Abstract

Multi-color photons are prominent candidates for carrying quantum information, as their unlimited dimensionality allows for novel qudit-based schemes. The generation and manipulation of such photons takes place in nonlinear optical media, and the coupling between the different frequency bins can be engineered to obtain the desired quantum state. Here, we propose the design of a frequency-domain Stern–Gerlach effect for photons, where quantum entanglement between the spatial and spectral degrees of freedom is manifested. In this scheme, orthogonal frequency-superposition states can be spatially separated, resulting in a direct projection of an input state onto the frequency-superposition basis. We analyze this phenomenon for two-color qubits and three-color qutrits, and present a generalized wavelength-domain analog of the Hong–Ou–Mandel interference with distinguishable photons. Our results pave the way toward realization of single-element, all-optically controlled spectral-to-spatial beam splitters and tritters that can benefit quantum information processing in the frequency domain.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  6. M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
    [Crossref]
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    [Crossref]
  9. P. Imany, O. D. Odele, M. S. Alshaykh, H. H. Lu, D. E. Leaird, and A. M. Weiner, “Frequency-domain Hong–Ou–Mandel interference with linear optics,” Opt. Lett. 43, 2760–2763 (2018).
    [Crossref]
  10. H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
    [Crossref]
  11. A. Karnieli and A. Arie, “All-optical Stern-Gerlach effect,” Phys. Rev. Lett. 120, 053901 (2018).
    [Crossref]
  12. R. W. Boyd, Nonlinear Optics (Academic, 2008).
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  14. More accurately, these relations are obtained by redefining ϕ12 as ϕ12→−π/2−ϕ12.
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    [Crossref]
  16. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
    [Crossref]
  17. A. Karnieli and A. Arie, “Fully controllable adiabatic geometric phase in nonlinear optics,” Opt. Express 26, 4920–4932 (2018).
    [Crossref]
  18. The nonlinear coupling, κ12, is proportional to the magnitude of the first Fourier component of the poling pattern [12], given by a1=(2/π)dij sin(πD), where dij is the ijth component of the nonlinear susceptibility tensor and D is the poling duty cycle. D can be varied along, say y, such that κ12 varies linearly with y, by letting D=(1/π)arcsin(y/W), where W is the poled region width. For further details, see Ref. [11].
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    [Crossref]
  20. M. Y. Niu, I. L. Chuang, and J. H. Shapiro, “Qudit-basis universal quantum computation using χ(2) interactions,” Phys. Rev. Lett. 120, 160502 (2018).
    [Crossref]
  21. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
    [Crossref]
  22. A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. 4, 355–373 (2010).
    [Crossref]
  23. R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
    [Crossref]
  24. D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
    [Crossref]
  25. A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
    [Crossref]
  26. A. Halevy, E. Megidish, T. Shacham, L. Dovrat, and H. S. Eisenberg, “Projection of two biphoton qutrits onto a maximally entangled state,” Phys. Rev. Lett. 106, 130502 (2011).
    [Crossref]
  27. S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
    [Crossref]
  28. Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9, 609 (2018).
    [Crossref]

2018 (6)

H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
[Crossref]

A. Karnieli and A. Arie, “All-optical Stern-Gerlach effect,” Phys. Rev. Lett. 120, 053901 (2018).
[Crossref]

M. Y. Niu, I. L. Chuang, and J. H. Shapiro, “Qudit-basis universal quantum computation using χ(2) interactions,” Phys. Rev. Lett. 120, 160502 (2018).
[Crossref]

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9, 609 (2018).
[Crossref]

A. Karnieli and A. Arie, “Fully controllable adiabatic geometric phase in nonlinear optics,” Opt. Express 26, 4920–4932 (2018).
[Crossref]

P. Imany, O. D. Odele, M. S. Alshaykh, H. H. Lu, D. E. Leaird, and A. M. Weiner, “Frequency-domain Hong–Ou–Mandel interference with linear optics,” Opt. Lett. 43, 2760–2763 (2018).
[Crossref]

