Abstract

The Hong-Ou-Mandel (HOM) effect ranks among the most notable quantum interference phenomena, and is central to many applications in quantum technologies. The fundamental effect appears when two independent and indistinguishable photons are superimposed on a beam splitter, which achieves a complete suppression of coincidences between the two output ports. Much less studied, however, is when the fields share coherence (continuous-wave lasers) or mode envelope properties (pulsed lasers). In this case, we expect the existence of two distinct and concurrent HOM interference regimes: the traditional HOM dip on the coherence length time scale, and a structured HOM interference pattern on the pulse length scale. We develop a theoretical framework that describes HOM interference for laser fields having arbitrary temporal waveforms and only partial overlap in time. We observe structured HOM interference from a continuous-wave laser via fast polarization modulation and time-resolved single photon detection fast enough to resolve these structured HOM dips.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
  2. G. Magyar and L. Mandel, “Interference Fringes Produced by Superposition of Two Independent Maser Light Beams,” Nature 198(4877), 255–256 (1963).
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    [Crossref]
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  6. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
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  7. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
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  8. H. Chen, T. Peng, S. Karmakar, Z. Xie, and Y. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A 84(3), 033835 (2011).
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  11. J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
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    [Crossref]
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    [Crossref]
  23. R. Tannous, Z. Ye, J. Jin, K. B. Kuntz, N. Lütkenhaus, and T. Jennewein, “Demonstration of a 6 state-4 state reference frame independent channel for quantum key distribution,” Appl. Phys. Lett. 115(21), 211103 (2019).
    [Crossref]
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    [Crossref]
  25. Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, M. B. Ward, and A. J. Shields, “Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications,” Phys. Rev. Appl. 2(6), 064006 (2014).
    [Crossref]

2019 (1)

R. Tannous, Z. Ye, J. Jin, K. B. Kuntz, N. Lütkenhaus, and T. Jennewein, “Demonstration of a 6 state-4 state reference frame independent channel for quantum key distribution,” Appl. Phys. Lett. 115(21), 211103 (2019).
[Crossref]

2017 (1)

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

2016 (1)

J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
[Crossref]

2015 (2)

J. Liu, M. Le, B. Bai, W. Wang, H. Chen, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference of two independent single-mode He-Ne lasers,” Opt. Commun. 350, 196–201 (2015).
[Crossref]

J.-P. Bourgoin, B. L. Higgins, N. Gigov, C. Holloway, C. J. Pugh, S. Kaiser, M. Cranmer, and T. Jennewein, “Free-space quantum key distribution to a moving receiver,” Opt. Express 23(26), 33437–33447 (2015).
[Crossref]

2014 (2)

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, M. B. Ward, and A. J. Shields, “Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications,” Phys. Rev. Appl. 2(6), 064006 (2014).
[Crossref]

Y.-S. Kim, O. Slattery, P. S. Kuo, and X. Tang, “Two-photon interference with continuous-wave multi-mode coherent light,” Opt. Express 22(3), 3611–3620 (2014).
[Crossref]

2013 (3)

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Theory of interferometric photon-correlation measurements: Differentiating coherent from chaotic light,” Phys. Rev. A 88(1), 013801 (2013).
[Crossref]

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Unequivocal differentiation of coherent and chaotic light through interferometric photon correlation measurements,” Phys. Rev. Lett. 110(16), 163603 (2013).
[Crossref]

Z. Yan, E. Meyer-Scott, J. P. Bourgoin, B. L. Higgins, N. Gigov, A. MacDonald, H. Hubel, and T. Jennewein, “Novel high-speed polarization source for decoy-state bb84 quantum key distribution over free space and satellite links,” J. Lightwave Technol. 31(9), 1399–1408 (2013).
[Crossref]

2012 (2)

J. H. Shapiro and E. Lantz, “Comment on “observation of anticorrelation in incoherent thermal light fields”,” Phys. Rev. A 85(5), 057801 (2012).
[Crossref]

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

2011 (1)

H. Chen, T. Peng, S. Karmakar, Z. Xie, and Y. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A 84(3), 033835 (2011).
[Crossref]

