L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. 12, 093027 (2010).

[CrossRef]

J. P. Torres, M. Hendrych, and A. Valencia, “Angular dispersion: an enabling tool in nonlinear and quantum optics,” Adv. Opt. Photon. 2, 319 (2010).

[CrossRef]

X. Shi, A. Valencia, M. Hendrych, and J. Torres, “Generation of indistinguishable and pure heralded single photons with tunable bandwidth,” Opt. Lett. 33, 875 (2008).

[CrossRef]
[PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).

[CrossRef]
[PubMed]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. 79, 135 (2007).

[CrossRef]

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).

[CrossRef]

M. Hendrych, M. Mičuda, and J. P. Torres, “Tunable control of the frequency correlations of entangled photons,” Opt. Lett. 32, 2339 (2007).

[CrossRef]

J. H. Eberly, “Schmidt analysis of pure-state entanglement,” Laser Phys. 16, 921 (2006).

[CrossRef]

A. B. U’Ren, R. K. Erdmann, M. de la Cruz-Gutierrez, and I. A. Walmsley, “Generation of Two-Photon States with an Arbitrary Degree of Entanglement Via Nonlinear Crystal Superlattices,” Phys. Rev. Lett. 97, 223602 (2006).

[CrossRef]

B. R. West and A. S. Helmy, “Dispersion tailoring of the quarter-wave Bragg reflection waveguide,” Opt. Express 14, 4073 (2006).

[CrossRef]
[PubMed]

S. Pezzagna, P. Vennéguès, N. Grandjean, A. D. Wieck, and J. Massies, “Submicron periodic poling and chemical patterning of GaN,” Appl. Phys. Lett. 87, 062106 (2005).

[CrossRef]

J. P. Torres, F. Macià, S. Carrasco, and L. Torner, “Engineering the frequency correlations of entangled two-photon states by achromatic phase matching,” Opt. Lett. 30, 314 (2005).

[CrossRef]
[PubMed]

P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, “Quantum gate characterization in an extended Hilbert space,” Phys. Rev. A 72, 032306 (2005).

[CrossRef]

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

[CrossRef]
[PubMed]

S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).

[CrossRef]

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion,” Phys. Rev. A 67, 053810 (2003).

[CrossRef]

A. B. U’Ren, K. Banaszek, and I. A. Walmsley, “Photon engineering for quantum information processing,” Quantum Inf. Comput. 3, 480 (2003).

Notice that here Ep(ωs + ωi) refers to the spectral amplitude of the pump beam at the input face of the waveguide, while the operators as(ωs0+Ωs) and ai(ωi0+Ωi) refer to the quantum state at the output face of the waveguide. Under these conditions (see [8]), the exponential factor contained in the biphoton amplitude is of the form exp(iskL/2). Different definitions of Ep and as and ai lead to slightly different expressions for the exponential factor of the biphoton amplitude. This is the case, for instance, in [6], where Ep refers to the spectral amplitude of the pump beam at the output face of the nonlinear crystal, while the same definition of the quantum operators as and ai is used. Now the exponential factor contained in the biphoton amplitude is of the form exp(−iΔkL/2). Indeed, when Ep and the quantum operators are referred to the center of the nonlinear crystal, there is no exponential factor at all [see S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003)]. Of course, all of these expressions are related and should yield the same result when calculating correlation functions, since the electric field operators should also be correspondingly modified in each case.

[CrossRef]
[PubMed]

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237 (2002).

[CrossRef]

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

[CrossRef]

G. M. Laws, E. C. Larkins, I. Harrison, C. Molloy, and D. Somerford, “Improved refractive index formulas for the AlxGa1−xN and InyGa1−yN alloys,” J. Appl. Phys. 89, 1108 (2001).

[CrossRef]

A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. 63, 415 (1995).

[CrossRef]

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237 (2002).

[CrossRef]

A. B. U’Ren, K. Banaszek, and I. A. Walmsley, “Photon engineering for quantum information processing,” Quantum Inf. Comput. 3, 480 (2003).

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion,” Phys. Rev. A 67, 053810 (2003).

[CrossRef]

J. P. Torres, F. Macià, S. Carrasco, and L. Torner, “Engineering the frequency correlations of entangled two-photon states by achromatic phase matching,” Opt. Lett. 30, 314 (2005).

[CrossRef]
[PubMed]

S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).

[CrossRef]

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).

