Abstract

In this paper, we consider the effects of nonlinear phase modulation on frequency conversion by four-wave mixing (Bragg scattering) in the low-conversion regime. We derive the Green functions for this process using the time-domain collision method, for partial collisions, in which the four fields interact at the beginning or the end of the fiber, and complete collisions, in which the four fields interact at the midpoint of the fiber. If the Green function is separable, there is only one output Schmidt mode, which is free from temporal entanglement. We find that nonlinear phase modulation always chirps the input and output Schmidt modes and renders the Green function formally nonseparable. However, by pre-chirping the pumps, one can reduce the chirps of the Schmidt modes and enable approximate separability. Thus, even in the presence of nonlinear phase modulation, frequency conversion with arbitrary pulse reshaping is possible, as predicted previously [Opt. Express 20, 8367–8396 (2012)].

© 2012 OSA

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  7. W. Wasilewski and M. G. Raymer, “Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions,” Phys. Rev. A73, 063816 (2006).
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  28. T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett.16, 551–553 (2004).
    [CrossRef]
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    [CrossRef]
  31. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett.105, 093604 (2010).
    [CrossRef] [PubMed]
  32. H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express19, 17876–17907 (2011).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  34. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
    [CrossRef]
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    [CrossRef]
  36. C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express20, 6881–6886 (2012).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  40. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
    [CrossRef] [PubMed]

2012 (3)

2011 (8)

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express19, 13770–13778 (2011).
[CrossRef] [PubMed]

H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express19, 17876–17907 (2011).
[CrossRef] [PubMed]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett.23, 109–111 (2011).
[CrossRef]

D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett.106, 130501 (2011).
[CrossRef] [PubMed]

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper–engineered frequency conversion in nonlinear optical waveguides,” New J. Phys.13, 065029 (2011).
[CrossRef]

C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London)469, 508–511 (2011).
[CrossRef]

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

K. Srinivasan and M. G. Raymer, “Quantum frequency translation of single-photon states,” Opt. Photon. News22(12), 39 (2011).
[CrossRef]

2010 (2)

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett.105, 093604 (2010).
[CrossRef] [PubMed]

Y. Ding and Z. Y. Ou, “Frequency downconversion for a quantum network,” Opt. Lett.35, 2591–2593 (2010).
[CrossRef] [PubMed]

2008 (1)

H. J. Kimble, “The quantum internet,” Nature (London)453, 1023–1030 (2008).
[CrossRef]

2007 (1)

P. Kok, 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]

2006 (4)

W. Wasilewski and M. G. Raymer, “Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions,” Phys. Rev. A73, 063816 (2006).
[CrossRef]

M. Tsang and D. Psaltis, “Propagation of temporal entanglement,” Phys. Rev. A73, 013822 (2006).
[CrossRef]

A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express14, 8989–8994 (2006).
[CrossRef] [PubMed]

D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express14, 8995–8999 (2006).
[CrossRef] [PubMed]

2005 (5)

2004 (4)

A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency up-conversion,” J. Mod. Opt.51, 1433–1445 (2004).

M. A. Albota and F. N. C. Wong, “Efficient single-photon counting at 1.55 μm by means of frequency upconversion,” Opt. Lett.29, 1449–1451 (2004).
[CrossRef] [PubMed]

R. V. Roussev, C. Langrock, J. R. Kurz, and M. M. Fejer, “Periodically poled lithium niobate waveguide sum-frequency generator for efficient single-photon detection at communication wavelengths,” Opt. Lett.29, 1518–1520 (2004).
[CrossRef] [PubMed]

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett.16, 551–553 (2004).
[CrossRef]

2002 (1)

K. Uesaka, K. K. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quant.8, 560–568 (2002).
[CrossRef]

2001 (2)

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
[CrossRef]

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

2000 (1)

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
[CrossRef] [PubMed]

1996 (1)

1994 (1)

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon Technol. Lett.6, 1451–1453 (1994).
[CrossRef]

1992 (1)

J. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett.68, 2153–2156 (1992).
[CrossRef] [PubMed]

1987 (1)

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

1982 (1)

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature (London)299, 802–803 (1982).
[CrossRef]

1963 (1)

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev.129, 481–485 (1963).
[CrossRef]

1961 (1)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev.124, 1646–1653 (1961).
[CrossRef]

Afzelius, M.