2007 (1)

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79(2), 555–609 (2007).
[Crossref]

2004 (1)

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93(7), 070503 (2004).
[Crossref]

1989 (1)

1988 (1)

Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, “Observation of beating between blue and green light,” Opt. Commun. 69(1), 1–5 (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(18), 2044–2046 (1987).
[Crossref]

1986 (1)

H. Paul, “Interference between independent photons,” Rev. Mod. Phys. 58(1), 209–231 (1986).
[Crossref]

1977 (1)

J. Olivero and R. Longbothum, “J. Quant. Spectrosc. Radiat. Transfer,” J. of Quant. Spectr. and Rad. Trans. 17(2), 233–236 (1977).
[Crossref]

1967 (1)

R. L. Pfleegor and L. Mandel, “Interference of Independent Photon Beams,” Phys. Rev. 159(5), 1084–1088 (1967).
[Crossref]

1963 (3)

G. Magyar and L. Mandel, “Interference Fringes Produced by Superposition of Two Independent Maser Light Beams,” Nature 198(4877), 255–256 (1963).
[Crossref]

R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10(3), 84–86 (1963).
[Crossref]

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10(7), 277–279 (1963).
[Crossref]

Abram, I.

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Theory of interferometric photon-correlation measurements: Differentiating coherent from chaotic light,” Phys. Rev. A 88(1), 013801 (2013).
[Crossref]

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Unequivocal differentiation of coherent and chaotic light through interferometric photon correlation measurements,” Phys. Rev. Lett. 110(16), 163603 (2013).
[Crossref]

Agne, S.

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

Anisimova, E.

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

Bai, B.

J. Liu, M. Le, B. Bai, W. Wang, H. Chen, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference of two independent single-mode He-Ne lasers,” Opt. Commun. 350, 196–201 (2015).
[Crossref]

Bartlett, S. D.

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79(2), 555–609 (2007).
[Crossref]

Beveratos, A.

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Unequivocal differentiation of coherent and chaotic light through interferometric photon correlation measurements,” Phys. Rev. Lett. 110(16), 163603 (2013).
[Crossref]

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Theory of interferometric photon-correlation measurements: Differentiating coherent from chaotic light,” Phys. Rev. A 88(1), 013801 (2013).
[Crossref]

Bourgoin, J. P.

Bourgoin, J.-P.

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

J.-P. Bourgoin, B. L. Higgins, N. Gigov, C. Holloway, C. J. Pugh, S. Kaiser, M. Cranmer, and T. Jennewein, “Free-space quantum key distribution to a moving receiver,” Opt. Express 23(26), 33437–33447 (2015).
[Crossref]

Braive, R.

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Unequivocal differentiation of coherent and chaotic light through interferometric photon correlation measurements,” Phys. Rev. Lett. 110(16), 163603 (2013).
[Crossref]

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Theory of interferometric photon-correlation measurements: Differentiating coherent from chaotic light,” Phys. Rev. A 88(1), 013801 (2013).
[Crossref]

Chen, H.

J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
[Crossref]

J. Liu, M. Le, B. Bai, W. Wang, H. Chen, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference of two independent single-mode He-Ne lasers,” Opt. Commun. 350, 196–201 (2015).
[Crossref]

H. Chen, T. Peng, S. Karmakar, Z. Xie, and Y. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A 84(3), 033835 (2011).
[Crossref]

Chen, Z.-B.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Choi, E.

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

Cranmer, M.

Dynes, J. F.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, M. B. Ward, and A. J. Shields, “Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications,” Phys. Rev. Appl. 2(6), 064006 (2014).
[Crossref]

Fröhlich, B.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, M. B. Ward, and A. J. Shields, “Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications,” Phys. Rev. Appl. 2(6), 064006 (2014).
[Crossref]

Gage, E. C.

Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, “Fourth-order interference technique for determining the coherence time of a light beam,” J. Opt. Soc. Am. B 6(1), 100–103 (1989).
[Crossref]

Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, “Observation of beating between blue and green light,” Opt. Commun. 69(1), 1–5 (1988).
[Crossref]

Gao, H.