2017 (2)

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

J. M. Lukens and P. Lougovski, “Frequency-encoded photonic qubits for scalable quantum information processing,” Optica 4, 8–16 (2017).
[Crossref]

2016 (3)

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

P. Treutlein, “Viewpoint: Photon qubit is made of two colors,” Physics 9, 135 (2016).
[Crossref]

2011 (1)

A. Halevy, E. Megidish, T. Shacham, L. Dovrat, and H. S. Eisenberg, “Projection of two biphoton qutrits onto a maximally entangled state,” Phys. Rev. Lett. 106, 130502 (2011).
[Crossref]

2010 (1)

A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. 4, 355–373 (2010).
[Crossref]

2009 (2)

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

2008 (1)

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

2005 (1)

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref]

2002 (2)

D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

1997 (1)

M. A. M. Marte and S. Stenholm, “Paraxial light and atom optics: the optical Schrödinger equation and beyond,” Phys. Rev. A 56, 2940–2953 (1997).
[Crossref]

1922 (1)

W. Gerlach and O. Stern, “Der experimentelle nachweiss der richtungsquantelung im magnetfeld,” Z. Phys. 9, 349–352 (1922).
[Crossref]

Alshaykh, M. S.

Arie, A.

A. Karnieli and A. Arie, “Fully controllable adiabatic geometric phase in nonlinear optics,” Opt. Express 26, 4920–4932 (2018).
[Crossref]

A. Karnieli and A. Arie, “All-optical Stern-Gerlach effect,” Phys. Rev. Lett. 120, 053901 (2018).
[Crossref]

A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. 4, 355–373 (2010).
[Crossref]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref]

Azana, J.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Bahabad, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref]

Bello, L.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9, 609 (2018).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

Bruß, D.

D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref]

Caspani, L.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Chiao, R. Y.

J. C. Garrison and R. Y. Chiao, Quantum Optics (Oxford University, 2008).

Chu, S. T.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Chuang, I. L.

M. Y. Niu, I. L. Chuang, and J. H. Shapiro, “Qudit-basis universal quantum computation using χ(2) interactions,” Phys. Rev. Lett. 120, 160502 (2018).
[Crossref]

Cino, A.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Clemmen, S.

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

S. Clemmen, R. Van Laer, A. Farsi, J. S. Levy, M. Lipson, and A. Gaeta, “Towards frequency-coded q-dit manipulation using coherent four-wave mixing,” in Conference on Lasers and Electro-Optics (QELS) (2012), paper QM2H.6.

Cortes, L. R.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Dovrat, L.

A. Halevy, E. Megidish, T. Shacham, L. Dovrat, and H. S. Eisenberg, “Projection of two biphoton qutrits onto a maximally entangled state,” Phys. Rev. Lett. 106, 130502 (2011).
[Crossref]

Eisenberg, H. S.

A. Halevy, E. Megidish, T. Shacham, L. Dovrat, and H. S. Eisenberg, “Projection of two biphoton qutrits onto a maximally entangled state,” Phys. Rev. Lett. 106, 130502 (2011).
[Crossref]

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

Farsi, A.

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

C. Joshi, A. Farsi, and A. Gaeta, “Hong-Ou-Mandel interference in the frequency domain,” in Conference on Lasers and Electro-Optics (CLEO): QELS_Fundamental Science (2017), paper FF2E.3.

S. Clemmen, R. Van Laer, A. Farsi, J. S. Levy, M. Lipson, and A. Gaeta, “Towards frequency-coded q-dit manipulation using coherent four-wave mixing,” in Conference on Lasers and Electro-Optics (QELS) (2012), paper QM2H.6.

Fedrizzi, A.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

Gaeta, A.

