Abstract

The feasibility of using a parametric down-conversion process to generate squeezed electromagnetic states in three dimensional photonic crystal microcavity structures is investigated for the first time. The spectrum of the squeezed light is theoretically calculated by using an open cavity quantum mechanical formalism. The cavity communicates with two main channels, which model vertical radiation losses and coupling into a single-mode waveguide respectively. The amount of squeezing is determined by the correlation functions relating the field quadratures of light coupled into the waveguide. All of the relevant model parameters are realistically estimated for structures made in Al0.3Ga0.7As, using finite-difference time-domain simulations. Squeezing up to ~30% below the shot noise level is predicted for 10 mW average power, 80 MHz repetition, 500 ps excitation pulses using in a [111] oriented wafer.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. J. Shields, "Semiconductor quantum light sources," Nature Photon. 1, 215-223 (2007).
    [CrossRef]
  2. M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, "A single-photon server with just one atom," Nature Phys. 3, 253-255 (2007).
    [CrossRef]
  3. M. Dusek, N. Lutkenhaus, and M. Hendrych, "Quantum Cryptography," Progress in Optics, E. Wolf, ed., (Elsevier, 2006), Vol. 49.
  4. E. Knill, L. Laflamme, and G. J. Milburn, "Efficient linear optics quantum computation," Nature 409, 46-52 (2001).
    [CrossRef] [PubMed]
  5. S. L. Braunstein and A. K. Pati, Quantum information with continuous variables (Kluwer Academic Publishers, 2003).
  6. A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
    [CrossRef] [PubMed]
  7. S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77513-577 (2005).
    [CrossRef]
  8. J. A. Gaj,  et al., "Semiconductor heterostructures for spintronics and quantum information," C. R. Physique 8, 243-252 (2007).
    [CrossRef]
  9. G. Burkard, "Spin qubits: Connect the dots," Nature Phys. 2, 807-808 (2006).
    [CrossRef]
  10. M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).
  11. J. P. Karr, A. Bass, R. Houdre, and E. Giacobino, "Squeezing in semiconductor microcavities in the strongcoupling regime," Phys. Rev. A 69,031802(R) (2004).
  12. A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Theory of spontaneous parametric downconversion from photonic crystals," Phys. Rev. A 70, 043810 (2004).
    [CrossRef]
  13. G. Weihs, "Parametric down-conversion in photonic crystal waveguides," Int. J. Mod. Phys. B 20, 1543-1550 (2006).
    [CrossRef]
  14. L. Xiao, Y. Wang, W. Zhang, Y. Huang, and J. Peng, "A 2-D photonic crystal based source of polarization entangled photon pairs with high nonlinear conversion efficiency and without walk-off compensation," Opt. Commun. 272, 525-528 (2007).
    [CrossRef]
  15. C. Viviescas and G. Hackenbroich, "Field quantization for open optical cavities," Phys. Rev. A 67, 013805 (2003).
    [CrossRef]
  16. H. J. Kimble, Fundamental Systems in Quantum Optics (Elsevier Science Publishing, 1992) Chap. 10.
  17. H. Carmichael, An open systems approach to quantum optics (Springer-Verlag, 1993).
  18. S. Reynaud, C. Fabre, and E. Giacobino, " Quantum fluctuations in a 2-mode parametric oscillator," J. Opt. Soc. Am. B 4, 152-1524 (1987).
    [CrossRef]
  19. C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, "Squeezing in detuned degenerate optical parametric oscillators," Quantum Opt. 2, 159-187 (1990).
    [CrossRef]
  20. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2005).
  21. M. G. Banaee, A. G. Pattantyus-Abraham, M. W. McCutcheon, G. W. Rieger, and J. F. Young, "Efficient coupling of photonic crystal microcavity modes to a ridge waveguide," Appl. Phys. Lett. 90, 193106 (2007).
    [CrossRef]
  22. K. Srinivasan and O. Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10, 670-684 (2002).
    [PubMed]
  23. Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
    [CrossRef] [PubMed]
  24. A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, "Efficient photonic crystal cavity-waveguide couplers," Appl. Phys. Lett. 90, 073102 (2007).
    [CrossRef]
  25. H. Feshbach, "A unified theory of nuclear reactions. II," Ann. Phys. 19, 287-313 (1962).
    [CrossRef]
  26. S. Nordholm and S. A. Rice, "A quantum ergodic theory approach to unimolecular fragmentation," J. Chem. Phys. 62, 157-168 (1975).
    [CrossRef]
  27. M. Shapiro and P. Brumer, "Quantum control of bound and continuum state dynamics," Phys. Rep. 425, 195-264 (2006).
    [CrossRef]
  28. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).
  29. M. Hillery and L. Mlodinow, "Quantized fields in a nonlinear dielectric medium: a microscopic approach," Phys. Rev. A 55, 678-689 (1997).
    [CrossRef]
  30. M. Hillery, Quantum Squeezing (Springer-Verlag, 2004), edited by P. D. Drummond, and Z. Ficek, chap. 2.
  31. Z. Y. Ou, S. F. Pereira, and H. J. Kimble, "Realization of the Einstein-Podolsky-Rosen paradox for continuous variables in nondegenerate parametric amplification," Appl. Phys. B 55, 265-278 (1992).
    [CrossRef]
  32. M. D. Reid and P.D. Drummond, "Quantum correlations of phase in nondegenerate parametric oscillation," Phys. Rev. Lett,  60, 2731-2733 (1988).
    [CrossRef] [PubMed]
  33. M. D. Reid and P. D. Drummond, "Correlations in nondegenerate parametric oscillation II, below threshold results," Phys. Rev. A 41, 3930-3949 (1990).
    [CrossRef] [PubMed]
  34. Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, "Design of a channel drop filter by using a donortype cavity with high-quality factor in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 82, 1341-1343 (2003).
    [CrossRef]
  35. S. Adachi, GaAs and Related Materials: Bulk Semiconducting and Superlattice Properties (World Scientific Publishing Company, 1994).
    [CrossRef]
  36. Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, "Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002).
    [PubMed]
  37. T. Asano, Bong-Shik Song, Y. Akahane, and S. Noda, "Ultrahigh-Q Nanocavities in Two-Dimensional Photonic Crystal Slabs," IEEE J. Sel. Top. Quantum Electron. 12, 1123-1134 (2006).
    [CrossRef]