C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London)469, 508–511 (2011).
[CrossRef]

Albota, M. A.

Alibart, O.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London)437, 116–120 (2005).
[CrossRef]

Baldi, P.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London)437, 116–120 (2005).
[CrossRef]

Brecht, B.

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper–engineered frequency conversion in nonlinear optical waveguides,” New J. Phys.13, 065029 (2011).
[CrossRef]

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express19, 13770–13778 (2011).
[CrossRef] [PubMed]

Bussières, F.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London)469, 508–511 (2011).
[CrossRef]

Cargill, D. S.

Centanni, J. C.

Chen, J.

Christ, A.

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper–engineered frequency conversion in nonlinear optical waveguides,” New J. Phys.13, 065029 (2011).
[CrossRef]

Clausen, C.

C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London)469, 508–511 (2011).
[CrossRef]

Corney, J.

D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett.106, 130501 (2011).
[CrossRef] [PubMed]

de Riedmatten, H.

C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London)469, 508–511 (2011).
[CrossRef]

Ding, Y.

Dowling, J. P.

P. Kok, 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]

Eberly, J. H.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
[CrossRef] [PubMed]

Eckstein, A.

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper–engineered frequency conversion in nonlinear optical waveguides,” New J. Phys.13, 065029 (2011).
[CrossRef]

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express19, 13770–13778 (2011).
[CrossRef] [PubMed]

Fejer, M. M.

Gbur, G. J.

G. J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge University Press, Cambridge, 2011).

George, M.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

Gisin, N.

C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London)469, 508–511 (2011).
[CrossRef]

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London)437, 116–120 (2005).
[CrossRef]

Gnauck, A. H.

Goh, C. S.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett.16, 551–553 (2004).
[CrossRef]

Gordon, J. P.

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev.129, 481–485 (1963).
[CrossRef]

Grice, W. P.

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
[CrossRef]

Halder, M.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London)437, 116–120 (2005).
[CrossRef]

Harvey, J. D.

Hong, C. K.

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

Huang, J.

J. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett.68, 2153–2156 (1992).
[CrossRef] [PubMed]

Inoue, K.

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon Technol. Lett.6, 1451–1453 (1994).
[CrossRef]

Jin, J.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

Jopson, R. M.

Kazovsky, L. G.

K. Uesaka, K. K. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quant.8, 560–568 (2002).
[CrossRef]

M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky, “Widely tunable spectrum translation and wavelength exchange by four-wave mixing in optical fibers,” Opt. Lett.21, 1906–1908 (1996).
[CrossRef] [PubMed]

Kielpinski, D.

D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett.106, 130501 (2011).
[CrossRef] [PubMed]

Kikuchi, K.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett.16, 551–553 (2004).
[CrossRef]

Kimble, H. J.

H. J. Kimble, “The quantum internet,” Nature (London)453, 1023–1030 (2008).
[CrossRef]

Knill, E.

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

Kok, P.

P. Kok, 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]

Kumar, P.

Kurz, J. R.

Kwiat, P. G.

A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency up-conversion,” J. Mod. Opt.51, 1433–1445 (2004).

Laflamme, R.

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

Langrock, C.

Law, C. K.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
[CrossRef] [PubMed]

Lee, K. F.

Li, X.

Louisell, W. H.

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev.129, 481–485 (1963).
[CrossRef]

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev.124, 1646–1653 (1961).
[CrossRef]

Mandel, L.

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

Marhic, M. E.

K. Uesaka, K. K. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quant.8, 560–568 (2002).
[CrossRef]

M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky, “Widely tunable spectrum translation and wavelength exchange by four-wave mixing in optical fibers,” Opt. Lett.21, 1906–1908 (1996).
[CrossRef] [PubMed]

McGuinness, H. J.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett.23, 109–111 (2011).
[CrossRef]

H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express19, 17876–17907 (2011).
[CrossRef] [PubMed]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett.105, 093604 (2010).
[CrossRef] [PubMed]

McKinstrie, C. J.

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express20, 8367–8396 (2012).
[CrossRef] [PubMed]

C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express20, 6881–6886 (2012).
[CrossRef] [PubMed]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A85, 053829 (2012).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett.23, 109–111 (2011).
[CrossRef]

H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express19, 17876–17907 (2011).
[CrossRef] [PubMed]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett.105, 093604 (2010).
[CrossRef] [PubMed]

D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express14, 8995–8999 (2006).
[CrossRef] [PubMed]

A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express14, 8989–8994 (2006).
[CrossRef] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13, 4986–5012 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express13, 9131–9142 (2005).
[CrossRef] [PubMed]

Méchin, D.

