Complete three photon Hong-Ou-Mandel interference at a three port device

Simon Mährlein; Joachim von Zanthier; Girish S. Agarwal

doi:10.1364/OE.23.015833

Optics Express
Vol. 23,
Issue 12,
pp. 15833-15847
(2015)
•https://doi.org/10.1364/OE.23.015833

Complete three photon Hong-Ou-Mandel interference at a three port device

Simon Mährlein, Joachim von Zanthier, and Girish S. Agarwal

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Simon Mährlein, Joachim von Zanthier, and Girish S. Agarwal, "Complete three photon Hong-Ou-Mandel interference at a three port device," Opt. Express 23, 15833-15847 (2015)

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Table of Contents Category
- Quantum Optics
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Integrated optics devices (130.3120)
- Coherent optical effects (270.1670)
- Multiphoton processes (270.4180)
- Quantum information and processing (270.5585)

About this Article
History
- Original Manuscript: March 19, 2015
- Revised Manuscript: April 24, 2015
- Manuscript Accepted: April 29, 2015
- Published: June 8, 2015

Abstract

We report the possibility of completely destructive interference of three indistinguishable photons on a three port device providing a generalisation of the well known Hong-Ou-Mandel interference of two indistinguishable photons on a two port device. Our analysis is based on the underlying mathematical framework of SU(3) transformations rather than SU(2) transformations. We show the completely destructive three photon interference for a large range of parameters of the three port device and point out the physical origin of such interference in terms of the contributions from different quantum paths. As each output port can deliver zero to three photons the device generates higher dimensional entanglement. In particular, different forms of entangled states of qudits can be generated depending on the device parameters. Our system is different from a symmetric three port beam splitter which does not exhibit a three photon Hong-Ou-Mandel interference.

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References

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Article Order
Year
Author
Publication

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A. Rai, G. S. Agarwal, and J. H. H. Perk, “Transport and quantum walk of nonclassical light in coupled waveguides,” Phys. Rev. A 78, 042304 (2008).
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Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
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M. S. Tame, K. R. McEnery, S. K. Ozdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nature Physics 9, 329 (2013).
[Crossref]
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B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32(10), 1320 (2007).
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K. Suzuki, V. Sharma, J. G. Fujimoto, E. P. Ippen, and Y. Nasu, “Characterization of symmetric [3×3] directional couplers fabricated by direct writing with a femtosecond laser oscillator,” Opt. Express 14(6), 2335 (2006).
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[Crossref]
Y. L. Lim and A. Beige, “Generalized Hong-Ou-Mandel experiments with bosons and fermions,” New J. Phys. 7, 155 (2005).
[Crossref]
M. C. Tichy, M. Tiersch, F. De Melo, F. Mintert, and A. Buchleitner, “Zero-transmission law for multiport beam splitters,” Phys. Rev. Lett. 104, 220405 (2010).
[Crossref] [PubMed]
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[Crossref]
M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]
S.-H. Tan, Y. Y. Gao, H. De Guise, and B. C. Sanders, “SU(3) quantum interferometry with single-photon input pulses,” Phys. Rev. Lett. 110, 113603 (2013).
[Crossref] [PubMed]
G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger, “Two-photon interference in optical fiber multiports,” Phys. Rev. A 54, 893 (1996).
[Crossref] [PubMed]
A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref] [PubMed]
T. Meany, M. Delanty, S. Gross, G. D. Marshall, M. J. Steel, and M. J. Withford, “Non-classical interference in integrated 3D multiports,” Opt. Express 20(24), 26895 (2012).
[Crossref] [PubMed]
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798 (2013).
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[Crossref]
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[Crossref]
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2014 (4)

S. D. Gupta and G. S. Agarwal, “Two-photon quantum interference in plasmonics: theory and applications,” Opt. Lett. 39, 390 (2014).
[Crossref] [PubMed]

G. Di Martino, Y. Sonnefraud, M. S. Tame, S. Kéna-Cohen, F. Dieleman, S. K. Özdemir, M. S. Kim, and S. A. Maier, “Observation of quantum interference in the plasmonic Hong-Ou-Mandel effect,” Phys. Rev. Applied 1, 034004 (2014).
[Crossref]

J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photonics 8, 317 (2014).
[Crossref]

P. Gold, A. Thoma, S. Maier, S. Reitzenstein, C. Schneider, S. Höfling, and M. Kamp, “Two-photon interference from remote quantum dots with inhomogeneously broadened linewidths,” Phys. Rev. B 89, 035313 (2014).
[Crossref]

2013 (7)

M. S. Tame, K. R. McEnery, S. K. Ozdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nature Physics 9, 329 (2013).
[Crossref]

S.-H. Tan, Y. Y. Gao, H. De Guise, and B. C. Sanders, “SU(3) quantum interferometry with single-photon input pulses,” Phys. Rev. Lett. 110, 113603 (2013).
[Crossref] [PubMed]

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794 (2013).
[Crossref]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798 (2013).
[Crossref]

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nature Photonics 7, 540 (2013).
[Crossref]

A. Crespi, R. Osellame, R. Ramponi, J. B. Daniel, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Integrated multimode interferometers with arbitrary designs for photonic boson sampling,” Nature Photonics 7, 545 (2013).
[Crossref]