J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
[Crossref]

Gigov, N.

Glauber, R. J.

R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10(3), 84–86 (1963).
[Crossref]

Hennrich, M.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93(7), 070503 (2004).
[Crossref]

Higgins, B. L.

Holloway, C.

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(18), 2044–2046 (1987).
[Crossref]

Hubel, H.

Jennewein, T.

R. Tannous, Z. Ye, J. Jin, K. B. Kuntz, N. Lütkenhaus, and T. Jennewein, “Demonstration of a 6 state-4 state reference frame independent channel for quantum key distribution,” Appl. Phys. Lett. 115(21), 211103 (2019).
[Crossref]

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

J.-P. Bourgoin, B. L. Higgins, N. Gigov, C. Holloway, C. J. Pugh, S. Kaiser, M. Cranmer, and T. Jennewein, “Free-space quantum key distribution to a moving receiver,” Opt. Express 23(26), 33437–33447 (2015).
[Crossref]

Z. Yan, E. Meyer-Scott, J. P. Bourgoin, B. L. Higgins, N. Gigov, A. MacDonald, H. Hubel, and T. Jennewein, “Novel high-speed polarization source for decoy-state bb84 quantum key distribution over free space and satellite links,” J. Lightwave Technol. 31(9), 1399–1408 (2013).
[Crossref]

Jin, J.

R. Tannous, Z. Ye, J. Jin, K. B. Kuntz, N. Lütkenhaus, and T. Jennewein, “Demonstration of a 6 state-4 state reference frame independent channel for quantum key distribution,” Appl. Phys. Lett. 115(21), 211103 (2019).
[Crossref]

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

Kaiser, S.

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

J.-P. Bourgoin, B. L. Higgins, N. Gigov, C. Holloway, C. J. Pugh, S. Kaiser, M. Cranmer, and T. Jennewein, “Free-space quantum key distribution to a moving receiver,” Opt. Express 23(26), 33437–33447 (2015).
[Crossref]

Karmakar, S.

H. Chen, T. Peng, S. Karmakar, Z. Xie, and Y. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A 84(3), 033835 (2011).
[Crossref]

Kim, Y.-S.

Kuhn, A.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93(7), 070503 (2004).
[Crossref]

Kuntz, K. B.

R. Tannous, Z. Ye, J. Jin, K. B. Kuntz, N. Lütkenhaus, and T. Jennewein, “Demonstration of a 6 state-4 state reference frame independent channel for quantum key distribution,” Appl. Phys. Lett. 115(21), 211103 (2019).
[Crossref]

Kuo, P. S.

Lantz, E.

J. H. Shapiro and E. Lantz, “Comment on “observation of anticorrelation in incoherent thermal light fields”,” Phys. Rev. A 85(5), 057801 (2012).
[Crossref]

Le, M.

J. Liu, M. Le, B. Bai, W. Wang, H. Chen, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference of two independent single-mode He-Ne lasers,” Opt. Commun. 350, 196–201 (2015).
[Crossref]

Lebreton, A.

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Theory of interferometric photon-correlation measurements: Differentiating coherent from chaotic light,” Phys. Rev. A 88(1), 013801 (2013).
[Crossref]

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Unequivocal differentiation of coherent and chaotic light through interferometric photon correlation measurements,” Phys. Rev. Lett. 110(16), 163603 (2013).
[Crossref]

Legero, T.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93(7), 070503 (2004).
[Crossref]

Li, F. L.

J. Liu, M. Le, B. Bai, W. Wang, H. Chen, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference of two independent single-mode He-Ne lasers,” Opt. Commun. 350, 196–201 (2015).
[Crossref]

Li, F.-L.

J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
[Crossref]

Liu, J.

J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
[Crossref]

J. Liu, M. Le, B. Bai, W. Wang, H. Chen, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference of two independent single-mode He-Ne lasers,” Opt. Commun. 350, 196–201 (2015).
[Crossref]

Longbothum, R.

