Abstract

The design, fabrication, and detailed calibration of essential building blocks towards fully integrated linear-optics quantum computation are discussed. Photonic devices are made from silicon nitride rib waveguides, where measurements on ring resonators show small propagation losses. Directional couplers are designed to be insensitive to fabrication variations. Their offset and coupling lengths are measured, as well as the phase difference between the transmitted and reflected light. With careful calibrations, the insertion loss of the directional couplers is found to be small. Finally, an integrated controlled-NOT circuit is characterized by measuring the transmission through different combinations of inputs and outputs. The gate fidelity for the CNOT operation with this circuit is estimated to be 99.81% after post selection. This high fidelity is due to our robust design, good fabrication reproducibility, and extensive characterizations.

© 2016 Optical Society of America

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  1. A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
    [Crossref]
  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]
  3. 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]
  4. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University).
  5. J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett. 93, 080502 (2004).
    [Crossref]
  6. P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
    [Crossref] [PubMed]
  7. T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of quantum error correction using linear optics,” Phys. Rev. A 71, 052332 (2005).
    [Crossref]
  8. R. Okamoto, J. L. OBrien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn controlled-NOT photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA 108, 10067–10071 (2011).
    [Crossref] [PubMed]
  9. W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
    [Crossref]
  10. 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]
  11. B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129 (2005).
    [Crossref]
  12. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
    [Crossref] [PubMed]
  13. 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, 794–798 (2013).
    [Crossref]
  14. J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
    [Crossref] [PubMed]
  15. C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, “Superconducting nanowire single-photon detectors: physics and applications,” Supercond. Sci. Technol. 25, 063001 (2012).
    [Crossref]
  16. W. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat Commun 3, 1325 (2012).
    [Crossref] [PubMed]
  17. C. Schuck, W. H. Pernice, X. Ma, and H. X. Tang, “Optical time domain reflectometry with low noise waveguide-coupled superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 102, 191104 (2013).
    [Crossref]
  18. J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
    [Crossref]
  19. C. Schuck, W. H. P. Pernice, and H. X. Tang, “Waveguide integrated low noise NbTiN nanowire single-photon detectors with milli-Hz dark count rate,” Sci. Rep. 3, 1–6 (2013).
    [Crossref]
  20. M. Poot and H. X. Tang, “Broadband nanoelectromechanical phase shifting of light on a chip,” Appl. Phys. Lett. 104, 061101 (2014).
    [Crossref]
  21. M. Poot and H. X. Tang, “High-quality opto-electromechanical resonators,” In preparation (2015).
  22. M. Poot, K. Y. Fong, and H. X. Tang, “Classical non-gaussian state preparation through squeezing in an opto-electromechanical resonator,” Phys. Rev. A 90, 063809 (2014).
    [Crossref]
  23. M. Poot, K. Y. Fong, and H. X. Tang, “Deep feedback-stabilized parametric squeezing in an opto-electromechanical system,” New J. Phys. 17, 043056 (2015).
    [Crossref]
  24. P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
    [Crossref]
  25. R. St-Gelais, B. Guha, L. Zhu, S. Fan, and M. Lipson, “Demonstration of strong near-field radiative heat transfer between integrated nanostructures,” Nano Lett. 14, 6971–6975 (2014). PMID: .
    [Crossref] [PubMed]
  26. C. Schuck, X. Guo, L. Fan, X. Ma, M. Poot, and H. X. Tang, “Quantum interference in heterogeneous superconducting-photonic circuits on a silicon chip,” Nat Commun 7, 10352 (2016).
    [Crossref] [PubMed]
  27. C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14, 095014 (2012).
    [Crossref]
  28. C. Xiong, W. Pernice, K. K. Ryu, C. Schuck, K. Y. Fong, T. Palacios, and H. X. Tang, “Integrated gan photonic circuits on silicon (100) for second harmonic generation,” Opt. Express 19, 10462–10470 (2011).
    [Crossref] [PubMed]
  29. C. Wang, M. J. Burek, Z. Lin, H. A. Atikian, V. Venkataraman, I.-C. Huang, P. Stark, and M. Lončar, “Integrated high quality factor lithium niobate microdisk resonators,” Opt. Express 22, 30924–30933 (2014).
    [Crossref]
  30. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388–12393 (2006).
    [Crossref] [PubMed]
  31. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
    [Crossref]
  32. A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, “On-chip optical squeezing,” Phys. Rev. Applied 3, 044005 (2015).
    [Crossref]
  33. A. Gondarenko, J. S. Levy, and M. Lipson, “High confinement micron-scale silicon nitride high q ring resonator,” Opt. Express 17, 11366–11370 (2009).
    [Crossref] [PubMed]
  34. M.-C. Tien, J. F. Bauters, M. J. R. Heck, D. T. Spencer, D. J. Blumenthal, and J. E. Bowers, “Ultra-high quality factor planar Si3N4 ring resonators on si substrates,” Opt. Express 19, 13551–13556 (2011).
    [Crossref] [PubMed]
  35. D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
    [Crossref]
  36. M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
    [Crossref]
  37. N. Gruhler, C. Benz, H. Jang, J.-H. Ahn, R. Danneau, and W. H. P. Pernice, “High-quality Si3N4 circuits as a platform for graphene-based nanophotonic devices,” Opt. Express 21, 31678–31689 (2013).
    [Crossref]
  38. W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
    [Crossref]
  39. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightw. Technol. 20, 1968–1975 (2002).
    [Crossref]
  40. P. Rath, N. Gruhler, S. Khasminskaya, C. Nebel, C. Wild, and W. H. P. Pernice, “Waferscale nanophotonic circuits made from diamond-on-insulator substrates,” Opt. Express 21, 11031–11036 (2013).
    [Crossref] [PubMed]
  41. K. Y. Fong, M. Poot, and H. X. Tang, “Nano-optomechanical resonators in microfluidics,” Nano Lett. 15, 6116–6120 (2015).
    [Crossref] [PubMed]
  42. 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]
  43. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [Crossref]
  44. T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical controlled-not gate in the coincidence basis,” Phys. Rev. A 65, 062324 (2002).
    [Crossref]
  45. Y. Zhang, S. Yang, A. E.-J. Lim, G.-Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss y-junction for submicron silicon waveguide,” Opt. Express 21, 1310–1316 (2013).
    [Crossref] [PubMed]
  46. Z. Lu, H. Yun, Y. Wang, Z. Chen, F. Zhang, N. A. F. Jaeger, and L. Chrostowski, “Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control,” Opt. Express 23, 3795–3808 (2015).
    [Crossref] [PubMed]
  47. In practice, the transmissions are not determined in this step-by-step process, but instead all measured (logarithmic) transmission data is fitted simultaneously by using 15-th order polynomials in λ for Ti←2(λ) and Y(λ). This is done through linear fitting via LU decomposition. The final results do not depend significantly on the order of the polynomial as long as the order is high enough to follow the overall profile, but low enough to filter out the fine fringes. The small upturns near 1570 nm in the curves shown in Fig. 4(a) are artifacts of the polynomial fits due the end of the data range.
  48. The maximum transmission in Fig. 4(b) is ∼ −32dB. This value is lower than typical for our SiN devices (∼−15dB) since the fiber array was kept far away from the sample to prevent any accidental touching or scratching while scanning the entire chip. Note that none of the results presented here depend on the absolute transmission.
  49. M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a bell-state filter by quantum process tomography,” Phys. Rev. Lett. 91, 120402 (2003).
    [Crossref] [PubMed]
  50. The convention where the optical field is proportional to exp(−iωt) is used. In this case the phase of light propagating in, say, the +x direction is ϕp = +2πneffx/λ. Since the outer arms are the reference arms, the phase difference is defined as Δϕ = ϕinner − ϕouter. The outer arm is longer than the inner arm and hence Δϕ < 0. Since ∂ϕp/∂λ = −ϕp/λ (without dispersion), the phase difference increases with increasing wavelength.
  51. Although in a general beam splitter any value of the phase difference is allowed (with constraints on their combination), for a symmetric design such as our directional coupler, only −π/2 and +π/2 are possible [1].
  52. S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, and A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
    [Crossref] [PubMed]
  53. B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
    [Crossref] [PubMed]
  54. J. Skaar, J. C. García Escartín, and H. Landro, “Quantum mechanical description of linear optics,” American Journal of Physics 72, 1385–1391 (2004).
    [Crossref]
  55. U. Leonhardt and A. Neumaier, “Explicit effective hamiltonians for general linear quantum-optical networks,” Journal of Optics B: Quantum and Semiclassical Optics 6, L1 (2004).
    [Crossref]