N. Spagnolo, C. Vitelli, L. Aparo, P. Mataloni, F. Sciarrino, A. Crespi, R. Ramponi, and R. Osellame, “Three-photon bosonic coalescence in an integrated tritter,” Nature Communications 4, 1606 (2013).
[Crossref] [PubMed]

2012 (3)

S. Tanzilli, A. Martin, F. Kaiser, M. P. De Micheli, O. Alibart, and D.B. Ostrowsky, “On the genesis and evolution of integrated quantum optics,” Laser & Photon. Rev. 6, 115 (2012).
[Crossref]

J. Hofmann, M. Krug, N. Ortegel, L. Gerard, M. Weber, W. Rosenfeld, and H. Weinfurter, “Heralded entanglement between widely separated atoms,” Science 337, 72 (2012).
[Crossref] [PubMed]

T. Meany, M. Delanty, S. Gross, G. D. Marshall, M. J. Steel, and M. J. Withford, “Non-classical interference in integrated 3D multiports,” Opt. Express 20(24), 26895 (2012).
[Crossref] [PubMed]

2011 (3)

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref] [PubMed]

E. Poem and Y. Silberberg, “Photon correlations in multimode waveguides,” Phys. Rev. A 84, 041805 (2011).
[Crossref]

R. Wiegner, J. von Zanthier, and G. S. Agarwal, “Quantum interference and non-locality of independent photons from disparate sources,” J. Phys. B 44, 055501 (2011).
[Crossref]

2010 (4)

M. C. Tichy, M. Tiersch, F. De Melo, F. Mintert, and A. Buchleitner, “Zero-transmission law for multiport beam splitters,” Phys. Rev. Lett. 104, 220405 (2010).
[Crossref] [PubMed]

R. B. Patel, A. J. Bennett, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields, “Two-photon interference of the emission from electrically tunable remote quantum dots,” Nature Photonics 4, 632 (2010).
[Crossref]

J. Gillet, G. S. Agarwal, and T. Bastin, “Tunable entanglement, antibunching, and saturation effects in dipole blockade,” Phys. Rev. A 81, 013837 (2010).
[Crossref]

R. Wiegner, C. Thiel, J. von Zanthier, and G. S. Agarwal, “Creating path entanglement and violating Bell inequalities by independent photon sources,” Phys. Rev. A 374, 3405 (2010).

2009 (2)

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[Crossref] [PubMed]

K. Laiho, K. N. Cassemiro, and Ch. Silberhorn, “Producing high fidelity single photons with optimal brightness via waveguided parametric down-conversion,” Opt. Express 17(25), 22823 (2009).
[Crossref]

2008 (3)

X. Li, L. Yang, L. Cui, Z. Y. Ou, and D. Yu, “Observation of quantum interference between a single-photon state and a thermal state generated in optical fibers,” Opt. Express 16(17), 12505 (2008).
[Crossref] [PubMed]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646 (2008).
[Crossref] [PubMed]

A. Rai, G. S. Agarwal, and J. H. H. Perk, “Transport and quantum walk of nonclassical light in coupled waveguides,” Phys. Rev. A 78, 042304 (2008).
[Crossref]

2007 (2)

P. Maunz, D. L. Moehring, S. Olmschenk, K. C. Younge, D. N. Matsukevich, and C. Monroe, “Quantum interference of photon pairs from two remote trapped atomic ions,” Nature Physics 3, 538 (2007).
[Crossref]

B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32(10), 1320 (2007).
[Crossref] [PubMed]

2006 (1)

K. Suzuki, V. Sharma, J. G. Fujimoto, E. P. Ippen, and Y. Nasu, “Characterization of symmetric [3×3] directional couplers fabricated by direct writing with a femtosecond laser oscillator,” Opt. Express 14(6), 2335 (2006).
[Crossref] [PubMed]

2005 (2)

Y. L. Lim and A. Beige, “Generalized Hong-Ou-Mandel experiments with bosons and fermions,” New J. Phys. 7, 155 (2005).
[Crossref]

H. Wang and T. Kobayashi, “Phase measurement at the Heisenberg limit with three photons,” Phys. Rev. A 71, 021802 (2005).
[Crossref]

2000 (1)

R. Campos, “Three-photon Hong-Ou-Mandel interference at a multiport mixer,” Phys. Rev. A 62, 013809 (2000).
[Crossref]

1999 (2)

J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photonics Technol. Lett. 11, 212 (1999).
[Crossref]

Z. Y. Ou, J.-K. Rhee, and L. J. Wang, “Observation of four-photon interference with a beam splitter by pulsed parametric down-conversion,” Phys. Rev. Lett. 83, 959 (1999).
[Crossref]

1996 (1)

G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger, “Two-photon interference in optical fiber multiports,” Phys. Rev. A 54, 893 (1996).
[Crossref] [PubMed]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

1988 (1)

Y. H. Shih and C. O. Alley, “New Type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921 (1988).
[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 (1987).
[Crossref] [PubMed]

Aaronson, S.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794 (2013).
[Crossref]

Agarwal, G. S.