J. Olivero and R. Longbothum, “J. Quant. Spectrosc. Radiat. Transfer,” J. of Quant. Spectr. and Rad. Trans. 17(2), 233–236 (1977).
[Crossref]

Lu, C.-Y.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Lucamarini, M.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, M. B. Ward, and A. J. Shields, “Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications,” Phys. Rev. Appl. 2(6), 064006 (2014).
[Crossref]

Lütkenhaus, N.

R. Tannous, Z. Ye, J. Jin, K. B. Kuntz, N. Lütkenhaus, and T. Jennewein, “Demonstration of a 6 state-4 state reference frame independent channel for quantum key distribution,” Appl. Phys. Lett. 115(21), 211103 (2019).
[Crossref]

MacDonald, A.

Magill, B. E.

Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, “Fourth-order interference technique for determining the coherence time of a light beam,” J. Opt. Soc. Am. B 6(1), 100–103 (1989).
[Crossref]

Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, “Observation of beating between blue and green light,” Opt. Commun. 69(1), 1–5 (1988).
[Crossref]

Magyar, G.

G. Magyar and L. Mandel, “Interference Fringes Produced by Superposition of Two Independent Maser Light Beams,” Nature 198(4877), 255–256 (1963).
[Crossref]

Makarov, V.

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

Mandel, L.

Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, “Fourth-order interference technique for determining the coherence time of a light beam,” J. Opt. Soc. Am. B 6(1), 100–103 (1989).
[Crossref]

Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, “Observation of beating between blue and green light,” Opt. Commun. 69(1), 1–5 (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(18), 2044–2046 (1987).
[Crossref]

R. L. Pfleegor and L. Mandel, “Interference of Independent Photon Beams,” Phys. Rev. 159(5), 1084–1088 (1967).
[Crossref]

G. Magyar and L. Mandel, “Interference Fringes Produced by Superposition of Two Independent Maser Light Beams,” Nature 198(4877), 255–256 (1963).
[Crossref]

Meyer-Scott, E.

Olivero, J.

J. Olivero and R. Longbothum, “J. Quant. Spectrosc. Radiat. Transfer,” J. of Quant. Spectr. and Rad. Trans. 17(2), 233–236 (1977).
[Crossref]

Ou, Z. Y.

Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, “Fourth-order interference technique for determining the coherence time of a light beam,” J. Opt. Soc. Am. B 6(1), 100–103 (1989).
[Crossref]

Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, “Observation of beating between blue and green light,” Opt. Commun. 69(1), 1–5 (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(18), 2044–2046 (1987).
[Crossref]

Pan, J.-W.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Paul, H.

H. Paul, “Interference between independent photons,” Rev. Mod. Phys. 58(1), 209–231 (1986).
[Crossref]

Peng, T.

H. Chen, T. Peng, S. Karmakar, Z. Xie, and Y. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A 84(3), 033835 (2011).
[Crossref]

Pfleegor, R. L.

R. L. Pfleegor and L. Mandel, “Interference of Independent Photon Beams,” Phys. Rev. 159(5), 1084–1088 (1967).
[Crossref]

Pugh, C. J.

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

J.-P. Bourgoin, B. L. Higgins, N. Gigov, C. Holloway, C. J. Pugh, S. Kaiser, M. Cranmer, and T. Jennewein, “Free-space quantum key distribution to a moving receiver,” Opt. Express 23(26), 33437–33447 (2015).
[Crossref]

Rempe, G.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93(7), 070503 (2004).
[Crossref]

Robert-Philip, I.

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Unequivocal differentiation of coherent and chaotic light through interferometric photon correlation measurements,” Phys. Rev. Lett. 110(16), 163603 (2013).
[Crossref]

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Theory of interferometric photon-correlation measurements: Differentiating coherent from chaotic light,” Phys. Rev. A 88(1), 013801 (2013).
[Crossref]

Rudolph, T.

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79(2), 555–609 (2007).
[Crossref]

Sagnes, I.