2016 (1)

C. Schuck, X. Guo, L. Fan, X. Ma, M. Poot, and H. X. Tang, “Quantum interference in heterogeneous superconducting-photonic circuits on a silicon chip,” Nat Commun 7, 10352 (2016).
[Crossref] [PubMed]

2015 (5)

M. Poot, K. Y. Fong, and H. X. Tang, “Deep feedback-stabilized parametric squeezing in an opto-electromechanical system,” New J. Phys. 17, 043056 (2015).
[Crossref]

A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, “On-chip optical squeezing,” Phys. Rev. Applied 3, 044005 (2015).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

K. Y. Fong, M. Poot, and H. X. Tang, “Nano-optomechanical resonators in microfluidics,” Nano Lett. 15, 6116–6120 (2015).
[Crossref] [PubMed]

Z. Lu, H. Yun, Y. Wang, Z. Chen, F. Zhang, N. A. F. Jaeger, and L. Chrostowski, “Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control,” Opt. Express 23, 3795–3808 (2015).
[Crossref] [PubMed]

2014 (5)

P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
[Crossref]

R. St-Gelais, B. Guha, L. Zhu, S. Fan, and M. Lipson, “Demonstration of strong near-field radiative heat transfer between integrated nanostructures,” Nano Lett. 14, 6971–6975 (2014). PMID: .
[Crossref] [PubMed]

M. Poot and H. X. Tang, “Broadband nanoelectromechanical phase shifting of light on a chip,” Appl. Phys. Lett. 104, 061101 (2014).
[Crossref]

M. Poot, K. Y. Fong, and H. X. Tang, “Classical non-gaussian state preparation through squeezing in an opto-electromechanical resonator,” Phys. Rev. A 90, 063809 (2014).
[Crossref]

C. Wang, M. J. Burek, Z. Lin, H. A. Atikian, V. Venkataraman, I.-C. Huang, P. Stark, and M. Lončar, “Integrated high quality factor lithium niobate microdisk resonators,” Opt. Express 22, 30924–30933 (2014).
[Crossref]

2013 (8)

C. Schuck, W. H. P. Pernice, and H. X. Tang, “Waveguide integrated low noise NbTiN nanowire single-photon detectors with milli-Hz dark count rate,” Sci. Rep. 3, 1–6 (2013).
[Crossref]

N. Gruhler, C. Benz, H. Jang, J.-H. Ahn, R. Danneau, and W. H. P. Pernice, “High-quality Si3N4 circuits as a platform for graphene-based nanophotonic devices,” Opt. Express 21, 31678–31689 (2013).
[Crossref]

C. Schuck, W. H. Pernice, X. Ma, and H. X. Tang, “Optical time domain reflectometry with low noise waveguide-coupled superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 102, 191104 (2013).
[Crossref]

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, 794–798 (2013).
[Crossref]

Y. Zhang, S. Yang, A. E.-J. Lim, G.-Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss y-junction for submicron silicon waveguide,” Opt. Express 21, 1310–1316 (2013).
[Crossref] [PubMed]

P. Rath, N. Gruhler, S. Khasminskaya, C. Nebel, C. Wild, and W. H. P. Pernice, “Waferscale nanophotonic circuits made from diamond-on-insulator substrates,” Opt. Express 21, 11031–11036 (2013).
[Crossref] [PubMed]

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, and A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref] [PubMed]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

2012 (4)

C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, “Superconducting nanowire single-photon detectors: physics and applications,” Supercond. Sci. Technol. 25, 063001 (2012).
[Crossref]

W. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat Commun 3, 1325 (2012).
[Crossref] [PubMed]

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14, 095014 (2012).
[Crossref]

2011 (5)

C. Xiong, W. Pernice, K. K. Ryu, C. Schuck, K. Y. Fong, T. Palacios, and H. X. Tang, “Integrated gan photonic circuits on silicon (100) for second harmonic generation,” Opt. Express 19, 10462–10470 (2011).
[Crossref] [PubMed]

