Abstract

We study quantum frequency translation and two-color photon interference enabled by the Bragg scattering four-wave mixing process in optical fiber. Using realistic model parameters, we computationally and analytically determine the Green function and Schmidt modes for cases with various pump-pulse lengths. These cases can be categorized as either “non-discriminatory” or “discriminatory” in regards to their propensity to exhibit high-efficiency translation or high-visibility two-photon interference for many different shapes of input wave packets or for only a few input wave packets, respectively. Also, for a particular case, the Schmidt mode set was found to be nearly equal to a Hermite-Gaussian function set. The methods and results also apply with little modification to frequency conversion by sum-frequency conversion in optical crystals.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency upconversion,” J. Mod. Opt. 51, 1433–1445 (2004).
  2. S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
    [CrossRef] [PubMed]
  3. 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]
  4. 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]
  5. 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]
  6. 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,” Photon. Technol. Lett. 23, 109–111 (2011).
    [CrossRef]
  7. J. M. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. 68, 2153–2156 (1992).
    [CrossRef] [PubMed]
  8. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
    [CrossRef] [PubMed]
  9. M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photon. 4, 786–791 (2010).
    [CrossRef]
  10. 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]
  11. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
    [CrossRef] [PubMed]
  12. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
    [CrossRef]
  13. N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008).
    [CrossRef] [PubMed]
  14. J. L. O’Brien, “Quantum computing over the rainbow,” Physics 1, 23 (2008).
    [CrossRef]
  15. A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2010)
    [CrossRef]
  16. J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol. 8, 478–491 (1990).
    [CrossRef]
  17. M. E. Marhic, “Coherent optical CDMA networks,” J. Lightwave Technol. 11, 854–864 (1993).
    [CrossRef]
  18. 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]
  19. D. Kielpinski, A.F. Corney, and H.M. Wiseman, “Quantum optical waveform conversion” Phys. Rev Lett. 106, 130501 (2011).
    [CrossRef] [PubMed]
  20. H. J. McGuinness, “The creation and frequency translation of single-photon states of light in optical fiber,” Ph.D. thesis, University of Oregon, Eugene, Oregon (2011).
  21. 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]
  22. R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier Science, 2008).
  23. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
    [CrossRef] [PubMed]
  24. A. Ekert and P. L. Knight, “Entangled quantum systems and the schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
    [CrossRef]
  25. P. D. Drummond and C. W. Gardiner, “Generalized P-representations in quantum optics,” J. Phys. A 13, 2353–2368 (1980).
    [CrossRef]
  26. C. W. Gardiner, Quantum Noise (Springer, 1992).
  27. 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]
  28. W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A,  73, 063816 (2006).
    [CrossRef]
  29. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Elsevier Science, 2008).
  30. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge University Press, 1992).
  31. P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
    [CrossRef] [PubMed]
  32. G. K. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express 13, 8662–8670 (2005).
    [CrossRef] [PubMed]
  33. F. G. Mehler, “Über die entwicklung einer funktion von beliebig vielen variablen nach Laplaceschen functionen höherer ordnung,” J. Reine Angew. Math. 66, 161–176 (1866).
    [CrossRef]
  34. P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

2011

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,” 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]

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

2010

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

M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photon. 4, 786–791 (2010).
[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]

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2010)
[CrossRef]

2008

N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008).
[CrossRef] [PubMed]

J. L. O’Brien, “Quantum computing over the rainbow,” Physics 1, 23 (2008).
[CrossRef]

2007

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

2006

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]

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A,  73, 063816 (2006).
[CrossRef]

A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. 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]

2005

2004

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

C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
[CrossRef] [PubMed]

2003

P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

2001

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

1995

A. Ekert and P. L. Knight, “Entangled quantum systems and the schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[CrossRef]

1994

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]

1993

M. E. Marhic, “Coherent optical CDMA networks,” J. Lightwave Technol. 11, 854–864 (1993).
[CrossRef]

1992

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

1990

J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol. 8, 478–491 (1990).
[CrossRef]

1980

P. D. Drummond and C. W. Gardiner, “Generalized P-representations in quantum optics,” J. Phys. A 13, 2353–2368 (1980).
[CrossRef]

1866

F. G. Mehler, “Über die entwicklung einer funktion von beliebig vielen variablen nach Laplaceschen functionen höherer ordnung,” J. Reine Angew. Math. 66, 161–176 (1866).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Elsevier Science, 2008).