[CrossRef]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, 2000).

A. B. U’Ren, R. K. Erdmann, M. de la Cruz-Gutierrez, and I. A. Walmsley, “Generation of Two-Photon States with an Arbitrary Degree of Entanglement Via Nonlinear Crystal Superlattices,” Phys. Rev. Lett. 97, 223602 (2006).

[CrossRef]

Notice that here Ep(ωs + ωi) refers to the spectral amplitude of the pump beam at the input face of the waveguide, while the operators as(ωs0+Ωs) and ai(ωi0+Ωi) refer to the quantum state at the output face of the waveguide. Under these conditions (see [8]), the exponential factor contained in the biphoton amplitude is of the form exp(iskL/2). Different definitions of Ep and as and ai lead to slightly different expressions for the exponential factor of the biphoton amplitude. This is the case, for instance, in [6], where Ep refers to the spectral amplitude of the pump beam at the output face of the nonlinear crystal, while the same definition of the quantum operators as and ai is used. Now the exponential factor contained in the biphoton amplitude is of the form exp(−iΔkL/2). Indeed, when Ep and the quantum operators are referred to the center of the nonlinear crystal, there is no exponential factor at all [see S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003)]. Of course, all of these expressions are related and should yield the same result when calculating correlation functions, since the electric field operators should also be correspondingly modified in each case.

[CrossRef]
[PubMed]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. 79, 135 (2007).

[CrossRef]

J. H. Eberly, “Schmidt analysis of pure-state entanglement,” Laser Phys. 16, 921 (2006).

[CrossRef]

A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. 63, 415 (1995).

[CrossRef]

A. B. U’Ren, R. K. Erdmann, M. de la Cruz-Gutierrez, and I. A. Walmsley, “Generation of Two-Photon States with an Arbitrary Degree of Entanglement Via Nonlinear Crystal Superlattices,” Phys. Rev. Lett. 97, 223602 (2006).

[CrossRef]

S. Pezzagna, P. Vennéguès, N. Grandjean, A. D. Wieck, and J. Massies, “Submicron periodic poling and chemical patterning of GaN,” Appl. Phys. Lett. 87, 062106 (2005).

[CrossRef]

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

[CrossRef]

G. M. Laws, E. C. Larkins, I. Harrison, C. Molloy, and D. Somerford, “Improved refractive index formulas for the AlxGa1−xN and InyGa1−yN alloys,” J. Appl. Phys. 89, 1108 (2001).

[CrossRef]

J. P. Torres, M. Hendrych, and A. Valencia, “Angular dispersion: an enabling tool in nonlinear and quantum optics,” Adv. Opt. Photon. 2, 319 (2010).

[CrossRef]

P. Abolghasem, M. Hendrych, X. Shi, J. P. Torres, and A. Helmy, “Bandwidth control of paired photons generated in monolithic Bragg reflection waveguides,” Opt. Lett. 34, 2000 (2009).

[CrossRef]
[PubMed]

X. Shi, A. Valencia, M. Hendrych, and J. Torres, “Generation of indistinguishable and pure heralded single photons with tunable bandwidth,” Opt. Lett. 33, 875 (2008).

[CrossRef]
[PubMed]

M. Hendrych, M. Mičuda, and J. P. Torres, “Tunable control of the frequency correlations of entangled photons,” Opt. Lett. 32, 2339 (2007).

[CrossRef]

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley-IEEE Press, 2002).

A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. 63, 415 (1995).

[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. 79, 135 (2007).

[CrossRef]

G. M. Laws, E. C. Larkins, I. Harrison, C. Molloy, and D. Somerford, “Improved refractive index formulas for the AlxGa1−xN and InyGa1−yN alloys,” J. Appl. Phys. 89, 1108 (2001).

[CrossRef]

G. M. Laws, E. C. Larkins, I. Harrison, C. Molloy, and D. Somerford, “Improved refractive index formulas for the AlxGa1−xN and InyGa1−yN alloys,” J. Appl. Phys. 89, 1108 (2001).

[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).

[CrossRef]
[PubMed]

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237 (2002).

[CrossRef]

S. Pezzagna, P. Vennéguès, N. Grandjean, A. D. Wieck, and J. Massies, “Submicron periodic poling and chemical patterning of GaN,” Appl. Phys. Lett. 87, 062106 (2005).

[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. 79, 135 (2007).