Mejling, L.

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express20, 8367–8396 (2012).
[CrossRef] [PubMed]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A85, 053829 (2012).
[CrossRef]

Milburn, G. J.

P. Kok, 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 (London)409, 46–52 (2001).
[CrossRef]

Nemoto, K.

P. Kok, 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]

Oblak, D.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

Ou, Z. Y.

Y. Ding and Z. Y. Ou, “Frequency downconversion for a quantum network,” Opt. Lett.35, 2591–2593 (2010).
[CrossRef] [PubMed]

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

Park, Y.

Provo, R.

Psaltis, D.

M. Tsang and D. Psaltis, “Propagation of temporal entanglement,” Phys. Rev. A73, 013822 (2006).
[CrossRef]

Radic, S.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett.23, 109–111 (2011).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett.105, 093604 (2010).
[CrossRef] [PubMed]

A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express14, 8989–8994 (2006).
[CrossRef] [PubMed]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express13, 9131–9142 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13, 4986–5012 (2005).
[CrossRef] [PubMed]

Ralph, T. C.

P. Kok, 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]

Raymer, M. G.

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A85, 053829 (2012).
[CrossRef]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express20, 8367–8396 (2012).
[CrossRef] [PubMed]

K. Srinivasan and M. G. Raymer, “Quantum frequency translation of single-photon states,” Opt. Photon. News22(12), 39 (2011).
[CrossRef]

H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express19, 17876–17907 (2011).
[CrossRef] [PubMed]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett.23, 109–111 (2011).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett.105, 093604 (2010).
[CrossRef] [PubMed]

W. Wasilewski and M. G. Raymer, “Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions,” Phys. Rev. A73, 063816 (2006).
[CrossRef]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13, 4986–5012 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express13, 9131–9142 (2005).
[CrossRef] [PubMed]

I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science307, 1733–1734 (2005).
[CrossRef] [PubMed]

Ricken, R.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

Rottwitt, K.

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A85, 053829 (2012).
[CrossRef]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express20, 8367–8396 (2012).
[CrossRef] [PubMed]

Roussev, R. V.

Saglamyurek, E.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

Sangouard, N.

C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London)469, 508–511 (2011).
[CrossRef]

Set, S. Y.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett.16, 551–553 (2004).
[CrossRef]

Siegman, A. E.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev.124, 1646–1653 (1961).
[CrossRef]

Silberhorn, C.

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express19, 13770–13778 (2011).
[CrossRef] [PubMed]

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper–engineered frequency conversion in nonlinear optical waveguides,” New J. Phys.13, 065029 (2011).
[CrossRef]

Sinclair, N.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

Slater, J.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

Sohler, W.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

Srinivasan, K.

K. Srinivasan and M. G. Raymer, “Quantum frequency translation of single-photon states,” Opt. Photon. News22(12), 39 (2011).
[CrossRef]

Suche, H.

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper–engineered frequency conversion in nonlinear optical waveguides,” New J. Phys.13, 065029 (2011).
[CrossRef]

Tanemura, T.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett.16, 551–553 (2004).
[CrossRef]

Tanzilli, S.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London)437, 116–120 (2005).
[CrossRef]

Tittel, W.

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London)437, 116–120 (2005).
[CrossRef]

Tsang, M.

M. Tsang and D. Psaltis, “Propagation of temporal entanglement,” Phys. Rev. A73, 013822 (2006).
[CrossRef]

U’Ren, A. B.

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
[CrossRef]

Uesaka, K.

K. Uesaka, K. K. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quant.8, 560–568 (2002).
[CrossRef]

Usmani, I.

C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London)469, 508–511 (2011).
[CrossRef]

Vandevender, A. P.

A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency up-conversion,” J. Mod. Opt.51, 1433–1445 (2004).

Voss, P. L.

Walker, L. R.

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev.129, 481–485 (1963).
[CrossRef]

Walmsley, I. A.

I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science307, 1733–1734 (2005).
[CrossRef] [PubMed]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
[CrossRef]

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
[CrossRef] [PubMed]

Wasilewski, W.