Alibart, O.

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

Baldi, P.

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

Banaszek, K.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A,  73, 063816 (2006).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier Science, 2008).

Brecht, B.

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

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2010)
[CrossRef]

Centanni, J. C.

Chen, A. Y. H.

Christ, A.

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

Corney, A.F.

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

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Drummond, P. D.

P. D. Drummond and C. W. Gardiner, “Generalized P-representations in quantum optics,” J. Phys. A 13, 2353–2368 (1980).
[CrossRef]

Eckstein, A.

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

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2010)
[CrossRef]

Ekert, A.

A. Ekert and P. L. Knight, “Entangled quantum systems and the schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[CrossRef]

Feschbach, H.

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

Flammia, S. T.

N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge University Press, 1992).

Gardiner, C. W.

P. D. Drummond and C. W. Gardiner, “Generalized P-representations in quantum optics,” J. Phys. A 13, 2353–2368 (1980).
[CrossRef]

C. W. Gardiner, Quantum Noise (Springer, 1992).

Gisin, N.

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

Gnauck, A. H.

Ha, S. W.

Halder, M.

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

Harvey, J. D.

Heritage, J. P.

J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol. 8, 478–491 (1990).
[CrossRef]

Huang, J. M.

J. M. 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]

Joly, N.

Jopson, R. M.

Kielpinski, D.

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

Knight, P. L.

A. Ekert and P. L. Knight, “Entangled quantum systems and the schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[CrossRef]

Knill, E.

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

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Kruhlak, R. J.

Kumar, P.

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

Kwiat, P. G.

A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency upconversion,” 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 409, 46–52 (2001).
[CrossRef] [PubMed]

Leonhardt, R.

Lvovsky, A. I.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A,  73, 063816 (2006).
[CrossRef]

Ma, L.

M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photon. 4, 786–791 (2010).
[CrossRef]

Marhic, M. E.

M. E. Marhic, “Coherent optical CDMA networks,” J. Lightwave Technol. 11, 854–864 (1993).
[CrossRef]

McGuinness, H. J.

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

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

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, “The creation and frequency translation of single-photon states of light in optical fiber,” Ph.D. thesis, University of Oregon, Eugene, Oregon (2011).

McKinstrie, C. J.

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

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

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]

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. 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]

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, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
[CrossRef] [PubMed]

Méchin, D.

Mehler, F. G.

F. G. Mehler, “Über die entwicklung einer funktion von beliebig vielen variablen nach Laplaceschen functionen höherer ordnung,” J. Reine Angew. Math. 66, 161–176 (1866).
[CrossRef]

Menicucci, N. C.

N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008).
[CrossRef] [PubMed]

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

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

Morse, P. M.

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

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Murdoch, S. G.

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

O’Brien, J. L.

J. L. O’Brien, “Quantum computing over the rainbow,” Physics 1, 23 (2008).
[CrossRef]

Pfister, O.

N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge University Press, 1992).

Provo, R.

Radic, S.

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

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

A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express 14, 8989–8994 (2006).
[CrossRef] [PubMed]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
[CrossRef] [PubMed]

Radzewicz, C.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A,  73, 063816 (2006).
[CrossRef]

Rakher, M. T.

M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photon. 4, 786–791 (2010).
[CrossRef]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Raymer, M. G.