S. Clemmen, R. Van Laer, A. Farsi, J. S. Levy, M. Lipson, and A. Gaeta, “Towards frequency-coded q-dit manipulation using coherent four-wave mixing,” in Conference on Lasers and Electro-Optics (QELS) (2012), paper QM2H.6.

C. Joshi, A. Farsi, and A. Gaeta, “Hong-Ou-Mandel interference in the frequency domain,” in Conference on Lasers and Electro-Optics (CLEO): QELS_Fundamental Science (2017), paper FF2E.3.

Gaeta, A. L.

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

Garrison, J. C.

J. C. Garrison and R. Y. Chiao, Quantum Optics (Oxford University, 2008).

Gerlach, W.

W. Gerlach and O. Stern, “Der experimentelle nachweiss der richtungsquantelung im magnetfeld,” Z. Phys. 9, 349–352 (1922).
[Crossref]

Halevy, A.

A. Halevy, E. Megidish, T. Shacham, L. Dovrat, and H. S. Eisenberg, “Projection of two biphoton qutrits onto a maximally entangled state,” Phys. Rev. Lett. 106, 130502 (2011).
[Crossref]

Ikuta, R.

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

Imany, P.

Imoto, N.

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

Joshi, C.

C. Joshi, A. Farsi, and A. Gaeta, “Hong-Ou-Mandel interference in the frequency domain,” in Conference on Lasers and Electro-Optics (CLEO): QELS_Fundamental Science (2017), paper FF2E.3.

Karnieli, A.

A. Karnieli and A. Arie, “Fully controllable adiabatic geometric phase in nonlinear optics,” Opt. Express 26, 4920–4932 (2018).
[Crossref]

A. Karnieli and A. Arie, “All-optical Stern-Gerlach effect,” Phys. Rev. Lett. 120, 053901 (2018).
[Crossref]

Koashi, M.

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

Kobayashi, T.

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

Kues, M.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Langford, N. K.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

Leaird, D. E.

H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
[Crossref]

P. Imany, O. D. Odele, M. S. Alshaykh, H. H. Lu, D. E. Leaird, and A. M. Weiner, “Frequency-domain Hong–Ou–Mandel interference with linear optics,” Opt. Lett. 43, 2760–2763 (2018).
[Crossref]

Levy, J. S.

S. Clemmen, R. Van Laer, A. Farsi, J. S. Levy, M. Lipson, and A. Gaeta, “Towards frequency-coded q-dit manipulation using coherent four-wave mixing,” in Conference on Lasers and Electro-Optics (QELS) (2012), paper QM2H.6.

Lifshitz, R.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref]

Lipson, M.

S. Clemmen, R. Van Laer, A. Farsi, J. S. Levy, M. Lipson, and A. Gaeta, “Towards frequency-coded q-dit manipulation using coherent four-wave mixing,” in Conference on Lasers and Electro-Optics (QELS) (2012), paper QM2H.6.

Little, B. E.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Lougovski, P.

H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
[Crossref]

J. M. Lukens and P. Lougovski, “Frequency-encoded photonic qubits for scalable quantum information processing,” Optica 4, 8–16 (2017).
[Crossref]

Lu, H. H.

P. Imany, O. D. Odele, M. S. Alshaykh, H. H. Lu, D. E. Leaird, and A. M. Weiner, “Frequency-domain Hong–Ou–Mandel interference with linear optics,” Opt. Lett. 43, 2760–2763 (2018).
[Crossref]

H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
[Crossref]

Lukens, J. M.

H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
[Crossref]

J. M. Lukens and P. Lougovski, “Frequency-encoded photonic qubits for scalable quantum information processing,” Optica 4, 8–16 (2017).
[Crossref]

Macchiavello, C.

D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref]

Marte, M. A. M.

M. A. M. Marte and S. Stenholm, “Paraxial light and atom optics: the optical Schrödinger equation and beyond,” Phys. Rev. A 56, 2940–2953 (1997).
[Crossref]

Megidish, E.