S. D. Gupta and G. S. Agarwal, “Two-photon quantum interference in plasmonics: theory and applications,” Opt. Lett. 39, 390 (2014).
[Crossref] [PubMed]

R. Wiegner, J. von Zanthier, and G. S. Agarwal, “Quantum interference and non-locality of independent photons from disparate sources,” J. Phys. B 44, 055501 (2011).
[Crossref]

J. Gillet, G. S. Agarwal, and T. Bastin, “Tunable entanglement, antibunching, and saturation effects in dipole blockade,” Phys. Rev. A 81, 013837 (2010).
[Crossref]

R. Wiegner, C. Thiel, J. von Zanthier, and G. S. Agarwal, “Creating path entanglement and violating Bell inequalities by independent photon sources,” Phys. Rev. A 374, 3405 (2010).

A. Rai, G. S. Agarwal, and J. H. H. Perk, “Transport and quantum walk of nonclassical light in coupled waveguides,” Phys. Rev. A 78, 042304 (2008).
[Crossref]

G. S. Agarwal, Quantum Optics (Cambridge University, 2012), Sec. (5.7).
[Crossref]

Alibart, O.

S. Tanzilli, A. Martin, F. Kaiser, M. P. De Micheli, O. Alibart, and D.B. Ostrowsky, “On the genesis and evolution of integrated quantum optics,” Laser & Photon. Rev. 6, 115 (2012).
[Crossref]

Alley, C. O.

Aparo, L.

Atwater, H. A.

J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photonics 8, 317 (2014).
[Crossref]

Barbieri, M.

Bastin, T.

J. Gillet, G. S. Agarwal, and T. Bastin, “Tunable entanglement, antibunching, and saturation effects in dipole blockade,” Phys. Rev. A 81, 013837 (2010).
[Crossref]

Beige, A.

Y. L. Lim and A. Beige, “Generalized Hong-Ou-Mandel experiments with bosons and fermions,” New J. Phys. 7, 155 (2005).
[Crossref]

Bennett, A. J.

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

Bromberg, Y.

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[Crossref] [PubMed]

Broome, M. A.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794 (2013).
[Crossref]

Buchleitner, A.

M. C. Tichy, M. Tiersch, F. De Melo, F. Mintert, and A. Buchleitner, “Zero-transmission law for multiport beam splitters,” Phys. Rev. Lett. 104, 220405 (2010).
[Crossref] [PubMed]

Campos, R.

R. Campos, “Three-photon Hong-Ou-Mandel interference at a multiport mixer,” Phys. Rev. A 62, 013809 (2000).
[Crossref]

Cassemiro, K. N.

Chaboyer, Z.

Z. Chaboyer, T. Meany, L. G. Helt, M. J. Withford, and M. J. Steel, “Tuneable quantum interference in a 3D integrated circuit,” arXiv:1409.4908 [quant-ph] (2014).

Crespi, A.

Cryan, M. J.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646 (2008).
[Crossref] [PubMed]

Cui, L.

Dakic, B.

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nature Photonics 7, 540 (2013).
[Crossref]

Daniel, J. B.

Datta, A.

De Guise, H.

S.-H. Tan, Y. Y. Gao, H. De Guise, and B. C. Sanders, “SU(3) quantum interferometry with single-photon input pulses,” Phys. Rev. Lett. 110, 113603 (2013).
[Crossref] [PubMed]

De Melo, F.

M. C. Tichy, M. Tiersch, F. De Melo, F. Mintert, and A. Buchleitner, “Zero-transmission law for multiport beam splitters,” Phys. Rev. Lett. 104, 220405 (2010).
[Crossref] [PubMed]

De Micheli, M. P.

S. Tanzilli, A. Martin, F. Kaiser, M. P. De Micheli, O. Alibart, and D.B. Ostrowsky, “On the genesis and evolution of integrated quantum optics,” Laser & Photon. Rev. 6, 115 (2012).
[Crossref]

Delanty, M.

T. Meany, M. Delanty, S. Gross, G. D. Marshall, M. J. Steel, and M. J. Withford, “Non-classical interference in integrated 3D multiports,” Opt. Express 20(24), 26895 (2012).
[Crossref] [PubMed]

Di Martino, G.

Dieleman, F.

Dove, J.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794 (2013).
[Crossref]

Fakonas, J. S.

J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photonics 8, 317 (2014).
[Crossref]

Farrer, I.

Fedrizzi, A.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794 (2013).
[Crossref]

Fujimoto, J. G.

Galvão, E. F.

Gao, Y. Y.

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B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32(10), 1320 (2007).
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B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32(10), 1320 (2007).
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R. Wiegner, C. Thiel, J. von Zanthier, and G. S. Agarwal, “Creating path entanglement and violating Bell inequalities by independent photon sources,” Phys. Rev. A 374, 3405 (2010).

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T. Meany, M. Delanty, S. Gross, G. D. Marshall, M. J. Steel, and M. J. Withford, “Non-classical interference in integrated 3D multiports,” Opt. Express 20(24), 26895 (2012).
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Younge, K. C.

Yu, D.

Yu, S.