W. Wasilewski and M. G. Raymer, “Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions,” Phys. Rev. A73, 063816 (2006).
[CrossRef]

Wiseman, H.

D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett.106, 130501 (2011).
[CrossRef] [PubMed]

Wong, F. N. C.

Wong, K. K.

K. Uesaka, K. K. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quant.8, 560–568 (2002).
[CrossRef]

Wootters, W. K.

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature (London)299, 802–803 (1982).
[CrossRef]

Yang, F. S.

Yariv, A.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev.124, 1646–1653 (1961).
[CrossRef]

Yu, M.

Zbinden, H.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London)437, 116–120 (2005).
[CrossRef]

Zurek, W. H.

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature (London)299, 802–803 (1982).
[CrossRef]

IEEE J. Sel. Top. Quant. (1)

K. Uesaka, K. K. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quant.8, 560–568 (2002).
[CrossRef]

IEEE Photon Technol. Lett. (1)

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon Technol. Lett.6, 1451–1453 (1994).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett.23, 109–111 (2011).
[CrossRef]

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett.16, 551–553 (2004).
[CrossRef]

J. Mod. Opt. (1)

A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency up-conversion,” J. Mod. Opt.51, 1433–1445 (2004).

Nature (London) (6)

C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature (London)469, 508–511 (2011).
[CrossRef]

E. Saglamyurek, N. Sinclair, J. Jin, J. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature (London)469, 512–515 (2011).
[CrossRef]

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature (London)299, 802–803 (1982).
[CrossRef]

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature (London)437, 116–120 (2005).
[CrossRef]

H. J. Kimble, “The quantum internet,” Nature (London)453, 1023–1030 (2008).
[CrossRef]

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

New J. Phys. (1)

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper–engineered frequency conversion in nonlinear optical waveguides,” New J. Phys.13, 065029 (2011).
[CrossRef]

Opt. Express (9)

X. Li, P. L. Voss, J. Chen, K. F. Lee, and P. Kumar, “Measurement of co- and cross-polarized Raman spectra in silica fiber for small detunings,” Opt. Express13, 2236–2244 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13, 4986–5012 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express13, 9131–9142 (2005).
[CrossRef] [PubMed]

A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express14, 8989–8994 (2006).
[CrossRef] [PubMed]

D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express14, 8995–8999 (2006).
[CrossRef] [PubMed]

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express19, 13770–13778 (2011).
[CrossRef] [PubMed]

H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express19, 17876–17907 (2011).
[CrossRef] [PubMed]

C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express20, 6881–6886 (2012).
[CrossRef] [PubMed]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express20, 8367–8396 (2012).
[CrossRef] [PubMed]

Opt. Lett. (4)

Opt. Photon. News (1)

K. Srinivasan and M. G. Raymer, “Quantum frequency translation of single-photon states,” Opt. Photon. News22(12), 39 (2011).
[CrossRef]

Phys. Rev. (2)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev.124, 1646–1653 (1961).
[CrossRef]

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev.129, 481–485 (1963).
[CrossRef]

Phys. Rev. A (4)

W. Wasilewski and M. G. Raymer, “Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions,” Phys. Rev. A73, 063816 (2006).
[CrossRef]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
[CrossRef]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A85, 053829 (2012).
[CrossRef]

M. Tsang and D. Psaltis, “Propagation of temporal entanglement,” Phys. Rev. A73, 013822 (2006).
[CrossRef]

Phys. Rev. Lett. (5)

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett.105, 093604 (2010).
[CrossRef] [PubMed]

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
[CrossRef] [PubMed]

D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett.106, 130501 (2011).
[CrossRef] [PubMed]

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

J. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett.68, 2153–2156 (1992).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

P. Kok, 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]

Science (1)

I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science307, 1733–1734 (2005).
[CrossRef] [PubMed]

Other (1)

G. J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge University Press, Cambridge, 2011).

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Figures (16)

Fig. 1
Fig. 1

(a) A demonstration of the frequency placement of the two pumps p and q along with the sidebands r and s for frequency conversion in the near-conversion regime. The zero-dispersion frequency is denoted by ω0. (b) Frequency conversion in the far-conversion regime where the pumps are farther from each other. In both figures arrows pointing down mean that photons are destroyed in those particular modes, whereas arrows pointing up mean photons are created. The directions of the arrows are reversible.