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Unequivocal differentiation of coherent and chaotic light through interferometric photon correlation measurements,” Phys. Rev. Lett. 110(16), 163603 (2013).
[Crossref]

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Theory of interferometric photon-correlation measurements: Differentiating coherent from chaotic light,” Phys. Rev. A 88(1), 013801 (2013).
[Crossref]

Scully, M.

M. Scully, Quantum Optics (Cambridge University, 1997).

Shapiro, J. H.

J. H. Shapiro and E. Lantz, “Comment on “observation of anticorrelation in incoherent thermal light fields”,” Phys. Rev. A 85(5), 057801 (2012).
[Crossref]

Shields, A. J.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, M. B. Ward, and A. J. Shields, “Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications,” Phys. Rev. Appl. 2(6), 064006 (2014).
[Crossref]

Shih, Y.

H. Chen, T. Peng, S. Karmakar, Z. Xie, and Y. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A 84(3), 033835 (2011).
[Crossref]

Slattery, O.

Spekkens, R. W.

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79(2), 555–609 (2007).
[Crossref]

Sudarshan, E. C. G.

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10(7), 277–279 (1963).
[Crossref]

Sultana, N.

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

Tang, X.

Tannous, R.

R. Tannous, Z. Ye, J. Jin, K. B. Kuntz, N. Lütkenhaus, and T. Jennewein, “Demonstration of a 6 state-4 state reference frame independent channel for quantum key distribution,” Appl. Phys. Lett. 115(21), 211103 (2019).
[Crossref]

Wang, W.

J. Liu, M. Le, B. Bai, W. Wang, H. Chen, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference of two independent single-mode He-Ne lasers,” Opt. Commun. 350, 196–201 (2015).
[Crossref]

Ward, M. B.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, M. B. Ward, and A. J. Shields, “Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications,” Phys. Rev. Appl. 2(6), 064006 (2014).
[Crossref]

Wei, D.

J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
[Crossref]

Weinfurter, H.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Wilk, T.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93(7), 070503 (2004).
[Crossref]

Xie, Z.

H. Chen, T. Peng, S. Karmakar, Z. Xie, and Y. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A 84(3), 033835 (2011).
[Crossref]

Xu, Z.

J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
[Crossref]

J. Liu, M. Le, B. Bai, W. Wang, H. Chen, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference of two independent single-mode He-Ne lasers,” Opt. Commun. 350, 196–201 (2015).
[Crossref]

Yan, Z.

Ye, Z.

R. Tannous, Z. Ye, J. Jin, K. B. Kuntz, N. Lütkenhaus, and T. Jennewein, “Demonstration of a 6 state-4 state reference frame independent channel for quantum key distribution,” Appl. Phys. Lett. 115(21), 211103 (2019).
[Crossref]

Yuan, Z. L.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, M. B. Ward, and A. J. Shields, “Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications,” Phys. Rev. Appl. 2(6), 064006 (2014).
[Crossref]

Zeilinger, A.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Zheng, H.

J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
[Crossref]

Zhou, Y.

J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
[Crossref]

J. Liu, M. Le, B. Bai, W. Wang, H. Chen, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference of two independent single-mode He-Ne lasers,” Opt. Commun. 350, 196–201 (2015).
[Crossref]

Zukowski, M.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Appl. Phys. Lett. (1)

R. Tannous, Z. Ye, J. Jin, K. B. Kuntz, N. Lütkenhaus, and T. Jennewein, “Demonstration of a 6 state-4 state reference frame independent channel for quantum key distribution,” Appl. Phys. Lett. 115(21), 211103 (2019).
[Crossref]

Chin. Phys. B (1)

J. Liu, D. Wei, H. Chen, Y. Zhou, H. Zheng, H. Gao, F.-L. Li, and Z. Xu, “Second-order interference of two independent and tunable single-mode continuous-wave lasers,” Chin. Phys. B 25(3), 034203 (2016).
[Crossref]

J. Lightwave Technol. (1)

J. of Quant. Spectr. and Rad. Trans. (1)

J. Olivero and R. Longbothum, “J. Quant. Spectrosc. Radiat. Transfer,” J. of Quant. Spectr. and Rad. Trans. 17(2), 233–236 (1977).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature (1)

