Abstract

A novel quantum mechanical formulation of the bi-photon wavefunction and spectra resulting from four-wave mixing is developed for azimuthally symmetric systems. Numerical calculations are performed verifying the use of the angular group velocity and angular group velocity dispersion in such systems, as opposed their commonly used linear counterparts. The dispersion profile and bi-photon spectra of two illustrative examples are given, emphasizing the physical origin of the effects leading to the conditions for angular momentum and energy conservation. A scheme is proposed in which widely spaced narrowband entangled photons may be produced through a four-wave mixing process in a chip-scale ring resonator. The entangled photon pairs are found to conserve energy and momentum in the four-wave mixing interaction, even though both photon modes lie in spectral regions of steep angular group velocity dispersion.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
    [CrossRef]
  2. T. C. Ralph and G. J. Pryde, “Optical quantum computation,” in Prog. Optics,vol. 54, E. Wolf, ed. (Elsevier Science2010), pp. 209–269.
    [CrossRef]
  3. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
    [CrossRef] [PubMed]
  4. A. Haase, N. Piro, J. Eschner, and M. W. Mitchell, “Tunable narrowband entangled photon pair source for resonant single-photon single-atom interaction,” Opt. Lett. 34, 55–57 (2009).
    [CrossRef]
  5. K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. 11, 013049 (2009).
    [CrossRef]
  6. Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
    [CrossRef]
  7. S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, P. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558–16570 (2009).
    [CrossRef] [PubMed]
  8. L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous four-wave mixing in microring resonators,” Opt. Lett. 35, 3006–3008 (2010).
    [CrossRef] [PubMed]
  9. J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a Silicon-on-Insulator micro-resonator,” Opt. Express 19, 1470–1483 (2011).
    [CrossRef] [PubMed]
  10. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
    [CrossRef]
  11. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388–12393 (2006).
    [CrossRef] [PubMed]
  12. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. I. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
    [CrossRef]
  13. X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053601 (2005).
    [CrossRef] [PubMed]
  14. H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-i. Itabashi, “Generation of polarization entangledphoton pairs using silicon wirewaveguide,” Opt. Express 16, 5721–5727 (2008).
    [CrossRef] [PubMed]
  15. K.-i. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-i. Itabashi, “Generation of high-purity entangled photon pairs using silicon wire waveguide,” Opt. Express 16, 20368–20373 (2008).
    [CrossRef] [PubMed]
  16. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
    [CrossRef]
  17. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4, 37–40 (2010).
    [CrossRef]
  18. L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
    [CrossRef]
  19. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
    [CrossRef] [PubMed]
  20. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based optical frequency combs,” Science 332, 555 –559 (2011).
    [CrossRef] [PubMed]
  21. F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
    [CrossRef]
  22. A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100GHz repetition rates,” Opt. Lett. 37, 875–877 (2012).
    [CrossRef] [PubMed]
  23. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
    [CrossRef]
  24. A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16, 4881–4887 (2008).
    [CrossRef] [PubMed]
  25. Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
    [CrossRef]
  26. Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. 104, 103902 (2010).
    [CrossRef] [PubMed]
  27. M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518–3523 (2009).
    [CrossRef]
  28. D. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).
  29. T. Bååk, “Silicon oxynitride; a material for GRIN optics,” Appl. Optics 21, 1069–1072 (1982).
    [CrossRef]
  30. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965).
    [CrossRef]
  31. D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University Press, 2008).
    [CrossRef]
  32. P. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Optic. 42, 1991–2004 (1995).
    [CrossRef]
  33. B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
    [CrossRef]

2012

2011

J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a Silicon-on-Insulator micro-resonator,” Opt. Express 19, 1470–1483 (2011).
[CrossRef] [PubMed]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based optical frequency combs,” Science 332, 555 –559 (2011).
[CrossRef] [PubMed]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

2010

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4, 37–40 (2010).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
[CrossRef]

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
[CrossRef] [PubMed]

L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous four-wave mixing in microring resonators,” Opt. Lett. 35, 3006–3008 (2010).
[CrossRef] [PubMed]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. 104, 103902 (2010).
[CrossRef] [PubMed]

2009

M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518–3523 (2009).
[CrossRef]

S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, P. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558–16570 (2009).
[CrossRef] [PubMed]

A. Haase, N. Piro, J. Eschner, and M. W. Mitchell, “Tunable narrowband entangled photon pair source for resonant single-photon single-atom interaction,” Opt. Lett. 34, 55–57 (2009).
[CrossRef]

K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. 11, 013049 (2009).
[CrossRef]

2008

2007

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. I. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[CrossRef]

2006

2005

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

2003

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[CrossRef] [PubMed]

2002

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[CrossRef]

2000

1999

Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
[CrossRef]

1997

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
[CrossRef]

1995

P. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Optic. 42, 1991–2004 (1995).
[CrossRef]

1991

B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
[CrossRef]

1982

T. Bååk, “Silicon oxynitride; a material for GRIN optics,” Appl. Optics 21, 1069–1072 (1982).
[CrossRef]

1965

Absil, P. P.

