Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime

L. Mejling; C. J. McKinstrie; M. G. Raymer; K. Rottwitt

doi:10.1364/OE.20.008367

Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt

Optics Express
Vol. 20,
Issue 8,
pp. 8367-8396
(2012)
•https://doi.org/10.1364/OE.20.008367

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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. Express 20, 8367-8396 (2012)

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Related Topics
Table of Contents Category
- Nonlinear Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Nonlinear optics, fibers (060.4370)
- Nonlinear optics, four-wave mixing (190.4380)
- Quantum information and processing (270.5585)

About this Article
History
- Original Manuscript: February 13, 2012
- Revised Manuscript: March 16, 2012
- Manuscript Accepted: March 16, 2012
- Published: March 26, 2012

Accessible

Open Access

Abstract

In this paper we consider frequency translation enabled by Bragg scattering, a four-wave mixing process. First we introduce the theoretical background of the Green function formalism and the Schmidt decomposition. Next the Green functions for the low-conversion regime are derived perturbatively in the frequency domain, using the methods developed for three-wave mixing, then transformed to the time domain. These results are also derived and verified using an alternative time-domain method, the results of which are more general. For the first time we include the effects of convecting pumps, a more realistic assumption, and show that separability and arbitrary reshaping is possible. This is confirmed numerically for Gaussian pumps as well as higher-order Hermite-Gaussian pumps.

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References

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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]
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[Crossref]
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]
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[Crossref]
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[Crossref] [PubMed]
A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241–1249 (2004).
[Crossref]
A. V. Rodionov and A. S. Chirkin, “Entangled photon states in consecutive nonlinear optical interactions,” JETP Lett. 79, 253–256 and 582 (2004).
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[Crossref]
R. C. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 30, 2635–2637 (2005).
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K. N. Cassemiro, A. S. Villar, P. Valente, M. Martinelli, and P. Nussenzveig, “Experimental observation of three-color optical quantum correlations,” Opt. Lett. 32, 695–697 (2007).
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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. A 73, 063816 (2006).
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D. Kielpinski, J. Corney, and H. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett. 106, 130501 (2011).
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A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
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[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
Y. Ding and Z. Y. Ou, “Frequency downconversion for a quantum network,” Opt. Lett. 35, 2591–2593 (2010).
[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]
C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[Crossref] [PubMed]
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]
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]
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. Quantum Electron. 8, 560–568 (2002).
[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]
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. Express 14, 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. Express 14, 8995–8999 (2006).
[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).
[PubMed]
C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004).
[Crossref] [PubMed]
C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[Crossref] [PubMed]
M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[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. Express 19, 17876–17907 (2011).
[Crossref] [PubMed]
K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992).
[Crossref]
M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003).
[Crossref]
C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (2006).
[Crossref] [PubMed]
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. Express 13, 2236–2244 (2005).
[Crossref] [PubMed]
R. Loudon, The Quantum Theory of Light, 3rd. ed. (Oxford University Press, 2000).
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]
C. J McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282, 583–593 (2009).
[Crossref]
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[PubMed]
W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001).
[Crossref]
A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).
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C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express 20, 6881–6886 (2012).
[Crossref]
F. G. Mehler, “Über die Entwicklung einer Funktion von beliebig vielen Variablen nach Laplaceshen Functionen höherer Ordnung,” Journal für die reine und angewandte Mathematik, 161–176 (1866).
[Crossref] [PubMed]
P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 781 and 786.
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[Crossref]

2012 (2)

C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron. 18, 958–969 (2012).
[Crossref]

C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express 20, 6881–6886 (2012).
[Crossref]

2011 (7)

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 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. Express 19, 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]

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]

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

2010 (3)

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).
[PubMed]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[Crossref]

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

2009 (1)

C. J McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282, 583–593 (2009).
[Crossref]

2008 (1)

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

2007 (1)

K. N. Cassemiro, A. S. Villar, P. Valente, M. Martinelli, and P. Nussenzveig, “Experimental observation of three-color optical quantum correlations,” Opt. Lett. 32, 695–697 (2007).
[Crossref] [PubMed]

2006 (5)

C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (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. Express 14, 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. Express 14, 8995–8999 (2006).
[Crossref] [PubMed]

A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig, “Direct production of tripartite pump-signal-idler entanglement in the above-threshold optical parametric oscillator,” Phys. Rev. Lett. 97, 140504 (2006).
[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. A 73, 063816 (2006).
[Crossref]

2005 (7)

I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005).
[Crossref] [PubMed]

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]

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

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. Express 13, 2236–2244 (2005).
[Crossref] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[Crossref] [PubMed]

R. C. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 30, 2635–2637 (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. Express 13, 9131–9142 (2005).
[Crossref] [PubMed]

2004 (7)

C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004).
[Crossref] [PubMed]

A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241–1249 (2004).
[Crossref]

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]

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

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]

O. Pfister, S. Feng, G. Jennings, R. C. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302 (2004).
[Crossref]

2003 (1)

M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003).
[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. Quantum Electron. 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. A 64, 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)

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]

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 (2)

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

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992).
[Crossref]

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]

1907 (1)

E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen,” Mathematische Annalen 63, 433–476 (1907).
[Crossref]

1866 (1)

F. G. Mehler, “Über die Entwicklung einer Funktion von beliebig vielen Variablen nach Laplaceshen Functionen höherer Ordnung,” Journal für die reine und angewandte Mathematik, 161–176 (1866).
[Crossref] [PubMed]

1806 (1)

M. P. des Chênes, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaire du second ordre, à coefficients constants,” Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savans, et lus dans ses assembleés. Sciences, mathématiques et physiques, 638–648 (1806).
[PubMed]

Afzelius, M.

Albota, M. A.