[CrossRef]

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).

[CrossRef]

G. M. Laws, E. C. Larkins, I. Harrison, C. Molloy, and D. Somerford, “Improved refractive index formulas for the AlxGa1−xN and InyGa1−yN alloys,” J. Appl. Phys. 89, 1108 (2001).

[CrossRef]

Notice that here Ep(ωs + ωi) refers to the spectral amplitude of the pump beam at the input face of the waveguide, while the operators as(ωs0+Ωs) and ai(ωi0+Ωi) refer to the quantum state at the output face of the waveguide. Under these conditions (see [8]), the exponential factor contained in the biphoton amplitude is of the form exp(iskL/2). Different definitions of Ep and as and ai lead to slightly different expressions for the exponential factor of the biphoton amplitude. This is the case, for instance, in [6], where Ep refers to the spectral amplitude of the pump beam at the output face of the nonlinear crystal, while the same definition of the quantum operators as and ai is used. Now the exponential factor contained in the biphoton amplitude is of the form exp(−iΔkL/2). Indeed, when Ep and the quantum operators are referred to the center of the nonlinear crystal, there is no exponential factor at all [see S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003)]. Of course, all of these expressions are related and should yield the same result when calculating correlation functions, since the electric field operators should also be correspondingly modified in each case.

[CrossRef]
[PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).

[CrossRef]
[PubMed]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. 79, 135 (2007).

[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. 79, 135 (2007).

[CrossRef]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, 2000).

P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, “Quantum gate characterization in an extended Hilbert space,” Phys. Rev. A 72, 032306 (2005).

[CrossRef]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. 12, 093027 (2010).

[CrossRef]

Notice that here Ep(ωs + ωi) refers to the spectral amplitude of the pump beam at the input face of the waveguide, while the operators as(ωs0+Ωs) and ai(ωi0+Ωi) refer to the quantum state at the output face of the waveguide. Under these conditions (see [8]), the exponential factor contained in the biphoton amplitude is of the form exp(iskL/2). Different definitions of Ep and as and ai lead to slightly different expressions for the exponential factor of the biphoton amplitude. This is the case, for instance, in [6], where Ep refers to the spectral amplitude of the pump beam at the output face of the nonlinear crystal, while the same definition of the quantum operators as and ai is used. Now the exponential factor contained in the biphoton amplitude is of the form exp(−iΔkL/2). Indeed, when Ep and the quantum operators are referred to the center of the nonlinear crystal, there is no exponential factor at all [see S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003)]. Of course, all of these expressions are related and should yield the same result when calculating correlation functions, since the electric field operators should also be correspondingly modified in each case.

[CrossRef]
[PubMed]

S. Pezzagna, P. Vennéguès, N. Grandjean, A. D. Wieck, and J. Massies, “Submicron periodic poling and chemical patterning of GaN,” Appl. Phys. Lett. 87, 062106 (2005).

[CrossRef]

P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, “Quantum gate characterization in an extended Hilbert space,” Phys. Rev. A 72, 032306 (2005).

[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. 79, 135 (2007).

[CrossRef]

P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, “Quantum gate characterization in an extended Hilbert space,” Phys. Rev. A 72, 032306 (2005).

[CrossRef]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. 12, 093027 (2010).

[CrossRef]

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

[CrossRef]
[PubMed]

P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, “Quantum gate characterization in an extended Hilbert space,” Phys. Rev. A 72, 032306 (2005).

[CrossRef]

S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).

[CrossRef]

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion,” Phys. Rev. A 67, 053810 (2003).

[CrossRef]

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237 (2002).

[CrossRef]

S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).

[CrossRef]

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion,” Phys. Rev. A 67, 053810 (2003).

[CrossRef]

P. Abolghasem, M. Hendrych, X. Shi, J. P. Torres, and A. Helmy, “Bandwidth control of paired photons generated in monolithic Bragg reflection waveguides,” Opt. Lett. 34, 2000 (2009).

[CrossRef]
[PubMed]

X. Shi, A. Valencia, M. Hendrych, and J. Torres, “Generation of indistinguishable and pure heralded single photons with tunable bandwidth,” Opt. Lett. 33, 875 (2008).

[CrossRef]
[PubMed]

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).

[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).

[CrossRef]
[PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).