Akiba, K.

K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. 11, 013049 (2009).
[CrossRef]

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Arikawa, M.

K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. 11, 013049 (2009).
[CrossRef]

Azaa, J.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
[CrossRef] [PubMed]

Bååk, T.

T. Bååk, “Silicon oxynitride; a material for GRIN optics,” Appl. Optics 21, 1069–1072 (1982).
[CrossRef]

Baets, R. G.

Barnett, S. M.

B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
[CrossRef]

Baumberg, J. J.

B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
[CrossRef]

Benson, O.

M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518–3523 (2009).
[CrossRef]

Boca, A.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[CrossRef] [PubMed]

Bogaerts, W.

Boozer, A. D.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[CrossRef] [PubMed]

Bowen, W. P.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[CrossRef] [PubMed]

Cardenas, J.

Chembo, Y. K.

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. 104, 103902 (2010).
[CrossRef] [PubMed]

Chen, J.

Chen, L.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Cho, P. S.

Chou, C. W.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[CrossRef] [PubMed]

Chu, S.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
[CrossRef]

Chu, S. T.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
[CrossRef] [PubMed]

Cirac, J. I.

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
[CrossRef]

Clemmen, S.

Del’Haye, P.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based optical frequency combs,” Science 332, 555 –559 (2011).
[CrossRef] [PubMed]

Duan, L.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[CrossRef] [PubMed]

Duchesne, D.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
[CrossRef]

Emplit, P.

Eschner, J.

Fan, J.

Ferdous, F.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Ferrera, M.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
[CrossRef]

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
[CrossRef] [PubMed]

Fiorentino, M.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[CrossRef]

Foster, M. A.

Fukuda, H.

Gaeta, A. L.

Gondarenko, A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4, 37–40 (2010).
[CrossRef]

Haase, A.

Harada, K.-i.

Helt, L. G.

Ho, P.

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based optical frequency combs,” Science 332, 555 –559 (2011).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Hryniewicz, J. V.

Huttner, B.

B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
[CrossRef]

Huy, K. P.

Itabashi, S. I.

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. I. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[CrossRef]

Itabashi, S.-i.

Johnson, A. R.

Joneckis, L. G.

Kashiwagi, K.

K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. 11, 013049 (2009).
[CrossRef]

Kimble, H. J.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[CrossRef] [PubMed]

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
[CrossRef]

Kippenberg, T. J.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based optical frequency combs,” Science 332, 555 –559 (2011).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Koch, L.

M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518–3523 (2009).
[CrossRef]

Kozuma, M.

K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. 11, 013049 (2009).
[CrossRef]

Kumar, P.

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388–12393 (2006).
[CrossRef] [PubMed]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[CrossRef]

Kuzmich, A.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[CrossRef] [PubMed]

Leaird, D. E.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Lee, K. F.

Levine, Z. H.

Levy, J. S.

A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100GHz repetition rates,” Opt. Lett. 37, 875–877 (2012).
[CrossRef] [PubMed]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4, 37–40 (2010).
[CrossRef]

Li, X.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

Lipson, M.

Liscidini, M.

Little, B.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
[CrossRef]

Little, B. E.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
[CrossRef] [PubMed]

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
[CrossRef]

Lu, Y. J.

Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
[CrossRef]

Mabuchi, H.

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
[CrossRef]

Malitson, I. H.

Massar, S.

Miao, H.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Migdall, A. L.

Milburn, G. J.

D. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

Miller, D. A. B.

D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University Press, 2008).
[CrossRef]

Milonni, P.

P. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Optic. 42, 1991–2004 (1995).
[CrossRef]

Mitchell, M. W.

Morandotti, R.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
[CrossRef] [PubMed]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
[CrossRef]

Moss, D.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
[CrossRef]

Moss, D. J.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
[CrossRef] [PubMed]

Okawachi, Y.

Ou, Z. Y.

Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
[CrossRef]

Park, Y.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
[CrossRef] [PubMed]

Piro, N.