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]

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]

Allevi, A.

Andreoni, A.

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]

Banaszek, K.

Bondani, M.

Brecht, B.

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
[Crossref] [PubMed]

Bussières, F.

Cargill, D. S.

C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express 20, 6881–6886 (2012).
[Crossref]

Cassemiro, K. N.

Centanni, J. C.

Chen, J.

Chirkin, A. S.

A. V. Rodionov and A. S. Chirkin, “Entangled photon states in consecutive nonlinear optical interactions,” JETP Lett. 79, 253–256 and 582 (2004).

Christ, A.

Clausen, C.

Corney, J.

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

de Riedmatten, H.

des Chênes, M. P.

Ding, Y.

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

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.

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
[Crossref] [PubMed]

Erdmann, R.

Fabre, C.

Fejer, M. M.

Feng, S.

O. Pfister, S. Feng, G. Jennings, R. C. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302 (2004).
[Crossref]

Ferraro, A.

Feschbach, H.

P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 781 and 786.

George, M.

Gisin, N.

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.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gordon, J. P.

C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron. 18, 958–969 (2012).
[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]

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. A 64, 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.

D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express 14, 8995–8999 (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. Express 13, 9131–9142 (2005).
[Crossref] [PubMed]

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]

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1990).

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]

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992).
[Crossref]

Jennings, G.

O. Pfister, S. Feng, G. Jennings, R. C. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302 (2004).
[Crossref]

Jin, J.

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1990).

Kielpinski, D.

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

Kikuchi, K.

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]

Kogelnik, H.

C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (2006).
[Crossref] [PubMed]

C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004).
[Crossref] [PubMed]

Kumar, P.

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

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.

Loudon, R.

R. Loudon, The Quantum Theory of Light, 3rd. ed. (Oxford University Press, 2000).

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.

M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003).
[Crossref]

Martinelli, M.

McGuinness, H. J.

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).
[PubMed]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[Crossref]

McKinstrie, C. J

C. J McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282, 583–593 (2009).
[Crossref]

McKinstrie, C. J.

C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron. 18, 958–969 (2012).
[Crossref]

C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express 20, 6881–6886 (2012).
[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).
[PubMed]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[Crossref]

C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (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. Express 14, 8995–8999 (2006).
[Crossref] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 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. Express 13, 9131–9142 (2005).
[Crossref] [PubMed]

C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004).
[Crossref] [PubMed]

Méchin, D.

D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express 14, 8995–8999 (2006).
[Crossref] [PubMed]

Mehler, F. G.

Milburn, G. J.

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

Moiseiwitsch, B. L.

B. L. Moiseiwitsch, Integral Equations (Dover, 2005).

Morse, P. M.

P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 781 and 786.

Nussenzveig, P.

Oblak, D.

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]

Paris, M. G. A.

Park, Y.

Pfister, O.

R. C. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 30, 2635–2637 (2005).
[Crossref] [PubMed]

O. Pfister, S. Feng, G. Jennings, R. C. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302 (2004).
[Crossref]

Pooser, R. C.

R. C. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 30, 2635–2637 (2005).
[Crossref] [PubMed]

O. Pfister, S. Feng, G. Jennings, R. C. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302 (2004).
[Crossref]

Provo, R.

D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express 14, 8995–8999 (2006).
[Crossref] [PubMed]

Puddu, E.

Radic, S.

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).
[PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 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. Express 13, 9131–9142 (2005).
[Crossref] [PubMed]

C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004).
[Crossref] [PubMed]

Raymer, M. G.

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).
[PubMed]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[Crossref]

I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (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. Express 13, 9131–9142 (2005).
[Crossref] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[Crossref] [PubMed]

Ricken, R.

Rodionov, A. V.

A. V. Rodionov and A. S. Chirkin, “Entangled photon states in consecutive nonlinear optical interactions,” JETP Lett. 79, 253–256 and 582 (2004).

Roussev, R. V.

Saglamyurek, E.

Sangouard, N.

Schenato, L.

C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (2006).
[Crossref] [PubMed]

Schmidt, E.

E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen,” Mathematische Annalen 63, 433–476 (1907).
[Crossref]

Set, S. Y.

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. Express 19, 13770–13778 (2011).
[Crossref] [PubMed]

Simmons, G. F.

G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd. ed. (McGraw-Hill, 1991).

Sinclair, N.

Slater, J.

Sohler, W.

Suche, H.

Tanemura, T.

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.

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]

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. A 64, 063815 (2001).
[Crossref]

Uesaka, K.

Usmani, I.

Valente, P.

van Enk, S. J.

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[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).

Villar, A. S.

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,” Science 307, 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. A 64, 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.

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974), Chap. 2.

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.

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]

Wong, K. K.

Wong, K. K. Y.

M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003).
[Crossref]

Wootters, W. K.

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

Xie, D.

O. Pfister, S. Feng, G. Jennings, R. C. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302 (2004).
[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.

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[Crossref] [PubMed]

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. Quantum Electron. (1)

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron. 18, 958–969 (2012).
[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)

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).

J. Opt. Soc. Am. B (2)

M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003).
[Crossref]

JETP Lett. (1)

A. V. Rodionov and A. S. Chirkin, “Entangled photon states in consecutive nonlinear optical interactions,” JETP Lett. 79, 253–256 and 582 (2004).

