Abstract

We theoretically investigate quantum interference of two single photons at a lossy asymmetric beam splitter, the most general passive 2×2 optical circuit. The losses in the circuit result in a non-unitary scattering matrix with a non-trivial set of constraints on the elements of the scattering matrix. Our analysis using the noise operator formalism shows that the loss allows tunability of quantum interference to an extent not possible with a lossless beam splitter. Our theoretical studies support the experimental demonstrations of programmable quantum interference in highly multimodal systems such as opaque scattering media and multimode fibers.

© 2016 Optical Society of America

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References

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  1. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
    [Crossref] [PubMed]
  2. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
    [Crossref]
  3. M. Reck, A. Zeilinger, H.J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
    [Crossref] [PubMed]
  4. A. Politi, M.J. Cryan, J.G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
    [Crossref] [PubMed]
  5. R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic Hong-Ou-Mandel experiment,” Nature 520, 66–68 (2015).
    [Crossref] [PubMed]
  6. T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
    [Crossref]
  7. H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
    [Crossref]
  8. J. R. Petta, H. Lu, and A. C. Gossard, “A coherent beam splitter for electronic spin states,” Science 327, 669–672 (2010).
    [Crossref] [PubMed]
  9. C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013).
    [Crossref]
  10. R. W. Heeres, L. P. Kouwenhoven, and V. Zwiller, “Quantum interference in plasmonic circuits,” Nat. Nanotechnol. 8, 719–722 (2013).
    [Crossref] [PubMed]
  11. J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photon. 8, 317–320 (2014).
    [Crossref]
  12. T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
    [Crossref] [PubMed]
  13. L. Baldacci, S. Zanotto, G. Biasiol, L. Sorba, and A. Tredicucci, “Interferometric control of absorption in thin plasmonic metamaterials: general two port theory and broadband operation,” Opt. Exp. 23, 9202–9210 (2015).
    [Crossref]
  14. S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
    [Crossref]
  15. T. Gruner and D.-G. Welsch, “Quantum-optical input-output relations for dispersive and lossy multilayer dielectric plates,” Phys. Rev. A 54, 1661–1677 (1996).
    [Crossref] [PubMed]
  16. J. Jeffers, “Interference and the lossless lossy beam splitter,” J. Mod. Opt. 47, 1819–1824 (2000).
    [Crossref]
  17. S. Scheel, L. Knöll, T. Opatrný, and D.-G. Welsch, “Entanglement transformation at absorbing and amplifying four-port devices,” Phys. Rev. A 62, 043803 (2000).
    [Crossref]
  18. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
    [Crossref] [PubMed]
  19. A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
    [Crossref]
  20. I. A. Walmsley, “Quantum optics: Science and technology in a new light,” Science 348, 525–530 (2015).
    [Crossref] [PubMed]
  21. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).
  22. B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306–4322 (1992).
    [Crossref] [PubMed]
  23. J. Zmuidzinas, “Thermal noise and correlations in photon detection,” App. Opt. 42, 4989–5008 (2003).
    [Crossref]
  24. P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
    [Crossref]
  25. W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997).
    [Crossref]
  26. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
    [Crossref] [PubMed]
  27. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nature Photon. 6, 283–292 (2012).
    [Crossref]
  28. S. R. Huisman, T. J. Huisman, S. A. Goorden, A. P. Mosk, and P. W. H. Pinkse, “Programming balanced optical beam splitters in white paint,” Opt. Express 22, 8320–8332 (2014).
    [Crossref] [PubMed]
  29. S. R. Huisman, T. J. Huisman, T. A. W. Wolterink, A. P. Mosk, and P. W. H. Pinkse, “Programmable multiport optical circuits in opaque scattering materials,” Opt. Express 23, 3102–3116 (2015).
    [Crossref] [PubMed]
  30. J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
    [Crossref] [PubMed]
  31. N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

2016 (2)

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

2015 (6)