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,” Photon. Technol. Lett. 23, 109–111 (2011).
[CrossRef]

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

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]

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]

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, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
[CrossRef] [PubMed]

Russell, P. S. J.

P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

Salehi, J. A.

J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol. 8, 478–491 (1990).
[CrossRef]

Silberhorn, C.

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

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2010)
[CrossRef]

Slattery, O.

M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photon. 4, 786–791 (2010).
[CrossRef]

Srinivasan, K.

M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photon. 4, 786–791 (2010).
[CrossRef]

Suche, H.

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

Tang, X.

M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photon. 4, 786–791 (2010).
[CrossRef]

Tanzilli, S.

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

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge University Press, 1992).

Tittel, W.

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

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 upconversion,” J. Mod. Opt. 51, 1433–1445 (2004).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge University Press, 1992).

Wasilewski, W.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A,  73, 063816 (2006).
[CrossRef]

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]

Weiner, A. M.

J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol. 8, 478–491 (1990).
[CrossRef]

Wiseman, H.M.

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

Wong, G. K.

Zbinden, H.

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

Am. J. Phys.

A. Ekert and P. L. Knight, “Entangled quantum systems and the schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[CrossRef]

IEEE Photon. Technol. Lett.

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]

J. Lightwave Technol.

J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol. 8, 478–491 (1990).
[CrossRef]

M. E. Marhic, “Coherent optical CDMA networks,” J. Lightwave Technol. 11, 854–864 (1993).
[CrossRef]

J. Mod. Opt.

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

J. Phys. A

P. D. Drummond and C. W. Gardiner, “Generalized P-representations in quantum optics,” J. Phys. A 13, 2353–2368 (1980).
[CrossRef]

J. Reine Angew. Math.

F. G. Mehler, “Über die entwicklung einer funktion von beliebig vielen variablen nach Laplaceschen functionen höherer ordnung,” J. Reine Angew. Math. 66, 161–176 (1866).
[CrossRef]

Nature

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

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

Nature Photon.

M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photon. 4, 786–791 (2010).
[CrossRef]

New J. Phys.

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

Opt. Commun.

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

Photon. Technol. Lett.

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,” Photon. Technol. Lett. 23, 109–111 (2011).
[CrossRef]

Phys. Rev Lett.

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

Phys. Rev. A

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]

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A,  73, 063816 (2006).
[CrossRef]

Phys. Rev. Lett.

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

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

N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008).
[CrossRef] [PubMed]

Physics

J. L. O’Brien, “Quantum computing over the rainbow,” Physics 1, 23 (2008).
[CrossRef]

Rev. Mod. Phys.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Science

P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

Other

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Elsevier Science, 2008).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge University Press, 1992).

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

H. J. McGuinness, “The creation and frequency translation of single-photon states of light in optical fiber,” Ph.D. thesis, University of Oregon, Eugene, Oregon (2011).

C. W. Gardiner, Quantum Noise (Springer, 1992).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier Science, 2008).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (22)

Fig. 1
Fig. 1

Quantum frequency translation of wavepacket states between green and blue spectral regions. The states being translated could be coherent, squeezed, or number states.

Fig. 2
Fig. 2

BS process where the pumps have frequencies ωp and ωq and the signals have frequencies ωg and ωb , where ωp + ωg = ωq + ωb . The arrows symbolize either annihilation (down) or creation (up) of a photon in that mode. The frequency ωA is the average value of all the frequencies and Δω is the frequency separation between the pump fields and also between the signal fields.

Fig. 3
Fig. 3

The D parameter for the photonic crystal fiber used in simulations from 600 nm to 1600 nm. The dashed line is at D = 0 for reference.

Fig. 4
Fig. 4

Phase-matching function sinc(ΔβL/2) for the fiber used in the simulation, where the pump wavelengths were fixed and the “green” and “blue” fields frequencies were varied in an energy-conserving manner. The horizontal axis is the “green” frequency variation from its corresponding central wavelength value of 673 nm. For this plot the fiber length is L = 20m, and the FWHM of the function is approximately 0.3 Trad/s (1012 rad/s).