Fig. 2
Fig. 2

Illustration of the characteristic coordinates. In this picture pump p and the signal propagates along the x-axis for fixed y, whereas pump q and the idler propagates along the y-axis for fixed x. The fiber input corresponds to x + y = 0 and is denoted by the dashed line. The blue and red lines represent idler and signal rays respectively.

Fig. 3
Fig. 3

Illustrations of the two pump phases ϕp, subfigure (a), and ϕq, subfigure (b), as a function of time for four fiber lengths. The two pumps are identical Gaussians, timed in such a way that they overlap at βl/τ = 2. In both plots βr = −βs = β and τ is the pump width of the identical Gaussian pumps.

Fig. 4
Fig. 4

Generation of an idler from a pulsed signal. The gray area shows the area of the high pump power region. The upward and downward diagonal lines are the characteristics of the idler and the signal respectively. The output idler at time t is generated by a collision with the signal occurring at the point c.

Fig. 5
Fig. 5

An asymmetric collision at the fiber input point zi = 0 and βl/τ = 4 for two equal Gaussian pumps. In (a) contours of the magnitude of Grs are plotted. The magnitudes of the Green functions with and without NPM are identical, as expected. (b) shows contours of the phase of the Green function. In both plots the white lines denote the area of causality, i.e. where the step functions are nonzero. (c) shows the Schmidt coefficients where crosses are the coefficients for the Green function with NPM whereas open circles are for the results without NPM. (d) The absolute values of the lowest-order Schmidt modes are the solid curves. The dashed curves are the phases of the Schmidt modes with NPM. The red curves are the input modes and the blue ones are the output modes. The strength parameter was γ̄ = 0.25.

Fig. 6
Fig. 6

An asymmetric collision at the fiber output zi = l and βl/τ = 4 for two equal Gaussian pumps. In (a) contours of the magnitude of the Green function are plotted while (b) shows contours of the phase of the Green function. (c) shows the Schmidt coefficients where crosses are the coefficients with NPM and open circles are the ones without NPM. (d) The absolute values of the lowest-order Schmidt modes are solid lines whereas the dashed lines are the phases of the Schmidt modes with NPM. The red curves are for the input mode and the blue ones for the output mode. In all the simulations γ̄ = 0.25.

Fig. 7
Fig. 7

In both plots the squares of the ratios of the first and the second Schmidt coefficients are plotted for identical Gaussian pumps as functions of the normalized interaction distance in the fiber. Subfigure (a) shows the ratio for a relatively short fiber with βl/τ = 1, whereas subfigure (b) shows the ratio for βl/τ = 4. For both figures γ̄ = 0.25.

Fig. 8
Fig. 8

A symmetric collision for βl/τ = 1 and identical Gaussian pumps. In (a) the magnitude of the Green function is plotted, while (b) shows contours of the phase. (c) Plot of the Schmidt coefficients where crosses are the coefficients with NPM and open circles are the ones without NPM. (d) The (common) absolute value of the lowest-order Schmidt mode is the solid black curve and the dashed curves the phases of the Schmidt modes. Since the collision is symmetric, with identical pumps, the magnitudes of the input and output modes are identical. The red curve is the phase of the input Schmidt mode and the blue curve is the phase of the output mode. Also the absolute values of the lowest-order mode with and without NPM were indistinguishable. We used the strength parameter γ̄ = 0.25.

Fig. 9
Fig. 9

A symmetric collision for βl/τ = 4 for Gaussian pumps. In (a) the magnitude is plotted. (b) shows contours of the phase of the Green function. (c) Shows the Schmidt coefficients crosses are the coefficients with NPM. (d) The solid curve is the absolute value of the lowest-order Schmidt modes, whereas the dashed curve is the phase of the Schmidt modes with NPM. Again the magnitude of the output and input modes are identical. The red curve is for the phase of the input mode whereas the blue curve is for the phase of the output Schmidt mode. In all the simulations γ̄ = 0.25.

Fig. 10
Fig. 10

In (a) we consider a long fiber, βl/τ = 4 and plot the relative difference of the first Schmidt coefficient with and without NPM (open circles) and the square of the ratio of the first and second Schmidt coefficients with NPM (crosses) as functions of the strength parameter for identical Gaussian pumps and a symmetric collision. (b) shows the same results, as (a) but plotted as functions of the fiber length with γ̄ = 0.25.