G. Magyar and L. Mandel, “Interference Fringes Produced by Superposition of Two Independent Maser Light Beams,” Nature 198(4877), 255–256 (1963).
[Crossref]

Opt. Commun. (2)

Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, “Observation of beating between blue and green light,” Opt. Commun. 69(1), 1–5 (1988).
[Crossref]

J. Liu, M. Le, B. Bai, W. Wang, H. Chen, Y. Zhou, F. L. Li, and Z. Xu, “The second-order interference of two independent single-mode He-Ne lasers,” Opt. Commun. 350, 196–201 (2015).
[Crossref]

Opt. Express (2)

Phys. Rev. (1)

R. L. Pfleegor and L. Mandel, “Interference of Independent Photon Beams,” Phys. Rev. 159(5), 1084–1088 (1967).
[Crossref]

Phys. Rev. A (3)

H. Chen, T. Peng, S. Karmakar, Z. Xie, and Y. Shih, “Observation of anticorrelation in incoherent thermal light fields,” Phys. Rev. A 84(3), 033835 (2011).
[Crossref]

J. H. Shapiro and E. Lantz, “Comment on “observation of anticorrelation in incoherent thermal light fields”,” Phys. Rev. A 85(5), 057801 (2012).
[Crossref]

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Theory of interferometric photon-correlation measurements: Differentiating coherent from chaotic light,” Phys. Rev. A 88(1), 013801 (2013).
[Crossref]

Phys. Rev. Appl. (1)

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, M. B. Ward, and A. J. Shields, “Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications,” Phys. Rev. Appl. 2(6), 064006 (2014).
[Crossref]

Phys. Rev. Lett. (5)

A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, “Unequivocal differentiation of coherent and chaotic light through interferometric photon correlation measurements,” Phys. Rev. Lett. 110(16), 163603 (2013).
[Crossref]

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93(7), 070503 (2004).
[Crossref]

R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10(3), 84–86 (1963).
[Crossref]

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10(7), 277–279 (1963).
[Crossref]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref]

Quantum Sci. Technol. (1)

C. J. Pugh, S. Kaiser, J.-P. Bourgoin, J. Jin, N. Sultana, S. Agne, E. Anisimova, V. Makarov, E. Choi, B. L. Higgins, and T. Jennewein, “Airborne demonstration of a quantum key distribution receiver payload,” Quantum Sci. Technol. 2(2), 024009 (2017).
[Crossref]

Rev. Mod. Phys. (3)

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79(2), 555–609 (2007).
[Crossref]

H. Paul, “Interference between independent photons,” Rev. Mod. Phys. 58(1), 209–231 (1986).
[Crossref]

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Other (1)

M. Scully, Quantum Optics (Cambridge University, 1997).

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Figures (4)