A. Halevy, E. Megidish, T. Shacham, L. Dovrat, and H. S. Eisenberg, “Projection of two biphoton qutrits onto a maximally entangled state,” Phys. Rev. Lett. 106, 130502 (2011).
[Crossref]

Michael, Y.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9, 609 (2018).
[Crossref]

Miki, S.

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

Morandotti, R.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Moss, D. J.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Niu, M. Y.

M. Y. Niu, I. L. Chuang, and J. H. Shapiro, “Qudit-basis universal quantum computation using χ(2) interactions,” Phys. Rev. Lett. 120, 160502 (2018).
[Crossref]

Odele, O. D.

H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
[Crossref]

P. Imany, O. D. Odele, M. S. Alshaykh, H. H. Lu, D. E. Leaird, and A. M. Weiner, “Frequency-domain Hong–Ou–Mandel interference with linear optics,” Opt. Lett. 43, 2760–2763 (2018).
[Crossref]

Oron, D.

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

Pe’er, A.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9, 609 (2018).
[Crossref]

Peters, N. A.

H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
[Crossref]

Ramelow, S.

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

Ratschbacher, L.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

Reimer, C.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Rosenbluh, M.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9, 609 (2018).
[Crossref]

Roztocki, P.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Sciara, S.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge, 2001).

Shacham, T.

A. Halevy, E. Megidish, T. Shacham, L. Dovrat, and H. S. Eisenberg, “Projection of two biphoton qutrits onto a maximally entangled state,” Phys. Rev. Lett. 106, 130502 (2011).
[Crossref]

Shaked, Y.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9, 609 (2018).
[Crossref]

Shapiro, J. H.

M. Y. Niu, I. L. Chuang, and J. H. Shapiro, “Qudit-basis universal quantum computation using χ(2) interactions,” Phys. Rev. Lett. 120, 160502 (2018).
[Crossref]

Silberberg, Y.

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

Stenholm, S.

M. A. M. Marte and S. Stenholm, “Paraxial light and atom optics: the optical Schrödinger equation and beyond,” Phys. Rev. A 56, 2940–2953 (1997).
[Crossref]

Stern, O.

W. Gerlach and O. Stern, “Der experimentelle nachweiss der richtungsquantelung im magnetfeld,” Z. Phys. 9, 349–352 (1922).
[Crossref]

Suchowski, H.

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

Takashi, Y.

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

Terai, H.

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

Treutlein, P.

P. Treutlein, “Viewpoint: Photon qubit is made of two colors,” Physics 9, 135 (2016).
[Crossref]

Van Laer, R.

S. Clemmen, R. Van Laer, A. Farsi, J. S. Levy, M. Lipson, and A. Gaeta, “Towards frequency-coded q-dit manipulation using coherent four-wave mixing,” in Conference on Lasers and Electro-Optics (QELS) (2012), paper QM2H.6.

Vaziri, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

Vered, R. Z.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9, 609 (2018).
[Crossref]

Voloch, N.

A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. 4, 355–373 (2010).
[Crossref]

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

Weihs, G.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

Weiner, A. M.

H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
[Crossref]

P. Imany, O. D. Odele, M. S. Alshaykh, H. H. Lu, D. E. Leaird, and A. M. Weiner, “Frequency-domain Hong–Ou–Mandel interference with linear optics,” Opt. Lett. 43, 2760–2763 (2018).
[Crossref]

Wetzel, B.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Yamashita, T.

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

Yasui, S.

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

Zeilinger, A.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

Zhamg, Y.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge, 2001).