Fig. 11
Fig. 11

A symmetric collision for βl/τ = 4 and Gaussian pumps of very different width where τq = 100 and τp = 1. (a) The magnitude of the Green function is shown. In (b) contours of the phase of the Green function are plotted. (c) The Schmidt coefficients, where crosses are the coefficients with NPM and open circles are the ones without NPM. (d) The solid curves are the absolute values of the lowest-order Schmidt modes, whereas the dashed curves are the phases of the Schmidt modes with NPM. Red curves are for the input modes whereas blue curves are for the output modes. In all the simulations γ̄ = 0.25.

Fig. 12
Fig. 12

The case where pump q is a Hermite–Gaussian pump of order 1 (HG1) and pump p is a Gaussian (HG0) for βl/τ = 4. In (a) contours of the magnitude of the Green function is plotted. (b) shows contours of the phase of the Green function. (c) The Schmidt coefficients where crosses are the coefficients with NPM and open circles are the ones without NPM. (d) The solid curves are the absolute values of the lowest-order Schmidt modes, whereas the dashed curves are the phases of the Schmidt modes with NPM. Here red curves are for the input modes whereas blue curves are for the output modes. The jump of π in the phase of the output mode is due to the Schmidt mode changing its sign at this point. Again we have used γ̄ = 0.25.

Fig. 13
Fig. 13

A plot of the nonseparability function Θ(T,T′) for βl/τ = 4 and identical Gaussian pumps. The white circles represent contours of the magnitude of the Green function. Subfigure (a) is for an asymmetric collision at the fiber input, whereas subfigure (b) is for a symmetric collision.

Fig. 14
Fig. 14

Plots of the input and output modes, with and without pump pre-chirps, for two identical Gaussian pumps. In both plots the solid black curves are the magnitudes of the lowest-order Schmidt modes, whereas the dashed and dotted curves are the phases with and without pre-chirps, respectively. The red curves are for the input modes and the blue curves are for the output modes. Notice that the magnitudes of the Schmidt modes are not changed by the pre-chirps and are thus the same for both cases. In (a) the fiber length is βl/τ = 1 and in (b) the length is βl/τ = 4. For both simulations γ̄ = 0.25.

Fig. 15
Fig. 15

Comparisons of numerical and perturbative solutions of the governing equations for a long fiber, βl/τ = 4, and a symmetric collision, zi = l/2. The plots show the output idler for a Gaussian signal and identical Gaussian pumps. In (a) the conversion strength γ̄ = 0.11, whereas in (b) γ̄ = 0.25. In both plots the solid curves are magnitudes and the dashed curves are phases. The red curves represent perturbative solutions of the governing equations, whereas and the blue ones represent numerical solutions.

Fig. 16
Fig. 16

Comparisons of numerical and perturbative solutions of the governing equations for a long fiber, βl/τ = 4, and a symmetric collision, zi = l/2. In this case γ̄ = 0.25 for both simulations. In (a) the inputs are all Gaussians, where pump q is 100 times wider than the input signal and pump p, whereas in (b) pump q is a Hermite–Gaussian of order 1 with the same FWHM as the signal and pump p. The red curves represent the perturbative solutions of the governing equations whereas the blue curves represent the numerical solutions.

Equations (50)