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

L. Baldacci, S. Zanotto, G. Biasiol, L. Sorba, and A. Tredicucci, “Interferometric control of absorption in thin plasmonic metamaterials: general two port theory and broadband operation,” Opt. Exp. 23, 9202–9210 (2015).
[Crossref]

R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic Hong-Ou-Mandel experiment,” Nature 520, 66–68 (2015).
[Crossref] [PubMed]

I. A. Walmsley, “Quantum optics: Science and technology in a new light,” Science 348, 525–530 (2015).
[Crossref] [PubMed]

S. R. Huisman, T. J. Huisman, T. A. W. Wolterink, A. P. Mosk, and P. W. H. Pinkse, “Programmable multiport optical circuits in opaque scattering materials,” Opt. Express 23, 3102–3116 (2015).
[Crossref] [PubMed]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

2014 (2)

2013 (2)

C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013).
[Crossref]

R. W. Heeres, L. P. Kouwenhoven, and V. Zwiller, “Quantum interference in plasmonic circuits,” Nat. Nanotechnol. 8, 719–722 (2013).
[Crossref] [PubMed]

2012 (2)

A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nature Photon. 6, 283–292 (2012).
[Crossref]

2010 (1)

J. R. Petta, H. Lu, and A. C. Gossard, “A coherent beam splitter for electronic spin states,” Science 327, 669–672 (2010).
[Crossref] [PubMed]

2008 (1)

A. Politi, M.J. Cryan, J.G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[Crossref] [PubMed]

2007 (2)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
[Crossref] [PubMed]

2003 (1)

J. Zmuidzinas, “Thermal noise and correlations in photon detection,” App. Opt. 42, 4989–5008 (2003).
[Crossref]

2001 (1)

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

2000 (2)

J. Jeffers, “Interference and the lossless lossy beam splitter,” J. Mod. Opt. 47, 1819–1824 (2000).
[Crossref]

S. Scheel, L. Knöll, T. Opatrný, and D.-G. Welsch, “Entanglement transformation at absorbing and amplifying four-port devices,” Phys. Rev. A 62, 043803 (2000).
[Crossref]

1998 (1)

S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[Crossref]

1997 (1)

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997).
[Crossref]

1996 (1)

T. Gruner and D.-G. Welsch, “Quantum-optical input-output relations for dispersive and lossy multilayer dielectric plates,” Phys. Rev. A 54, 1661–1677 (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, 58–61 (1994).
[Crossref] [PubMed]

1992 (1)

B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306–4322 (1992).
[Crossref] [PubMed]

1987 (1)

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

1964 (1)

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
[Crossref]

Aspect, A.

R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic Hong-Ou-Mandel experiment,” Nature 520, 66–68 (2015).
[Crossref] [PubMed]

Aspuru-Guzik, A.

A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

Atwater, H. A.

J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photon. 8, 317–320 (2014).
[Crossref]

Baehr-Jones, T.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

Baldacci, L.

L. Baldacci, S. Zanotto, G. Biasiol, L. Sorba, and A. Tredicucci, “Interferometric control of absorption in thin plasmonic metamaterials: general two port theory and broadband operation,” Opt. Exp. 23, 9202–9210 (2015).
[Crossref]

Barbieri, M.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

Barnett, S. M.

S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[Crossref]

B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306–4322 (1992).
[Crossref] [PubMed]

Bernstein, H.J.

M. Reck, A. Zeilinger, H.J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (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, 58–61 (1994).
[Crossref] [PubMed]

Biasiol, G.

L. Baldacci, S. Zanotto, G. Biasiol, L. Sorba, and A. Tredicucci, “Interferometric control of absorption in thin plasmonic metamaterials: general two port theory and broadband operation,” Opt. Exp. 23, 9202–9210 (2015).
[Crossref]

Blais, A.

C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013).
[Crossref]

Boiron, D.