Journal für die reine und angewandte Mathematik (1)

Laser Phys. (1)

Mathematische Annalen (1)

E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen,” Mathematische Annalen 63, 433–476 (1907).
[Crossref]

Nature (London) (6)

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]

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

New J. Phys. (1)

Opt. Commun. (2)

C. J McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282, 583–593 (2009).
[Crossref]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[Crossref]

Opt. Express (10)

C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004).
[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. Express 13, 9131–9142 (2005).
[Crossref] [PubMed]

C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (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. Express 14, 8995–8999 (2006).
[Crossref] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[Crossref] [PubMed]

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
[Crossref] [PubMed]

C. J. McKinstrie and D. S. Cargill, “Simultaneous frequency conversion, regeneration and reshaping of optical signals,” Opt. Express 20, 6881–6886 (2012).
[Crossref]

Opt. Lett. (6)

R. C. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 30, 2635–2637 (2005).
[Crossref] [PubMed]

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

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]

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 (3)

O. Pfister, S. Feng, G. Jennings, R. C. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302 (2004).
[Crossref]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001).
[Crossref]

Phys. Rev. Lett. (6)

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]

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).
[PubMed]

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

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

Science (1)

I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005).
[Crossref] [PubMed]

Sciences, mathématiques et physiques (1)

Other (7)

P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 781 and 786.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974), Chap. 2.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd. ed. (McGraw-Hill, 1991).

R. Loudon, The Quantum Theory of Light, 3rd. ed. (Oxford University Press, 2000).

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1990).

B. L. Moiseiwitsch, Integral Equations (Dover, 2005).

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Fig. 1 (a) The placement in the frequency spectrum of the two pumps p and q along with the sidebands s and i for frequency conversion in the near-conversion regime. ω₀ is the zero-dispersion frequency. In this case the two pumps are closely spaced in frequency. (b) Illustration of frequency conversion in the far-conversion regime. Here the pumps are farther from each other. Arrows pointing up denote creation of photons and arrows pointing down destruction of photons.

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Fig. 2 A characteristic diagram for the generation of an idler from pulsed pumps for β_r = −β_s. The solid diagonal lines show the influence from the signal at the input point (t′, 0) whereas the dashed lines show the domain that influences the output idler at (t, l) from the time-dependent pumps.

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Fig. 3 Characteristic diagrams for (a) idler generation from a pulsed signal and (b) generation of a signal from a pulsed idler. 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 (signal) at time t is generated by a collision with the signal (idler) occurring at the point c.

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Fig. 4 Numerical studies of the Green function G_rs. In all the plots β = 1, γ̄ = 0.1. The two pumps are normalized Gaussians with a root-mean-square width τ = 1. The white lines denote the cut-off due to the step-functions, but they only apply to the Green functions without the Gaussian approximation. For the approximate Green functions the height h = 1.28 was used. (a) The absolute value of the step-function Green function for βl/τ = 1, i.e. the non-separable case. (b) The Green function with the Gaussian approximation for βl/τ = 1. (c) The absolute value of G_rs for βl/τ = 2.2768, the case where it is expected to be separable. The shape of the function implies that no separability is attainable. (d) The Gaussian approximation for the separable fiber-length, and it is indeed seen to be separable.

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Fig. 5 (a) The Schmidt coefficients for the four plots in Fig. 4. Diamonds and crosses are for the separable and non-separable fiber length respectively with the Gaussian window whereas squares and open circles are the Schmidt coefficients with the rectangle window for the separable and non-separable length respectively. Notice that numerical study for the separable fiber-length, shows that the Gaussian approximation leads to separability, whereas the one with step-functions does not. (b) The first two Schmidt modes. The dashed curves are the Schmidt modes for the Gaussian Green function and the solid ones for the step-function window. The black curves are the zeroth-order modes and the blue curves the first-order modes. The normalized length of the fiber is one in this numerical study.

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Fig. 6 (a) A plot of the Schmidt coefficients for a long fiber βl/τ = 10. The crosses and circles are the Schmidt coefficients for the Gaussian window for h = 1 and h = 1.28 respectively. The diamonds are for the rectangular window. (b) The first two Schmidt modes. The dashed curves are the Schmidt modes for the Gaussian window while the step-function Schmidt modes are solid. Again the black curves are the zeroth-order modes and the blue curves the first-order modes. Notice that since the Schmidt modes are normalized they are identical for both heights in the Gaussian approximation.

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Fig. 7 Numerical studies of the Green function G_rs when including the convecting pumps. In all the plots β_r = 1 = −β_s, γ̄ = 0.1. The numerical study uses Gaussian pumps with a root-mean square width τ = 1. In (a) and (b) white lines denote the cut-off due to the step-functions. (a) The absolute value of the Heaviside Green function for βl/τ = 1. (b) The Green function for βl/τ = 3. (c) A plot of the Schmidt coefficients for the two fiber lengths where the crosses are for the shorter length and the circles for βl/τ. For βl/τ = 3 the function is almost separable which is seen since we only have one dominating Schmidt mode. (d) The two first output Schmidt modes for the two lengths. The solid curves are for βl/τ = 1 and the dashed ones for βl/τ = 3. The lowest-order Schmidt mode is plotted in black and the next one in blue. For βl/τ = 3 the first output mode corresponds to the shape of pump q centered on β_rl/2 like expected. The second output mode has another shape, but its Schmidt coefficient is almost zero, so this mode is negligible.

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Fig. 8 The square of the first two Schmidt coefficients as a function of the interaction distance z_i. The triangles and circles correspond to λ₀ and λ₁ respectively for βl/τ = 1. The dashed line is the square of the lowest-order Schmidt coefficient squared and the solid line the square of the next Schmidt coefficient for βl/τ = 3.

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Fig. 9 Plots of the square of the lowest-order Schmidt coefficient as a function of the length of the fiber. (a) For w = ξ = 0.3858, the value giving the same FWHM in the frequency-domain. The solid line is the square of the analytic lowest-order Schmidt coefficient, Eq. (79) with h = [2/(wπ)]^1/2 ≈ 1.28. The crosses are the numerically found values for the Gaussian approximated Green function, which agrees with the analytic value. The open circles are also found numerically, but are for the convecting Green function with step-function window. Finally the dots is Eq. (79) with h = 1. (b) The same curves as the left panel but for h = 1 and w = 0.7213 which is the width giving the same FWHM in the time-domain.