2007

A. J. Shields, "Semiconductor quantum light sources," Nature Photon. 1, 215-223 (2007).
[CrossRef]

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, "A single-photon server with just one atom," Nature Phys. 3, 253-255 (2007).
[CrossRef]

J. A. Gaj,  et al., "Semiconductor heterostructures for spintronics and quantum information," C. R. Physique 8, 243-252 (2007).
[CrossRef]

L. Xiao, Y. Wang, W. Zhang, Y. Huang, and J. Peng, "A 2-D photonic crystal based source of polarization entangled photon pairs with high nonlinear conversion efficiency and without walk-off compensation," Opt. Commun. 272, 525-528 (2007).
[CrossRef]

M. G. Banaee, A. G. Pattantyus-Abraham, M. W. McCutcheon, G. W. Rieger, and J. F. Young, "Efficient coupling of photonic crystal microcavity modes to a ridge waveguide," Appl. Phys. Lett. 90, 193106 (2007).
[CrossRef]

A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, "Efficient photonic crystal cavity-waveguide couplers," Appl. Phys. Lett. 90, 073102 (2007).
[CrossRef]

2006

G. Weihs, "Parametric down-conversion in photonic crystal waveguides," Int. J. Mod. Phys. B 20, 1543-1550 (2006).
[CrossRef]

G. Burkard, "Spin qubits: Connect the dots," Nature Phys. 2, 807-808 (2006).
[CrossRef]

M. Shapiro and P. Brumer, "Quantum control of bound and continuum state dynamics," Phys. Rep. 425, 195-264 (2006).
[CrossRef]

T. Asano, Bong-Shik Song, Y. Akahane, and S. Noda, "Ultrahigh-Q Nanocavities in Two-Dimensional Photonic Crystal Slabs," IEEE J. Sel. Top. Quantum Electron. 12, 1123-1134 (2006).
[CrossRef]

T. Asano, Bong-Shik Song, Y. Akahane, and S. Noda, "Ultrahigh-Q Nanocavities in Two-Dimensional Photonic Crystal Slabs," IEEE J. Sel. Top. Quantum Electron. 12, 1123-1134 (2006).
[CrossRef]