M.-C. Tien, J. F. Bauters, M. J. R. Heck, D. T. Spencer, D. J. Blumenthal, and J. E. Bowers, “Ultra-high quality factor planar Si3N4 ring resonators on si substrates,” Opt. Express 19, 13551–13556 (2011).
[Crossref] [PubMed]

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[Crossref] [PubMed]

R. Okamoto, J. L. OBrien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn controlled-NOT photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA 108, 10067–10071 (2011).
[Crossref] [PubMed]

2010 (1)

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

2009 (1)

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]

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[Crossref]

2006 (1)

2005 (4)

B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129 (2005).
[Crossref]

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of quantum error correction using linear optics,” Phys. Rev. A 71, 052332 (2005).
[Crossref]

M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
[Crossref]

2004 (3)

J. Skaar, J. C. García Escartín, and H. Landro, “Quantum mechanical description of linear optics,” American Journal of Physics 72, 1385–1391 (2004).
[Crossref]

U. Leonhardt and A. Neumaier, “Explicit effective hamiltonians for general linear quantum-optical networks,” Journal of Optics B: Quantum and Semiclassical Optics 6, L1 (2004).
[Crossref]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

2003 (1)

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a bell-state filter by quantum process tomography,” Phys. Rev. Lett. 91, 120402 (2003).
[Crossref] [PubMed]

2002 (3)

T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical controlled-not gate in the coincidence basis,” Phys. Rev. A 65, 062324 (2002).
[Crossref]

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightw. Technol. 20, 1968–1975 (2002).
[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]

1994 (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]

W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[Crossref]

1981 (1)

A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
[Crossref]

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, 794–798 (2013).
[Crossref]

Ahn, J.-H.

Aspelmeyer, M.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

Atikian, H. A.

Baehr-Jones, T.

Baets, R.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

Barbieri, M.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

Bauters, J. F.

Beetz, J.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Bell, T. B.

T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical controlled-not gate in the coincidence basis,” Phys. Rev. A 65, 062324 (2002).
[Crossref]

Bellutti, P.

M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
[Crossref]

Benz, C.

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]

Bienstman, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

Blumenthal, D. J.

Bogaerts, W.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

Bowers, J. E.

Brink, D.

P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
[Crossref]

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, 794–798 (2013).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, and A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref] [PubMed]

Burek, M. J.

Carolan, J.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Chen, Y.-A.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Chen, Z.

Chen, Z.-B.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Chrostowski, L.

Chuang, I. L.

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

Claes, T.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

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]

Dai, H.-N.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Daldosso, N.

M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
[Crossref]

Dalton, L. R.

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightw. Technol. 20, 1968–1975 (2002).
[Crossref]

Danneau, R.

De Heyn, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

De Mesel, K.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

De Vos, K.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

Diewald, S.

P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
[Crossref]

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, 794–798 (2013).
[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]

Dumon, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

Dutt, A.

A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, “On-chip optical squeezing,” Phys. Rev. Applied 3, 044005 (2015).
[Crossref]

Eisaman, M. D.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[Crossref] [PubMed]

Ellenor, C. W.

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a bell-state filter by quantum process tomography,” Phys. Rev. Lett. 91, 120402 (2003).
[Crossref] [PubMed]

Fan, J.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[Crossref] [PubMed]

Fan, L.

C. Schuck, X. Guo, L. Fan, X. Ma, M. Poot, and H. X. Tang, “Quantum interference in heterogeneous superconducting-photonic circuits on a silicon chip,” Nat Commun 7, 10352 (2016).
[Crossref] [PubMed]

Fan, S.

R. St-Gelais, B. Guha, L. Zhu, S. Fan, and M. Lipson, “Demonstration of strong near-field radiative heat transfer between integrated nanostructures,” Nano Lett. 14, 6971–6975 (2014). PMID: .
[Crossref] [PubMed]

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, 794–798 (2013).
[Crossref]

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, and A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref] [PubMed]

Fickler, R.

Fiore, A.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Fong, K. Y.

M. Poot, K. Y. Fong, and H. X. Tang, “Deep feedback-stabilized parametric squeezing in an opto-electromechanical system,” New J. Phys. 17, 043056 (2015).
[Crossref]

K. Y. Fong, M. Poot, and H. X. Tang, “Nano-optomechanical resonators in microfluidics,” Nano Lett. 15, 6116–6120 (2015).
[Crossref] [PubMed]

M. Poot, K. Y. Fong, and H. X. Tang, “Classical non-gaussian state preparation through squeezing in an opto-electromechanical resonator,” Phys. Rev. A 90, 063809 (2014).
[Crossref]

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14, 095014 (2012).
[Crossref]

C. Xiong, W. Pernice, K. K. Ryu, C. Schuck, K. Y. Fong, T. Palacios, and H. X. Tang, “Integrated gan photonic circuits on silicon (100) for second harmonic generation,” Opt. Express 19, 10462–10470 (2011).
[Crossref] [PubMed]

Foster, M. A.

Franson, J. D.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of quantum error correction using linear optics,” Phys. Rev. A 71, 052332 (2005).
[Crossref]

Frucci, G.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Fukuda, H.

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[Crossref]

Gaeta, A. L.

Gaggero, A.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Galland, C.

Gao, W.-B.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

García Escartín, J. C.

J. Skaar, J. C. García Escartín, and H. Landro, “Quantum mechanical description of linear optics,” American Journal of Physics 72, 1385–1391 (2004).
[Crossref]

Gates, J. C.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

Gilchrist, A.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Goebel, A. M.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Goltsman, G.

W. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat Commun 3, 1325 (2012).
[Crossref] [PubMed]

Gondarenko, A.

Gruhler, N.

Guha, B.

R. St-Gelais, B. Guha, L. Zhu, S. Fan, and M. Lipson, “Demonstration of strong near-field radiative heat transfer between integrated nanostructures,” Nano Lett. 14, 6971–6975 (2014). PMID: .
[Crossref] [PubMed]

Guo, X.

C. Schuck, X. Guo, L. Fan, X. Ma, M. Poot, and H. X. Tang, “Quantum interference in heterogeneous superconducting-photonic circuits on a silicon chip,” Nat Commun 7, 10352 (2016).
[Crossref] [PubMed]

Hadfield, R. H.