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Fig. 10 Plots of the Green function G_rs with different pump widths in this case τ_q = τ_p/2. In (a) βl/τ_p = 1 whereas (b) considers βl/τ_p = 3. The function is elongated in the t′ direction since pump p has the largest width. (c) shows the Schmidt coefficients where open circles are for the longer fiber-length. The long fiber Green function is still separable even though the pumps have different widths. In (d) the two lowest-order Schmidt modes for the two lengths is plotted. The black curves are the lowest-order Schmidt mode and the blue ones the next one. Dashed curves are for the longer fiber.

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Fig. 11 Plot of the lowest-order Schmidt coefficient as a function of the aspect ratio. The crosses are for βl/τ_p = 2 and the open circles for the variable length 2τ_q/τ_p. (b) The Schmidt coefficients for a pulse with a very narrow aspect ratio (τ_q = τ_p/100) for two different fiber lengths βl/τ_p = 0.5 (crosses) and 3 (open circles).

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Fig. 12 Plots of the limits of the square of the lowest-order Schmidt coefficient as a function of the aspect ratio. (a) This figure shows convecting pumps where βl/τ_p = 2, w = 0.3858 and the open circles are for the rectangular windows and filled circles and crosses are for the Gaussian window with h = 1 and h = 1.28 respectively. (b) The same functions as the left panel but for the the width yielding the same FWHM in the time-domain, w = 0.7213.

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Fig. 13 Numerical study of the Green function G_rs with pump p as HG0 and pump q as HG1. In all the cases γ̄ = 0.1. (a) A plot of the Green function for βl/τ = 1. (b) The Green function for the long fiber interaction, βl/τ = 4. (c) The first 10 Schmidt coefficients for the two fiber lengths, where the open circles are for the longer case. The longer one is clearly separable. (d) The first two output Schmidt modes for each of the numerical studies, again black is for the lowest-order mode and blue for the next one, while dashed curves are the ones for the longer fiber. The long fiber first-order Schmidt mode is a HG1 centered on βl/τ = 2 like expected, whereas the one for the shorter fiber is slightly distorted.

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Fig. 14 Plots of the Green function G_rs where both pumps are HG1 functions and γ̄ = 0.1. (a) A contour plot of the Green function for the fiber length βl/τ = 1. (b) The contour plot, for βl/τ = 5. (c) The computed first 10 Schmidt coefficients for the two fiber lengths, open circles are again for the longer fiber which is clearly separable. (d) A plot of the lowest-order output Schmidt mode as well as the theoretical HG1 modes. The black curves are for the shorter fiber and the dashed curves are the numerically found Schmidt modes. The input modes have not been plotted since they coincided (as expected) with the output modes.

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Equations (117)

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(\partial_{z} + β_{r} \partial_{t}) A_{r} = i γ_{pq} A_{s},

(\partial_{z} + β_{s} \partial_{t}) A_{s} = i γ_{pq}^{*} A_{r},

\begin{matrix} [{\hat{a}}_{i} (t), {\hat{a}}_{j} (t^{'})] = 0 & and & [{\hat{a}}_{i} (t), {\hat{a}}_{j}^{†} (t^{'})] = δ_{i j} δ (t - t^{'}), \end{matrix}

A_{j} (t, l) = \sum_{k} \int_{- \infty}^{\infty} G_{j k} (t, l; t^{'}, 0) A_{k} (t^{'}, 0) d t^{'} .

A_{r} (t) = \int_{- \infty}^{\infty} G_{rr} (t; t^{'}) A_{r} (t^{'}) d t^{'} + \int_{- \infty}^{\infty} G_{rs} (t; t^{'}) A_{s} (t^{'}) d t^{'},

G (t; t^{'}) = \sum_{n} v_{n} (t) λ_{n}^{1 / 2} u_{n} (t^{'}),

\int_{- \infty}^{\infty} K_{1} (t, t^{'}) v_{n} (t^{'}) d t^{'} = λ_{n} v_{n} (t),

\int_{- \infty}^{\infty} K_{2} (t, t^{'}) u_{n} (t^{'}) d t^{'} = λ_{n} u_{n} (t),

[\begin{array}{l} G_{rr} & G_{rs} \\ G_{sr} & G_{ss} \end{array}],

[\begin{array}{l} G_{rr} & G_{rs} \\ G_{sr} & G_{ss} \end{array}] = \sum_{n} [\begin{array}{l} v_{r, n} (t) τ_{n} u_{r, n}^{*} (t^{'}) & v_{r, n} (t) ρ_{n} u_{s, n}^{*} (t^{'}) \\ - v_{s, n} (t) ρ_{n}^{*} u_{r, n}^{*} (t^{'}) & v_{s, n} (t) τ_{n}^{*} u_{s, n}^{*} (t^{'}) \end{array}] .

ℱ {A (t)} = {(2 π)}^{- 1 / 2} \int_{- \infty}^{\infty} A (t) exp (i ω t) d t,

ℱ^{- 1} {A (ω)} = {(2 π)}^{- 1 / 2} \int_{- \infty}^{\infty} A (ω) exp (- i ω t) d ω .

A_{r} (ω_{r}) = \int_{- \infty}^{\infty} G_{rs} (ω_{r}; t^{'}) A_{s} (t^{'}) d t^{'} .