R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic Hong-Ou-Mandel experiment,” Nature 520, 66–68 (2015).
[Crossref] [PubMed]

Bolduc, E.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

Boller, K.-J.

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

Carolan, J.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Cheneau, M.

R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic Hong-Ou-Mandel experiment,” Nature 520, 66–68 (2015).
[Crossref] [PubMed]

Couteau, C.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

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, 646–649 (2008).
[Crossref] [PubMed]

Ctistis, G.

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

Defienne, H.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Eichler, C.

C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013).
[Crossref]

Englund, D.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

Faccio, D.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

Fakonas, J. S.

J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photon. 8, 317–320 (2014).
[Crossref]

Fink, J. M.

C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013).
[Crossref]

Fink, M.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nature Photon. 6, 283–292 (2012).
[Crossref]

Gatti, A.

S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[Crossref]

Gerry, C. C.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

Gigan, S.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

Goorden, S. A.

Gossard, A. C.

J. R. Petta, H. Lu, and A. C. Gossard, “A coherent beam splitter for electronic spin states,” Science 327, 669–672 (2010).
[Crossref] [PubMed]

Grice, W. P.

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997).
[Crossref]

Gruner, T.

T. Gruner and D.-G. Welsch, “Quantum-optical input-output relations for dispersive and lossy multilayer dielectric plates,” Phys. Rev. A 54, 1661–1677 (1996).
[Crossref] [PubMed]

Harris, N. C.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

Harrold, C.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Hashimoto, T.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Heeres, R. W.

R. W. Heeres, L. P. Kouwenhoven, and V. Zwiller, “Quantum interference in plasmonic circuits,” Nat. Nanotechnol. 8, 719–722 (2013).
[Crossref] [PubMed]

Heitz, J. J. F.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

Hochberg, M.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

Hong, C. K.

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

Huisman, S. R.

Huisman, T. J.

Huttner, B.

B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306–4322 (1992).
[Crossref] [PubMed]

Imanaliev, A.

R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic Hong-Ou-Mandel experiment,” Nature 520, 66–68 (2015).
[Crossref] [PubMed]

Itoh, M.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Jeffers, J.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

J. Jeffers, “Interference and the lossless lossy beam splitter,” J. Mod. Opt. 47, 1819–1824 (2000).
[Crossref]

S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[Crossref]

Kelaita, Y. A.

J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photon. 8, 317–320 (2014).
[Crossref]

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
[Crossref]

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
[Crossref]

Knight, P. L.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

Knill, E.

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

Knöll, L.

S. Scheel, L. Knöll, T. Opatrný, and D.-G. Welsch, “Entanglement transformation at absorbing and amplifying four-port devices,” Phys. Rev. A 62, 043803 (2000).
[Crossref]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Kouwenhoven, L. P.

R. W. Heeres, L. P. Kouwenhoven, and V. Zwiller, “Quantum interference in plasmonic circuits,” Nat. Nanotechnol. 8, 719–722 (2013).
[Crossref] [PubMed]

Laflamme, R.

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

Lagendijk, A.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nature Photon. 6, 283–292 (2012).
[Crossref]

Lahini, Y.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

Laing, A.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Lang, C.

C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013).
[Crossref]

Leach, J.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

Lee, H.

J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photon. 8, 317–320 (2014).
[Crossref]

Lerosey, G.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nature Photon. 6, 283–292 (2012).
[Crossref]

Lloyd, S.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

Lopes, R.

R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic Hong-Ou-Mandel experiment,” Nature 520, 66–68 (2015).
[Crossref] [PubMed]

Loudon, R.

S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[Crossref]

Lu, H.

J. R. Petta, H. Lu, and A. C. Gossard, “A coherent beam splitter for electronic spin states,” Science 327, 669–672 (2010).
[Crossref] [PubMed]

Mandel, L.

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

Marshall, G. D.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Martín-López, E.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Matsuda, N.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Matthews, J. C. F.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

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

Mosk, A. P.

