]

Brunner, N.

D. Cavalcanti, N. Brunner, P. Skrzypczyk, A. Salles, and V. Scarani, “Large violation of Bell inequalities using both particle and wave measurements,” Phys. Rev. A 84, 022105 (2011).
[Crossref]

Cai, X. D.

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Carlton, M. C.

J. Zhang, D. L. Matthias, and M. C. Carlton, “Mixing nonclassical pure states in a linear-optical network almost always generates modal entanglement,” Phys. Rev. A 88, 044301 (2013).
[Crossref]

Caspani, L.

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “Quantum imaging and spatio-temporal correlations,” Opt. Spectrosc. 111, 505–509 (2011).
[Crossref]

Cavalcanti, D.

D. Cavalcanti, N. Brunner, P. Skrzypczyk, A. Salles, and V. Scarani, “Large violation of Bell inequalities using both particle and wave measurements,” Phys. Rev. A 84, 022105 (2011).
[Crossref]

Chan, P.

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, and W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[Crossref]

Chen, J. F.

C. Liu, J. F. Chen, S. C. Zhang, S. Y. Zhou, Y. H. Kim, M. M. T. Loy, G. K. L. Wong, and S. W. Du, “Two-photon interferences with degenerate and nondegenerate paired photons,” Phys. Rev. A 85, 021803 (2012).
[Crossref]

Chen, M. C.

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Chiao, R. Y.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer,” Phys. Rev. A 45, 6659–6665 (1992).
[Crossref]

Cho, K.

K. Cho and J. Noh, “Temporal ghost imaging of a time object, dispersion cancelation, and nonlocal time lens with bi-photon state,” Opt. Commun. 285, 1275–1282 (2012).
[Crossref]

Cho, Y. W.

Clark, A. S.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor quantum metrology with entangled photons,” Phys. Rev. Lett. 111, 093603 (2013).
[Crossref]

derholm, J. S.

S. Mori, J. S. derholm, N. Namekata, and S. Inoue, “On the distribution of 1550-nm photon pairs efficiently generated using a periodically poled lithium niobate waveguide,” Opt. Commun. 264, 156–162 (2006).
[Crossref]

Du, S. W.

C. Liu, J. F. Chen, S. C. Zhang, S. Y. Zhou, Y. H. Kim, M. M. T. Loy, G. K. L. Wong, and S. W. Du, “Two-photon interferences with degenerate and nondegenerate paired photons,” Phys. Rev. A 85, 021803 (2012).
[Crossref]

Feng, Y. H.

Flagg, E. B.

E. B. Flagg, S. V. Polyakov, T. Thomay, and G. S. Solomon, “Dynamics of nonclassical light from a single solid-state quantum emitter,” Phys. Rev. Lett. 109, 163601 (2012).
[Crossref]

Föhlisch, A.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Franson, J. D.

J. D. Franson, “Nonlocal cancellation of dispersion,” Phys. Rev. A 45, 3126–3132 (1992).
[Crossref]

Gatti, A.

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “Quantum imaging and spatio-temporal correlations,” Opt. Spectrosc. 111, 505–509 (2011).
[Crossref]

Gelmukhanov, F.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Gu, M.

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Gui, Y. Z.

J. Qiu, Y. S. Zhang, G. Y. Xiang, S. S. Han, and Y. Z. Gui, “Unified view of the second-order and the fourth-order interferences in a single interferometer,” Opt. Commun. 336, 9–13 (2015).
[Crossref]

Han, S. S.

J. Qiu, Y. S. Zhang, G. Y. Xiang, S. S. Han, and Y. Z. Gui, “Unified view of the second-order and the fourth-order interferences in a single interferometer,” Opt. Commun. 336, 9–13 (2015).
[Crossref]

Hennies, F.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Hennrich, M.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004).
[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]

Humphreys, P. C.

P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X. M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computing in a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013).
[Crossref]

Inoue, S.