2005

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77513-577 (2005).
[CrossRef]

2004

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Theory of spontaneous parametric downconversion from photonic crystals," Phys. Rev. A 70, 043810 (2004).
[CrossRef]

2003

C. Viviescas and G. Hackenbroich, "Field quantization for open optical cavities," Phys. Rev. A 67, 013805 (2003).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, "Design of a channel drop filter by using a donortype cavity with high-quality factor in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 82, 1341-1343 (2003).
[CrossRef]

2002

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, "Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002).
[PubMed]

K. Srinivasan and O. Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10, 670-684 (2002).
[PubMed]

2001

E. Knill, L. Laflamme, and G. J. Milburn, "Efficient linear optics quantum computation," Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

1998

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
[CrossRef] [PubMed]

1997

M. Hillery and L. Mlodinow, "Quantized fields in a nonlinear dielectric medium: a microscopic approach," Phys. Rev. A 55, 678-689 (1997).
[CrossRef]

1992

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, "Realization of the Einstein-Podolsky-Rosen paradox for continuous variables in nondegenerate parametric amplification," Appl. Phys. B 55, 265-278 (1992).
[CrossRef]

1990

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, "Squeezing in detuned degenerate optical parametric oscillators," Quantum Opt. 2, 159-187 (1990).
[CrossRef]

M. D. Reid and P. D. Drummond, "Correlations in nondegenerate parametric oscillation II, below threshold results," Phys. Rev. A 41, 3930-3949 (1990).
[CrossRef] [PubMed]

1988

M. D. Reid and P.D. Drummond, "Quantum correlations of phase in nondegenerate parametric oscillation," Phys. Rev. Lett,  60, 2731-2733 (1988).
[CrossRef] [PubMed]

1987

S. Reynaud, C. Fabre, and E. Giacobino, " Quantum fluctuations in a 2-mode parametric oscillator," J. Opt. Soc. Am. B 4, 152-1524 (1987).
[CrossRef]

1975

S. Nordholm and S. A. Rice, "A quantum ergodic theory approach to unimolecular fragmentation," J. Chem. Phys. 62, 157-168 (1975).
[CrossRef]

1962

H. Feshbach, "A unified theory of nuclear reactions. II," Ann. Phys. 19, 287-313 (1962).
[CrossRef]

Abram, I.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, "Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002).
[PubMed]

Aers, G. C.

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

Akahane, Y.

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, "Design of a channel drop filter by using a donortype cavity with high-quality factor in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 82, 1341-1343 (2003).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Asano, T.

T. Asano, Bong-Shik Song, Y. Akahane, and S. Noda, "Ultrahigh-Q Nanocavities in Two-Dimensional Photonic Crystal Slabs," IEEE J. Sel. Top. Quantum Electron. 12, 1123-1134 (2006).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, "Design of a channel drop filter by using a donortype cavity with high-quality factor in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 82, 1341-1343 (2003).
[CrossRef]

Banaee, M. G.

M. G. Banaee, A. G. Pattantyus-Abraham, M. W. McCutcheon, G. W. Rieger, and J. F. Young, "Efficient coupling of photonic crystal microcavity modes to a ridge waveguide," Appl. Phys. Lett. 90, 193106 (2007).
[CrossRef]

Bong-Shik Song, T.

T. Asano, Bong-Shik Song, Y. Akahane, and S. Noda, "Ultrahigh-Q Nanocavities in Two-Dimensional Photonic Crystal Slabs," IEEE J. Sel. Top. Quantum Electron. 12, 1123-1134 (2006).
[CrossRef]

Braunstein, S. L.

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77513-577 (2005).
[CrossRef]

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
[CrossRef] [PubMed]

Brumer, P.

M. Shapiro and P. Brumer, "Quantum control of bound and continuum state dynamics," Phys. Rep. 425, 195-264 (2006).
[CrossRef]

Burkard, G.

G. Burkard, "Spin qubits: Connect the dots," Nature Phys. 2, 807-808 (2006).
[CrossRef]

Cheung, I. W.

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

Dalacu, D.

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

Drummond, P. D.

M. D. Reid and P. D. Drummond, "Correlations in nondegenerate parametric oscillation II, below threshold results," Phys. Rev. A 41, 3930-3949 (1990).
[CrossRef] [PubMed]

Drummond, P.D.