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G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger, “Two-photon interference in optical fiber multiports,” Phys. Rev. A 54, 893 (1996).
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IEEE Photonics Technol. Lett. (1)

J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photonics Technol. Lett. 11, 212 (1999).
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J. Phys. B (1)

R. Wiegner, J. von Zanthier, and G. S. Agarwal, “Quantum interference and non-locality of independent photons from disparate sources,” J. Phys. B 44, 055501 (2011).
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Laser & Photon. Rev. (1)

S. Tanzilli, A. Martin, F. Kaiser, M. P. De Micheli, O. Alibart, and D.B. Ostrowsky, “On the genesis and evolution of integrated quantum optics,” Laser & Photon. Rev. 6, 115 (2012).
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Nat. Commun. (1)

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref] [PubMed]

Nature Communications (1)

Nature Photonics (4)

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nature Photonics 7, 540 (2013).
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J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photonics 8, 317 (2014).
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Nature Physics (2)

M. S. Tame, K. R. McEnery, S. K. Ozdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nature Physics 9, 329 (2013).
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New J. Phys. (1)

Y. L. Lim and A. Beige, “Generalized Hong-Ou-Mandel experiments with bosons and fermions,” New J. Phys. 7, 155 (2005).
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Opt. Express (4)

T. Meany, M. Delanty, S. Gross, G. D. Marshall, M. J. Steel, and M. J. Withford, “Non-classical interference in integrated 3D multiports,” Opt. Express 20(24), 26895 (2012).
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Opt. Lett. (2)

S. D. Gupta and G. S. Agarwal, “Two-photon quantum interference in plasmonics: theory and applications,” Opt. Lett. 39, 390 (2014).
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B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32(10), 1320 (2007).
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Phys. Rev. A (7)

H. Wang and T. Kobayashi, “Phase measurement at the Heisenberg limit with three photons,” Phys. Rev. A 71, 021802 (2005).
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R. Campos, “Three-photon Hong-Ou-Mandel interference at a multiport mixer,” Phys. Rev. A 62, 013809 (2000).
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J. Gillet, G. S. Agarwal, and T. Bastin, “Tunable entanglement, antibunching, and saturation effects in dipole blockade,” Phys. Rev. A 81, 013837 (2010).
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R. Wiegner, C. Thiel, J. von Zanthier, and G. S. Agarwal, “Creating path entanglement and violating Bell inequalities by independent photon sources,” Phys. Rev. A 374, 3405 (2010).

A. Rai, G. S. Agarwal, and J. H. H. Perk, “Transport and quantum walk of nonclassical light in coupled waveguides,” Phys. Rev. A 78, 042304 (2008).
[Crossref]

E. Poem and Y. Silberberg, “Photon correlations in multimode waveguides,” Phys. Rev. A 84, 041805 (2011).
[Crossref]

G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger, “Two-photon interference in optical fiber multiports,” Phys. Rev. A 54, 893 (1996).
[Crossref] [PubMed]

Phys. Rev. Applied (1)

Phys. Rev. B (1)

Phys. Rev. Lett. (7)

Z. Y. Ou, J.-K. Rhee, and L. J. Wang, “Observation of four-photon interference with a beam splitter by pulsed parametric down-conversion,” Phys. Rev. Lett. 83, 959 (1999).
[Crossref]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[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 (1987).
[Crossref] [PubMed]

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

S.-H. Tan, Y. Y. Gao, H. De Guise, and B. C. Sanders, “SU(3) quantum interferometry with single-photon input pulses,” Phys. Rev. Lett. 110, 113603 (2013).
[Crossref] [PubMed]

M. C. Tichy, M. Tiersch, F. De Melo, F. Mintert, and A. Buchleitner, “Zero-transmission law for multiport beam splitters,” Phys. Rev. Lett. 104, 220405 (2010).
[Crossref] [PubMed]

Science (4)

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794 (2013).
[Crossref]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646 (2008).
[Crossref] [PubMed]

J. Hofmann, M. Krug, N. Ortegel, L. Gerard, M. Weber, W. Rosenfeld, and H. Weinfurter, “Heralded entanglement between widely separated atoms,” Science 337, 72 (2012).
[Crossref] [PubMed]

Other (5)

Z. Chaboyer, T. Meany, L. G. Helt, M. J. Withford, and M. J. Steel, “Tuneable quantum interference in a 3D integrated circuit,” arXiv:1409.4908 [quant-ph] (2014).

V. Tamma and S. Laibacher, “Multiboson correlation interferometry with arbitrary single-photon pure states,” arXiv:1410.8121 [quant-ph] (2014).

G. S. Agarwal, Quantum Optics (Cambridge University, 2012), Sec. (5.7).
[Crossref]

S. Scheel, “Permanents in linear optical networks,” arXiv:quant-ph/0406127 (2004).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd edition (Wiley, 2007), Sec. (8.5).

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Fig. 1 Scheme of a 3 × 3 waveguide array with continuous coupling between the inner mode and the outer modes

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Fig. 2 Three examples for different indistinguishable quantum paths leading to the same output state |300⟩ in (a), |021⟩ in (b) and |111⟩ in (c)

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Fig. 3 Plot of the three photon coincidence probability |c ¹¹¹|² and the HOM contour, where the condition |c ¹¹¹|²=0 is fulfilled.

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Fig. 4 Plots of the HOM contour and the coordinates of the investigated states (red crosses, green dots and blue diamonds), which display bipartite entanglement (a) or possibly tripartite entanglement (b)/(c).

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Tables (5)

Table 1 Table of coefficients for equation (16). Coefficients, which are not listed, are equal to 0.

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Table 2 Table of coefficients for equation (17). Coefficients, which are not listed, are equal to 0.

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Table 3 Table of coefficients for equation (18). Coefficients, which are not listed, are equal to 0.

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Table 4 Table of coefficients for equation (18). Coefficients, which are not listed, are equal to 0.

