Abstract

Generation of entangled photons in nonlinear media constitutes a basic building block of modern photonic quantum technology. Current optical materials are severely limited in their ability to produce three or more entangled photons in a single event due to weak nonlinearities and challenges achieving phase-matching. We use integrated nanophotonics to enhance nonlinear interactions and develop protocols to design multimode waveguides that enable sustained phase-matching for third-order spontaneous parametric down-conversion (TOSPDC). We predict a generation efficiency of 0.13 triplets/s/mW of pump power in TiO2-based integrated waveguides, an order of magnitude higher than previous theoretical and experimental demonstrations. We experimentally verify our device design methods in TiO2 waveguides using third-harmonic generation (THG), the reverse process of TOSPDC that is subject to the same phase-matching constraints. We finally discuss the effect of finite detector bandwidth and photon losses on the energy-time coherence properties of the expected TOSPDC source.

© 2016 Optical Society of America

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References

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2015 (4)

M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015).
[Crossref] [PubMed]

C. C. Evans, K. Shtyrkova, O. Reshef, M. Moebius, J. D. B. Bradley, S. Griesse-Nascimento, E. Ippen, and E. Mazur, “Multimode phase-matched third-harmonic generation in sub-micrometer-wide anatase TiO2 waveguides,” Opt. Express 23, 7832 (2015).
[Crossref] [PubMed]

O. Reshef, K. Shtyrkova, M. G. Moebius, S. Griesse-Nascimento, S. Spector, C. C. Evans, E. Ippen, and E. Mazur, “Polycrystalline anatase titanium dioxide micro-ring resonators with negative thermo-optic coefficient,” J. Opt. Soc. Am. B: Opt. Phys. 32, 2288–2293 (2015).
[Crossref]

C. C. Evans, C. Liu, and J. Suntivich, “Low-loss titanium dioxide waveguides and resonators using a dielectric lift-off fabrication process,” Opt. Express 23, 11160 (2015).
[Crossref] [PubMed]

2014 (1)

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

2013 (4)

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
[Crossref] [PubMed]

C. C. Evans, K. Shtyrkova, J. D. B. Bradley, O. Reshef, E. Ippen, and E. Mazur, “Spectral broadening in anatase titanium dioxide waveguides at telecommunication and near-visible wavelengths,” Opt. Express 21, 18582–18591 (2013).
[Crossref] [PubMed]

T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B: Opt. Phys. 30, 505–511 (2013).
[Crossref]

2012 (4)

2011 (5)

V. Giovannetti and S. Lloyd, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (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, 71101–71125 (2011).
[Crossref]

M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011).
[Crossref]

S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett. 36, 3000–3002 (2011).
[Crossref] [PubMed]

S. K. Das, C. Schwanke, A. Pfuch, W. Seeber, M. Bock, G. Steinmeyer, T. Elsaesser, and R. Grunwald, “Highly efficient THG in TiO2 nanolayers for third-order pulse characterization,” Opt. Express 19, 16985–16995 (2011).
[Crossref] [PubMed]

2010 (4)

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
[Crossref] [PubMed]

H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
[Crossref] [PubMed]

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

2009 (3)

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

J. Wen and M. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A: At. Mol. Opt. Phys. 79, 025802 (2009).
[Crossref]

2007 (5)

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[Crossref]

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

K. Preston, B. Schmidt, and M. Lipson, “Polysilicon photonic resonators for large-scale 3D integration of optical networks,” Opt. Express 15, 17283 (2007).
[Crossref] [PubMed]

Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: role of Raman scattering and pump polarization,” Phys. Rev. A: At. Mol. Opt. Phys. 75, 023803 (2007).
[Crossref]

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

2005 (1)

M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
[Crossref]

2004 (2)

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[Crossref]

2003 (1)

Y. Shih, “Entangled biphoton source–roperty and preparation,” Rep. Prog. Phys. 66, 1009–1044 (2003).
[Crossref]

2002 (2)

L. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176 (2002).
[Crossref]

Z. Zhu and T. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002).
[Crossref] [PubMed]

1999 (1)

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

1997 (1)

1995 (1)

Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995).
[Crossref]

1990 (1)

M. Sheik-Bahae, D. Hagan, and E. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Agrawal, G. P.

Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: role of Raman scattering and pump polarization,” Phys. Rev. A: At. Mol. Opt. Phys. 75, 023803 (2007).
[Crossref]

Ali-Khan, I.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

Bachor, H.-a.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Bao, X.-H.