M. D. Reid and P.D. Drummond, "Quantum correlations of phase in nondegenerate parametric oscillation," Phys. Rev. Lett,  60, 2731-2733 (1988).
[CrossRef] [PubMed]

Dumeige, Y.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, "Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002).
[PubMed]

Englund, D.

A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, "Efficient photonic crystal cavity-waveguide couplers," Appl. Phys. Lett. 90, 073102 (2007).
[CrossRef]

Fabre, C.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, "Squeezing in detuned degenerate optical parametric oscillators," Quantum Opt. 2, 159-187 (1990).
[CrossRef]

S. Reynaud, C. Fabre, and E. Giacobino, " Quantum fluctuations in a 2-mode parametric oscillator," J. Opt. Soc. Am. B 4, 152-1524 (1987).
[CrossRef]

Faraon, A.

A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, "Efficient photonic crystal cavity-waveguide couplers," Appl. Phys. Lett. 90, 073102 (2007).
[CrossRef]

Feshbach, H.

H. Feshbach, "A unified theory of nuclear reactions. II," Ann. Phys. 19, 287-313 (1962).
[CrossRef]

Frederick, S.

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

Fuchs, C. A.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
[CrossRef] [PubMed]

Furusawa, A.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
[CrossRef] [PubMed]

Fushman, I.

A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, "Efficient photonic crystal cavity-waveguide couplers," Appl. Phys. Lett. 90, 073102 (2007).
[CrossRef]

Gaj, J. A.

J. A. Gaj,  et al., "Semiconductor heterostructures for spintronics and quantum information," C. R. Physique 8, 243-252 (2007).
[CrossRef]

Giacobino, E.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, "Squeezing in detuned degenerate optical parametric oscillators," Quantum Opt. 2, 159-187 (1990).
[CrossRef]

S. Reynaud, C. Fabre, and E. Giacobino, " Quantum fluctuations in a 2-mode parametric oscillator," J. Opt. Soc. Am. B 4, 152-1524 (1987).
[CrossRef]

Hackenbroich, G.

C. Viviescas and G. Hackenbroich, "Field quantization for open optical cavities," Phys. Rev. A 67, 013805 (2003).
[CrossRef]

Heidmann, A.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, "Squeezing in detuned degenerate optical parametric oscillators," Quantum Opt. 2, 159-187 (1990).
[CrossRef]

Hijlkema, M.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, "A single-photon server with just one atom," Nature Phys. 3, 253-255 (2007).
[CrossRef]

Hillery, M.

M. Hillery and L. Mlodinow, "Quantized fields in a nonlinear dielectric medium: a microscopic approach," Phys. Rev. A 55, 678-689 (1997).
[CrossRef]

Huang, Y.

L. Xiao, Y. Wang, W. Zhang, Y. Huang, and J. Peng, "A 2-D photonic crystal based source of polarization entangled photon pairs with high nonlinear conversion efficiency and without walk-off compensation," Opt. Commun. 272, 525-528 (2007).
[CrossRef]

Kaige, W.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, "Squeezing in detuned degenerate optical parametric oscillators," Quantum Opt. 2, 159-187 (1990).
[CrossRef]

Kimble, H. J.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
[CrossRef] [PubMed]

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, "Realization of the Einstein-Podolsky-Rosen paradox for continuous variables in nondegenerate parametric amplification," Appl. Phys. B 55, 265-278 (1992).
[CrossRef]

Knill, E.

E. Knill, L. Laflamme, and G. J. Milburn, "Efficient linear optics quantum computation," Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

Kuhn, A.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, "A single-photon server with just one atom," Nature Phys. 3, 253-255 (2007).
[CrossRef]

Laflamme, L.

E. Knill, L. Laflamme, and G. J. Milburn, "Efficient linear optics quantum computation," Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

Levenson, A.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, "Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002).
[PubMed]

Lugiato, L.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, "Squeezing in detuned degenerate optical parametric oscillators," Quantum Opt. 2, 159-187 (1990).
[CrossRef]

McCutcheon, M. W.

M. G. Banaee, A. G. Pattantyus-Abraham, M. W. McCutcheon, G. W. Rieger, and J. F. Young, "Efficient coupling of photonic crystal microcavity modes to a ridge waveguide," Appl. Phys. Lett. 90, 193106 (2007).
[CrossRef]

McCutcheon, M.W.