Fig. 5
Fig. 5

Singular values, V matrix (columns of which are coefficients in the HG basis for individual Schmidt modes), and w matrix of Ggb Green function relating to the case with long pumps (1000 ps) having peak powers of 400 mW and a fiber length of 20 m. (a) Absolute value of the V matrix (input Schmidt modes). (b) Absolute value of the w matrix (output Schmidt modes). (c) Squared singular values ρ 2 (red) and τ 2 (blue) indexed by the corresponding Schmidt mode number.

Fig. 6
Fig. 6

Translation efficiency as a function of length along the fiber relating to the case with long pumps (1000 ps) having peak powers of 400 mW. The red, green, and blue lines relate to the first, second, and third green Schmidt modes, respectively.

Fig. 7
Fig. 7

Amplitude and phase in both frequency and time of the green 673-nm input Schmidt modes relating to the case with long pumps (1000 ps) having peak powers of 400 mW. The red, green, and blue lines relate to the first, second, and third Schmidt modes, respectively. Note: Trad/s indicates 1012 rad/s.

Fig. 8
Fig. 8

Singular values, V matrix, and w matrix of Ggb Green function relating to the case with short pumps (70 ps) having peak powers of 400 mW. (a) Absolute value of the V matrix. (b) Absolute value of the w matrix. (c) Squared singular values ρ 2 (red) and τ 2 (blue) index by the corresponding Schmidt mode number.

Fig. 9
Fig. 9

Translation efficiency as a function of length along the fiber relating to the case with short pumps (70 ps) having peak powers of 400 mW. The red, green, and blue lines relate to the first, second, and third green Schmidt modes, respectively.

Fig. 10
Fig. 10

Amplitude and phase in both frequency and time of the green 673-nm input Schmidt modes relating to the case with short pumps (70 ps) having peak powers of 400 mW.

Fig. 11
Fig. 11

Parameter study relating to frequency translation in the case with long pumps (1000 ps) having peak powers of 400 mW and fiber length 20 m. (a) The maximal translation efficiency achieved in the fiber as a function of signal pulse width. (b) The length at which the maximal translation efficiency is achieved as a function of signal pulse width.

Fig. 12
Fig. 12

Parametric study relating to the case with short pumps (70 ps) having peak powers of 400 mW and fiber length 20 m. (a) The maximal translation efficiency achieved in the fiber as a function of signal pulse width. (b) The length at which the maximal translation efficiency is achieved as a function of signal pulse width.

Fig. 13
Fig. 13

Singular values, V matrix, and w matrix of Ggb Green function relating to the case with long pumps (1000 ps) having peak powers of 200 mW optimized for good HOM interference. (a) Absolute value of the V matrix. (b) Absolute value of the w matrix. (c) Squared singular values ρ 2 (red) and τ 2 (blue) index by the corresponding Schmidt mode number.

Fig. 14
Fig. 14

HOM singular values 2ρnτn and P 11 values relating to the case with long pumps (1000 ps) having peak powers of 200 mW. (a) HOM singular value index by Schmidt mode number. (b) Two-photon coincidence count probability P 11 as a function of length for the first three input Schmidt modes.The red, green, and blue lines relate to the first, second, and third Schmidt modes, respectively.

Fig. 15
Fig. 15

Amplitude and phase in both frequency and time of the first three green 673-nm input Schmidt modes relating to the case with long pumps (1000 ps) having peak powers of 200 mW. The red, green, and blue lines relate to the first, second, and third Schmidt modes, respectively.

Fig. 16
Fig. 16

Singular values, V matrix, and w matrix of Ggb Green function relating to the case with short pumps (70 ps) having peak powers of 200 mW, optimized for good HOM interference. (a) Absolute value of the V matrix. (b) Absolute value of the w matrix. (c) Squared singular values ρ 2 (red) and τ 2 (blue) indexed by the corresponding Schmidt mode number.