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Bencheikh, K.

S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett. 36, 3000–3002 (2011).
[Crossref] [PubMed]

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

Berardi, V.

M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
[Crossref]

Beveratos, A.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Bloch, J.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Bock, M.

Boulanger, B.

S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett. 36, 3000–3002 (2011).
[Crossref] [PubMed]

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

Bowen, W. P.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Boyd, R.

R. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, Burlington, USA, 2008).

Bradley, J. D. B.

Brambilla, E.

L. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176 (2002).
[Crossref]

Brambilla, G.

T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B: Opt. Phys. 30, 505–511 (2013).
[Crossref]

Brendel, J.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Broadbent, C. J.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

Broderick, N. G. R.

T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B: Opt. Phys. 30, 505–511 (2013).
[Crossref]

Brown, T.

Browne, D. E.

M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015).
[Crossref] [PubMed]

Bufetov, I. A.

Burgess, I. B.

Chekhova, M.

M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
[Crossref]

Choy, J. T.

Corcoran, B.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

Corona, M.

M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011).
[Crossref]

Daria, V.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Das, S. K.

Deotare, P. B.

Dianov, E. M.

Douady, J.

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
[Crossref]

Dousse, A.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
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Suffczynski, J.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

Suntivich, J.

Symul, T.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Taylor, M. A.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Thew, R.

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[Crossref]

Tittel, W.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Tsuchiya, T.

Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995).
[Crossref]

U’Ren, A. B.

M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011).
[Crossref]

Van Stryland, E.

M. Sheik-Bahae, D. Hagan, and E. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Verma, V. B.

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

Vitullo, D. L. P.

M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
[Crossref] [PubMed]

Voisin, P.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
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Vuckovic, J.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Watanabe, Y.

Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995).
[Crossref]

Wen, J.

J. Wen and M. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A: At. Mol. Opt. Phys. 79, 025802 (2009).
[Crossref]

White, T. P.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

Widom, J. R.

M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
[Crossref] [PubMed]

Yaman, F.

Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: role of Raman scattering and pump polarization,” Phys. Rev. A: At. Mol. Opt. Phys. 75, 023803 (2007).
[Crossref]

Yan, Z.

L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012).
[Crossref]

Yuan, Z.-S.

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Zbinden, H.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Zhang, J.

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Zhu, Z.

Appl. Phys. Lett. (1)

Y. Watanabe, M. Ohnishi, and T. Tsuchiya, “Measurement of nonlinear absorption and refraction in titanium dioxide single crystal by using a phase distortion method,” Appl. Phys. Lett. 66, 3431 (1995).
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C. R. Phys. (1)

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C. R. Phys. 8, 206–220 (2007).
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J. Opt. B: Quantum Semiclassical Opt. (1)

L. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176 (2002).
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J. Opt. Soc. Am. B: Opt. Phys. (2)

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J. Phys. Chem. B (1)

M. G. Raymer, A. H. Marcus, J. R. Widom, and D. L. P. Vitullo, “Entangled photon-pair two-dimensional fluorescence spectroscopy,” J. Phys. Chem. B 117, 15559–15575 (2013).
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Nat. Mater. (1)

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
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Nat. Photonics (6)

V. Giovannetti and S. Lloyd, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-a. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[Crossref]

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

Nat. Phys. (1)

L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2012).
[Crossref]

Nature (3)

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaitre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[Crossref] [PubMed]

H. Hubel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010).
[Crossref] [PubMed]

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45 (2010).
[Crossref] [PubMed]

Opt. Express (8)