Mower, J.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

O’Brien, J. L.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[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, 646–649 (2008).
[Crossref] [PubMed]

Oguma, M.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Opatrný, T.

S. Scheel, L. Knöll, T. Opatrný, and D.-G. Welsch, “Entanglement transformation at absorbing and amplifying four-port devices,” Phys. Rev. A 62, 043803 (2000).
[Crossref]

Ou, Z. Y.

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

Petta, J. R.

J. R. Petta, H. Lu, and A. C. Gossard, “A coherent beam splitter for electronic spin states,” Science 327, 669–672 (2010).
[Crossref] [PubMed]

Pinkse, P. W. H.

Politi, A.

A. Politi, M.J. Cryan, J.G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[Crossref] [PubMed]

Prabhu, M.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Rarity, J.G.

A. Politi, M.J. Cryan, J.G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[Crossref] [PubMed]

Reck, M.

M. Reck, A. Zeilinger, H.J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref] [PubMed]

Roger, T.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

Russell, N. J.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Scheel, S.

S. Scheel, L. Knöll, T. Opatrný, and D.-G. Welsch, “Entanglement transformation at absorbing and amplifying four-port devices,” Phys. Rev. A 62, 043803 (2000).
[Crossref]

Shadbolt, P. J.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Silverstone, J. W.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Smith, B. J.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

Soci, C.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

Sorba, L.

L. Baldacci, S. Zanotto, G. Biasiol, L. Sorba, and A. Tredicucci, “Interferometric control of absorption in thin plasmonic metamaterials: general two port theory and broadband operation,” Opt. Exp. 23, 9202–9210 (2015).
[Crossref]

Sparrow, C.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Steffen, L.

C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013).
[Crossref]

Steinbrecher, G. R.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

Thompson, M. G.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Tredicucci, A.

L. Baldacci, S. Zanotto, G. Biasiol, L. Sorba, and A. Tredicucci, “Interferometric control of absorption in thin plasmonic metamaterials: general two port theory and broadband operation,” Opt. Exp. 23, 9202–9210 (2015).
[Crossref]

Uppu, R.

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

Valente, J.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

Vellekoop, I. M.

Vezzoli, S.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

Vos, W. L.

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

Wallraff, A.

C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013).
[Crossref]

Walmsley, I. A.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

I. A. Walmsley, “Quantum optics: Science and technology in a new light,” Science 348, 525–530 (2015).
[Crossref] [PubMed]

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997).
[Crossref]

Walther, P.

A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

Welsch, D.-G.

S. Scheel, L. Knöll, T. Opatrný, and D.-G. Welsch, “Entanglement transformation at absorbing and amplifying four-port devices,” Phys. Rev. A 62, 043803 (2000).
[Crossref]

T. Gruner and D.-G. Welsch, “Quantum-optical input-output relations for dispersive and lossy multilayer dielectric plates,” Phys. Rev. A 54, 1661–1677 (1996).
[Crossref] [PubMed]

Westbrook, C. I.

R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic Hong-Ou-Mandel experiment,” Nature 520, 66–68 (2015).
[Crossref] [PubMed]

Wolterink, T. A. W.

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

S. R. Huisman, T. J. Huisman, T. A. W. Wolterink, A. P. Mosk, and P. W. H. Pinkse, “Programmable multiport optical circuits in opaque scattering materials,” Opt. Express 23, 3102–3116 (2015).
[Crossref] [PubMed]

Woolley, M. J.

C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013).
[Crossref]

Yu, S.

A. Politi, M.J. Cryan, J.G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[Crossref] [PubMed]

Zanotto, S.

L. Baldacci, S. Zanotto, G. Biasiol, L. Sorba, and A. Tredicucci, “Interferometric control of absorption in thin plasmonic metamaterials: general two port theory and broadband operation,” Opt. Exp. 23, 9202–9210 (2015).
[Crossref]

Zeilinger, A.