C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, “Superconducting nanowire single-photon detectors: physics and applications,” Supercond. Sci. Technol. 25, 063001 (2012).
[Crossref]

Harrold, C.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Hashimoto, T.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Heck, M. J. R.

Heidrich, N.

P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
[Crossref]

Hfling, S.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Hochberg, M.

Hofmann, H. F.

R. Okamoto, J. L. OBrien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn controlled-NOT photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA 108, 10067–10071 (2011).
[Crossref] [PubMed]

Huang, I.-C.

Huang, W.-P.

Humphreys, P. C.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

Itabashi, S.-i.

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[Crossref]

Itoh, M.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Jacobs, B. C.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of quantum error correction using linear optics,” Phys. Rev. A 71, 052332 (2005).
[Crossref]

Jaeger, N. A. F.

Jahanmirinejad, S.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

James, D. F. V.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Jang, H.

Jin, X.-M.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

Kamp, M.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Khasminskaya, S.

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]

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]

Kompocholis, C.

M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
[Crossref]

Krauss, T.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

Kumar, P.

Kumar Selvaraja, S.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

Kundys, D.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (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]

Laing, A.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Landro, H.

J. Skaar, J. C. García Escartín, and H. Landro, “Quantum mechanical description of linear optics,” American Journal of Physics 72, 1385–1391 (2004).
[Crossref]

Langford, N. K.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical controlled-not gate in the coincidence basis,” Phys. Rev. A 65, 062324 (2002).
[Crossref]

Lee, K. F.

Leonhardt, U.

U. Leonhardt and A. Neumaier, “Explicit effective hamiltonians for general linear quantum-optical networks,” Journal of Optics B: Quantum and Semiclassical Optics 6, L1 (2004).
[Crossref]

Leoni, R.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Lermer, M.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Levy, J. S.

Lewes-Malandrakis, G.

P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
[Crossref]

Li, M.

W. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat Commun 3, 1325 (2012).
[Crossref] [PubMed]

Lim, A. E.-J.

Lin, Z.

Lipson, M.

A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, “On-chip optical squeezing,” Phys. Rev. Applied 3, 044005 (2015).
[Crossref]

R. St-Gelais, B. Guha, L. Zhu, S. Fan, and M. Lipson, “Demonstration of strong near-field radiative heat transfer between integrated nanostructures,” Nano Lett. 14, 6971–6975 (2014). PMID: .
[Crossref] [PubMed]

A. Gondarenko, J. S. Levy, and M. Lipson, “High confinement micron-scale silicon nitride high q ring resonator,” Opt. Express 17, 11366–11370 (2009).
[Crossref] [PubMed]

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388–12393 (2006).
[Crossref] [PubMed]

Lo, G.-Q.

Loncar, M.

Lounis, B.

B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129 (2005).
[Crossref]

Lu, C.-Y.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Lu, Z.

Lui, A.

M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
[Crossref]

Luke, K.

A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, “On-chip optical squeezing,” Phys. Rev. Applied 3, 044005 (2015).
[Crossref]

Ma, X.

C. Schuck, X. Guo, L. Fan, X. Ma, M. Poot, and H. X. Tang, “Quantum interference in heterogeneous superconducting-photonic circuits on a silicon chip,” Nat Commun 7, 10352 (2016).
[Crossref] [PubMed]

C. Schuck, W. H. Pernice, X. Ma, and H. X. Tang, “Optical time domain reflectometry with low noise waveguide-coupled superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 102, 191104 (2013).
[Crossref]

Manipatruni, S.

A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, “On-chip optical squeezing,” Phys. Rev. Applied 3, 044005 (2015).
[Crossref]

Marshall, G. D.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Martn-Lpez, E.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Matsuda, N.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Matthews, J. C. F.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Mattioli, F.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Melchiorri, M.

M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
[Crossref]

Metcalf, B. J.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

Migdall, A.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[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]

Minaeva, O.

W. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat Commun 3, 1325 (2012).
[Crossref] [PubMed]

Mitchell, M. W.

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a bell-state filter by quantum process tomography,” Phys. Rev. Lett. 91, 120402 (2003).
[Crossref] [PubMed]

Moerman, I.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

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]

Natarajan, C. M.

C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, “Superconducting nanowire single-photon detectors: physics and applications,” Supercond. Sci. Technol. 25, 063001 (2012).
[Crossref]

Nebel, C.

P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
[Crossref]

P. Rath, N. Gruhler, S. Khasminskaya, C. Nebel, C. Wild, and W. H. P. Pernice, “Waferscale nanophotonic circuits made from diamond-on-insulator substrates,” Opt. Express 21, 11031–11036 (2013).
[Crossref] [PubMed]

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]

Neumaier, A.

U. Leonhardt and A. Neumaier, “Explicit effective hamiltonians for general linear quantum-optical networks,” Journal of Optics B: Quantum and Semiclassical Optics 6, L1 (2004).
[Crossref]

Nielsen, M. A.

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

Nussenzveig, P.

A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, “On-chip optical squeezing,” Phys. Rev. Applied 3, 044005 (2015).
[Crossref]

O’Brien, J. L.

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]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

OBrien, J. L.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

R. Okamoto, J. L. OBrien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn controlled-NOT photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA 108, 10067–10071 (2011).
[Crossref] [PubMed]

Oguma, M.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Okamoto, R.

R. Okamoto, J. L. OBrien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn controlled-NOT photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA 108, 10067–10071 (2011).
[Crossref] [PubMed]

Orrit, M.

B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129 (2005).
[Crossref]

Palacios, T.

Pan, J.-W.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Pavesi, L.

M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
[Crossref]

Peng, C.-Z.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Pernice, W.

W. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat Commun 3, 1325 (2012).
[Crossref] [PubMed]

C. Xiong, W. Pernice, K. K. Ryu, C. Schuck, K. Y. Fong, T. Palacios, and H. X. Tang, “Integrated gan photonic circuits on silicon (100) for second harmonic generation,” Opt. Express 19, 10462–10470 (2011).
[Crossref] [PubMed]

Pernice, W. H.

C. Schuck, W. H. Pernice, X. Ma, and H. X. Tang, “Optical time domain reflectometry with low noise waveguide-coupled superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 102, 191104 (2013).
[Crossref]

Pernice, W. H. P.