A_{r} (ω_{r}) = {(2 π)}^{- 1} \iint_{- \infty}^{\infty} d t^{'} d ω_{s} \int_{- \infty}^{\infty} d ω G_{rs} (ω_{r}; ω_{s} - ω) A_{s} (ω) exp (- i ω_{s} t^{'})

A_{r} (ω_{r}) = \int_{- \infty}^{\infty} G_{rs} (ω_{r}; - ω_{s}) A_{s} (ω_{s}) d ω_{s} .

A_{s} (ω_{s}) = \int_{- \infty}^{\infty} G_{sr} (ω_{s}; - ω_{r}) A_{r} (ω_{r}) d ω_{r} .

G_{rs} (ω_{r}; - ω_{s}) = {(2 π)}^{- 1} \iint_{- \infty}^{\infty} G_{rs} (t; t^{'}) exp (i ω_{r} t - i ω_{s} t^{'}) d t d t^{'},

G_{rs} (t; t^{'}) = {(2 π)}^{- 1} \iint_{- \infty}^{\infty} G_{rs} (ω_{r}; - ω_{s}) exp (- i ω_{r} t + i ω_{s} t^{'}) d ω_{r} d ω_{s} .

G_{rs} (t; t^{'}) = \sum_{n = 0}^{\infty} λ_{n}^{1 / 2} v_{n} (t) u_{n}^{*} (t^{'}),

\begin{array}{l} G_{rs} (ω_{r}; - ω_{s}) = \sum_{n = 0}^{\infty} λ_{n}^{1 / 2} \int_{- \infty}^{\infty} v_{n} (t) exp (i ω_{r} t) {(2 π)}^{- 1 / 2} d t \\ \times \int_{- \infty}^{\infty} {[u_{n} (t^{'}) exp (i ω_{s} t_{s})]}^{*} {(2 π)}^{- 1 / 2} d t^{'}, \end{array}

G_{rs} (ω_{r}; - ω_{s}) = \sum_{n = 0}^{\infty} λ_{n}^{1 / 2} v_{n} (ω_{r}) u_{n}^{*} (ω_{s}) .

\partial_{z} A_{r} (ω_{r}, z) = i β_{r} (ω_{r}) A_{r} (ω_{r}, z) + i {(2 π)}^{- 1 / 2} \int_{\infty}^{\infty} γ_{pq} (ω_{r} - ω_{s}, z) A_{s} (ω_{s}) d ω_{s},

\partial_{z} A_{s} (ω_{s}, z) = i β_{s} (ω_{s}) A_{s} (ω_{s}, z) + i {(2 π)}^{- 1 / 2} \int_{\infty}^{\infty} γ_{pq}^{*} (ω_{s} - ω_{r}, z) A_{r} (ω_{r}) d ω_{r} .

\begin{matrix} B_{r} (ω_{r}, l) \approx i {(2 π)}^{- 1 / 2} \int_{0}^{l} \int_{- \infty}^{\infty} γ_{pq} (ω_{r} - ω_{s}, z) exp (i z ω_{s} β_{s}) \\ \times exp (- i z ω_{r} β_{r}) B_{s} (ω_{s}, 0) d ω_{s} d z, \end{matrix}

G_{rs} (ω_{r}; - ω_{s}) = i {(2 π)}^{- 1 / 2} \int_{0}^{l} γ_{pq} (ω_{r} - ω_{s}, z) exp [i β_{r} (l - z) ω_{r} + i β_{s} z ω_{s}] d z,

G_{sr} (ω_{s}; - ω_{r}) = i {(2 π)}^{- 1 / 2} \int_{0}^{l} γ_{pq}^{*} (ω_{s} - ω_{r}, z) exp [i β_{s} (l - z) ω_{s} + i β_{r} z ω_{r}] d z .

A_{p} (t) = A_{q} (t) = {(π τ_{0}^{2})}^{- 1 / 4} exp [- t^{2} / (2 τ_{0}^{2})],

γ_{pq} (ω) = γ {(2 π)}^{- 1 / 2} exp (- σ^{2} ω^{2} / 2),

G_{rs} (ω_{r}; - ω_{s}) = i γ_{0} l exp [i β_{r} ω_{r} l - i l δ_{0} / 2 - σ^{2} {(ω_{r} - ω_{s})}^{2} / 2] sinc [l δ_{0} / 2],

G_{rs} (ω_{r}; - ω_{s}) = i γ_{0} l exp [i l (β_{r} ω_{r} + β_{s} ω_{s}) / 2 - σ^{2} {(ω_{r} - ω_{s})}^{2} / 2 - {(α_{r} ω_{r} - α_{s} ω_{s})}^{2} / 2],

G_{rs} (ω_{r}; - ω_{s}) = i \sum_{n = 0}^{\infty} exp (i l β_{r} ω_{r} / 2 + i l β_{s} ω_{s} / 2) λ_{n}^{1 / 2} ϕ_{n} (τ_{r} ω_{r}) ϕ_{n}^{*} (τ_{s} ω_{s}),

ϕ_{n} (a x) = a^{1 / 2} H_{n} (a x) exp [- {(a x)}^{2} / 2] / [π^{1 / 4} {(n! 2^{n})}^{1 / 2}],

λ_{n} = \frac{π {(γ_{0} l)}^{2}}{τ_{r} τ_{s}} (1 - μ^{2}) μ^{2 n} .

μ = \frac{{[(σ^{2} / α_{r}^{2}) (σ^{2} + α_{s}^{2})]}^{1 / 2} - σ α_{rs}}{σ^{2} + α_{r} α_{s}},

τ_{r} = {(σ | α_{r} - α_{s} |)}^{1 / 2} {(\frac{σ^{2} + α_{r}^{2}}{σ^{2} + α_{s}^{2}})}^{1 / 4},

τ_{s} = {(σ | α_{r} - α_{s} |)}^{1 / 2} {(\frac{σ^{2} + α_{s}^{2}}{σ^{2} + α_{r}^{2}})}^{1 / 4} .