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Table 5 Table of coefficients for equation (18). Coefficients, which are not listed, are equal to 0.

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Equations (18)

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{\hat{H}}_{int} = ℏ (g_{1} {\hat{a}}_{1} {\hat{a}}_{2}^{†} + g_{2} {\hat{a}}_{2} {\hat{a}}_{3}^{†} + g_{1}^{*} {\hat{a}}_{2} {\hat{a}}_{1}^{†} + g_{2}^{*} {\hat{a}}_{3} {\hat{a}}_{2}^{†}) .

\frac{d}{d t} \vec{a} (t) = i \underset{M}{\underset{︸}{(\begin{matrix} 0 & g_{1} & 0 \\ g_{1}^{*} & 0 & g_{2} \\ 0 & g_{2}^{*} & 0 \end{matrix}) \vec{a} (t)}},

V = (\begin{matrix} \cos^{2} (θ) \cos (G) + \sin^{2} (θ) & - i \cos (θ) \sin (G) e^{i ψ} & \cos (θ) \sin (θ) e^{i (φ + ψ)} (\cos (G) - 1) \\ - i \cos (θ) \sin (G) e^{- i ψ} & \cos (G) & - i \sin (θ) \sin (G) e^{i φ} \\ \cos (θ) \sin (θ) e^{- i (φ + ψ)} (\cos (G) - 1) & - i \sin (θ) \sin (G) e^{- i φ} & \sin^{2} (θ) \cos (G) + \cos^{2} (θ) \end{matrix}) .

| ψ_{out} 〉 = \sum_{l = 1}^{3} V_{1 l} {\hat{a}}_{l}^{†} (t) \cdot \sum_{m = 1}^{3} V_{2 m} {\hat{a}}_{m}^{†} (t) \cdot \sum_{n = 1}^{3} V_{3 n} {\hat{a}}_{n}^{†} (t) | 0 〉,

\begin{array}{r} | ψ_{out} 〉 = c_{300} | 3, 0, 0 〉 + c_{030} | 0, 3, 0 〉 + \dots \\ + c_{210} | 2, 1, 0 〉 + \dots + c_{111} | 1, 1, 1 〉, \end{array}

Perm = \sum_{σ} ∐_{j = 1}^{N} A_{j σ (j)},

c_{k l m} = \frac{{Perm}^{{k, l, m}}}{\sqrt{k! l! m!}} .

V^{{3, 0, 0} =} (\begin{matrix} V_{11} & V_{11} & V_{11} \\ V_{21} & V_{21} & V_{21} \\ V_{31} & V_{31} & V_{31} \end{matrix}),

Perm V^{{3, 0, 0}} = 6 V_{11} V_{21} V_{31} .

V^{{0, 2, 1}} = (\begin{matrix} V_{12} & V_{12} & V_{13} \\ V_{22} & V_{22} & V_{23} \\ V_{32} & V_{32} & V_{33} \end{matrix})

Perm V^{{0, 2, 1}} = 2 (V_{12} V_{22} V_{33} + V_{12} V_{23} V_{32} + V_{13} V_{22} V_{32}) .

\begin{array}{r} Perm V^{{1, 1, 1}} = V_{11} V_{22} V_{33} + V_{12} V_{23} V_{31} + V_{13} V_{21} V_{32} \\ + V_{11} V_{23} V_{32} + V_{12} V_{21} V_{33} + V_{13} V_{22} V_{31} . \end{array}

c_{111} \overset{!}{=} 0

\begin{array}{r} c_{111} = V_{11} V_{22} V_{33} + V_{12} V_{23} V_{31} + V_{13} V_{21} V_{32} \\ + V_{11} V_{23} V_{32} + V_{12} V_{21} V_{33} + V_{13} V_{22} V_{31} . \end{array}

θ (G) = n π \pm arcsec [4 / \sqrt{8 \pm \frac{\sqrt{2} \csc^{4} (\frac{G}{2}) \sqrt{\sin^{4} (\frac{G}{2}) (20 \cos (G) + 3 (8 \cos (2 G) + 4 \cos (3 G) + 3 \cos (4 G) + 5))}}{3 \cos (G) + 2}}]

| ψ_{out} 〉 = \frac{1}{\sqrt{2}} (| 2_{j}, 0_{k} 〉 + | 0_{j}, 2_{k} 〉) | 1_{l} 〉

\begin{array}{r} | ψ_{out} 〉 = \frac{\sqrt{3}}{4} (| 3_{j}, 0_{k} 〉 + | 0_{j}, 3_{k} 〉) | 0_{l} 〉 \\ + \frac{1}{4} (| 2_{j}, 1_{k} 〉 + | 1_{j}, 2_{k} 〉) | 0_{l} 〉 \\ + \frac{1}{2} (| 1_{j}, 0_{k} 〉 + | 0_{j}, 1_{k} 〉) | 2_{l} 〉 \end{array}

\begin{array}{l} | ψ_{out} 〉 = \frac{1}{3 \sqrt{2}} (2 | 3_{j}, 0_{k}, 0_{l} 〉 + | 0_{j}, 3_{k}, 0_{l} 〉 + | 0_{j}, 0_{k}, 3_{l} 〉) \\ + \frac{1}{\sqrt{6}} (| 1_{j}, 0_{l} 〉 + | 0_{j}, 1_{l} 〉) | 2_{k} 〉 \\ + \frac{1}{\sqrt{6}} (| 1_{j}, 0_{l} 〉 + | 0_{j}, 1_{k} 〉) | 2_{l} 〉 \end{array}