P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
[Crossref]

C. Schuck, W. H. P. Pernice, and H. X. Tang, “Waveguide integrated low noise NbTiN nanowire single-photon detectors with milli-Hz dark count rate,” Sci. Rep. 3, 1–6 (2013).
[Crossref]

N. Gruhler, C. Benz, H. Jang, J.-H. Ahn, R. Danneau, and W. H. P. Pernice, “High-quality Si3N4 circuits as a platform for graphene-based nanophotonic devices,” Opt. Express 21, 31678–31689 (2013).
[Crossref]

P. Rath, N. Gruhler, S. Khasminskaya, C. Nebel, C. Wild, and W. H. P. Pernice, “Waferscale nanophotonic circuits made from diamond-on-insulator substrates,” Opt. Express 21, 11031–11036 (2013).
[Crossref] [PubMed]

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14, 095014 (2012).
[Crossref]

Pittman, T. B.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of quantum error correction using linear optics,” Phys. Rev. A 71, 052332 (2005).
[Crossref]

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]

Polyakov, S. V.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[Crossref] [PubMed]

Poot, M.

C. Schuck, X. Guo, L. Fan, X. Ma, M. Poot, and H. X. Tang, “Quantum interference in heterogeneous superconducting-photonic circuits on a silicon chip,” Nat Commun 7, 10352 (2016).
[Crossref] [PubMed]

M. Poot, K. Y. Fong, and H. X. Tang, “Deep feedback-stabilized parametric squeezing in an opto-electromechanical system,” New J. Phys. 17, 043056 (2015).
[Crossref]

K. Y. Fong, M. Poot, and H. X. Tang, “Nano-optomechanical resonators in microfluidics,” Nano Lett. 15, 6116–6120 (2015).
[Crossref] [PubMed]

M. Poot, K. Y. Fong, and H. X. Tang, “Classical non-gaussian state preparation through squeezing in an opto-electromechanical resonator,” Phys. Rev. A 90, 063809 (2014).
[Crossref]

M. Poot and H. X. Tang, “Broadband nanoelectromechanical phase shifting of light on a chip,” Appl. Phys. Lett. 104, 061101 (2014).
[Crossref]

M. Poot and H. X. Tang, “High-quality opto-electromechanical resonators,” In preparation (2015).

Pryde, G. J.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Pucker, G.

M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
[Crossref]

Rabiei, P.

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightw. Technol. 20, 1968–1975 (2002).
[Crossref]

Rahimi-Keshari, S.

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, and A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (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, 794–798 (2013).
[Crossref]

Ralph, T. C.

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, 794–798 (2013).
[Crossref]

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, and A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref] [PubMed]

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]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical controlled-not gate in the coincidence basis,” Phys. Rev. A 65, 062324 (2002).
[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]

Rath, P.

P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
[Crossref]

P. Rath, N. Gruhler, S. Khasminskaya, C. Nebel, C. Wild, and W. H. P. Pernice, “Waferscale nanophotonic circuits made from diamond-on-insulator substrates,” Opt. Express 21, 11031–11036 (2013).
[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]

Resch, K. J.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

Rudolph, T.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

Russell, N. J.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Ryu, K. K.

Sahin, D.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Sanjines, R.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Sbrana, F.

M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
[Crossref]

Schenck, E.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

Schmidt, B. S.

Schneider, S.

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a bell-state filter by quantum process tomography,” Phys. Rev. Lett. 91, 120402 (2003).
[Crossref] [PubMed]

Schuck, C.

C. Schuck, X. Guo, L. Fan, X. Ma, M. Poot, and H. X. Tang, “Quantum interference in heterogeneous superconducting-photonic circuits on a silicon chip,” Nat Commun 7, 10352 (2016).
[Crossref] [PubMed]

C. Schuck, W. H. P. Pernice, and H. X. Tang, “Waveguide integrated low noise NbTiN nanowire single-photon detectors with milli-Hz dark count rate,” Sci. Rep. 3, 1–6 (2013).
[Crossref]

C. Schuck, W. H. Pernice, X. Ma, and H. X. Tang, “Optical time domain reflectometry with low noise waveguide-coupled superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 102, 191104 (2013).
[Crossref]

W. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat Commun 3, 1325 (2012).
[Crossref] [PubMed]

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14, 095014 (2012).
[Crossref]

C. Xiong, W. Pernice, K. K. Ryu, C. Schuck, K. Y. Fong, T. Palacios, and H. X. Tang, “Integrated gan photonic circuits on silicon (100) for second harmonic generation,” Opt. Express 19, 10462–10470 (2011).
[Crossref] [PubMed]

Sergienko, A.

W. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat Commun 3, 1325 (2012).
[Crossref] [PubMed]

Shadbolt, P. J.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Sharping, J. E.

Silverstone, J. W.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Skaar, J.

J. Skaar, J. C. García Escartín, and H. Landro, “Quantum mechanical description of linear optics,” American Journal of Physics 72, 1385–1391 (2004).
[Crossref]

Smith, B. J.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

Smith, P. G.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

Sparrow, C.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Spencer, D. T.

Sprengers, J. P.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

Spring, J. B.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

Stark, P.

Steier, W. H.

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightw. Technol. 20, 1968–1975 (2002).
[Crossref]

Steinberg, A. M.

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a bell-state filter by quantum process tomography,” Phys. Rev. Lett. 91, 120402 (2003).
[Crossref] [PubMed]

Steven Kolthammer, W.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

St-Gelais, R.

R. St-Gelais, B. Guha, L. Zhu, S. Fan, and M. Lipson, “Demonstration of strong near-field radiative heat transfer between integrated nanostructures,” Nano Lett. 14, 6971–6975 (2014). PMID: .
[Crossref] [PubMed]

Sun, X.

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14, 095014 (2012).
[Crossref]

Taillaert, D.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

Takesue, H.

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[Crossref]

Takeuchi, S.

R. Okamoto, J. L. OBrien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn controlled-NOT photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA 108, 10067–10071 (2011).
[Crossref] [PubMed]

Tang, H.

W. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat Commun 3, 1325 (2012).
[Crossref] [PubMed]

Tang, H. X.