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

Meriadec, C.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, "Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002).
[PubMed]

Milburn, G. J.

E. Knill, L. Laflamme, and G. J. Milburn, "Efficient linear optics quantum computation," Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

Mlodinow, L.

M. Hillery and L. Mlodinow, "Quantized fields in a nonlinear dielectric medium: a microscopic approach," Phys. Rev. A 55, 678-689 (1997).
[CrossRef]

Mochizuki, M.

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, "Design of a channel drop filter by using a donortype cavity with high-quality factor in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 82, 1341-1343 (2003).
[CrossRef]

Monnier, P.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, "Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002).
[PubMed]

Noda, S.

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, "Design of a channel drop filter by using a donortype cavity with high-quality factor in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 82, 1341-1343 (2003).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Nordholm, S.

S. Nordholm and S. A. Rice, "A quantum ergodic theory approach to unimolecular fragmentation," J. Chem. Phys. 62, 157-168 (1975).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, "Realization of the Einstein-Podolsky-Rosen paradox for continuous variables in nondegenerate parametric amplification," Appl. Phys. B 55, 265-278 (1992).
[CrossRef]

Painter, O.

Pattantyus-Abraham, A. G.

M. G. Banaee, A. G. Pattantyus-Abraham, M. W. McCutcheon, G. W. Rieger, and J. F. Young, "Efficient coupling of photonic crystal microcavity modes to a ridge waveguide," Appl. Phys. Lett. 90, 193106 (2007).
[CrossRef]

Peng, J.

L. Xiao, Y. Wang, W. Zhang, Y. Huang, and J. Peng, "A 2-D photonic crystal based source of polarization entangled photon pairs with high nonlinear conversion efficiency and without walk-off compensation," Opt. Commun. 272, 525-528 (2007).
[CrossRef]

Pereira, S. F.

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, "Realization of the Einstein-Podolsky-Rosen paradox for continuous variables in nondegenerate parametric amplification," Appl. Phys. B 55, 265-278 (1992).
[CrossRef]

Polzik, E. S.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
[CrossRef] [PubMed]

Poole, P. J.

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

Reid, M. D.

M. D. Reid and P. D. Drummond, "Correlations in nondegenerate parametric oscillation II, below threshold results," Phys. Rev. A 41, 3930-3949 (1990).
[CrossRef] [PubMed]

M. D. Reid and P.D. Drummond, "Quantum correlations of phase in nondegenerate parametric oscillation," Phys. Rev. Lett,  60, 2731-2733 (1988).
[CrossRef] [PubMed]

Rempe, G.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, "A single-photon server with just one atom," Nature Phys. 3, 253-255 (2007).
[CrossRef]

Reynaud, S.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, "Squeezing in detuned degenerate optical parametric oscillators," Quantum Opt. 2, 159-187 (1990).
[CrossRef]

S. Reynaud, C. Fabre, and E. Giacobino, " Quantum fluctuations in a 2-mode parametric oscillator," J. Opt. Soc. Am. B 4, 152-1524 (1987).
[CrossRef]

Rice, S. A.

S. Nordholm and S. A. Rice, "A quantum ergodic theory approach to unimolecular fragmentation," J. Chem. Phys. 62, 157-168 (1975).
[CrossRef]

Rieger, G. W.

M. G. Banaee, A. G. Pattantyus-Abraham, M. W. McCutcheon, G. W. Rieger, and J. F. Young, "Efficient coupling of photonic crystal microcavity modes to a ridge waveguide," Appl. Phys. Lett. 90, 193106 (2007).
[CrossRef]

Rieger, G.W.

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

Sagnes, I.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, "Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002).
[PubMed]

Saleh, B. E. A.

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Theory of spontaneous parametric downconversion from photonic crystals," Phys. Rev. A 70, 043810 (2004).
[CrossRef]

Sergienko, A. V.

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Theory of spontaneous parametric downconversion from photonic crystals," Phys. Rev. A 70, 043810 (2004).
[CrossRef]

Shapiro, M.

M. Shapiro and P. Brumer, "Quantum control of bound and continuum state dynamics," Phys. Rep. 425, 195-264 (2006).
[CrossRef]

Shields, A. J.