Fig. 1.
Fig. 1. Square HOM Waves: Experimental Setup and Concept. (a) A continuous-wave laser is attenuated and polarization modulated before being divided into two optical modes for the HOM interferometer. As explained in the main text, the fixed and variable time delays in the two paths are responsible for isolating the HOM interference and scanning the optical delay, respectively. (b) Adjusting the optical delay $\tau _{\textrm {Opt}}$ changes the polarization overlap pattern between the two input modes at the HOM beam splitter (blue dotted and black dashed traces represent the two inputs), which results in changes to the two-photon coincidences (red solid traces). Y-axes for plots in (b) refer to the normalized coincidences (red solid traces) only. VFA: variable fiber attenuator, FFD: fixed 2 km fiber delay line, VFSD: variable free-space delay line, BS: (HOM) 50:50 non-polarizing beam splitter, PMOD: polarization modulator, FBS: fiber beam splitter, TRSPD: time-resolving single-photon detectors. Note that the coherence time of the laser (microseconds) is much larger than the modulation period ($\sim$2.8 ns).
Fig. 2.
Fig. 2. Experimental Square Wave HOM Interference with Polarization Modulated CW Lasers. (a) Using time stamps of the modulation trigger as a stable time reference (indicated by the arrows in the lower plot), together with the photon detection times, we can selectively add up coincidences over time to form the resolved HOM square wave (see main text). (b) As expected from Eq. (19), a square wave pattern with double the modulation frequency emerges for certain optical delays $\tau _{\textrm {Opt}}$. Coincidences are detected within a $T_{\textrm {coin}}=312.5$ ps window, which approximates $\tau =0$ ns. The histogram bin size is 156.25 ps.
Fig. 3.
Fig. 3. Experimental Triangle Wave HOM Interference with CW Lasers. (a) Traditional HOM dips extracted for optical delays (retroreflector positions) ranging over one modulator period $T_{\textrm {Mod}}$. Black dots: data points, red solid line: fit. (b) Plot of HOM dip visibilities for optical delays ranging from $0$ to $7.14$ ns. The red curve is a fit described by Eq. (20) for $\tau \approx 0$. The period of the fitted waveform is $T=(2.80\pm 0.04$) ns, which matches the modulator period as expected. Compared with the measurements presented in Fig. 2, here our setup had an offset of the optical delay of (0.85$\pm$0.02) ns, which the fit takes into account.
Fig. 4.
Fig. 4. Effect of non-ideal optical delay and asymmetrical duty cycle settings. (a) and (b) are theory plots showing the predicted normalized coincidences (red solid line) and the polarization patterns (black dashed lines). (a) Non-ideal (offset) optical delay with a symmetrical 0.5 duty cycle, (b) non-ideal optical delay and an asymmetrical duty cycle of 0.7 are taken into account in Eq. (19) for the $\tau _{\textrm {Opt}}=T_{\textrm {Mod}}/2$ case. As a consequence, the extracted experimental HOM interference pattern (c) does not show a flat line at maximum coincidences. Note that for clarity, the x-axis for (a) and (b) graphs extend only over one modulator period, whereas the data in (c) extends over two modulator periods.

Equations (20)