\frac{π (1 - μ^{2})}{τ_{r} τ_{s}} = \frac{2 π}{{[(σ^{2} + α_{r}^{2}) (σ^{2} + α_{s}^{2})]}^{1 / 2} + σ α_{rs}},

λ_{0} = \frac{π γ_{0}^{2} l^{2}}{{(α_{r} α_{s})}^{1 / 2} α_{rs}},

τ_{r} = {(σ^{2} + α_{r}^{2})}^{1 / 2},

τ_{s} = {(σ^{2} + α_{s}^{2})}^{1 / 2},

G_{rs} (ω_{r}; - ω_{s}) = i \sum_{n = 0}^{\infty} λ_{n}^{1 / 2} v_{n} (ω_{r}) u_{n}^{*} (ω_{s}),

G_{rs} (t; t^{'}) = \frac{i γ}{π σ ξ^{1 / 2} | β_{rs} |} exp [- \frac{(σ^{2} + α_{s}^{2}) {\bar{t}}^{2} - 2 (σ^{2} + α_{r} α_{s}) \bar{t t^{'}} + (σ^{2} + α_{r}^{2}) {(\bar{t^{'}})}^{2}}{2 {(σ α_{rs})}^{2}}]

G_{rs} (t; t^{'}) = i \sum_{n = 0}^{\infty} λ_{n}^{1 / 2} ϕ_{n} (\frac{t - β_{r} l / 2}{τ_{r}}) ϕ_{n}^{*} (\frac{t^{'} + β_{s} l / 2}{τ_{s}}),

\begin{array}{l} G_{rs} (t; t^{'}) = i {(2 π)}^{- 3 / 2} \int_{0}^{l} \iint_{- \infty}^{\infty} γ_{pq} (ω_{r} - ω_{s}, z) \\ \times exp [- i (t - β_{r} l + β_{r} z) ω_{r} + i (t^{'} + β_{s} z) ω_{s}] d ω_{r} d ω_{s} d z . \end{array}

G_{rs} (t; t^{'}) = i {(2 π)}^{- 1 / 2} \int_{0}^{l} \int_{- \infty}^{\infty} γ_{pq} (t - β_{r} l + β_{r} z, z) exp [i ω_{s} (t^{'} + β_{s} z - t + β_{r} l - β_{r} z)] d ω_{s} d z .

G_{rs} (t; t^{'}) = i / | β_{rs} | γ_{pq} (\frac{β_{r} t^{'} - β_{s} [t - β_{r} l]}{β_{rs}}, \frac{t^{'} - t + β_{r} l}{β_{rs}}) H (t^{'} + β_{r} L - t) H (t - t^{'} - β_{s} L),

\begin{matrix} z_{c} = [t^{'} - (t - β_{r} l)] / β_{rs}, & t_{c} = [β_{r} t^{'} - β_{s} (t - β_{r} l)] / β_{rs}, \end{matrix}

G_{rs} (t; t^{'}) \approx [i γ_{pq} (z_{c}, t_{c}) / | β_{rs} |] H (t^{'} + β_{r} l - t) H (t - t^{'} - β_{s} l) .

\begin{matrix} z_{c} = [(t - β_{s} l) - t^{'}] / β_{rs}, & t_{c} = [β_{r} (t - β_{s} l) - β_{s} t^{'})] / β_{rs}, \end{matrix}

G_{sr} (t, t^{'}) \approx [i γ_{pq}^{*} (z_{c}, t_{c}) / | β_{rs} |] H (t^{'} + β_{r} l - t) H (t - t^{'} - β_{s} l) .

β_{r} t^{'} - β_{s} (t - β_{r} l) = β_{r} (t^{'} + β_{s} l / 2) - β_{s} (t - β_{r} l / 2) = β_{r} \bar{t^{'}} - β_{s} \bar{t},

H (t - t^{'} - β_{s} l) H (t^{'} + β_{r} l - t) = rect [(t - t_{a}) / (β_{rs} l)] = rect [(\bar{t} - \bar{t^{'}}) / (β_{rs} l)],

rect (x) = {\begin{array}{l} 1, & if | x | < 1 / 2; \\ 1 / 2, & if x = \pm 1 / 2; \\ 0, & otherwise. \end{array}

A_{p} (t) A_{q}^{*} (t) = exp [- t^{2} / (2 σ^{2})] / {(2 π σ^{2})}^{1 / 2} .

G (t; t^{'}) = \frac{i γ}{{(2 π)}^{1 / 2} σ | β_{rs} |} exp [- \frac{{(β_{r} \bar{t^{'}} - β_{s} \bar{t})}^{2}}{2 {(β_{rs} σ)}^{2}}] rect (\frac{\bar{t} - \bar{t^{'}}}{β_{rs} l}) .

h exp [- \frac{{(\bar{t} - \bar{t^{'}})}^{2}}{2 w {(β_{rs} l / 2)}^{2}}],

G_{rs} (t; t^{'}) \approx \frac{i γ h}{{(2 π)}^{1 / 2} σ | β_{rs} |} exp [- \frac{(σ^{2} + α_{s}^{2}) \bar{t^{2}} - 2 (σ^{2} + α_{r} α_{s}) \bar{t t^{'}} + {(σ^{2} + α_{r}^{2})}^{2} {(\bar{t^{'}})}^{2}}{2 {(σ α_{rs})}^{2}}],

λ_{0} = \frac{{(γ h l)}^{2} w}{4 [{(α_{r}^{2} + σ^{2})}^{1 / 2} {(α_{s}^{2} + σ^{2})}^{1 / 2} + α_{rs} σ]},

μ_{r} = \frac{1}{{(α_{rs} σ)}^{1 / 2}} {(\frac{α_{s}^{2} + σ^{2}}{α_{r}^{2} + σ^{2}})}^{1 / 4} = \frac{1}{τ_{r}},

μ_{s} = \frac{1}{{(α_{rs} σ)}^{1 / 2}} {(\frac{α_{r}^{2} + σ^{2}}{α_{s}^{2} + σ^{2}})}^{1 / 4} = \frac{1}{τ_{s}},

λ_{0} \to {(γ h l)}^{2} w / (4 σ^{2}) = {(γ h l)}^{2} w / (2 τ^{2}),

μ_{j} \to 1 / {(α_{rs} σ)}^{1 / 2} = 2^{1 / 4} / {(α_{rs} τ)}^{1 / 2} .