C. C. Evans, J. D. B. Bradley, E. A. Marti-Panameno, and E. Mazur, “Mixed two- and three-photon absorption in bulk rutile (TiO2) around 800 nm,” Opt. Express 20, 3118–3128 (2012).
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C. C. Evans, K. Shtyrkova, J. D. B. Bradley, O. Reshef, E. Ippen, and E. Mazur, “Spectral broadening in anatase titanium dioxide waveguides at telecommunication and near-visible wavelengths,” Opt. Express 21, 18582–18591 (2013).
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J. D. B. Bradley, C. C. Evans, J. T. Choy, O. Reshef, P. B. Deotare, F. Parsy, K. C. Phillips, M. Loncar, and E. Mazur, “Submicrometer-wide amorphous and polycrystalline anatase TiO2 waveguides for microphotonic devices,” Opt. Express 20, 23821–23831 (2012).
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K. Preston, B. Schmidt, and M. Lipson, “Polysilicon photonic resonators for large-scale 3D integration of optical networks,” Opt. Express 15, 17283 (2007).
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C. C. Evans, K. Shtyrkova, O. Reshef, M. Moebius, J. D. B. Bradley, S. Griesse-Nascimento, E. Ippen, and E. Mazur, “Multimode phase-matched third-harmonic generation in sub-micrometer-wide anatase TiO2 waveguides,” Opt. Express 23, 7832 (2015).
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Z. Zhu and T. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002).
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S. K. Das, C. Schwanke, A. Pfuch, W. Seeber, M. Bock, G. Steinmeyer, T. Elsaesser, and R. Grunwald, “Highly efficient THG in TiO2 nanolayers for third-order pulse characterization,” Opt. Express 19, 16985–16995 (2011).
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C. C. Evans, C. Liu, and J. Suntivich, “Low-loss titanium dioxide waveguides and resonators using a dielectric lift-off fabrication process,” Opt. Express 23, 11160 (2015).
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Opt. Lett. (3)

Phys. Rep. (1)

Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Phys. Rep. 497, 1–40 (2010).
[Crossref]

Phys. Rev. A: At. Mol. Opt. Phys. (4)

Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: role of Raman scattering and pump polarization,” Phys. Rev. A: At. Mol. Opt. Phys. 75, 023803 (2007).
[Crossref]

M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source,” Phys. Rev. A: At. Mol. Opt. Phys. 84, 033823 (2011).
[Crossref]

M. Chekhova, O. Ivanova, V. Berardi, and A. Garuccio, “Spectral properties of three-photon entangled states generated via three-photon parametric down-conversion in a χ(3) medium,” Phys. Rev. A: At. Mol. Opt. Phys. 72, 023818 (2005).
[Crossref]

J. Wen and M. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A: At. Mol. Opt. Phys. 79, 025802 (2009).
[Crossref]

Phys. Rev. Lett. (5)

M. Sheik-Bahae, D. Hagan, and E. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation,” Phys. Rev. Lett. 115, 020502 (2015).
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Rep. Prog. Phys. (1)

Y. Shih, “Entangled biphoton source–roperty and preparation,” Rep. Prog. Phys. 66, 1009–1044 (2003).
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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, 71101–71125 (2011).
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Other (2)

S. Krapick and C. Silberhorn, “Analysis of photon triplet generation in pulsed cascaded parametric down-conversion sources,” arXiv.org/abs/1506.07655 (2015).

R. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, Burlington, USA, 2008).

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic of higher-order mode phase matching using calculated dispersion data for a fully-etched 550 × 360 nm TiO2 waveguide with SiO2 cladding. Shifting the pump mode from λp to 3λp gives a point of intersection where the effective indices of the TM00 signal and TM02 pump modes are the same. Phase matching is achieved at λp = 532 nm. Guided modes reach a cutoff wavelength when their effective index is equal to the cladding SiO2 index (gray region). (b) Schematic of the waveguide cross section and tunable parameters. (c) 3D schematic of an integrated device with input and output coupling.

Fig. 2
Fig. 2

We visualize the joint spectral intensity as an intensity plot of frequency versus frequency for several values of the pump detuning Δλp from the PPM wavelength. Panels (a), (b) and (c) correspond to Δp = 0, Δp = −2 nm and Δp = −0.5 nm, respectively. We assume a waveguide length L = 2 mm and use dispersion values for the phase matching point shown in Fig. 1(a). Fixing one of the three signal wavelengths (ωs) on the horizontal axis, the two other signal wavelengths (ωr and ωi) are given by the two values on the vertical axis that intersect the ellipse. Phase matching is achieved along the perimeter of the ellipse and the thickness is determined by the pump and signal interaction length, based on Eq. (1). By collapsing the plots in (a), (b), and (c) onto the horizontal axis, we generate the spectra shown in (d). All three spectra are normalized by the total conversion efficiency. Panel (e) shows signal spectrum bandwidth as a function of pump detuning Δωp. The dashed black curve gives the bandwidth defined by Eq. (2). Red and blue curves are numerically obtained signal bandwidths for waveguide lengths L = 2 mm and L =20 mm, respectively. Panel (f) zooms near the PPM point.