Fig. 17
Fig. 17

HOM singular values and P 11 relating to the case with short pumps (70 ps) having peak powers of 200 mW. (a) HOM singular values indexed by Schmidt mode number. (b) Two-photon coincidence count probability P 11 values as a function of length along the fiber for the first three input Schmidt modes. The red, green, and blue lines relate to the first, second, and third Schmidt modes, respectively.

Fig. 18
Fig. 18

Amplitude and phase in both frequency and time of the green 673-nm input Schmidt modes relating to the case with short pumps (70 ps) having peak powers of 200 mW. The red, green, and blue lines relate to the first, second, and third Schmidt modes, respectively.

Fig. 19
Fig. 19

Parametric study relating to the case with long pumps (1000 ps) having peak powers of 200 mW. (a) The minimal P 11 value achieved in the fiber as a function of signal pulse duration. (b) The length at which the minimal P 11 value is achieved as a function of signal pulse duration.

Fig. 20
Fig. 20

Parametric study relating to the case with short pumps (70 ps) having peak powers of 200 mW. (a) The minimal P 11 value achieved in the fiber as a function of signal pulse duration. (b) The length at which the minimal P 11 value is achieved as a function of signal pulse duration.

Fig. 21
Fig. 21

Approximate Schmidt modes for the faster signal plotted as functions of time and frequency, for long pump pulses. The red, green, and blue lines denote the first, second and third input Schmidt modes, respectively.

Fig. 22
Fig. 22

Approximate Schmidt modes for the faster signal plotted as functions of time and frequency, for short pump pulses. The red, green, and blue lines denote the first, second and third input Schmidt modes, respectively.

Equations (56)

Equations on this page are rendered with MathJax. Learn more.