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( z + β p t ) A p ( z , t ) = i γ [ | A p ( z , t ) | 2 + 2 | A q ( z , t ) | 2 ] A p ( z , t ) ,
( z + β q t ) A q ( z , t ) = i γ [ 2 | A p ( z , t ) | 2 + | A q ( z , t ) | 2 ] A q ( z , t ) .
x A p ( x , y ) = i γ ¯ [ F p ( y ) + 2 F q ( x ) ] A p ( x , y ) ,
y A q ( x , y ) = i γ ¯ [ 2 F p ( y ) + F q ( x ) ] A q ( x , y ) ,
ϕ p ( x , y ) = γ ¯ F p ( y ) ( x x 0 ) + 2 γ ¯ x 0 x F q ( s ) d s + ϕ p 0 ,
ϕ q ( x , y ) = γ ¯ F q ( x ) ( y y 0 ) + 2 γ ¯ y 0 y F p ( s ) d s + ϕ q 0 ,
( z + β r t ) A r ( z , t ) = 2 i γ [ | A p ( z , t ) | 2 + | A q ( z , t ) | 2 ] A r ( z , t ) + 2 i γ A p ( z , t ) A q * ( z , t ) A s ( z , t ) ,
( z + β s t ) A s ( z , t ) = 2 i γ [ | A p ( z , t ) | 2 + | A q ( z , t ) | 2 ] A s ( z , t ) + 2 i γ A p * ( z , t ) A q ( z , t ) A r ( z , t ) ,
y A r ( x , y ) = 2 i γ ¯ [ F p ( y ) + F q ( x ) ] A r ( x , y ) + 2 i γ ¯ A p ( x , y ) A q * ( x , y ) A s ( x , y ) ,
x A s ( x , y ) = 2 i γ ¯ [ F p ( y ) + F q ( x ) ] A s ( x , y ) + 2 i γ ¯ A p * ( x , y ) A q ( x , y ) A r ( x , y ) .
ϕ s ( x , y ) = 2 γ ¯ F p ( y ) ( x x 0 ) + 2 γ ¯ x 0 x F q ( s ) d s + ϕ s 0 ,
ϕ p ( x , y 0 ) = γ ¯ F p ( y 0 ) ( x + y 0 ) + 2 γ ¯ y 0 x F q ( s ) d s ,
ϕ s ( x , y 0 ) = 2 γ ¯ F p ( y 0 ) ( x + y 0 ) + 2 γ ¯ y 0 x F q ( s ) d s .
ϕ q ( x , y 0 ) = γ ¯ F q ( x ) ( y 0 + x ) + 2 γ ¯ x y 0 F p ( s ) d s .
A s ( x , y 0 ) = δ ( y y 0 ) exp [ i ϕ s ( x , y 0 ) ] .
y A r ( x , y ) = 2 i γ ¯ A p ( x , y ) A q * ( x , y ) A s ( x , y ) ,
A r ( x , y 0 ) = 2 i γ ¯ a p ( y 0 ) a q ( x ) exp [ i ϕ p ( x , y 0 ) i ϕ q ( x , y 0 ) + i ϕ s ( x , y 0 ) ] .
ϕ r ( x , y ) = 2 γ ¯ F q ( x ) ( y y 0 ) + 2 γ ¯ y 0 y F p ( s ) d s ,
A r ( x , y ) = 2 i γ ¯ a p ( y 0 ) a q ( x ) exp [ i ϕ p ( x , y 0 ) i ϕ q ( x , y 0 ) + i ϕ r ( x , y ) + i ϕ s ( x , y 0 ) ] .
G rs ( x , y ) = 2 i γ ¯ a p ( y 0 ) a q ( x ) exp [ i ϕ p ( x , y 0 ) i ϕ q ( x , y 0 ) + i ϕ r ( x , y ) + i ϕ s ( x , y 0 ) ] × H ( x + y 0 ) H ( y y 0 ) ,
ϕ p ( z , t ) = γ F p ( t β s z ) z + 2 γ ¯ t β r z t β s z F q ( s ) d s + ϕ p 0 ,
ϕ q ( z , t ) = γ F q ( t β r z ) z + 2 γ ¯ t β r z t β s z F p ( s ) d s + ϕ q 0 .
ϕ s ( z , t ) = 2 γ F p ( t β s z ) z + 2 γ ¯ t β r z t β s z F q ( s ) d s .
z c = [ t ( t β r z ) ] / β rs , t c = [ β r t β s ( t β r z ) ] / β rs ,
t c β r z c = t β r z , t c β s z c = t .
ϕ r ( z , t ) = 2 γ F q ( t c β r z c ) ( z z c ) + 2 γ ¯ t c β s z c t β s z F p ( s ) d s .
G rs ( t ; t ) = 2 i γ ¯ a p ( z c , t c ) a q ( z c , t c ) exp [ i ϕ p ( z c , t c ) i ϕ q ( z c , t c ) + i ϕ r ( z , t ) + i ϕ s ( z c , t c ) ] × H ( t t + β r z ) H ( t β s z t ) .