Fig. 3
Fig. 3

(a) Signal intensity as a function of device length for pump and signal loss values, respectively, of (i) 16 and 4 dB/cm, (ii) 16 and 12 dB/cm, (iii) 28 and 4 dB/cm, and (iv) 28 and 12 dB/cm, representing losses on the lower and upper extremes for TiO2 devices. Circles show the full quantum prediction, including signal dispersion, from Eq. (6) and lines show the prediction from Eq. (7). All curves are normalized by the maximum signal. (b) Ideal waveguide length plotted as a function of pump and signal losses. Current polycrystalline anatase TiO2 waveguide losses give optimal device lengths Lopt = 1 – 4 mm. White denotes Lopt > 8 mm.

Fig. 4
Fig. 4

(top) The effective nonlinearity γ as a function of waveguide width and thickness for the best phase-matching point. The highest γ = 1100 W−1km−1 is achieved in a 600 × 300 nm waveguide for TE signal and a 600 × 400 nm waveguide for TM signal. However, these dimensions do not achieve phase-matching. Lines mark regions with high which do achieve phase-matching. (bottom) Figure of merit as a function of waveguide dimensions. The best combination of phase matching and γ are achieved at 600 × 245 nm (γ = 908 W−1km−1) for TE signal and 550 × 360 nm (γ = 674 W−1km−1) for TM signal.

Fig. 5
Fig. 5

Experimental demonstration of THG in an integrated waveguide with two phase-matching points within the infrared pump bandwidth (black curve). The calculated THG signal (dashed curve) shows agreement with the measured THG signal (solid green curve). The inset shows a top-down image of scattered THG signal from the waveguide.

Fig. 6
Fig. 6

Triple and two-photon coincidence signals for detectors placed at equal distances from the TOSPDC output. Time delays τ12 = t1t2 and τ32 = t3t2 are given in units of the characteristic timescale τ 0 = D s L / 2. Panels (a) and (b) are G(3) functions for the detector filter bandwidths σ = 0.2ν0 and σ = 5ν0, respectively, with a loss mismatch parameter ΔαL ≡ (αp − 3αs) L = 0.1. Panels (d) and (e) correspond to the G(3) functions for ΔαL = 1 and ΔαL = 10, respectively, with a detector bandwidth σ = 5ν0. Panel (c) is the G(2) function for several values of σ with fixed ΔαL = 0.1 and (f) is the G(2) signal for several values of ΔαL with fixed σ = 5ν0. Frequency is in units of ν0 = 1/τ0, L is the waveguide length and Ds is the group velocity dispersion (GVD) of the signal guiding mode. Plots are normalized to their maximum values.

Fig. 7
Fig. 7

Examples of (a) a signal mode power density profile and (b) pump mode power density profile. These mode profiles are for a TM00 signal mode and TM02 pump mode phase matched at 532 nm in a 550 × 360 nm TiO2 waveguide with SiO2 cladding (first presented in Fig. 1 of the main text).

Fig. 8
Fig. 8

Examples of a) E-field orientation, b) crystal axes orientation and c) crystal axes orientation with a rotation by an arbitrary angle in the xy plane.

Tables (1)

Tables Icon

Table 1 Waveguide Parameters for Phase-matching Regions With High Figure of Merit . es is the Signal Mode Polarization and γ the Effective Nonlinearity.

Equations (46)