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E ^ 3 ( t ) = 1 2 ( ζ 1 ( t ) a ^ 1 ( t ) + ζ 2 ( t ) a ^ 2 ( t ) ) E ^ 4 ( t ) = 1 2 ( ζ 1 ( t ) a ^ 1 ( t ) ζ 2 ( t ) a ^ 2 ( t ) ) .
G ( 2 x ) ( t 3 , t 4 ) := E ^ 3 ( t 3 ) E ^ 4 ( t 4 ) E ^ 4 + ( t 4 ) E ^ 3 + ( t 3 ) ,
G ( 2 x ) ( t 3 , t 4 ) = 1 4 | ζ 1 ( t 3 ) ζ 1 ( t 4 ) | 2 a ^ 1 ( t 3 ) a ^ 1 ( t 4 ) a ^ 1 ( t 4 ) a ^ 1 ( t 3 ) + 1 4 | ζ 2 ( t 3 ) ζ 2 ( t 4 ) | 2 a ^ 2 ( t 3 ) a ^ 2 ( t 4 ) a ^ 2 ( t 4 ) a ^ 2 ( t 3 ) + 1 4 | ζ 1 ( t 3 ) | 2 | ζ 2 ( t 4 ) | 2 a ^ 1 ( t 3 ) a ^ 1 ( t 3 ) a ^ 2 ( t 4 ) a ^ 2 ( t 4 ) + 1 4 | ζ 1 ( t 4 ) | 2 | ζ 2 ( t 3 ) | 2 a ^ 1 ( t 4 ) a ^ 1 ( t 4 ) a ^ 2 ( t 3 ) a ^ 2 ( t 3 ) 1 2 R e { ζ 1 ( t 3 ) ζ 1 ( t 4 ) ζ 2 ( t 3 ) ζ 2 ( t 4 ) a ^ 1 ( t 3 ) a ^ 1 ( t 4 ) a ^ 2 ( t 4 ) a ^ 2 ( t 3 ) } .
ρ ^ = k = 1 , 2 C k d α k P ( α k ) | α k α k | ,
G ( 2 x ) ( t 3 , t 4 ) = 1 4 | ζ 1 ( t 3 ) ζ 1 ( t 4 ) | 2 | α 1 ( t 3 ) | 2 | α 1 ( t 4 ) | 2 α 1 + 1 4 | ζ 2 ( t 3 ) ζ 2 ( t 4 ) | 2 | α 2 ( t 3 ) | 2 | α 2 ( t 4 ) | 2 α 2 + 1 4 | ζ 1 ( t 3 ) | 2 | ζ 2 ( t 4 ) | 2 | α 1 ( t 3 ) | 2 α 1 | α 2 ( t 4 ) | 2 α 2 + 1 4 | ζ 1 ( t 4 ) | 2 | ζ 2 ( t 3 ) | 2 | α 1 ( t 4 ) | 2 α 1 | α 2 ( t 3 ) | 2 α 2 1 2 R e { ζ 1 ( t 3 ) ζ 1 ( t 4 ) ζ 2 ( t 3 ) ζ 2 ( t 4 ) G 1 ( 1 ) ( τ ) G 2 ( 1 ) ( τ ) } ,
f ( α k , α k ) α k = C k d α k P ( α k ) f ( α k , α k ) .
G k ( 1 ) ( τ ) = α k ( t ) α k ( t + τ ) α k
| α k ( t ) | 2 α k = G k ( 1 ) ( 0 ) I 0 .
| α k ( t ) | 2 | α k ( t ) | 2 α k = I 0 2 .
G ( 1 ) ( τ ) = exp ( | τ | τ coh τ 2 τ coh 2 ) exp ( i ω 0 τ ) ,
G ( 2 x ) ( τ ) = 1 1 2 exp [ 2 | τ | τ coh 2 τ 2 τ coh 2 ] .
V HOM = G ( 2 x ) ( τ ) max G ( 2 x ) ( τ ) min G ( 2 x ) ( τ ) max
R ( 2 x ) ( t 0 , τ , Δ T ) = η 1 η 2 t 0 t 0 + Δ T d t 1 t 0 + τ t 0 + τ + Δ T d t 2 G ( 2 x ) ( t 1 , t 2 ) ,
R ( 2 x ) ( t 0 , τ , Δ T ) η 1 η 2 ( Δ T ) 2 G ( 2 x ) ( t 0 , t 0 + τ ) ,
G ( 2 x ) ( t 3 , t 4 ) = G H , H ( 2 x ) ( t 3 , t 4 ) + G H , V ( 2 x ) ( t 3 , t 4 ) + G V , H ( 2 x ) ( t 3 , t 4 ) + G V , V ( 2 x ) ( t 3 , t 4 ) ,
G σ , σ ( 2 x ) ( t 0 ) = 1 4 ( ζ σ , 1 ( t 0 ) 4 + ζ σ , 2 ( t 0 ) 4 ) G σ , σ ( 2 x ) ( t 0 ) = 1 4 ( ζ σ , 1 ( t 0 ) 2 ζ σ , 1 ( t 0 ) 2 + ζ σ , 2 ( t 0 ) 2 ζ σ , 2 ( t 0 ) 2 + ζ σ , 1 ( t 0 ) 2 ζ σ , 2 ( t 0 ) 2 + ζ σ , 2 ( t 0 ) 2 ζ σ , 1 ( t 0 ) 2 ) ,
ζ H , 1 ( t ) = SW 0 1 ( t τ Opt T Mod ) ζ H , 2 ( t ) = SW 0 1 ( t T Mod ) ,
ζ V , 1 ( t ) = SW 0 1 ( t T Mod / 2 τ Opt T Mod ) ζ V , 2 ( t ) = SW 0 1 ( t T Mod / 2 T Mod ) .
G ( 2 x ) ( t 0 , τ Opt ) = 1 4 ( 3 SW 1 1 [ t 0 T Mod ] SW 1 1 [ t 0 τ Opt T Mod ] ) .
G ( 2 x ) ( τ Opt ) = T M ( 1 1 2 TW 0 1 [ τ Opt T Mod 3 4 ] ) ,