[CrossRef]
[PubMed]

G. M. Laws, E. C. Larkins, I. Harrison, C. Molloy, and D. Somerford, “Improved refractive index formulas for the AlxGa1−xN and InyGa1−yN alloys,” J. Appl. Phys. 89, 1108 (2001).

[CrossRef]

S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).

[CrossRef]

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion,” Phys. Rev. A 67, 053810 (2003).

[CrossRef]

J. P. Torres, F. Macià, S. Carrasco, and L. Torner, “Engineering the frequency correlations of entangled two-photon states by achromatic phase matching,” Opt. Lett. 30, 314 (2005).

[CrossRef]
[PubMed]

S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).

[CrossRef]

J. P. Torres, M. Hendrych, and A. Valencia, “Angular dispersion: an enabling tool in nonlinear and quantum optics,” Adv. Opt. Photon. 2, 319 (2010).

[CrossRef]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. 12, 093027 (2010).

[CrossRef]

P. Abolghasem, M. Hendrych, X. Shi, J. P. Torres, and A. Helmy, “Bandwidth control of paired photons generated in monolithic Bragg reflection waveguides,” Opt. Lett. 34, 2000 (2009).

[CrossRef]
[PubMed]

M. Hendrych, M. Mičuda, and J. P. Torres, “Tunable control of the frequency correlations of entangled photons,” Opt. Lett. 32, 2339 (2007).

[CrossRef]

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).

[CrossRef]

J. P. Torres, F. Macià, S. Carrasco, and L. Torner, “Engineering the frequency correlations of entangled two-photon states by achromatic phase matching,” Opt. Lett. 30, 314 (2005).

[CrossRef]
[PubMed]

S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).

[CrossRef]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. 12, 093027 (2010).

[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).

[CrossRef]
[PubMed]

A. B. U’Ren, R. K. Erdmann, M. de la Cruz-Gutierrez, and I. A. Walmsley, “Generation of Two-Photon States with an Arbitrary Degree of Entanglement Via Nonlinear Crystal Superlattices,” Phys. Rev. Lett. 97, 223602 (2006).

[CrossRef]

A. B. U’Ren, K. Banaszek, and I. A. Walmsley, “Photon engineering for quantum information processing,” Quantum Inf. Comput. 3, 480 (2003).

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

[CrossRef]

J. P. Torres, M. Hendrych, and A. Valencia, “Angular dispersion: an enabling tool in nonlinear and quantum optics,” Adv. Opt. Photon. 2, 319 (2010).

[CrossRef]

X. Shi, A. Valencia, M. Hendrych, and J. Torres, “Generation of indistinguishable and pure heralded single photons with tunable bandwidth,” Opt. Lett. 33, 875 (2008).

[CrossRef]
[PubMed]

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).

[CrossRef]

S. Pezzagna, P. Vennéguès, N. Grandjean, A. D. Wieck, and J. Massies, “Submicron periodic poling and chemical patterning of GaN,” Appl. Phys. Lett. 87, 062106 (2005).

[CrossRef]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. 12, 093027 (2010).

[CrossRef]

Notice that here Ep(ωs + ωi) refers to the spectral amplitude of the pump beam at the input face of the waveguide, while the operators as(ωs0+Ωs) and ai(ωi0+Ωi) refer to the quantum state at the output face of the waveguide. Under these conditions (see [8]), the exponential factor contained in the biphoton amplitude is of the form exp(iskL/2). Different definitions of Ep and as and ai lead to slightly different expressions for the exponential factor of the biphoton amplitude. This is the case, for instance, in [6], where Ep refers to the spectral amplitude of the pump beam at the output face of the nonlinear crystal, while the same definition of the quantum operators as and ai is used. Now the exponential factor contained in the biphoton amplitude is of the form exp(−iΔkL/2). Indeed, when Ep and the quantum operators are referred to the center of the nonlinear crystal, there is no exponential factor at all [see S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003)]. Of course, all of these expressions are related and should yield the same result when calculating correlation functions, since the electric field operators should also be correspondingly modified in each case.

[CrossRef]
[PubMed]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. 12, 093027 (2010).

[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).

[CrossRef]
[PubMed]

A. B. U’Ren, R. K. Erdmann, M. de la Cruz-Gutierrez, and I. A. Walmsley, “Generation of Two-Photon States with an Arbitrary Degree of Entanglement Via Nonlinear Crystal Superlattices,” Phys. Rev. Lett. 97, 223602 (2006).