Laser Photon. Rev. (1)

A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. 4, 355–373 (2010).
[Crossref]

Nat. Commun. (1)

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9, 609 (2018).
[Crossref]

Nat. Photonics (2)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, Y. Takashi, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

Nature (1)

M. Kues, C. Reimer, P. Roztocki, L. R. Cortes, S. Sciara, B. Wetzel, Y. Zhamg, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (1)

M. A. M. Marte and S. Stenholm, “Paraxial light and atom optics: the optical Schrödinger equation and beyond,” Phys. Rev. A 56, 2940–2953 (1997).
[Crossref]

Phys. Rev. A. (1)

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

Phys. Rev. Lett. (9)

H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
[Crossref]

A. Karnieli and A. Arie, “All-optical Stern-Gerlach effect,” Phys. Rev. Lett. 120, 053901 (2018).
[Crossref]

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

M. Y. Niu, I. L. Chuang, and J. H. Shapiro, “Qudit-basis universal quantum computation using χ(2) interactions,” Phys. Rev. Lett. 120, 160502 (2018).
[Crossref]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref]

D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

A. Halevy, E. Megidish, T. Shacham, L. Dovrat, and H. S. Eisenberg, “Projection of two biphoton qutrits onto a maximally entangled state,” Phys. Rev. Lett. 106, 130502 (2011).
[Crossref]

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

Physics (1)

P. Treutlein, “Viewpoint: Photon qubit is made of two colors,” Physics 9, 135 (2016).
[Crossref]

Z. Phys. (1)

W. Gerlach and O. Stern, “Der experimentelle nachweiss der richtungsquantelung im magnetfeld,” Z. Phys. 9, 349–352 (1922).
[Crossref]

Other (7)

C. Joshi, A. Farsi, and A. Gaeta, “Hong-Ou-Mandel interference in the frequency domain,” in Conference on Lasers and Electro-Optics (CLEO): QELS_Fundamental Science (2017), paper FF2E.3.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge, 2001).

S. Clemmen, R. Van Laer, A. Farsi, J. S. Levy, M. Lipson, and A. Gaeta, “Towards frequency-coded q-dit manipulation using coherent four-wave mixing,” in Conference on Lasers and Electro-Optics (QELS) (2012), paper QM2H.6.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

J. C. Garrison and R. Y. Chiao, Quantum Optics (Oxford University, 2008).

More accurately, these relations are obtained by redefining ϕ12 as ϕ12→−π/2−ϕ12.

The nonlinear coupling, κ12, is proportional to the magnitude of the first Fourier component of the poling pattern [12], given by a1=(2/π)dij sin(πD), where dij is the ijth component of the nonlinear susceptibility tensor and D is the poling duty cycle. D can be varied along, say y, such that κ12 varies linearly with y, by letting D=(1/π)arcsin(y/W), where W is the poled region width. For further details, see Ref. [11].

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Figures (5)