G	π (2m + 1)	$\frac{π}{4} (2 m + 1)$	$\frac{π}{4} (2 m + 1)$
θ	$\frac{π}{8} (2 n + 1)$	πn	$\frac{π}{2} (2 n + 1)$

c ₂₁₀	$\frac{{(- 1)}^{n + 1}}{\sqrt{2}} e^{- i (ψ + φ)}$	0	0
C ₀₁₂	$\frac{{(- 1)}^{n}}{\sqrt{2}} e^{i (ψ + φ)}$	0	0
c ₂₀₁	0	$\frac{i {(- 1)}^{n + m}}{\sqrt{2}} e^{- i ψ}$	0
c ₀₂₁	0	$\frac{i {(- 1)}^{n + m}}{\sqrt{2}} e^{i ψ}$	0
c ₁₂₀	0	0	$\frac{i {(- 1)}^{n + m + 1}}{\sqrt{2}} e^{- i φ}$
c ₁₀₂	0	0	$\frac{i {(- 1)}^{n + m + 1}}{\sqrt{2}} e^{i φ}$

G	$2 π m + 2 \arctan (\sqrt{5 + 2 \sqrt{3}})$
θ	$π n + \arctan (1 + \sqrt{3})$	$π n - \arctan (1 + \sqrt{3})$

c ₃₀₀	$i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{- i (2 ψ + φ)}$	$- i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{- i (2 ψ + φ)}$
c ₀₃₀	$\frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$	$- \frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$
c ₀₀₃	$- i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{ψ + 2 φ}$	$- i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{ψ + 2 φ}$
c ₂₁₀	$\frac{1}{\sqrt{6}} e^{- i (ψ + φ)}$	$- \frac{1}{\sqrt{6}} e^{- i (ψ + φ)}$
c ₂₀₁	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i ψ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i ψ}$
c ₁₂₀	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$
c ₀₂₁	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$

G	$2 π m - 2 \arctan (\sqrt{5 + 2 \sqrt{3}})$
θ	$π n + \arctan (1 + \sqrt{3})$	$π n - \arctan (1 + \sqrt{3})$

c ₃₀₀	$- i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{- i (2 ψ + φ)}$	$i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{- i (2 ψ + φ)}$
c ₀₃₀	$\frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$	$- \frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$
c ₀₀₃	$i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{ψ + 2 φ}$	$i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{ψ + 2 φ}$
c ₂₁₀	$\frac{1}{\sqrt{6}} e^{- i (ψ + φ)}$	$- \frac{1}{\sqrt{6}} e^{- i (ψ + φ)}$
c ₂₀₁	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i ψ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i ψ}$
c ₁₂₀	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$
c ₀₂₁	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$

G	$2 π m + 2 \arctan (\sqrt{5 - 2 \sqrt{3}})$
θ	$π n + \arctan (1 - \sqrt{3})$	$π n - \arctan (1 - \sqrt{3})$

c ₃₀₀	$- i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{- i (2 ψ + φ)}$	$i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{- i (2 ψ + φ)}$
c ₀₃₀	$- \frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$	$\frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$
c ₀₀₃	$i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{ψ + 2 φ}$	$i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{ψ + 2 φ}$
c ₂₁₀	$- \frac{1}{\sqrt{6}} e^{- i (ψ + φ)}$	$\frac{1}{\sqrt{6}} e^{- i (ψ + φ)}$
c ₂₀₁	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i ψ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i ψ}$
c ₁₂₀	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$
c ₀₂₁	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$

G	$2 π m - 2 \arctan (\sqrt{5 - 2 \sqrt{3}})$
θ	$π n + \arctan (1 - \sqrt{3})$	$π n - \arctan (1 - \sqrt{3})$

c ₃₀₀	$i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{- i (2 ψ + φ)}$	$- i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{- i (2 ψ + φ)}$
c ₀₃₀	$- \frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$	$\frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$
c ₀₀₃	$- i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{ψ + 2 φ}$	$- i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{ψ + 2 φ}$
c ₂₁₀	$- \frac{1}{\sqrt{6}} e^{- i (ψ + φ)}$	$\frac{1}{\sqrt{6}} e^{- i (ψ + φ)}$
c ₂₀₁	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i ψ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i ψ}$
c ₁₂₀	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$
c ₀₂₁	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$