G rs ( t ; t ) = 2 i γ ¯ a p ( t ) a q ( t β r z ) exp { 3 i γ ¯ [ t ( t β r z ) ] [ F p ( t ) F q ( t β r z ) ] } × exp { 4 i γ ¯ t β r z t [ F q ( s ) F p ( s ) ] d s + 2 i γ F q ( t β r z ) z } × exp [ 2 i γ ¯ t β r z t β s z F p ( s ) d s + i ϕ p 0 i ϕ q 0 ] H ( t t + β r z ) H ( t β s z t ) .
G rs ( t ; t ) = 2 i γ ¯ a p ( t ) a q ( t β r z ) exp [ i θ ( z , t , t ) ] H ( t t + β r z ) H ( t β s z t ) ,
θ ( z , t , t ) = 3 γ ¯ [ t ( t β r z ) ] [ F p ( t ) F q ( t β r z ) ] + 4 γ ¯ t β r z t [ F q ( s ) F p ( s ) ] d s + 2 γ F q ( t β r z ) z + 2 γ ¯ t β r z t β s z F p ( s ) d s + ϕ p 0 ϕ q 0 .
G rs ( t ; t ) = n v n ( t ) λ n 1 / 2 u n * ( t ) ,
a j ( z , t ) = ( τ 2 π ) 1 / 4 exp { [ t β j ( z z i ) ] 2 / ( 2 τ 2 ) } , j { p , q }
G rs ( t ; t ) = 2 i γ ¯ a p ( t + β s z i ) a q [ t β r ( l z i ) ] exp [ i θ ( l , t , t ) ] H ( t t + β r l ) H ( t β s l t ) .
G rs ( t ; t ) = 2 i γ ¯ ( τ 2 π ) 1 / 2 exp [ t 2 + ( t β l ) 2 2 τ 2 + i θ ( l , t , t ) ] H ( t t + β l ) H ( t + β l t ) .
G rs ( t ; t ) = 2 i γ ¯ ( τ 2 π ) 1 / 2 exp [ t 2 + ( t β l ) 2 2 τ 2 + i θ ( l , t , t ) ] H ( t t + β l ) H ( t + β l t ) .
G rs ( t ; t ) = 2 i γ ¯ ( τ 2 π ) 1 / 2 exp { [ ( t β l / 2 ) 2 + ( t β l / 2 ) 2 ] / ( 2 τ 2 ) } × exp [ i θ ( l , t , t ) ] H ( t t + β l ) H ( t + β l t ) .
G rs ( t ; t ) = 2 i γ ¯ ( τ p τ q π ) 1 / 2 exp [ ( t β l / 2 ) 2 / ( 2 τ p ) ( t β l / 2 ) 2 / ( 2 τ q 2 ) ] × exp [ i θ ( l , t , t ) ] H ( t t + β l ) H ( t + β l t ) .
ψ n ( x ) = H n ( x ) exp ( x 2 / 2 ) π 1 / 4 ( 2 n n ! ) 1 / 2 ,
θ ( t , t ) = 3 γ z c [ F p ( t ) F q ( t β r l ) ] + 4 γ ¯ t F qp ( s ) d s 4 γ ¯ t β r l F qp ( s ) d s + 2 γ F q ( t β r l ) l + 2 γ ¯ t β r l t β s l F p ( s ) d s + ϕ p 0 ( t ) ϕ q 0 ( t β r l ) ,
Φ ( t , t ) = 3 γ z c [ F p ( t ) F q ( t β r l ) ] .
F p ( s ) = G p ( s + β s z i ) , F q ( s ) = G q ( s + β r z i ) ,
Φ ( t , t ) = 3 γ z c { G p ( t + β s z i ) G q [ t β r ( l z i ) ] } .
z c = [ t + β s z i t + β r ( l z i ) + β rs z i ] / β rs .
T = t + β s z i , T = t β r ( l z i ) ,
Φ ( T , T ) = 3 γ ¯ [ G p ( T ) G q ( T ) ] ( T T + β rs z i ) ,
= 3 γ ¯ [ G p ( T ) T + G q ( T ) T ] + 3 γ [ G p ( T ) G q ( T ) ] z i 3 γ ¯ [ G p ( T ) T + G q ( T ) T ] .
Θ ( T , T ) = 3 γ ¯ [ G p ( T ) T + G q ( T ) T ] .
ϕ p 0 ( τ s ) = 4 γ ¯ τ s F qp ( s ) d s 3 γ ¯ F p ( τ s ) ( τ s + β r z i ) ,
ϕ q 0 ( τ r ) = 4 γ ¯ τ r F qp ( s ) d s + 2 γ F q ( τ r ) l + 2 γ ¯ τ r τ r + β rs l F p ( s ) d s + 3 γ ¯ F q ( τ r ) ( τ r + β s z i ) .
A s ( 0 , t ) = 1 10 π 1 / 4 exp [ ( t + β s z i ) 2 2 ] .

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