C. Schuck, X. Guo, L. Fan, X. Ma, M. Poot, and H. X. Tang, “Quantum interference in heterogeneous superconducting-photonic circuits on a silicon chip,” Nat Commun 7, 10352 (2016).
[Crossref] [PubMed]

M. Poot, K. Y. Fong, and H. X. Tang, “Deep feedback-stabilized parametric squeezing in an opto-electromechanical system,” New J. Phys. 17, 043056 (2015).
[Crossref]

K. Y. Fong, M. Poot, and H. X. Tang, “Nano-optomechanical resonators in microfluidics,” Nano Lett. 15, 6116–6120 (2015).
[Crossref] [PubMed]

M. Poot, K. Y. Fong, and H. X. Tang, “Classical non-gaussian state preparation through squeezing in an opto-electromechanical resonator,” Phys. Rev. A 90, 063809 (2014).
[Crossref]

M. Poot and H. X. Tang, “Broadband nanoelectromechanical phase shifting of light on a chip,” Appl. Phys. Lett. 104, 061101 (2014).
[Crossref]

C. Schuck, W. H. P. Pernice, and H. X. Tang, “Waveguide integrated low noise NbTiN nanowire single-photon detectors with milli-Hz dark count rate,” Sci. Rep. 3, 1–6 (2013).
[Crossref]

C. Schuck, W. H. Pernice, X. Ma, and H. X. Tang, “Optical time domain reflectometry with low noise waveguide-coupled superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 102, 191104 (2013).
[Crossref]

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14, 095014 (2012).
[Crossref]

C. Xiong, W. Pernice, K. K. Ryu, C. Schuck, K. Y. Fong, T. Palacios, and H. X. Tang, “Integrated gan photonic circuits on silicon (100) for second harmonic generation,” Opt. Express 19, 10462–10470 (2011).
[Crossref] [PubMed]

M. Poot and H. X. Tang, “High-quality opto-electromechanical resonators,” In preparation (2015).

Tanner, M. G.

C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, “Superconducting nanowire single-photon detectors: physics and applications,” Supercond. Sci. Technol. 25, 063001 (2012).
[Crossref]

Thomas-Peter, N.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

Thompson, M. G.

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

Tien, M.-C.

Tokura, Y.

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[Crossref]

Tsuchizawa, T.

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[Crossref]

Turner, A. C.

Ummethala, S.

P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
[Crossref]

Van Daele, P.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

Van Thourhout, D.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

Van Vaerenbergh, T.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

Vedral, V.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

Venkataraman, V.

Verstuyft, S.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

Wagenknecht, C.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Walmsley, I. A.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

Walther, P.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

Wang, C.

Wang, Y.

Watanabe, T.

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[Crossref]

Weinfurter, H.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

White, A. G.

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, 794–798 (2013).
[Crossref]

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, and A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref] [PubMed]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical controlled-not gate in the coincidence basis,” Phys. Rev. A 65, 062324 (2002).
[Crossref]

Wild, C.

Xiong, C.

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14, 095014 (2012).
[Crossref]

C. Xiong, W. Pernice, K. K. Ryu, C. Schuck, K. Y. Fong, T. Palacios, and H. X. Tang, “Integrated gan photonic circuits on silicon (100) for second harmonic generation,” Opt. Express 19, 10462–10470 (2011).
[Crossref] [PubMed]

Yamada, K.

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[Crossref]

Yang, S.

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]

Yun, H.

Zeilinger, A.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

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]

A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
[Crossref]

Zhang, C.

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightw. Technol. 20, 1968–1975 (2002).
[Crossref]

Zhang, F.

Zhang, Q.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Zhang, Y.

Zhao, B.

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Zhu, L.

R. St-Gelais, B. Guha, L. Zhu, S. Fan, and M. Lipson, “Demonstration of strong near-field radiative heat transfer between integrated nanostructures,” Nano Lett. 14, 6971–6975 (2014). PMID: .
[Crossref] [PubMed]

Am. J. Phys. (1)

A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
[Crossref]

American Journal of Physics (1)

J. Skaar, J. C. García Escartín, and H. Landro, “Quantum mechanical description of linear optics,” American Journal of Physics 72, 1385–1391 (2004).
[Crossref]

Appl. Phys. Lett. (6)

C. Schuck, W. H. Pernice, X. Ma, and H. X. Tang, “Optical time domain reflectometry with low noise waveguide-coupled superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 102, 191104 (2013).
[Crossref]

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Hfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).
[Crossref]

M. Poot and H. X. Tang, “Broadband nanoelectromechanical phase shifting of light on a chip,” Appl. Phys. Lett. 104, 061101 (2014).
[Crossref]

P. Rath, S. Ummethala, S. Diewald, G. Lewes-Malandrakis, D. Brink, N. Heidrich, C. Nebel, and W. H. P. Pernice, “Diamond electro-optomechanical resonators integrated in nanophotonic circuits,” Appl. Phys. Lett. 105, 251102 (2014).
[Crossref]

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[Crossref]

M. Melchiorri, N. Daldosso, F. Sbrana, L. Pavesi, G. Pucker, C. Kompocholis, P. Bellutti, and A. Lui, “Propagation losses of silicon nitride waveguides in the near-infrared range,” Appl. Phys. Lett. 86, 121111 (2005).
[Crossref]

J. Lightw. Technol. (1)

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightw. Technol. 20, 1968–1975 (2002).
[Crossref]

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

Journal of Optics B: Quantum and Semiclassical Optics (1)

U. Leonhardt and A. Neumaier, “Explicit effective hamiltonians for general linear quantum-optical networks,” Journal of Optics B: Quantum and Semiclassical Optics 6, L1 (2004).
[Crossref]

Laser & Photonics Reviews (1)

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews 6, 47–73 (2012).
[Crossref]

Nano Lett. (2)

K. Y. Fong, M. Poot, and H. X. Tang, “Nano-optomechanical resonators in microfluidics,” Nano Lett. 15, 6116–6120 (2015).
[Crossref] [PubMed]

R. St-Gelais, B. Guha, L. Zhu, S. Fan, and M. Lipson, “Demonstration of strong near-field radiative heat transfer between integrated nanostructures,” Nano Lett. 14, 6971–6975 (2014). PMID: .
[Crossref] [PubMed]

Nat Commun (3)