S. Mori, J. S. derholm, N. Namekata, and S. Inoue, “On the distribution of 1550-nm photon pairs efficiently generated using a periodically poled lithium niobate waveguide,” Opt. Commun. 264, 156–162 (2006).
[Crossref]

Jedrkiewicz, O.

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “Quantum imaging and spatio-temporal correlations,” Opt. Spectrosc. 111, 505–509 (2011).
[Crossref]

Jin, X. M.

P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X. M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computing in a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013).
[Crossref]

Kannan, S.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor quantum metrology with entangled photons,” Phys. Rev. Lett. 111, 093603 (2013).
[Crossref]

Karlsson, H. O.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Kennedy, B.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Kim, Y. H.

C. Liu, J. F. Chen, S. C. Zhang, S. Y. Zhou, Y. H. Kim, M. M. T. Loy, G. K. L. Wong, and S. W. Du, “Two-photon interferences with degenerate and nondegenerate paired photons,” Phys. Rev. A 85, 021803 (2012).
[Crossref]

S. Y. Baek, Y. W. Cho, and Y. H. Kim, “Nonlocal dispersion cancellation using entangled photons,” Opt. Express 17, 19241 (2009).
[Crossref]

Y. H. Kim, S. P. Kulik, and Y. H. Shih, “Bell-state preparation using pulsed nondegenerate two-photon entanglement,” Phys. Rev. A 63, 060301(R) (2001).
[Crossref]

Kimble, H. J.

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Quantum communication with correlated nonclassical states,” Phys. Rev. A 62, 042311 (2000).
[Crossref]

Klyshko, D. N.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[Crossref]

D. N. Klyshko, Photons and Nonlinear Optics (Gordon & Breach, 1988).

Kobayashi, T.

Y. Li and T. Kobayashi, “Multi-photon entangled states from two-crystal geometry parametric down-conversion and their application in quantum teleportation,” Opt. Commun. 244, 285–289 (2005).
[Crossref]

Kolthammer, W. S.

P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X. M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computing in a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013).
[Crossref]

Kuhn, A.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004).
[Crossref]

Kulik, S. P.

Y. H. Kim, S. P. Kulik, and Y. H. Shih, “Bell-state preparation using pulsed nondegenerate two-photon entanglement,” Phys. Rev. A 63, 060301(R) (2001).
[Crossref]

Kwiat, P. G.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer,” Phys. Rev. A 45, 6659–6665 (1992).
[Crossref]

Legero, T.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004).
[Crossref]

Li, J. F.

Li, L.

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Li, M.

Li, Y.

Y. Li and T. Kobayashi, “Multi-photon entangled states from two-crystal geometry parametric down-conversion and their application in quantum teleportation,” Opt. Commun. 244, 285–289 (2005).
[Crossref]

Liang, X. Y.

Lin, H.

Liu, C.

C. Liu, J. F. Chen, S. C. Zhang, S. Y. Zhou, Y. H. Kim, M. M. T. Loy, G. K. L. Wong, and S. W. Du, “Two-photon interferences with degenerate and nondegenerate paired photons,” Phys. Rev. A 85, 021803 (2012).
[Crossref]

Liu, N. L.

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Loy, M. M. T.

C. Liu, J. F. Chen, S. C. Zhang, S. Y. Zhou, Y. H. Kim, M. M. T. Loy, G. K. L. Wong, and S. W. Du, “Two-photon interferences with degenerate and nondegenerate paired photons,” Phys. Rev. A 85, 021803 (2012).
[Crossref]

Lu, C. Y.

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Lucio-Martinez, I.

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, and W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[Crossref]

Lugiato, L. A.

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “Quantum imaging and spatio-temporal correlations,” Opt. Spectrosc. 111, 505–509 (2011).
[Crossref]

Mandel, L.

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref]

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54–57 (1988).
[Crossref]

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]

Matthias, D. L.