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Δ PPM = 4 π L | D s |
δ s = 2 ( 2 3 ) 1 / 2 [ 2 ( v p 1 v s 1 ) D s Δ p + ( D p D s ) Δ p 2 ] 1 / 2 ,
Δ s 2 + Δ r 2 Δ r Δ s Δ p ( Δ r + Δ s ) = A p ,
γ = 3 χ ( 3 ) ω p 4 ε 0 c 2 n 2 η ,
n 2 = 3 χ ( 3 ) 4 n 0 2 ε 0 c
R 3 = 2 2 3 3 h ¯ c 3 n p 3 π 2 [ ω p 0 ] 2 γ 2 L 2 P ( ω 0 n 0 2 ( ω ) g ( ω 0 ) ) 3 e ( α p + 3 α s ) L / 2 d ν r d ν s | Φ ( ν r , ν s ) | 2 ,
N 3 0 L d N 3 ( z ) d z = ζ ˜ N p 0 α p 3 α s ( e 3 α s L e α p L ) ,
= γ 2 δ s ,
G ( 3 ) ( x 1 , x 2 , x 3 ) = E ( ) ( x 1 ) E ( ) ( x 2 ) E ( ) ( x 3 ) E ( + ) ( x 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) ,
G ( 3 ) ( x 1 , x 2 , x 3 ) = | 0 | E ( + ) ( x 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) | Ψ 3 | 2 .
G ( 2 ) ( x 1 , x 2 ) = d ω 3 | 0 | a ( ω 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) | Ψ 3 | 2 .
( Δ s 2 + Δ r 2 + Δ i 2 ) = 2 ( v p 1 v s 1 ) D s Δ p + ( D p D s ) Δ p 2 r p 2
x s 2 + x r 2 x r x s x r x s = a p .
δ s 2 3 × 2 r p = 2 ( 2 3 ) 1 / 2 [ 2 ( v p 1 v s 1 ) D s Δ p + ( D p D s ) Δ p 2 ] 1 / 2 ,
γ = 3 χ ( 3 ) ω p 4 ε 0 c 2 n 2 η ,
η i j k l = d x d y E p i * E s j E s k E s l [ | E p ( x , y ) | 2 d x d y ] 1 / 2 [ | E s ( x , y ) | 2 d x d y ] 3 / 2 ,
j j k k = k k j j = k k i i = i i k k = i i j j = j j i i ; j k k j = k j j k = k i i k = i k k i = i j j i = j i i j j k j k = k j k j = k i k i = i k i k = i j i j = j i j i ; i i i i = j j j j = k k k k = i i j j + i j i j + i j j i
η i i j j = η i j i j = η i j j i .
n 2 = 3 χ ( 3 ) 4 n 0 2 ε 0 c ,
H ^ 1 = 3 ε 0 χ ( 3 ) 4 d V E ^ p ( + ) ( r , t ) E ^ r ( ) ( r , t ) E ^ s ( ) ( r , t ) E ^ i ( ) ( r , t ) + H . c .
E p ( + ) ( r , t ) = A 0 A p ( x , y ) d ω p β ( ω p ) e [ i ( k p ( ω p ) z ω p t ) ] ,
| Ψ = [ 1 i h ¯ t 0 t d t H 1 ( t ) ] | 0 = | 0 + λ | Ψ 3 .
| Ψ 3 = d ω p β ( ω p ) k s , k r , k i ( ω r ) ( ω s ) ( ω i ) [ t 0 t d t d z ϕ ( t , z ) a ^ ( k r ) a ^ ( k s ) a ^ ( k i ) | 0 ] ,
λ = 3 ε 0 χ ( 3 ) A 0 4 h ¯ A eff ( δ k ) 3 / 2 ,
| Ψ 3 = d ω p β ( ω p ) k r , k s , k i ( ω r ) ( ω s ) ( ω i ) × [ t 0 t d t d z ϕ ( t , z ) e ( α p 3 α s ) z / 2 e ( α p 3 α s ) L / 4 × a ^ ( k r ) a ^ ( k s ) a ^ ( k i ) | 0 ] .
| Ψ 3 = ( 2 π λ L ( δ k ) 3 ) e ( α p + 3 α s ) L / 4 k r k s k i Φ ( ω r , ω s , ω i ) a ^ ( ω r ) a ^ ( ω s ) a ^ ( ω i ) | 0 ,
Φ ( ω r , ω s , ω i ) = β ( ω p ) ( ω r ) ( ω s ) ( ω i ) sinc [ { Δ k + i Δ α } L / 4 ] ,