G	$π (2 m + 1 - \frac{1}{3})$	$π (2 m + 1 + \frac{1}{3})$	$π (2 m + 1 - \frac{1}{3})$	$π (2 m + 1 + \frac{1}{3})$
θ	$π n + arc \cot (\sqrt{2})$		$π n - arccot (\sqrt{2})$

c ₀₃₀	$\frac{\sqrt{3}}{4} e^{i (ψ - φ)}$	$\frac{\sqrt{3}}{4} e^{i (ψ - φ)}$	$- \frac{\sqrt{3}}{4} e^{i (ψ - φ)}$	$- \frac{\sqrt{3}}{4} e^{i (ψ - φ)}$
c ₀₀₃	$i \frac{{(- 1)}^{n} \sqrt{3}}{4} e^{i (ψ + 2 φ)}$	$- i \frac{{(- 1)}^{n} \sqrt{3}}{4} e^{i (ψ + 2 φ)}$	$i \frac{{(- 1)}^{n} \sqrt{3}}{4} e^{i (ψ + 2 φ)}$	$- i \frac{{(- 1)}^{n} \sqrt{3}}{4} e^{i (ψ + 2 φ)}$
c ₂₁₀	$\frac{1}{2} e^{- i (ψ + φ)}$	$\frac{1}{2} e^{- i (ψ + φ)}$	$- \frac{1}{2} e^{- i (ψ + φ)}$	$\frac{1}{2} e^{- i (ψ + φ)}$
c ₂₀₁	$- i \frac{{(- 1)}^{n}}{2} e^{- i ψ}$	$i \frac{{(- 1)}^{n}}{2} e^{- i ψ}$	$- i \frac{{(- 1)}^{n}}{2} e^{- i ψ}$	$i \frac{{(- 1)}^{n}}{2} e^{- i ψ}$
c ₀₂₁	$i \frac{{(- 1)}^{n}}{4} e^{i ψ}$	$- i \frac{{(- 1)}^{n}}{4} e^{i ψ}$	$i \frac{{(- 1)}^{n}}{4} e^{i ψ}$	$- i \frac{{(- 1)}^{n}}{4} e^{i ψ}$
c ₀₁₂	$\frac{{(- 1)}^{n}}{4} e^{i (ψ + φ)}$	$\frac{{(- 1)}^{n}}{4} e^{i (ψ + φ)}$	$- \frac{{(- 1)}^{n}}{4} e^{i (ψ + φ)}$	$- \frac{{(- 1)}^{n}}{4} e^{i (ψ + φ)}$

G	$\frac{π}{2} (2 m + 1)$
θ	$\frac{π}{4} (2 (4 \tilde{n} + 0) + 1)$	$\frac{π}{4} (2 (4 \tilde{n} + 1) + 1)$	$\frac{π}{4} (2 (4 \tilde{n} + 2) + 1)$	$\frac{π}{4} (2 (4 \tilde{n} + 3) + 1)$

c ₃₀₀	$i \frac{{(- 1)}^{m} \sqrt{3}}{4} e^{- i (2 ψ + φ)}$	$i \frac{{(- 1)}^{m} \sqrt{3}}{4} e^{- i (2 ψ + φ)}$	$- i \frac{{(- 1)}^{m} \sqrt{3}}{4} e^{- i (2 ψ + φ)}$	$- i \frac{{(- 1)}^{m} \sqrt{3}}{4} e^{- i (2 ψ + φ)}$
c ₀₀₃	$i \frac{{(- 1)}^{m} \sqrt{3}}{4} e^{- i (ψ + 2 φ)}$	$- i \frac{{(- 1)}^{m} \sqrt{3}}{4} e^{- i (ψ + 2 φ)}$	$- i \frac{{(- 1)}^{m} \sqrt{3}}{4} e^{- i (ψ + 2 φ)}$	$- i \frac{{(- 1)}^{m} \sqrt{3}}{4} e^{- i (ψ + 2 φ)}$
c ₂₀₁	$- i \frac{{(- 1)}^{m}}{4} e^{- i ψ}$	$i \frac{{(- 1)}^{m}}{4} e^{- i ψ}$	$i \frac{{(- 1)}^{m}}{4} e^{- i ψ}$	$- i \frac{{(- 1)}^{m}}{4} e^{- i ψ}$
c ₁₂₀	$i \frac{{(- 1)}^{m}}{2} e^{- i φ}$	$i \frac{{(- 1)}^{m}}{2} e^{- i φ}$	$- i \frac{{(- 1)}^{m}}{2} e^{- i φ}$	$- i \frac{{(- 1)}^{m}}{2} e^{- i φ}$
c ₀₂₁	$i \frac{{(- 1)}^{m}}{2} e^{i ψ}$	$- i \frac{{(- 1)}^{m}}{2} e^{i ψ}$	$- i \frac{{(- 1)}^{m}}{2} e^{i ψ}$	$i \frac{{(- 1)}^{m}}{2} e^{i ψ}$
c ₁₀₂	$- i \frac{{(- 1)}^{m}}{4} e^{i φ}$	$- i \frac{{(- 1)}^{m}}{4} e^{i φ}$	$i \frac{{(- 1)}^{m}}{4} e^{i φ}$	$i \frac{{(- 1)}^{m}}{4} e^{i φ}$