J. Zhang, D. L. Matthias, and M. C. Carlton, “Mixing nonclassical pure states in a linear-optical network almost always generates modal entanglement,” Phys. Rev. A 88, 044301 (2013).
[Crossref]

McMillan, A.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor quantum metrology with entangled photons,” Phys. Rev. Lett. 111, 093603 (2013).
[Crossref]

Metcalf, B. J.

P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X. M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computing in a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013).
[Crossref]

Minaeva, O.

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[Crossref]

Monken, C. H.

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pdua, “Time control of two-photon interference,” Phys. Rev. A 69, 053817 (2004).
[Crossref]

Moore, M.

P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X. M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computing in a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013).
[Crossref]

Mori, S.

S. Mori, J. S. derholm, N. Namekata, and S. Inoue, “On the distribution of 1550-nm photon pairs efficiently generated using a periodically poled lithium niobate waveguide,” Opt. Commun. 264, 156–162 (2006).
[Crossref]

Namekata, N.

S. Mori, J. S. derholm, N. Namekata, and S. Inoue, “On the distribution of 1550-nm photon pairs efficiently generated using a periodically poled lithium niobate waveguide,” Opt. Commun. 264, 156–162 (2006).
[Crossref]

Noh, J.

K. Cho and J. Noh, “Temporal ghost imaging of a time object, dispersion cancelation, and nonlocal time lens with bi-photon state,” Opt. Commun. 285, 1275–1282 (2012).
[Crossref]

Olindo, C.

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pdua, “Time control of two-photon interference,” Phys. Rev. A 69, 053817 (2004).
[Crossref]

Ou, Z. Y.

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Quantum communication with correlated nonclassical states,” Phys. Rev. A 62, 042311 (2000).
[Crossref]

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref]

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54–57 (1988).
[Crossref]

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]

Z. Y. Ou, Multi-Photon Quantum Interference (Springer, 2007).

Pan, J. W.

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Pdua, S.

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pdua, “Time control of two-photon interference,” Phys. Rev. A 69, 053817 (2004).
[Crossref]

Pereira, S. F.

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Quantum communication with correlated nonclassical states,” Phys. Rev. A 62, 042311 (2000).
[Crossref]

Pietzsch, A.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Pittman, T. B.

D. V. Strekalov, T. B. Pittman, and Y. H. Shih, “What we can learn about single photons in a two-photon interference experiment,” Phys. Rev. A 57, 567–570 (1998).
[Crossref]

Polyakov, S. V.

E. B. Flagg, S. V. Polyakov, T. Thomay, and G. S. Solomon, “Dynamics of nonclassical light from a single solid-state quantum emitter,” Phys. Rev. Lett. 109, 163601 (2012).
[Crossref]

Qiu, J.

J. Qiu, Y. S. Zhang, G. Y. Xiang, S. S. Han, and Y. Z. Gui, “Unified view of the second-order and the fourth-order interferences in a single interferometer,” Opt. Commun. 336, 9–13 (2015).
[Crossref]

Rarity, J. G.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor quantum metrology with entangled photons,” Phys. Rev. Lett. 111, 093603 (2013).
[Crossref]

Rempe, G.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004).
[Crossref]

Rinkevicius, Z.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Rubenok, A.

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, and W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[Crossref]

Rubensson, J.-E.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Rubin, M. H.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[Crossref]

Sagioro, M. A.

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pdua, “Time control of two-photon interference,” Phys. Rev. A 69, 053817 (2004).
[Crossref]

Saleh, B. E. A.

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[Crossref]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref]

Salles, A.

D. Cavalcanti, N. Brunner, P. Skrzypczyk, A. Salles, and V. Scarani, “Large violation of Bell inequalities using both particle and wave measurements,” Phys. Rev. A 84, 022105 (2011).
[Crossref]

Scarani, V.

D. Cavalcanti, N. Brunner, P. Skrzypczyk, A. Salles, and V. Scarani, “Large violation of Bell inequalities using both particle and wave measurements,” Phys. Rev. A 84, 022105 (2011).
[Crossref]

Schlappa, J.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Schmitt, T.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Sergienko, A. V.