M. Reck, A. Zeilinger, H.J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref] [PubMed]

Zheludev, N. I.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

Zmuidzinas, J.

J. Zmuidzinas, “Thermal noise and correlations in photon detection,” App. Opt. 42, 4989–5008 (2003).
[Crossref]

Zwiller, V.

R. W. Heeres, L. P. Kouwenhoven, and V. Zwiller, “Quantum interference in plasmonic circuits,” Nat. Nanotechnol. 8, 719–722 (2013).
[Crossref] [PubMed]

App. Opt. (1)

J. Zmuidzinas, “Thermal noise and correlations in photon detection,” App. Opt. 42, 4989–5008 (2003).
[Crossref]

J. Mod. Opt. (1)

J. Jeffers, “Interference and the lossless lossy beam splitter,” J. Mod. Opt. 47, 1819–1824 (2000).
[Crossref]

Nat. Commun. (1)

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

R. W. Heeres, L. P. Kouwenhoven, and V. Zwiller, “Quantum interference in plasmonic circuits,” Nat. Nanotechnol. 8, 719–722 (2013).
[Crossref] [PubMed]

Nat. Phys. (2)

A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, “Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies,” Nat. Phys. 9, 345–348 (2013).
[Crossref]

Nature (2)

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

R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, “Atomic Hong-Ou-Mandel experiment,” Nature 520, 66–68 (2015).
[Crossref] [PubMed]

Nature Photon. (2)

J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nature Photon. 8, 317–320 (2014).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nature Photon. 6, 283–292 (2012).
[Crossref]

Opt. Exp. (1)

L. Baldacci, S. Zanotto, G. Biasiol, L. Sorba, and A. Tredicucci, “Interferometric control of absorption in thin plasmonic metamaterials: general two port theory and broadband operation,” Opt. Exp. 23, 9202–9210 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. (1)

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
[Crossref]

Phys. Rev. A (6)

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A 56, 1627–1634 (1997).
[Crossref]

B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306–4322 (1992).
[Crossref] [PubMed]

S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[Crossref]

T. Gruner and D.-G. Welsch, “Quantum-optical input-output relations for dispersive and lossy multilayer dielectric plates,” Phys. Rev. A 54, 1661–1677 (1996).
[Crossref] [PubMed]

S. Scheel, L. Knöll, T. Opatrný, and D.-G. Welsch, “Entanglement transformation at absorbing and amplifying four-port devices,” Phys. Rev. A 62, 043803 (2000).
[Crossref]

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

Phys. Rev. Lett. (2)

M. Reck, A. Zeilinger, H.J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref] [PubMed]

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

Rev. Mod. Phys. (1)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Sci. Adv. (1)

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

Science (4)

J. R. Petta, H. Lu, and A. C. Gossard, “A coherent beam splitter for electronic spin states,” Science 327, 669–672 (2010).
[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, 646–649 (2008).
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J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
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Other (2)

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” ar.Xiv: 1507.03406 (2015).

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

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Figures (5)