C. Schuck, X. Guo, L. Fan, X. Ma, M. Poot, and H. X. Tang, “Quantum interference in heterogeneous superconducting-photonic circuits on a silicon chip,” Nat Commun 7, 10352 (2016).
[Crossref] [PubMed]

W. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat Commun 3, 1325 (2012).
[Crossref] [PubMed]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. Steven Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat Commun 4, 1356 (2013).
[Crossref] [PubMed]

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]

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

New J. Phys. (2)

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14, 095014 (2012).
[Crossref]

M. Poot, K. Y. Fong, and H. X. Tang, “Deep feedback-stabilized parametric squeezing in an opto-electromechanical system,” New J. Phys. 17, 043056 (2015).
[Crossref]

Opt. Express (10)

C. Xiong, W. Pernice, K. K. Ryu, C. Schuck, K. Y. Fong, T. Palacios, and H. X. Tang, “Integrated gan photonic circuits on silicon (100) for second harmonic generation,” Opt. Express 19, 10462–10470 (2011).
[Crossref] [PubMed]

C. Wang, M. J. Burek, Z. Lin, H. A. Atikian, V. Venkataraman, I.-C. Huang, P. Stark, and M. Lončar, “Integrated high quality factor lithium niobate microdisk resonators,” Opt. Express 22, 30924–30933 (2014).
[Crossref]

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388–12393 (2006).
[Crossref] [PubMed]

P. Rath, N. Gruhler, S. Khasminskaya, C. Nebel, C. Wild, and W. H. P. Pernice, “Waferscale nanophotonic circuits made from diamond-on-insulator substrates,” Opt. Express 21, 11031–11036 (2013).
[Crossref] [PubMed]

N. Gruhler, C. Benz, H. Jang, J.-H. Ahn, R. Danneau, and W. H. P. Pernice, “High-quality Si3N4 circuits as a platform for graphene-based nanophotonic devices,” Opt. Express 21, 31678–31689 (2013).
[Crossref]

A. Gondarenko, J. S. Levy, and M. Lipson, “High confinement micron-scale silicon nitride high q ring resonator,” Opt. Express 17, 11366–11370 (2009).
[Crossref] [PubMed]

M.-C. Tien, J. F. Bauters, M. J. R. Heck, D. T. Spencer, D. J. Blumenthal, and J. E. Bowers, “Ultra-high quality factor planar Si3N4 ring resonators on si substrates,” Opt. Express 19, 13551–13556 (2011).
[Crossref] [PubMed]

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, and A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref] [PubMed]

Y. Zhang, S. Yang, A. E.-J. Lim, G.-Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss y-junction for submicron silicon waveguide,” Opt. Express 21, 1310–1316 (2013).
[Crossref] [PubMed]

Z. Lu, H. Yun, Y. Wang, Z. Chen, F. Zhang, N. A. F. Jaeger, and L. Chrostowski, “Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control,” Opt. Express 23, 3795–3808 (2015).
[Crossref] [PubMed]

Phys. Rev. A (3)

T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical controlled-not gate in the coincidence basis,” Phys. Rev. A 65, 062324 (2002).
[Crossref]

M. Poot, K. Y. Fong, and H. X. Tang, “Classical non-gaussian state preparation through squeezing in an opto-electromechanical resonator,” Phys. Rev. A 90, 063809 (2014).
[Crossref]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of quantum error correction using linear optics,” Phys. Rev. A 71, 052332 (2005).
[Crossref]

Phys. Rev. Applied (1)

A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, “On-chip optical squeezing,” Phys. Rev. Applied 3, 044005 (2015).
[Crossref]

Phys. Rev. Lett. (3)

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

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]

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a bell-state filter by quantum process tomography,” Phys. Rev. Lett. 91, 120402 (2003).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

R. Okamoto, J. L. OBrien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn controlled-NOT photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA 108, 10067–10071 (2011).
[Crossref] [PubMed]

Proceedings of the National Academy of Sciences (1)

W.-B. Gao, A. M. Goebel, C.-Y. Lu, H.-N. Dai, C. Wagenknecht, Q. Zhang, B. Zhao, C.-Z. Peng, Z.-B. Chen, Y.-A. Chen, and J.-W. Pan, “Teleportation-based realization of an optical quantum two-qubit entangling gate,” Proceedings of the National Academy of Sciences 107, 20869–20874 (2010).
[Crossref]

Quantum Electronics, IEEE Journal of (1)

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[Crossref]

Rep. Prog. Phys. (1)

B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129 (2005).
[Crossref]

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]

Rev. Sci. Instrum. (1)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[Crossref] [PubMed]

Sci. Rep. (1)

C. Schuck, W. H. P. Pernice, and H. X. Tang, “Waveguide integrated low noise NbTiN nanowire single-photon detectors with milli-Hz dark count rate,” Sci. Rep. 3, 1–6 (2013).
[Crossref]

Science (3)

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, 794–798 (2013).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martn-Lpez, 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. OBrien, 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]

Supercond. Sci. Technol. (1)

C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, “Superconducting nanowire single-photon detectors: physics and applications,” Supercond. Sci. Technol. 25, 063001 (2012).
[Crossref]

Other (6)

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

M. Poot and H. X. Tang, “High-quality opto-electromechanical resonators,” In preparation (2015).

The convention where the optical field is proportional to exp(−iωt) is used. In this case the phase of light propagating in, say, the +x direction is ϕp = +2πneffx/λ. Since the outer arms are the reference arms, the phase difference is defined as Δϕ = ϕinner − ϕouter. The outer arm is longer than the inner arm and hence Δϕ < 0. Since ∂ϕp/∂λ = −ϕp/λ (without dispersion), the phase difference increases with increasing wavelength.

Although in a general beam splitter any value of the phase difference is allowed (with constraints on their combination), for a symmetric design such as our directional coupler, only −π/2 and +π/2 are possible [1].

In practice, the transmissions are not determined in this step-by-step process, but instead all measured (logarithmic) transmission data is fitted simultaneously by using 15-th order polynomials in λ for Ti←2(λ) and Y(λ). This is done through linear fitting via LU decomposition. The final results do not depend significantly on the order of the polynomial as long as the order is high enough to follow the overall profile, but low enough to filter out the fine fringes. The small upturns near 1570 nm in the curves shown in Fig. 4(a) are artifacts of the polynomial fits due the end of the data range.