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[Crossref]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[Crossref]

Y. H. Shih and A. V. Sergienko, “Observation of quantum beating in a simple beam-splitting experiment: two-particle entanglement in spin and space-time,” Phys. Rev. A 50, 2564–2568 (1994).
[Crossref]

Shih, Y. H.

Y. H. Kim, S. P. Kulik, and Y. H. Shih, “Bell-state preparation using pulsed nondegenerate two-photon entanglement,” Phys. Rev. A 63, 060301(R) (2001).
[Crossref]

D. V. Strekalov, T. B. Pittman, and Y. H. Shih, “What we can learn about single photons in a two-photon interference experiment,” Phys. Rev. A 57, 567–570 (1998).
[Crossref]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[Crossref]

Y. H. Shih and A. V. Sergienko, “Observation of quantum beating in a simple beam-splitting experiment: two-particle entanglement in spin and space-time,” Phys. Rev. A 50, 2564–2568 (1994).
[Crossref]

Simon, D. S.

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[Crossref]

Skrzypczyk, P.

D. Cavalcanti, N. Brunner, P. Skrzypczyk, A. Salles, and V. Scarani, “Large violation of Bell inequalities using both particle and wave measurements,” Phys. Rev. A 84, 022105 (2011).
[Crossref]

Slater, J. A.

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, and W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[Crossref]

Solomon, G. S.

E. B. Flagg, S. V. Polyakov, T. Thomay, and G. S. Solomon, “Dynamics of nonclassical light from a single solid-state quantum emitter,” Phys. Rev. Lett. 109, 163601 (2012).
[Crossref]

Spring, J. B.

P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X. M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computing in a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013).
[Crossref]

Steinberg, A. M.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer,” Phys. Rev. A 45, 6659–6665 (1992).
[Crossref]

Strekalov, D. V.

D. V. Strekalov, T. B. Pittman, and Y. H. Shih, “What we can learn about single photons in a two-photon interference experiment,” Phys. Rev. A 57, 567–570 (1998).
[Crossref]

Strocov, V. N.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Su, L. B.

Su, Z. E.

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Sun, Y.-P.

A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H. O. Karlsson, T. Schmitt, V. N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and F. Gelmukhanov, “Spatial quantum beats in vibrational resonant inelastic soft x-ray scattering at dissociating states in oxygen,” Phys. Rev. Lett. 106, 153004 (2011).
[Crossref]

Teich, M. C.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref]

Thomay, T.

E. B. Flagg, S. V. Polyakov, T. Thomay, and G. S. Solomon, “Dynamics of nonclassical light from a single solid-state quantum emitter,” Phys. Rev. Lett. 109, 163601 (2012).
[Crossref]

Tittel, W.

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, and W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[Crossref]

Wadsworth, W. J.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor quantum metrology with entangled photons,” Phys. Rev. Lett. 111, 093603 (2013).
[Crossref]

Walmsley, I. A.

P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X. M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. Walmsley, “Linear optical quantum computing in a single spatial mode,” Phys. Rev. Lett. 111, 150501 (2013).
[Crossref]

Weedbrook, C.

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Wilk, T.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004).
[Crossref]

Wong, G. K. L.

C. Liu, J. F. Chen, S. C. Zhang, S. Y. Zhou, Y. H. Kim, M. M. T. Loy, G. K. L. Wong, and S. W. Du, “Two-photon interferences with degenerate and nondegenerate paired photons,” Phys. Rev. A 85, 021803 (2012).
[Crossref]

Xiang, G. Y.

J. Qiu, Y. S. Zhang, G. Y. Xiang, S. S. Han, and Y. Z. Gui, “Unified view of the second-order and the fourth-order interferences in a single interferometer,” Opt. Commun. 336, 9–13 (2015).
[Crossref]

Xu, J.

Zhang, A.

Zhang, H. Y.

Zhang, J.