Fig. 1.
Fig. 1. (a), (b) A two-level photon may occupy two different frequencies, coupled by an external pump field, and have a certain spatio-temporal envelope. (c) The qubit state is represented by a point on the Bloch sphere: pure frequency states are situated on the poles and equal-superposition states lie on the equator. The magnetic field analog, B, was chosen to point to the equator with an angle determined by the pump phase. Hence, the two eigenstates in the direction of B are superposition states, while the original state |ψ can be projected onto each of them by employing the proposed SG effect.
Fig. 2.
Fig. 2. (a) Single-photon SG interference and (b) its representation on the Bloch sphere. A single-photon wavepacket state in the idler frequency |ψi is incident on a pumped nonlinear SG crystal and transformed into two spatially separated orthogonal frequency superposition single-photon states. (c) Two-photon SG-HOM interference and (d) its representation on the Bloch sphere. Two distinguishable photon wavepacket states are incident on the crystal, and are deflected into a frequency superposition 2002 state, where |ψ±=(|ψi±|ψs)/2. (e) The nonlinearity gradient can be realized by transversely varying the QPM poling duty cycle, D; see Ref. [18].
Fig. 3.
Fig. 3. Simulated photodetection probability in PPLN with a SG poling pattern [with units mm1 on the left figures and mm2 on the right ones, normalized so that the total probability, by integrating over y (left) or xy (right), is 1]. Propagation along the z axis is presented to the left, while the far-field pattern is given to the right. All input states are Gaussian with a waist of 50 μm. (a), (b) The input state is |ψin=|ψi and is deflected into two discrete angles. (c), (d) The input state is an eigenstate |ψin=(|ψi|ψs)/2, and is deflected only to the left. (e), (f) For the second eigenstate, |ψin=(|ψi+|ψs)/2, the deflection is only to the right.
Fig. 4.
Fig. 4. Three possible configurations for frequency-domain photonic qutrits: (a) Λ scheme, (b) V scheme, and (c) nearest-level (ladder) scheme. (d) Illustration of the SG deflection for a single Λ scheme photon in the lowest frequency |ψin=|ψ1. The expressions for the eigenstates |ψ± and |ψ0 are given in Eq. (16).
Fig. 5.
Fig. 5. Simulated photodetection probability in PPLN with a quasiperiodic SG poling pattern (with units mm1 on the left figures and mm2 on the right ones, normalized as in Fig. 3). All input states are Gaussian with a waist of 50 μm. (a), (b) The input state is the first eigenstate |ψin=|ψ+ and is deflected to the right. (c), (d) The second eigenstate |ψin=|ψ0 remains undeflected. (e), (f) The third eigenstate, |ψin=|ψ, is deflected to the left. In (g), (h), the state |ψin=|ψ3 is projected onto the eigenstates |ψ+ and |ψ [see Eq. (16)]. The resulting deflection is only to the right or to the left. Finally, (i), (j) show the projection of |ψin=|ψ1 onto all three eigenstates. The resulting far field consists of all three possible deflections.

Equations (20)

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ϕj(r,t)=Qd3q(2π)3a(kj+q)eiq·riδωj(q)t.
HSFG=d3rj=1,2vgjϕj(izT22kj)ϕjd3rκ12(ieiϕ12ϕ1ϕ2ieiϕ12ϕ1ϕ2),
HSFG=v¯gd3rϕ(iz+12M1p2Σ·B)ϕ,
izϕ(r,η)=[12M1p2Σ·B(r)]ϕ(r,η).
izϕ˜±=[kT22k¯(B0iBky)]ϕ˜±.
exp(iH±z)=exp(±izB0iz3B26k¯)×exp(zBky)×exp(izkT22k¯iz2Bky2k¯).
D(α)=exp(αk¯Σ·B^ky),
ϕ˜out=U1exp(ασzky)Uϕ˜in.
(ϕ˜Rϕ˜L)=12(1eiϕ12eiϕ121)(ϕ˜iϕ˜s),
|ψout=12[(|ψi+|ψs2)R+(|ψi|ψs2)L],
|ψout=12[(|ψi+|ψs2)R2(|ψi|ψs2)L2],
HSFG=d3rj=1,2,3vgjϕj(izT22kj)ϕjd3r(jk)κjk(ieiϕjkϕjϕkieiϕjkϕjϕk).
(Ljk)mn=eiϕjkδjmδkneiϕjkδjnδkm.
(ϕ˜Rϕ˜Mϕ˜L)=(eiϕ21sinθ2cosθ2ieiϕ232ieiϕ31cosθieiϕ32sinθ0eiϕ21sinθ2cosθ2ieiϕ232)(ϕ˜1ϕ˜2ϕ˜3),
|ψout=12|ψ+R+12|ψ0M+12|ψL,
|ψ±=|ψ1+|ψ2±2|ψ32,|ψ0=|ψ1|ψ22
|ψout=122|ψ+ψ+R12|ψ0ψ0M+122|ψψL+12|ψ+R|ψL,
G(1)(r,t;r,t)=v¯g2ε0ncjψ|ϕj(r,t)ϕj(r,t)|ψ,
izΨ=[12M1p2Σ·B(r)]Ψ,
G(1)(r,t;r,t)=v¯g2ε0ncj|ψj|2(r,t)=v¯g2ε0ncΨΨ(r,t).

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