G	$π (2 m + 1 - \frac{1}{3})$	$π (2 m + 1 + \frac{1}{3})$	$π (2 m + 1 - \frac{1}{3})$	$π (2 m + 1 + \frac{1}{3})$
θ	$π n + \arctan (\sqrt{2})$		$π n - \arctan (\sqrt{2})$

c ₃₀₀	$i \frac{{(- 1)}^{n} \sqrt{3}}{4} e^{- i (2 ψ + φ)}$	$- i \frac{{(- 1)}^{n} \sqrt{3}}{4} e^{- i (2 ψ + φ)}$	$- i \frac{{(- 1)}^{n} \sqrt{3}}{4} e^{- i (2 ψ + φ)}$	$i \frac{{(- 1)}^{n} \sqrt{3}}{4} e^{- i (2 ψ + φ)}$
c ₀₀₃	$\frac{\sqrt{3}}{4} e^{i (ψ - φ)}$	$\frac{\sqrt{3}}{4} e^{i (ψ - φ)}$	$- \frac{\sqrt{3}}{4} e^{i (ψ - φ)}$	$- \frac{\sqrt{3}}{4} e^{i (ψ - φ)}$
c ₂₁₀	$\frac{1}{4} e^{- i (ψ + φ)}$	$\frac{1}{4} e^{- i (ψ + φ)}$	$- \frac{1}{4} e^{- i (ψ + φ)}$	$- \frac{1}{4} e^{- i (ψ + φ)}$
c ₁₂₀	$i \frac{{(- 1)}^{n}}{4} e^{- i φ}$	$- i \frac{{(- 1)}^{n}}{4} e^{- i φ}$	$- i \frac{{(- 1)}^{n}}{4} e^{- i φ}$	$i \frac{{(- 1)}^{n}}{4} e^{- i φ}$
c ₁₀₂	$- i \frac{{(- 1)}^{n}}{2} e^{i φ}$	$i \frac{{(- 1)}^{n}}{2} e^{i φ}$	$i \frac{{(- 1)}^{n}}{2} e^{i φ}$	$- i \frac{{(- 1)}^{n}}{2} e^{i φ}$
c ₀₁₂	$\frac{1}{2} e^{i (ψ + φ)}$	$\frac{1}{2} e^{i (ψ + φ)}$	$- \frac{1}{2} e^{i (ψ + φ)}$	$\frac{1}{2} e^{i (ψ + φ)}$

G	$2 π m + 2 \arctan (\sqrt{5 + 2 \sqrt{3}})$
θ	$π n + \arctan (\frac{1}{2} (1 - \sqrt{3}))$	$π n - \arctan (\frac{1}{2} (1 - \sqrt{3}))$

c ₃₀₀	$i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{- i (2 ψ + φ)}$	$- i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{- i (2 ψ + φ)}$
c ₀₃₀	$- \frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$	$\frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$
c ₀₀₃	$i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{ψ + 2 φ}$	$i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{ψ + 2 φ}$
c ₁₂₀	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i φ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i φ}$
c ₀₂₁	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$
c ₁₀₂	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$
c ₀₁₂	$- \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i (ψ + φ)}$	$\frac{{(- 1)}^{n}}{\sqrt{6}} e^{i (ψ + φ)}$

G	$2 π m - 2 \arctan (\sqrt{5 + 2 \sqrt{3}})$
θ	$π n + \arctan (\frac{1}{2} (1 - \sqrt{3}))$	$π n - \arctan (\frac{1}{2} (1 - \sqrt{3}))$

c ₃₀₀	$- i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{- i (2 ψ + φ)}$	$i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{- i (2 ψ + φ)}$
c ₀₃₀	$- \frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$	$\frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$
c ₀₀₃	$- i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{ψ + 2 φ}$	$- i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{ψ + 2 φ}$
c ₁₂₀	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i φ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i φ}$
c ₀₂₁	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$
c ₁₀₂	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$
c ₀₁₂	$- \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i (ψ + φ)}$	$\frac{{(- 1)}^{n}}{\sqrt{6}} e^{i (ψ + φ)}$

G	$2 π m + 2 \arctan (\sqrt{5 - 2 \sqrt{3}})$
θ	$π n + \arctan (\frac{1}{2} (1 + \sqrt{3}))$	$π n - \arctan (\frac{1}{2} (1 + \sqrt{3}))$

c ₃₀₀	$i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{- i (2 ψ + φ)}$	$- i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{- i (2 ψ + φ)}$
c ₀₃₀	$\frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$	$- \frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$
c ₀₀₃	$i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{ψ + 2 φ}$	$i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{ψ + 2 φ}$
c ₁₂₀	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i φ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i φ}$
c ₀₂₁	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$
c ₁₀₂	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$
c ₀₁₂	$\frac{{(- 1)}^{n}}{\sqrt{6}} e^{i (ψ + φ)}$	$- \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i (ψ + φ)}$

G	$2 π m - 2 \arctan (\sqrt{5 - 2 \sqrt{3}})$
θ	$π n + \arctan (\frac{1}{2} (1 + \sqrt{3}))$	$π n - \arctan (\frac{1}{2} (1 + \sqrt{3}))$

c ₃₀₀	$- i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{- i (2 ψ + φ)}$	$i \frac{{(- 1)}^{n} \sqrt{2}}{3} e^{- i (2 ψ + φ)}$
c ₀₃₀	$\frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$	$- \frac{1}{3 \sqrt{2}} e^{i (ψ - φ)}$
c ₀₀₃	$- i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{ψ + 2 φ}$	$- i \frac{{(- 1)}^{n}}{3 \sqrt{2}} e^{ψ + 2 φ}$
c ₁₂₀	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i φ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{- i φ}$
c ₀₂₁	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$	$- i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i ψ}$
c ₁₀₂	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$	$i \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i φ}$
c ₀₁₂	$\frac{{(- 1)}^{n}}{\sqrt{6}} e^{i (ψ + φ)}$	$- \frac{{(- 1)}^{n}}{\sqrt{6}} e^{i (ψ + φ)}$

Abstract

References

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