μ \to 1 - α_{rs} / σ,

λ_{0} \to {(γ h)}^{2} / | β_{r} β_{s} |,

μ_{r} \to {(α_{s} / α_{r})}^{1 / 2} / {(α_{rs} σ)}^{1 / 2},

μ_{s} \to {(α_{r} / α_{s})}^{1 / 2} / {(α_{rs} σ)}^{1 / 2} .

μ \to 1 - σ α_{rs} / | α_{r} α_{s} |,

A_{p} (z, t) = F_{p} [t - β_{s} (z - z_{i})],

A_{q} (z, t) = F_{q} [t - β_{r} (z - z_{i})],

t_{cr} - β_{s} (z_{cr} - z_{i}) = t^{'} + β_{s} z_{i},

t_{cr} - β_{r} (z_{cr} - z_{i}) = t - β_{r} (l - z_{i}),

\begin{array}{r} G_{rs} (t; t^{'}) = i \bar{γ} A_{q}^{*} [t - β_{r} (l - z_{i})] A_{p} [t^{'} + β_{s} z_{i}] \\ \times H (t^{'} + β_{r} l - t) H (t - t^{'} - β_{s} l), \end{array}

t_{cs} - β_{s} (z_{cs} - z_{i}) = t - β_{s} (l - z_{i}),

t_{cs} - β_{r} (z_{cs} - z_{i}) = t^{'} + β_{r} z_{i},

G_{sr} (t; t^{'}) = i \bar{γ} A_{p}^{*} [t - β_{s} (l - z_{i})] A_{q} [t^{'} + β_{r} z_{i}]

\times H (t^{'} + β_{r} l - t) H (t - t^{'} - β_{s} l) .

\begin{array}{l} G_{rs} (t; t^{'}) & = & \frac{i \bar{γ}}{\sqrt{π} τ} exp [- \frac{{(t - β_{r} l / 2)}^{2} + {(t^{'} + β_{s} l / 2)}^{2}}{2 τ^{2}}] \\ \times H (t^{'} + β_{r} l - t) H (t - t^{'} - β_{s} l) . \end{array}

G_{rs} (t; t^{'}) \approx \frac{i \bar{γ} h}{π^{1 / 2} τ} exp [- \frac{(τ^{2} + α_{rs}^{2}) \bar{t^{2}} - 2 τ^{2} \bar{t t^{'}} + (τ^{2} + α_{rs}^{2}) {(\bar{t^{'}})}^{2}}{2 {(τ α_{rs})}^{2}}],

λ_{0} = \frac{{(γ h l)}^{2} w}{2 [α_{rs}^{2} + τ^{2} + α_{rs} {(α_{rs}^{2} + 2 τ^{2})}^{1 / 2}]},

μ_{j} = \frac{[α_{rs}^{2} {(α_{rs}^{2} + 2 τ^{2})}^{1 / 4}]}{α_{rs} τ},

μ = \frac{α_{rs}^{2} + τ^{2} - α_{rs} {(α_{rs}^{2} + 2 τ^{2})}^{1 / 2}}{τ^{2}},

λ_{0} \to {(γ h l)}^{2} w / (2 τ^{2}),

μ_{j} \to 2^{1 / 4} / {(α_{rs} τ)}^{1 / 2} = 1 / {(α_{rs} σ)}^{1 / 2},

μ \to 1 - α_{rs} 2^{1 / 2} / τ = 1 - α_{rs} / σ .

\begin{matrix} λ_{0} \to {(γ h)}^{2} / β_{rs}^{2}, & μ_{j} \to 1 / τ, & μ \to 0. \end{matrix}

\begin{array}{l} G_{rs} (t; t^{'}) & = & \frac{i \bar{γ}}{{(π τ_{p} τ_{q})}^{1 / 2}} exp [- \frac{{(t - β_{r} l / 2)}^{2}}{2 τ_{q}^{2}} + \frac{{(t^{'} + β_{s} l / 2)}^{2}}{2 τ_{p}^{2}}] \\ \times H (t^{'} + β_{r} l - t) H (t - t^{'} - β_{s} l), \end{array}

\begin{matrix} λ_{0} \to {(γ h / β_{rs})}^{2} (α_{rs} / τ_{q}), & μ_{r} \to 1 / α_{rs}, & μ_{s} \to 1 / τ_{p} . \end{matrix}

exp [- \frac{t^{2} (x^{2} + y^{2})}{(1 - t^{2})} + \frac{2 t x y}{(1 - t^{2})}] = {(1 - t^{2})}^{1 / 2} \sum_{n = 0}^{\infty} \frac{t^{n}}{2^{n} n!} H_{n} (x) H_{n} (y),

H_{n} (x) = e^{x^{2}} {(- d_{x})}^{n} e^{- x^{2}}

= e^{x^{2}} \int_{- \infty}^{\infty} {(2 i k)}^{n} e^{- k^{2} - 2 i k x} d k / π^{1 / 2} .