Pryde, G. J.

T. C. Ralph and G. J. Pryde, “Optical quantum computation,” in Prog. Optics,vol. 54, E. Wolf, ed. (Elsevier Science2010), pp. 209–269.
[CrossRef]

Ralph, T. C.

T. C. Ralph and G. J. Pryde, “Optical quantum computation,” in Prog. Optics,vol. 54, E. Wolf, ed. (Elsevier Science2010), pp. 209–269.
[CrossRef]

Razzari, L.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
[CrossRef] [PubMed]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
[CrossRef]

Saha, K.

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Schmidt, B. S.

Scholz, M.

M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518–3523 (2009).
[CrossRef]

Sharping, J. E.

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388–12393 (2006).
[CrossRef] [PubMed]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[CrossRef]

Sipe, J. E.

Srinivasan, K.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Strekalov, D. V.

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. 104, 103902 (2010).
[CrossRef] [PubMed]

Takesue, H.

Tokura, Y.

Tsuchizawa, T.

Turner, A. C.

Turner-Foster, A. C.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4, 37–40 (2010).
[CrossRef]

Varghese, L. T.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Voss, P. L.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[CrossRef]

Walls, D.

D. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

Wang, J.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Watanabe, T.

Weiner, A. M.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Wilson, R. A.

Yamada, K.

Yang, Z.

Yu, N.

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. 104, 103902 (2010).
[CrossRef] [PubMed]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Zoller, P.

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
[CrossRef]

Appl. Optics

T. Bååk, “Silicon oxynitride; a material for GRIN optics,” Appl. Optics 21, 1069–1072 (1982).
[CrossRef]

Appl. Phys. Lett.

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. I. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[CrossRef]

Europhys. Lett.

B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
[CrossRef]

IEEE Photon. Technol. Lett.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[CrossRef]

J. Mod. Optic.

P. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Optic. 42, 1991–2004 (1995).
[CrossRef]

J. Opt. Soc. Am.

Nat. Commun

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010).
[CrossRef] [PubMed]

Nat. Photonics

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4, 37–40 (2010).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
[CrossRef]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Nature

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[CrossRef] [PubMed]

New J. Phys.

K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. 11, 013049 (2009).
[CrossRef]

Opt. Commun.

M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518–3523 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Phys. Rev. Lett.

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. 104, 103902 (2010).
[CrossRef] [PubMed]

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
[CrossRef]

Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
[CrossRef]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

Science

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based optical frequency combs,” Science 332, 555 –559 (2011).
[CrossRef] [PubMed]

Other

D. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University Press, 2008).
[CrossRef]

T. C. Ralph and G. J. Pryde, “Optical quantum computation,” in Prog. Optics,vol. 54, E. Wolf, ed. (Elsevier Science2010), pp. 209–269.
[CrossRef]

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 (3)

Fig. 1
Fig. 1

Ring resonator geometry and example radial field profiles (Eρ) of the first three (3) radial modes with odd vector symmetry about the z-axis (TE-like modes). As can be seen from each of the field profiles, the circumnavigating field near the outside of the mode travels a much greater linear distance than the field near the inside of the mode, leading to linear group and phase velocities that depend on the coordinate ρ. The angular group and phase velocities are therefore better suited to describing light propagation in this system.

Fig. 2
Fig. 2

Dispersion profile and bi-photon generation spectra of a 20 μm × 250 nm Si3N4 disk in air. (a) Mode frequency vs. angular wavenumber m, showing a transition from the air light line to the Si3N4 light line with increasing mode frequency. Sub-panels show the spatial profiles um(ρ, z) for two wavenumbers, showing the transition from the air to the dielectric. (b) FSR (fm) calculated using Eq. (3b)(blue line), and Eq. (4)(circles) and discrete derivative of the the FSR (Δfm)(red), all vs. angular wavenumber m. The dashed line indicates the location of the pump beam for the spectra shown in panel (c). (c) Calculated bi-photon signal spectra from Eq. (9),assuming a cavity Q of 105. The FSR walk-off Δfm of approximately −2 GHz/m causes nearly a complete linewidth mismatch in energy conservation by the second sideband, as shown in the zoomed in sub-panels.