J. Zhang, D. L. Matthias, and M. C. Carlton, “Mixing nonclassical pure states in a linear-optical network almost always generates modal entanglement,” Phys. Rev. A 88, 044301 (2013).
[Crossref]

Zhang, S. C.

C. Liu, J. F. Chen, S. C. Zhang, S. Y. Zhou, Y. H. Kim, M. M. T. Loy, G. K. L. Wong, and S. W. Du, “Two-photon interferences with degenerate and nondegenerate paired photons,” Phys. Rev. A 85, 021803 (2012).
[Crossref]

Zhang, Y. S.

J. Qiu, Y. S. Zhang, G. Y. Xiang, S. S. Han, and Y. Z. Gui, “Unified view of the second-order and the fourth-order interferences in a single interferometer,” Opt. Commun. 336, 9–13 (2015).
[Crossref]

Zheng, L. H.

Zhou, S. Y.

C. Liu, J. F. Chen, S. C. Zhang, S. Y. Zhou, Y. H. Kim, M. M. T. Loy, G. K. L. Wong, and S. W. Du, “Two-photon interferences with degenerate and nondegenerate paired photons,” Phys. Rev. A 85, 021803 (2012).
[Crossref]

Zhu, M. J.

X. D. Cai, C. Weedbrook, Z. E. Su, M. C. Chen, M. Gu, M. J. Zhu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Chin. Opt. Lett. (2)

Opt. Commun. (4)

J. Qiu, Y. S. Zhang, G. Y. Xiang, S. S. Han, and Y. Z. Gui, “Unified view of the second-order and the fourth-order interferences in a single interferometer,” Opt. Commun. 336, 9–13 (2015).
[Crossref]

Y. Li and T. Kobayashi, “Multi-photon entangled states from two-crystal geometry parametric down-conversion and their application in quantum teleportation,” Opt. Commun. 244, 285–289 (2005).
[Crossref]

S. Mori, J. S. derholm, N. Namekata, and S. Inoue, “On the distribution of 1550-nm photon pairs efficiently generated using a periodically poled lithium niobate waveguide,” Opt. Commun. 264, 156–162 (2006).
[Crossref]

K. Cho and J. Noh, “Temporal ghost imaging of a time object, dispersion cancelation, and nonlocal time lens with bi-photon state,” Opt. Commun. 285, 1275–1282 (2012).
[Crossref]

Opt. Express (1)

Opt. Spectrosc. (1)

A. Gatti, E. Brambilla, L. Caspani, O. Jedrkiewicz, and L. A. Lugiato, “Quantum imaging and spatio-temporal correlations,” Opt. Spectrosc. 111, 505–509 (2011).
[Crossref]

Phys. Rev. A (11)

J. Zhang, D. L. Matthias, and M. C. Carlton, “Mixing nonclassical pure states in a linear-optical network almost always generates modal entanglement,” Phys. Rev. A 88, 044301 (2013).
[Crossref]

D. Cavalcanti, N. Brunner, P. Skrzypczyk, A. Salles, and V. Scarani, “Large violation of Bell inequalities using both particle and wave measurements,” Phys. Rev. A 84, 022105 (2011).
[Crossref]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer,” Phys. Rev. A 45, 6659–6665 (1992).
[Crossref]

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Quantum communication with correlated nonclassical states,” Phys. Rev. A 62, 042311 (2000).
[Crossref]

D. V. Strekalov, T. B. Pittman, and Y. H. Shih, “What we can learn about single photons in a two-photon interference experiment,” Phys. Rev. A 57, 567–570 (1998).
[Crossref]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[Crossref]

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pdua, “Time control of two-photon interference,” Phys. Rev. A 69, 053817 (2004).
[Crossref]

J. D. Franson, “Nonlocal cancellation of dispersion,” Phys. Rev. A 45, 3126–3132 (1992).
[Crossref]

Y. H. Kim, S. P. Kulik, and Y. H. Shih, “Bell-state preparation using pulsed nondegenerate two-photon entanglement,” Phys. Rev. A 63, 060301(R) (2001).
[Crossref]