[CrossRef]

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

[CrossRef]
[PubMed]

A. B. U’Ren, K. Banaszek, and I. A. Walmsley, “Photon engineering for quantum information processing,” Quantum Inf. Comput. 3, 480 (2003).

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

[CrossRef]

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion,” Phys. Rev. A 67, 053810 (2003).

[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).

[CrossRef]
[PubMed]

S. Pezzagna, P. Vennéguès, N. Grandjean, A. D. Wieck, and J. Massies, “Submicron periodic poling and chemical patterning of GaN,” Appl. Phys. Lett. 87, 062106 (2005).

[CrossRef]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. 12, 093027 (2010).

[CrossRef]

A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. 63, 415 (1995).

[CrossRef]

S. Pezzagna, P. Vennéguès, N. Grandjean, A. D. Wieck, and J. Massies, “Submicron periodic poling and chemical patterning of GaN,” Appl. Phys. Lett. 87, 062106 (2005).

[CrossRef]

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237 (2002).

[CrossRef]

G. M. Laws, E. C. Larkins, I. Harrison, C. Molloy, and D. Somerford, “Improved refractive index formulas for the AlxGa1−xN and InyGa1−yN alloys,” J. Appl. Phys. 89, 1108 (2001).

[CrossRef]

J. H. Eberly, “Schmidt analysis of pure-state entanglement,” Laser Phys. 16, 921 (2006).

[CrossRef]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. 12, 093027 (2010).

[CrossRef]

M. Hendrych, M. Mičuda, and J. P. Torres, “Tunable control of the frequency correlations of entangled photons,” Opt. Lett. 32, 2339 (2007).

[CrossRef]

X. Shi, A. Valencia, M. Hendrych, and J. Torres, “Generation of indistinguishable and pure heralded single photons with tunable bandwidth,” Opt. Lett. 33, 875 (2008).

[CrossRef]
[PubMed]

P. Abolghasem, M. Hendrych, X. Shi, J. P. Torres, and A. Helmy, “Bandwidth control of paired photons generated in monolithic Bragg reflection waveguides,” Opt. Lett. 34, 2000 (2009).

[CrossRef]
[PubMed]

J. P. Torres, F. Macià, S. Carrasco, and L. Torner, “Engineering the frequency correlations of entangled two-photon states by achromatic phase matching,” Opt. Lett. 30, 314 (2005).

[CrossRef]
[PubMed]

P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, “Quantum gate characterization in an extended Hilbert space,” Phys. Rev. A 72, 032306 (2005).

[CrossRef]

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

[CrossRef]

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion,” Phys. Rev. A 67, 053810 (2003).

[CrossRef]

S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).

[CrossRef]

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).

[CrossRef]

Notice that here Ep(ωs + ωi) refers to the spectral amplitude of the pump beam at the input face of the waveguide, while the operators as(ωs0+Ωs) and ai(ωi0+Ωi) refer to the quantum state at the output face of the waveguide. Under these conditions (see [8]), the exponential factor contained in the biphoton amplitude is of the form exp(iskL/2). Different definitions of Ep and as and ai lead to slightly different expressions for the exponential factor of the biphoton amplitude. This is the case, for instance, in [6], where Ep refers to the spectral amplitude of the pump beam at the output face of the nonlinear crystal, while the same definition of the quantum operators as and ai is used. Now the exponential factor contained in the biphoton amplitude is of the form exp(−iΔkL/2). Indeed, when Ep and the quantum operators are referred to the center of the nonlinear crystal, there is no exponential factor at all [see S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003)]. Of course, all of these expressions are related and should yield the same result when calculating correlation functions, since the electric field operators should also be correspondingly modified in each case.

[CrossRef]
[PubMed]

A. B. U’Ren, R. K. Erdmann, M. de la Cruz-Gutierrez, and I. A. Walmsley, “Generation of Two-Photon States with an Arbitrary Degree of Entanglement Via Nonlinear Crystal Superlattices,” Phys. Rev. Lett. 97, 223602 (2006).

[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).

[CrossRef]
[PubMed]

A. B. U’Ren, K. Banaszek, and I. A. Walmsley, “Photon engineering for quantum information processing,” Quantum Inf. Comput. 3, 480 (2003).

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. 79, 135 (2007).

[CrossRef]

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

[CrossRef]
[PubMed]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, 2000).

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley-IEEE Press, 2002).