Fig. 1
Fig. 1 (a) depicts the schematic of a general 2×2 beam splitter with input ports a1 and a2 and output ports b1 and b2. The transmission-reflection amplitudes for light in input ports a1 and a2 are t-r and τ-ρ respectively. (b) illustrates the output power at ports b1 (orange curve) and b2 (blue curve) as phase θ is varied between 0 and 2π at input port a1. The phase between the peak amplitudes α is related to the phases of the reflection coefficients ϕ1 and ϕ2 as α = ϕ1 + ϕ2.
Fig. 2
Fig. 2 The figure depicts the allowed tunable width Δα around π. The anti-diagonal line (r2 +t2 = 1) separating the allowed from the forbidden region corresponds to lossless beam splitter. The red dashed line is the curve t + r = 1. Any lossy circuit that satisfies t + r ≤ 1 allows complete tunability of α ∈ [0,2π].
Fig. 3
Fig. 3 (a) The variation of the maximal coincidence rate maxαP(11,12) in a general beam splitter is shown as a function of reflectance and transmittance. The solid curves in (a) and (b) correspond to cross-sections along different imbalance values t2/r2. The dashed curve in (a) and (b) is the coincidence probability in a lossless beam splitter. The dotted curve in (a) and (b) depicts the coincidence probability of beam splitters with t + r = 1.
Fig. 4
Fig. 4 The figure depicts the coincidence probability P(11,12) as a function of delay time (δt) at various values of α in a lossy symmetric balanced beamspliter with t = τ = r = ρ = 0.5. The coincidence probability P(11,12) varies like a cosine with α for perfectly indistinguishable photons δt = 0. The conventional Hong-Ou-Mandel dip (red curve) is seen at α = π which becomes a peak at α = 0 or 2π. The triangular shape of the Hong-Ou-Mandel dip or peak is a consequence of the photon pair generation process.
Fig. 5
Fig. 5 Programmability of the coincidence rate ΔP(11,12) is depicted here together with few representative contours at values indicated beside them. The black dashed curve represents t + r = 1. The lossless beam splitters have ΔP(11,12) = 0, while the balanced lossy beam splitters satisfying t + r < 1 have maximal programmability with ΔP(11,12) = 2.

Equations (39)