Fig. 3
Fig. 3

Dispersion profile and bi-photon generation spectra of a Si3N4 ring in SiO2. The ring has a geometry of r = 20 μm, h = 750 nm, and w = 1.1 μm. (a) Mode frequency vs. angular wavenumber m, showing a transition from the low index light line to the high index light line with increasing mode frequency. Sub-panels show the spatial profiles um(ρ, z) for two wavenumbers, showing the transition. (b) FSR (fm) calculated using Eq. (3b)(blue line), and Eq. (4)(circles) and discrete derivative of the the FSR (Δfm)(red), all vs. angular wavenumber m. The two arrows indicates the location of the pump beam for the spectra shown in panels (c) and (d). The shaded yellow region indicates an area proportional to the spectral spacing between the pump mode and the signal and idler modes. Equal areas on either side of the pump indicate energy conservation. Two “special” modes which conserve energy far outside the flat dispersion bandwidth of the ring are highlighted with blue circles. (c) Bi-photon generation spectra when the pump beam is tuned to the mode closest to zero Δfm (m = 137)(d) Bi-photon generation spectra when the pump beam is tuned to the mode with slightly higher angular wavenumber (m = 145) than that with the minimum Δfm. A cavity Q of 105 is assumed.

Equations (53)

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

E ^ k ( r , t ) = i h ¯ ω k v g k 2 ε 0 v p k [ a ^ k ( t ) a ^ k ( t ) ] u k ( r ) ,
V ε ( r ) u k 1 ( r ) u k 2 ( r ) = δ k 1 , k 2
v p m = ω m m
v g m = ω m m Δ ω m Δ m = 2 π f m ,
v g m ϕ = S | ϕ ^ [ E m ( r ) × H m * ( r ) ] | 1 2 V ε 0 d [ ε ( r ) ω ] d ω | ω m | E m ( r ) | 2 + μ 0 | H m ( r ) | 2 ,
E ^ ( r , t ) = m i m π h ¯ f m ε 0 ( a ^ m a ^ m ) u m ( r ) .
p i ( 3 ) ( r ) = j k l χ i j k l ( 3 ) ( r ) E ^ p ( j ) ( + ) E ^ s ( k ) ( ) E ^ i ( l ) ( )
H int = ε 0 V χ ( 3 ) ( r ) E ^ p ( + ) E ^ p ( + ) E ^ s ( ) E ^ i ( ) .
| ψ = 1 i h ¯ d t H int | 0 = 2 i π 2 Δ m ρ , z χ ( 3 ) ( ρ , z ) ρ | E p ( ρ , z ) | 2 m + m f m + f m a ^ p a ^ p × Ω s a ^ ( ω m + + Ω s ) a ^ ( ω m + Δ 0 Ω s ) 2 κ 1 [ i Ω s + κ tot ] [ i ( Δ 0 Ω s ) + κ tot ] u m + . u m | 0 ,
ψ | a ^ ( ω s ) a ^ ( ω s ) | ψ = 4 π 4 Δ m 4 κ 1 2 m + m f m + f m | κ tot i ( ω s ω m + ) | 2 | κ tot + i ( ω s ω m Δ 0 ) | 2 × [ ρ , z χ ( 3 ) ( ρ , z ) ρ | E p ( ρ , z ) | 2 u m + u m ] 2 ,
ε Si 3 N 4 = 1 + C 1 λ 2 λ 2 C 2 2 ,
ε SiO 2 = 1 + C 1 λ 2 λ 2 C 2 2 + C 3 λ 2 λ 2 C 4 2 + C 5 λ 2 λ 2 C 5 2 ,
U vac = 1 2 V ε 0 | E m ( r ) | 2 + μ 0 | H m ( r ) | 2 = V ε 0 | E m ( r ) | 2
U ε = 1 2 V ε 0 ε | E m ( r ) | 2 + μ 0 μ | H m ( r ) | 2 = ε U vac .
1 2 V ε 0 ε | E m ( r ) | 2 = 1 2 V μ 0 μ | H m ( r ) | 2 ,