R 3 = R Ψ 3 | k a ^ ( k ) a ^ ( k ) | Ψ 3 = R ( 2 π λ L ) 2 ( 3 2 ) δ k 3 e ( α p + 3 α s ) L / 2 d ω r d ω s d ω i g ( ω r ) g ( ω s ) g ( ω i ) | Φ ( ω r , ω s , ω i ) | 2
Δ k = k ( ω s ) + k ( ω r ) + k ( ω i ) k ( ω p ) = D s 2 [ ν r 2 + ν s 2 + ( ν s + ν r ) 2 ] ,
R 3 = 2 2 3 2 h ¯ c 3 n p 3 π 2 ( ω p 0 ) 2 γ 2 L 2 p ( ω 0 n 0 2 ( ω ) g ( ω 0 ) ) 3 × e ( α p + 3 α s ) L / 2 d ν r d ν s | Φ ( ν r , ν s ) | 2 .
( 2 π λ L ) 2 3 2 δ k 3 = 3 2 ( 2 π ) 2 ε 0 3 c 3 n p 3 h ¯ 2 ( ω p 0 ) 2 2 γ 2 L 2 P | d ω p β ( ω p ) | 2 ,
d N 3 ( z ) = ζ ˜ N p ( z ) e 3 α s ( L z ) Δ z = ζ ˜ N p 0 e α p z e 3 α s ( L z ) d z ,
N 3 0 L d z N 3 ( z ) = ζ ˜ N p 0 α p 3 α s ( e 3 α s L e α p L ) .
d I ( z ) d z = α ( 1 ) I ( z ) α ( 2 ) [ I ( z ) ] 2
I ( z ) = α I 0 / ( α ( 1 ) + α ( 2 ) I 0 ) e α ( 1 ) z α ( 2 ) I 0 / ( α ( 1 ) + α ( 2 ) I 0 )
d N 3 ( z ) = ζ ˜ A mode I p 0 h ¯ ω p α p ( 1 ) / ( α p ( 1 ) + α p ( 2 ) I p 0 ) e α p ( 1 ) z α p ( 2 ) I p 0 / ( α p ( 1 ) + α p ( 2 ) I p 0 ) e 3 α s ( 1 ) ( L z ) d z .
α p ( 2 ) I p 0 α p ( 1 ) ( 1 e α p ( 1 ) L opt ) < 0.2
G ( 3 ) ( x 1 , x 2 , x 3 ) = Ψ 3 | E ( ) ( x 1 ) E ( ) ( x 2 ) E ( ) ( x 3 ) E ( + ) ( x 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) Ψ 3 | .
G 3 ( x 1 , x 2 , x 3 ) = | ψ 1 ( x 1 , x 2 , x 3 ) | 2 ,
ψ ( x 1 , x 2 , x 3 ) × d ω 1 d ω 2 d ω 3 d ω r d ω s f 1 ( ω 1 ) f 2 ( ω 2 ) f 3 ( ω 3 ) e i ω 1 x 1 i ω 2 x 2 i ω 3 x 3 × Φ ( ω r , ω s ) 0 | a ( ω 1 ) a ( ω 2 ) a ( ω 3 ) a ^ ( ω r ) a ^ ( ω s ) a ^ ( ω p ω r ω s ) | 0 .
ψ ( x 1 , x 2 , x 3 ) = e i ω p x 3 d ω r d ω s f ( ω r ) f ( ω s ) f ( ω p ω r ω s ) Φ ( ω r ω s ) e i ω r ( x 1 x 3 ) i ω s ( x 2 x 3 ) .
G ( 3 ) ( x 1 , x 2 , x 3 ) = | d ν r d ν s f r ( ν r ) f s ( ν s ) f i ( ν r + ν s ) Φ ( ν r , ν s ) e i ν r ( x 1 x 3 ) i ν s ( x 2 x 3 ) | 2 .
G ( 2 ) ( x 1 , x 2 ) = E ^ ( ) ( x 1 ) E ^ ( ) ( x 2 ) E ^ ( + ) ( x 2 ) E ^ ( + ) ( x 1 ) ,
G ( 2 ) ( x 1 , x 2 ) = d ω 3 | 0 | a ( ω 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) | Ψ 3 | 2 ,
0 | a ( ω 3 ) E ( + ) ( x 2 ) E ( + ) ( x 1 ) | Ψ 3 = d ω 1 d ω 2 d ω r d ω s Φ ( ω r , ω s ) f ( ω 1 ) f ( ω 2 ) e i ω 1 x 1 i ω 2 x 2 × 0 | a ( ω 3 ) a ( ω 1 ) a ( ω 2 ) a ( ω r ) a ( ω s ) a ( ω p ω r ω s ) | 0 .
G ( 2 ) ( x 1 , x 2 ) = d ν 3 | d ν r G ( ν r , ν r + ν 3 ) f 1 ( ν r ) f 2 ( ν r + ν 3 ) e i ν r ( x 1 x 2 ) | 2 .

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