\begin{array}{l} \sum_{n = 0}^{\infty} \frac{t^{n}}{2^{n} n!} H_{n} (x) H_{n} (y) & = \frac{e^{x^{2} + y^{2}}}{π} \iint_{- \infty}^{\infty} \sum_{n = 0}^{\infty} \frac{{(- 2 t k l)}^{n}}{n!} e^{- k^{2} - 2 i x k} e^{- l^{2} - 2 i y l} d k d l \\ = \frac{e^{x^{2} + y^{2}}}{π} \iint_{- \infty}^{\infty} e^{- k^{2} - 2 (t l + i x) k} e^{- l^{2} - 2 i y l} d k d l \\ = \frac{e^{y^{2}}}{π^{1 / 2}} \int_{- \infty}^{\infty} e^{- (1 - t^{2}) l^{2} - 2 i (y - t x) l} d l \\ = \frac{1}{{(1 - t^{2})}^{1 / 2}} exp [- \frac{t^{2} (x^{2} + y^{2})}{1 - t^{2}} + \frac{2 t x y}{1 - t^{2}}], \end{array}

exp [- \frac{(1 + t^{2}) (x^{2} + y^{2})}{2 (1 - t^{2})} + \frac{2 t x y}{(1 - t^{2})}] = {[π (1 - t^{2})]}^{1 / 2} \sum_{n = 0}^{\infty} t^{n} ψ_{n} (x) ψ_{n} (y),

ψ_{n} (x) = \frac{H_{n} (x) exp (- x^{2} / 2)}{π^{1 / 4} {(2^{n} n!)}^{1 / 2}} .

K (x, y) = \sum_{n = 0}^{\infty} λ_{n}^{1 / 2} v_{n} (x) u_{n}^{*} (y) .

λ u (x) = L_{u} u (x),

λ v (x) = L_{v} v (x),

L_{u} (x, x^{'}) = \int_{- \infty}^{\infty} K^{*} (x^{″}, x) K (x^{″}, x^{'}) d x^{″},

L_{v} (x, x^{'}) = \int_{- \infty}^{\infty} K (x, x^{″}) K^{*} (x^{'}, x^{″}) d x^{″},

\int_{- \infty}^{\infty} u_{m} (x) u_{n}^{*} (x) d x = δ_{m n} = \int_{- \infty}^{\infty} v_{m} (x) v_{n}^{*} (x) d x .

K (x, y) = exp [- \frac{(1 + t^{2}) (x^{2} + y^{2})}{2 (1 - t^{2})} + \frac{2 t x y}{(1 - t^{2})}] .

L (x, y) = {[\frac{π (1 - t^{2})}{(1 + t^{2})}]}^{1 / 2} exp [- \frac{(1 + t^{4}) (x^{2} + y^{2})}{2 (1 - t^{4})} + \frac{2 t^{2} x y}{(1 - t^{4})}] .

L (x, y) = π (1 - t^{2}) \sum_{n = 0}^{\infty} t^{2 n} ψ_{n} (x) ψ_{n} (y) .

λ_{n} = π (1 - t^{2}) t^{2 n} .

K (ω_{r}, ω_{s}) = exp [- (a ω_{r}^{2} + 2 b ω_{r} ω_{s} + c ω_{s}^{2}) / 2],

t = - [{(a c)}^{1 / 2} - {(a c - b^{2})}^{1 / 2}] / b,

τ_{r} = {[a (a c - b^{2}) / c]}^{1 / 4},

τ_{s} = {[c (a c - b^{2}) / a]}^{1 / 4} .

\begin{array}{l} K (ω_{r}, ω_{s}) & = \sum_{n = 0}^{\infty} λ_{n}^{1 / 2} ψ_{n} (τ_{r} ω_{r}) ψ_{n} (τ_{s} ω_{s}) \\ = \sum_{n = 0}^{\infty} {(λ_{n} / τ_{r} τ_{s})}^{1 / 2} τ_{r}^{1 / 2} ψ_{n} (τ_{r} ω_{r}) τ_{s}^{1 / 2} ψ_{n} (τ_{s} ω_{s}), \end{array}

\frac{π (1 - t^{2})}{τ_{r} τ_{s}} = \frac{2 π}{{(a c)}^{1 / 2} + {(a c - b^{2})}^{1 / 2}},

exp (- t^{2} + 2 t x - x^{2} / 2) = \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} H_{n} (x) exp (- x^{2} / 2) .

F (k) = {(2 π)}^{- 1 / 2} \int_{- \infty}^{\infty} F (x) exp (- i k x) d x,

F (x) = {(2 π)}^{1 / 2} \int_{- \infty}^{\infty} F (x) exp (i k x) d x,

\begin{array}{l} \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} \int_{- \infty}^{\infty} H_{n} (x) exp (- x^{2} / 2) exp (- i k x) d x / {(2 π)}^{1 / 2} \\ = exp [- {(- i t)}^{2} + 2 (- i t) k - k^{2} / 2] \\ = \sum_{n = 0}^{\infty} \frac{{(- i t)}^{n}}{n!} H_{n} (k) exp (- k^{2} / 2) . \end{array}

\int_{- \infty}^{\infty} ψ_{n} (x) exp (- i k x) d x / {(2 π)}^{1 / 2} = {(- i)}^{n} ψ_{n} (k),

F (ω_{r}, - ω_{s}) = exp [- (a ω_{r}^{2} + 2 b ω_{r} ω_{s} + c ω_{s}^{2}) / 2] .

F (t, t^{'}) = exp [- \frac{c t_{r}^{2} + 2 b t_{r} t_{s} + a t_{s}^{2}}{2 (a c - b^{2})}] / {(a c - b^{2})}^{1 / 2},

exp (- X^{t} M X / 2) \leftrightarrow exp (- K^{t} M^{- 1} K / 2) / {[det (M)]}^{1 / 2},

Abstract

References

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