= Δ β ( a g a g a b a b ) / 2 + κ a g a b + κ * a b a g ,
d d z a b ( g ) ( z ) = i [ a b ( g ) ( z ) , ] .
a g ( z ) = μ ( z ) a g ( 0 ) + ν ( z ) a b ( 0 ) ,
a b ( z ) = μ * ( z ) a b ( 0 ) ν * ( z ) a g ( 0 ) ,
μ ( z ) = cos ( k z ) + i Δ β sin ( k z ) / 2 k ,
ν ( z ) = i κ sin ( k z ) / k ,
| 1 , 1 in = ( [ | μ | 2 | ν | 2 ] | 1 , 1 out + 2 μ ν | 2 , 0 out 2 μ * ν * | 0 , 2 out ) .
a ( L , ω ) = d ω G * ( ω , ω ) a ( 0 , ω ) .
a ( 0 , ω ) = d ω G ( ω , ω ) a ( L , ω ) ,
| ψ = d ω A ( 0 , ω ) d ω G ( ω , ω ) a ( L , ω ) | v a c .
d ω G ( ω , ω ) G * ( ω , ω ) = δ ( ω ω ) .
[ G g g ( ω g , ω g ) G g b ( ω g , ω b ) G b g ( ω b , ω g ) G b b ( ω b , ω b ) ]
G g b ( ω , ω ) = n ρ n V n ( ω ) w n * ( ω ) ,
G g g ( ω , ω ) = n τ n V n ( ω ) υ n * ( ω ) ,
G b g ( ω , ω ) = n ρ n W n ( ω ) υ n * ( ω ) ,
G b b ( ω , ω ) = n τ n W n ( ω ) w n * ( ω ) ,
σ n = 2 τ n ρ n = 1 ,
| ψ = C 11 | 1 , 1 + C 20 | 2 , 0 + C 02 | 0 , 2 .
d ω g d ω b A g ( 0 , ω g ) A b ( 0 , ω b ) a g ( 0 , ω g ) a b ( 0 , ω b ) | v a c .
| ψ = [ d ω g a g ( L , ω g ) A g g ( L , ω g ) + d ω S a b ( L , ω b ) A b g ( L , ω b ) ] × [ d ω g a g ( L , ω g ) A g b ( L , ω g ) + d ω b a b ( L , ω b ) A b b ( L , ω b ) ] | v a c .
| ψ 11 = d ω g d ω b [ A g g ( L , ω g ) A b b ( L , ω b ) + A g b ( L , ω g ) A b g ( L , ω b ) ] | 1 ω g , 1 ω b
P 11 = d ω g d ω b [ | A g g ( L , ω g ) | 2 | A b b ( L , ω b ) | 2 + | A g b ( L , ω g ) | 2 | A b g ( L , ω b ) | 2 + 2 Re [ A g g ( L , ω g ) A b b ( L , ω b ) A g b * ( L , ω g ) A b g * ( L , ω b ) | ] ] .
z A ( z , t ) = i β ( i t ) A ( z , t ) + i γ | A ( z , t ) | 2 A ( z , t ) .
E ( z , t ) = j A j ( z , t ) e i ( β j z ω j t ) + c . c . ,
z A p = i β p ( i T ; ω p ) A p + ( i γ | A p | 2 + 2 i γ k p | A k | 2 ) A p + 2 i γ A g * A q A b exp [ i Δ β z ] z A q = i β p ( i T ; ω q ) A q + ( i γ | A q | 2 + 2 i γ k q | A k | 2 ) A q + 2 i γ A b * A p A g exp [ i Δ β z ] z A g = i β p ( i T ; ω g ) A g + ( i γ | A g | 2 + 2 i γ k g | A k | 2 ) A g + 2 i γ A p * A q A b exp [ i Δ β z ] z A b = i β p ( i T ; ω b ) A b + ( i γ | A b | 2 + 2 i γ k b | A k | 2 ) A b + 2 i γ A q * A p A g exp [ i Δ β z ] ,
β p ( i T ; Ω 0 ) = n = 1 β ( n ) ( Ω 0 ) ( i T ) n n ! β ( 1 ) ( ω p ) i T .
A ( T , z + Δ z ) exp ( Δ z 2 D ^ ) exp ( z z + Δ z N ^ d z ) exp ( Δ z 2 D ^ ) A ( T , z ) .
ψ n ( x ) = ( 1 2 n n ! π ) 1 / 2 e x 2 / 2 ( 1 ) n e x 2 d n d x n e x 2 ,
G = A o u t A i n 1 .
G s u b = A o u t , s u b A i n , s u b 1 ,
V n ( ω ) = j c j n ψ j ( ω ) .