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S = [ t ρ e i ϕ 2 r e i ϕ 1 τ ] .
| t E 1 + ρ e i ϕ 2 E 2 | 2 + | r E 1 e i ϕ 1 + τ E 2 | 2 | E 1 | 2 + | E 2 | 2 .
t ρ cos ( ϕ 2 δ ) + τ r cos ( ϕ 1 + δ ) ( 1 t 2 r 2 ) + | c | 2 ( 1 τ 2 ρ 2 ) 2 | c | .
t ρ cos ( ϕ 2 δ ) + τ r cos ( ϕ 1 + δ ) ( 1 t 2 r 2 ) ( 1 τ 2 ρ 2 ) .
t 2 ρ 2 + τ 2 r 2 + 2 τ ρ r t cos ( ϕ 1 + ϕ 2 ) sin ( δ + θ off ) ( 1 t 2 r 2 ) ( 1 τ 2 ρ 2 ) ,
t 2 ρ 2 + τ 2 r 2 + 2 τ ρ r t cos α ( 1 t 2 r 2 ) ( 1 τ 2 ρ 2 ) .
S sym-bal = t [ 1 i i 1 ]
| cos α 2 | 1 t 2 r 2 2 t r .
[ b ^ 1 ( ω ) b ^ 2 ( ω ) ] = [ t ( ω ) ρ ( ω ) e i ϕ 2 r ( ω ) e i ϕ 1 τ ( ω ) ] [ a ^ 1 ( ω ) a ^ 2 ( ω ) ] + [ F ^ 1 ( ω ) F ^ 2 ( ω ) ] .
[ a ^ i ( ω ) , a ^ j ( ω ) ] = 0 ; i , j { 1 , 2 } ,
[ a ^ i ( ω ) , a ^ j ( ω ) ] = δ i j δ ( ω ω ) ; i , j { 1 , 2 } ,
[ b ^ i ( ω ) , b ^ j ( ω ) ] = 0 ; i , j { 1 , 2 } ,
[ b ^ i ( ω ) , b ^ j ( ω ) ] = δ i j δ ( ω ω ) ; i , j { 1 , 2 } .
[ a ^ i ( ω ) , F ^ j ( ω ) ] = [ a ^ i ( ω ) , F ^ j ( ω ) ] = 0 ; i , j { 1 , 2 } ,
[ F ^ i ( ω ) , F ^ j ( ω ) ] = [ F ^ i ( ω ) , F ^ j ( ω ) ] = 0 ; i , j { 1 , 2 } ,
[ F ^ 1 ( ω ) , F ^ 1 ( ω ) ] = δ ( ω ω ) [ 1 t 2 ( ω ) ρ 2 ( ω ) ] ,
[ F ^ 2 ( ω ) , F ^ 2 ( ω ) ] = δ ( ω ω ) [ 1 τ 2 ( ω ) r 2 ( ω ) ] ,
[ F ^ 1 ( ω ) , F ^ 2 ( ω ) ] = δ ( ω ω ) [ t ( ω ) r ( ω ) e i ϕ 1 + ρ ( ω ) τ ( ω ) e i ϕ 2 ] ,
[ F ^ 2 ( ω ) , F ^ 1 ( ω ) ] = δ ( ω ω ) [ t ( ω ) r ( ω ) e i ϕ 1 + ρ ( ω ) τ ( ω ) e i ϕ 2 ] .
| Ψ = | 1 1 , 1 2 = 0 d ω 1 0 d ω 2 ψ ( ω 1 , ω 2 ) a ^ 1 ( ω 1 ) a ^ 2 ( ω 2 ) | 0 .
N ^ i ( ω ) = 0 d ω b ^ i ( ω ) b ^ i ( ω ) i { 1 , 2 } .
P ( 2 1 , 0 2 ) = 1 2 N ^ 1 ( N ^ 1 1 ) ,
P ( 0 1 , 2 2 ) = 1 2 N ^ 2 ( N ^ 2 1 ) ,
P ( 1 1 , 1 2 ) = N ^ 1 N ^ 2
P ( 1 1 , 0 2 ) = N ^ 1 N ^ 1 ( N ^ 1 1 ) N ^ 1 N ^ 2 ,
P ( 0 1 , 1 2 ) = N ^ 2 N ^ 2 ( N ^ 2 1 ) N ^ 1 N ^ 2
P ( 0 1 , 0 2 ) = 1 N ^ 1 N ^ 2 + N ^ 1 N ^ 2 + 1 2 N ^ 1 ( N ^ 1 1 ) + 1 2 N ^ 2 ( N ^ 2 1 )
N ^ 1 = t 2 + ρ 2 ,
N ^ 2 = τ 2 + r 2 ,
N ^ 1 ( N ^ 1 1 ) = 2 t 2 ρ 2 [ 1 + I overlap ( δ t ) ] ,
N ^ 1 ( N ^ 1 1 ) = 2 τ 2 r 2 [ 1 + I overlap ( δ t ) ] ,
N ^ 1 N ^ 2 = t 2 τ 2 + r 2 ρ 2 + 2 τ ρ t r I overlap ( δ t ) cos α ,
I overlap ( δ t ) = 0 d ω 1 0 d ω 2 ψ ( ω 1 , ω 2 ) ψ * ( ω 2 , ω 1 ) exp [ i ( ω 1 ω 2 ) δ t ] .
P ( 1 1 , 1 2 ) = t 4 + r 4 + 2 t 2 r 2 I overlap ( δ t ) cos α ,
P ( 2 1 , 0 2 ) = P ( 0 1 , 2 2 ) = t 2 r 2 [ 1 + I overlap ( δ t ) ] ,
P ( 1 1 , 0 2 ) = P ( 0 1 , 1 2 ) = t 2 + r 2 t 4 r 4 2 t 2 r 2 { 1 + I overlap ( δ t ) [ 1 + cos α ] } ,
P ( 0 1 , 0 2 ) = 1 2 ( t 2 + r 2 ) + t 4 + r 4 2 t 2 r 2 { 1 + I overlap ( δ t ) [ 1 + cos α ] } ,
ψ ( ω i , ω s ) = sinc ( k p k i k s 2 π Λ π L 2 ) exp { [ ( ω s + ω i ω p ) τ p 2 ] 2 } ,
Δ P ( 1 1 , 1 2 ) max α P ( 1 1 , 1 2 ) min α P ( 1 1 , 1 2 ) P ( 1 1 , 1 2 ; distinguisable ) ,

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