The maximum transmission in Fig. 4(b) is ∼ −32dB. This value is lower than typical for our SiN devices (∼−15dB) since the fiber array was kept far away from the sample to prevent any accidental touching or scratching while scanning the entire chip. Note that none of the results presented here depend on the absolute transmission.

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Figures (6)

Fig. 1
Fig. 1 (a) Optical micrograph of a ring resonator with a feeding waveguide and grating couplers (triangular structures). The spacing between the grating couplers is 250 um and the ring has a diameter of 110 um. (b) Optical transmission of a device with a waveguide width of 1 um and a separation between the ring and the feeding waveguide of 700 nm. The transmission is normalized to that of the grating couplers. The shaded area indicates the resonance highlighted in panel (c). (c) Zooms of resonances near 1552 nm with fits (dashed lines) for different separations between the waveguide and the ring on a linear scale. The curves are offset for clarity. (d) Linewidths extracted from the fits. The solid symbols are the linewidths due to the coupling between the feed waveguide and the ring, whereas the open circles are the internal linewidths. The colors are consistent between all panels.

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Fig. 2
Fig. 2 (a) Finite element simulation of a directional coupler (top view) when light is sent into port 1. The z-component of the magnetic field Hz is shown according to the color scale. The inset shows a zoom of Hz near the outputs (ports 2,3). (b) Measured atomic-force microscope profile (averaged) across the center of a directional coupler with 1000 nm wide waveguides and 400 nm separation between them. The image is colored to indicate the guiding and cladding layer. The distance between the top of the waveguide and the cladding is 330 nm. Note the difference in the horizontal and vertical scales. (c) Calculated effective refractive index of the even (green) mode and the odd (blue) mode versus the separation between the two waveguides. The insets show the Ey field of the modes. (d) Magnitudes of the elements of the scattering matrix for different Lint obtained from FEM simulations as in panel (a).

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Fig. 3
Fig. 3 (a) Optical dark-field image of a device to calibrate the coupling ratio of the directional coupler (dashed rectangle). The insets show two directional couplers with different interaction lengths. (b) Measured normalized cross power Pc/(Pt + Pc) at a wavelength of 1554 nm for different Lint, together with a fit of Eq. (3) (solid line) to the data. The data is obtained by measuring the transmission profiles with a tunable laser, followed by a 2 nm averaging. (c) The extracted wavelength dependence of the coupling length ℓc and the offset length ℓ0. (d) The cross power as a function of wavelength for directional couplers with two different Lint values. The dashed lines indicate 50/50 and 33/67 beam splitters.

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Fig. 4
Fig. 4 (a) Schematics of calibration devices to determine the insertion loss. Light is sent into port 2 (blue arrow) and detected at the other ports (orange). The three devices have different routings that enable extraction of the coupler transmissions Ti←2 and the insertion loss of the Y-splitter Y. The combinations that the light encounters are indicated at each output grating coupler. (b) Extracted transmission profiles Ti←2(λ) and Y(λ). (c) Insertion loss of the 22 directional couplers (same devices as Fig. 3, averaged over wavelength) and a box plot showing the median value (red), 25 and 75% percentile (box) and the extent (bars). The interaction length is varied from 0 to 50 μm when going from device D3 to AB3.

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Fig. 5
Fig. 5 (a) optical micrograph of a device to determine the phase difference between the two outputs of a directional coupler (b,d) schematics of the two type of interferometers. The left (right) MZI is indicated in blue (orange). The colorized electron micrograph on the right side of (d) shows the central area of a device with a directional coupler. (c,e) MZI fringes measured at the two outputs of the device (light blue, orange) together with fits (blue and black lines) to determine the periods and wavelength shift. The phase of the fringes relative to the fringes of the left MZI is indicated on the top axis. Panels (a) to (c) show a reference device, and (d) and (e) are for a device with a directional coupler with an interaction length of 8μm. (f) Phase difference between the two interferometers.

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Fig. 6
Fig. 6 (a) Schematic of a CNOT gate for linear-optics quantum computation. The port number and the cross-over ratios of the directional couplers are indicated. (b) Micrograph of a device to measure |S′41|2. The waveguide above the CNOT gate is used to locate the device as some combinations (including this one) have zero transmission. This auxiliary waveguide does not influence the actual CNOT circuit. (c) Measured normalized transmission matrix and that for an ideal CNOT gate (d). (e) Bar chart comparing the measured data [cf. (c)], the fit, and the transmission for an ideal CNOT gate [cf. (d)]. (f) Calculated fidelity, i.e. the probability of obtaining the right result after post selection vs. C1/2 and C2/3. The dot indicates the fitted values of (e), and the cross is located at the ideal values.

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Tables (1)

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Table 1 Changes of the coupling length δℓc from the nominal value ℓc = 37.47μm for variations of the device parameters. The values of the variations (third column) represent estimated fabrication uncertainties around the target values (second column). The “remaining thickness” is the thickness of the SiN remaining on top of the SiO2 after etching, far outside the coupler region. The “center thickness” is the thickness remaining in between the two waveguides, which is larger due to the narrow slot. All variations δp are small enough for the changes in coupling length to be proportional to the variation in the parameter p that caused it: δℓc ≈ ∂ℓc/∂p × δp.

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Equations (5)

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T ( λ ) = T 0 1 + F c sin 2 ( π [ λ − λ fp ] / FSR fp ) × ( 1 − w c w int ( w c + w int ) 2 / 4 + ( λ − λ 0 ) 2 ) ,
( a 2 a 3 ) out = ( S 21 S 24 S 31 S 34 )     ( a 1 a 4 ) in ↔ ( a 2 a 3 ) out ′ = ( S 11 ′ S 12 ′ S 21 ′ S 22 ′ )     ( a 1 a 2 ) in ′ .
P c P in ≡ C = | S 21 ′ | 2 = sin 2 ( π 2 L int + ℓ 0 ℓ c ) .
| ∂ C ∂ λ | = 1 ℓ c C ( 1 − C ) { π ∂ ℓ 0 ∂ λ − [ 2 π k ± arccos ( 1 − 2 C ) ] ∂ ℓ c ∂ λ } .
S ′ = ( t i c i c t ) ,

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