V ( ω g ) = t 0 1 / 2 ψ n ( t 0 ω g ) exp ( + i ω g β 1 L / 2 ) ,
W ( ω b ) = t 0 1 / 2 ψ n ( t 0 ω b ) exp ( i ω b β 1 L / 2 ) ,
v ( ω g ) = t 0 1 / 2 ψ n ( t 0 ω g ) exp ( i ω g β 1 L / 2 ) ,
w ( ω b ) = t 0 1 / 2 ψ n ( t 0 ω b ) exp ( + i ω b β 1 L / 2 ) .
P 20 = ψ 20 | | ψ 20 = v a c | ( d ω ¯ g d ω ¯ g a g ( ω ¯ g ) a g ( ω ¯ g ) A g g * ( L , ω ¯ g ) A g b * ( L , ω ¯ g ) ) × ( d ω g d ω g a g ( ω g ) a g ( ω g ) A g g ( L , ω g ) A g b ( L , ω g ) ) | v a c = v a c | d ω ¯ g d ω ¯ g d ω g d ω g A g g * ( L , ω ¯ g ) A g b * ( L , ω ¯ g ) A g g ( L , ω g ) A g b ( L , ω g ) a g ( ω ¯ g ) a g ( ω ¯ g ) a g ( ω g ) a g ( ω g ) | v a c ,
a g ( ω ¯ g ) a g ( ω g ) = δ ( ω ¯ g ω g ) + a g ( ω g ) a g ( ω ¯ g ) .
P 20 = | d ω A g g * ( L , ω ) A g b ( L , ω ) | 2 + d ω d ω | A g g ( L , ω ) | 2 | A g b ( L , ω ) | 2 .
z A g ( z , ω g ) = i k g ( ω g ) A g ( z , ω g ) + i γ ¯ ( z , ω g ω b ) A b ( z , ω b ) d ω b ,
z A b ( z , ω b ) = i k b ( ω b ) A b ( z , ω b ) + i γ ¯ * ( z , ω b ω g ) A g ( z , ω g ) d ω g ,
z B g ( z , ω g ) = i γ ¯ ( z , ω g ω b ) exp [ i k b ( ω b ) z i k g ( ω g ) z ] B b ( z , ω b ) d ω b ,
z B b ( z , ω b ) = i γ ¯ * ( z , ω b ω g ) exp [ i k g ( ω g ) z i k b ( ω b ) z ] B g ( z , ω g ) d ω g .
B g ( L , ω g ) B g ( 0 , ω g ) + i 0 L κ ( z , ω g , ω b ) B b ( 0 , ω b ) d z d ω b ,
B b ( L , ω b ) B b ( 0 , ω b ) + i 0 L κ ( z , ω b , ω g ) B g ( 0 , ω g ) d z d ω g ,
γ ¯ ( ω ) = [ γ p 0 σ ( 2 / π ) 1 / 2 ] exp ( σ 2 ω 2 / 2 ) ,
δ ( ω g , ω b ) = β 1 ( ω g + ω b ) + β 2 ( ω g 2 ω b 2 ) / 2 .
K ( ω g , ω b ) = [ γ p 0 L σ ( 2 / π ) 1 / 2 ] exp [ i β 1 L ( ω g + ω b ) / 2 i β 2 L ( ω g 2 ω b 2 ) / 4 ] × exp [ σ ( ω g ω b ) 2 / 2 ] sinc [ β 1 L ( ω g + ω b ) / 2 + β 2 L ( ω g 2 ω b 2 ) / 4 ] .
K ( ω g , ω b ) [ γ p 0 L σ ( 2 / π ) 1 / 2 ] exp [ i β 1 L ( ω g + ω b ) / 2 ] × exp [ σ 2 ( ω g ω b ) 2 / 2 β 2 ( ω g + ω b ) 2 / 2 ] ,
exp [ ( 1 + μ 2 ) ( x 2 + y 2 ) 2 ( 1 μ 2 ) + 2 μ x y ( 1 μ 2 ) ] = [ π ( 1 μ 2 ) ] 1 / 2 n = 0 | μ | n ψ n ( x ) ψ n ( y ) ,
K ( ω g , ω b ) = n = 0 λ n φ n ( ω g ) φ n ( ω b ) exp [ i β 1 L ( ω g + ω b ) / 2 ] ,
J g b ( ω g , ω b ) = i n = 0 λ n φ n ( ω g ) φ n ( ω b ) exp [ i β 1 L ( ω g ω b ) / 2 ] ,
v ( ω g ) = φ n ( ω g ) exp ( i β 1 L ω g / 2 ) ,
W ( ω b ) = φ n ( ω b ) exp ( i β 1 L ω b / 2 ) ,
J b g ( ω b , ω g ) = i n = 0 λ n φ n ( ω b ) φ n ( ω g ) exp [ i β 1 L ( ω b ω g ) / 2 ] ,
w ( ω b ) = φ n ( ω b ) exp ( + i β 1 L ω b / 2 ) ,
V ( ω g ) = φ n ( ω g ) exp ( + i β 1 L ω g / 2 ) ,

Metrics