U ε ( ω ) = 1 2 V ε 0 d [ ε ( ω ) ω ] d ω | ω m | E m ( r ) | 2 + μ 0 d [ μ ( ω ) ω ] d ω | ω m | H m ( r ) | 2 .
U ε ( ω ) = V ε 0 ( ε ( ω ) + 1 2 ω d ε ( ω ) d ω | ω m ) | E m ( r ) | 2 = V ε 0 ( n ( ω ) 2 + n ( ω ) ω d n ( ω ) d ω | ω m ) | E m ( r ) | 2 = ε ( ω ) v p m v g m U vac .
E ( r , t ) = m p m ( t ) D m u m ( r )
B ( r , t ) = m q m ( t ) D m c v m ( r ) .
V ε ( r ) u m 1 ( r ) u m 2 ( r ) = δ m 1 , m 2
V v m 1 ( r ) v m 2 ( r ) = δ m 1 , m 2 .
× × E ( r ) = ( ω c ) 2 ε ( r , ω ) E ( r ) .
Im [ ε ( ω ) ] = 2 ω π P 0 d ω Re [ ε ( ω ) ] ω 2 ω 2 .
× E m ( r , t ) = B m ( r , t ) t
× B m ( r , t ) = ε ( r ) c 2 E m ( r , t ) t
× u m ( r ) = ω m c v m ( r )
× v m ( r ) = ω m c ε ( r ) u m ( r ) ,
d q m ( t ) d t = ω m p m ( t )
d p m ( t ) d t = ω m q m ( t ) .
H = U ε ( r , ω ) = 1 2 v p m v g m V ε ( r ) ε 0 | E ( r , t ) | 2 + 1 μ 0 | B ( r , t ) | 2 = 1 2 ε 0 m v p m v g m D m 2 [ p m ( t ) 2 + q m ( t ) 2 ] ,
D m = v g m ω m v p m ε 0 ,
d p m ( t ) d t = H q m ( t )
d q m ( t ) d t = H p m ( t ) ,
V ε ( r ) u m 1 ( r ) u m 2 ( r ) = ω m 1 2 U ε ( r , ω m 1 ) δ m 1 , m 2
V v m 1 ( r ) v m 2 ( r ) = ω m 1 2 U ε ( r , ω m 1 ) δ m 1 , m 2 ,
H ^ m = h ¯ ω m [ a ^ m a ^ m + 1 2 ] ,
a ^ m 1 2 ( d d ξ m + ξ m )
a ^ m 1 2 ( d d ξ m + ξ m ) ,
E ^ m ( r , t ) = i h ¯ ω m v g m 2 ε 0 v p m [ a ^ m ( t ) a ^ m ( t ) ] u m ( r ) .
a ˙ = i Ω a a κ 0 a l = 1 , 2 κ l + i l = 1 , 2 2 κ l s l +
s l = s l + + i 2 κ l a ,
a ˙ = i Ω a a ( κ 0 + κ 1 + κ 2 ) + i 2 κ 1 s 1 +
s l = s l + + i 2 κ l a .
a s s = i 2 κ 1 i Ω + ( κ 0 + κ 1 + κ 2 ) s 1 +
s 1 = s 1 + + i 2 κ 1 a s s
s 2 = i 2 κ 2 a s s .
a ^ m ( t ) = 1 2 π a ^ m ( ω m + Ω ) i 2 κ 1 i Ω + ( κ 0 + κ 1 + κ 2 ) e i Ω t d Ω .
E ^ ( ) ( r , t ) = m h ¯ m π f m ε 0 1 2 π a ^ ( ω m + Ω ) i 2 κ 1 i Ω + ( κ 0 + κ 1 + κ 2 ) e i Ω t d Ω u m ( r ) e i m ϕ ω m t .
| ψ = 1 i h ¯ d t H int | 0 = ε 0 i h ¯ t V χ ( 3 ) ( r ) | E ^ p ( + ) | 2 E ^ s ( ) E ^ i ( ) | 0 = ε 0 i h ¯ t V χ ( 3 ) ( r ) | E p ( r ) | 2 a ^ p a ^ p [ i m s h ¯ π m s f m s ε 0 a ^ m s ( t ) u m s ( r ) ] [ i m i h ¯ π m i f m i ε 0 a ^ m i ( t ) u m i ( r ) ] | 0
| ψ = 2 i π 2 Δ m t ρ , z χ ( 3 ) ( r ) ρ | E p ( ρ , z ) | 2 m + m f m + f m a ^ p a ^ p a ^ m s ( t ) a ^ m i ( t ) u m + ( ρ , z ) u m ( ρ , z ) | 0
0 2 π d ϕ e i [ 2 m p ( m i + m s ) ] ϕ = 2 π δ 2 m p , m s + m i .
| ψ = 2 i π 2 Δ m ρ , z χ ( 3 ) ( ρ , z ) ρ | E p ( ρ , z ) | 2 m + m f m + f m a ^ p a ^ p × Ω s a ^ ( ω m + + Ω s ) a ^ ( ω m + Δ 0 Ω s ) 2 κ 1 [ i Ω s + κ tot ] [ i ( Δ 0 Ω s ) + κ tot ] u m + u m | 0
d t e i [ 2 ω m p ( ω m s + ω m i ) ] t = 2 π δ [ 2 ω m p ( ω m s + ω m i ) ] .

Metrics