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Figures (4)

Fig. 1.
Fig. 1. Schematic diagram of the scheme. Frequency anticorrelated photon pairs are generated from the spontaneous parametric downconversion source [nonlinear crystal (NLC)]. The signal and the idler photons are sent into an unbalanced MZ interferometer. In the signal arm, a tunable time delay τ1 is introduced outside the MZ interferometer. Photon pairs are combined at the last beam splitter (BS), and we can observe the interference of quantum beats by observing the coincidence count rates between detectors D1 and D2. IF1 and IF2 are filters with different central frequencies set in front of the detectors. M represents the reflecting mirrors.
Fig. 2.
Fig. 2. Normalized coincidence count rate, which shows three quantum beats with the same interval of τ2=6ps when the two filters in front of the detectors have different central frequencies. The three central dips are at the position of τ1=6ps, τ1=0ps, and τ1=6ps.
Fig. 3.
Fig. 3. Feynman’s path diagrams in different regions of τ1. (a) |τ1|0psτc, where each photon has two alternatives before arriving at the beam splitter; (b)||τ1|τ2|0psτc, where each photon only has one choice before arriving at the beam splitter in order to produce interference.
Fig. 4.
Fig. 4. Normalized coincidence count rate when the two filters have the same central frequencies. It shows three dips with the same interval of τ2=6ps. The three central dips are at the positions of τ1=6ps, τ1=0ps, and τ1=6ps.

Equations (9)

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|ψ=dωsdωiΦ(ωs,ωi)a^s(ωs)a^i(ωi)|0,
exp(iωsτ1)[1+exp(iωsτ2)][1+exp(iωiτ2)],
|ψ=dωsdωiΦ(ωs,ωi)exp(iωsτ1)[1+exp(iωsτ2)][1+exp(iωiτ2)]a^s(ωs)a^i(ωi)|0.
E^1(+)(t1)=dω1a^1(ω1)g1(ω1)exp(iω1t1),
E^2(+)(t2)=dω2a^2(ω2)g2(ω2)exp(iω2t2),
0|E^1(+)(t1)E^2(+)(t2)|ψ=0|dωsdωidω1dω2Φ(ωs,ωi)g1(ω1)g2(ω2)×exp(iω1t1)exp(iω2t2)exp(iωsτ1)×[1+exp(iωsτ2)][1+exp(iωiτ2)]×a^1(ω1)a^2(ω2)a^s(ωs)a^i(ωi)|0.
R(τ1,τ2)=dt1dt2G(2)(t1,t2)=dt1dt2|0|E^1(+)(t1)E^2(+)(t2)|ψ|2=dωsdωi{Φ(ωs,ωi)Φ*(ωs,ωi)Φ(ωs,ωi)×Φ*(ωi,ωs)exp[i(ωsωi)τ1]}[cos(ωsτ2)+1]×[cos(ωiτ2)+1]{exp[(ωsωa)2σ2]exp[(ωiωb)2σ2]+exp[(ωiωa)2σ2]×exp[(ωsωb)2σ2]}.
R(τ1,τ2)=dt1dt2G(2)(t1,t2)=dt1dt2|0|E^1(+)(t1)E^2(+)(t2)|ψ|2=dω{|f(ω)|2+|f(ω)|2[f(ω)f*(ω)×exp(2iωτ1)+c.c.]}[cos((ω0+ω)τ2)+1]×[cos((ω0ω)τ2)+1]{exp[(ω0+ωωa)2σ2]exp[(ω0ωωb)2σ2]+exp[(ω0ωωa)2σ2]exp[(ω0+ωωb)2σ2]}.
R(τ1,τ2)=1exp(σ2τ122)cos[(ωaωb)τ1]12exp[σ2(τ1τ2)22]cos[(ωaωb)(τ1τ2)]12exp[σ2(τ1+τ2)22]cos[(ωaωb)(τ1+τ2)].

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