A. J. Shields, "Semiconductor quantum light sources," Nature Photon. 1, 215-223 (2007).
[CrossRef]

Song, B. S.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Sorensen, J. L.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
[CrossRef] [PubMed]

Specht, H. P.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, "A single-photon server with just one atom," Nature Phys. 3, 253-255 (2007).
[CrossRef]

Srinivasan, K.

Tanaka, Y.

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, "Design of a channel drop filter by using a donortype cavity with high-quality factor in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 82, 1341-1343 (2003).
[CrossRef]

Teich, M. C.

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Theory of spontaneous parametric downconversion from photonic crystals," Phys. Rev. A 70, 043810 (2004).
[CrossRef]

Vadacchino, M.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, "Squeezing in detuned degenerate optical parametric oscillators," Quantum Opt. 2, 159-187 (1990).
[CrossRef]

Vamivakas, A. N.

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Theory of spontaneous parametric downconversion from photonic crystals," Phys. Rev. A 70, 043810 (2004).
[CrossRef]

van Loock, P.

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77513-577 (2005).
[CrossRef]

Vidakovic, P.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, "Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002).
[PubMed]

Viviescas, C.

C. Viviescas and G. Hackenbroich, "Field quantization for open optical cavities," Phys. Rev. A 67, 013805 (2003).
[CrossRef]

Vuckovic, J.

A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, "Efficient photonic crystal cavity-waveguide couplers," Appl. Phys. Lett. 90, 073102 (2007).
[CrossRef]

Waks, E.

A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, "Efficient photonic crystal cavity-waveguide couplers," Appl. Phys. Lett. 90, 073102 (2007).
[CrossRef]

Wang, Y.

L. Xiao, Y. Wang, W. Zhang, Y. Huang, and J. Peng, "A 2-D photonic crystal based source of polarization entangled photon pairs with high nonlinear conversion efficiency and without walk-off compensation," Opt. Commun. 272, 525-528 (2007).
[CrossRef]

Weber, B.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, "A single-photon server with just one atom," Nature Phys. 3, 253-255 (2007).
[CrossRef]

Webster, S. C.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, "A single-photon server with just one atom," Nature Phys. 3, 253-255 (2007).
[CrossRef]

Weihs, G.

G. Weihs, "Parametric down-conversion in photonic crystal waveguides," Int. J. Mod. Phys. B 20, 1543-1550 (2006).
[CrossRef]

Williams, R. L.

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

Xiao, L.

L. Xiao, Y. Wang, W. Zhang, Y. Huang, and J. Peng, "A 2-D photonic crystal based source of polarization entangled photon pairs with high nonlinear conversion efficiency and without walk-off compensation," Opt. Commun. 272, 525-528 (2007).
[CrossRef]

Young, J. F.

M. G. Banaee, A. G. Pattantyus-Abraham, M. W. McCutcheon, G. W. Rieger, and J. F. Young, "Efficient coupling of photonic crystal microcavity modes to a ridge waveguide," Appl. Phys. Lett. 90, 193106 (2007).
[CrossRef]

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

Zhang, W.

L. Xiao, Y. Wang, W. Zhang, Y. Huang, and J. Peng, "A 2-D photonic crystal based source of polarization entangled photon pairs with high nonlinear conversion efficiency and without walk-off compensation," Opt. Commun. 272, 525-528 (2007).
[CrossRef]

Ann. Phys.

H. Feshbach, "A unified theory of nuclear reactions. II," Ann. Phys. 19, 287-313 (1962).
[CrossRef]

Appl. Phys. B

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, "Realization of the Einstein-Podolsky-Rosen paradox for continuous variables in nondegenerate parametric amplification," Appl. Phys. B 55, 265-278 (1992).
[CrossRef]

Appl. Phys. Lett.

A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, "Efficient photonic crystal cavity-waveguide couplers," Appl. Phys. Lett. 90, 073102 (2007).
[CrossRef]

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, "Design of a channel drop filter by using a donortype cavity with high-quality factor in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 82, 1341-1343 (2003).
[CrossRef]

M. G. Banaee, A. G. Pattantyus-Abraham, M. W. McCutcheon, G. W. Rieger, and J. F. Young, "Efficient coupling of photonic crystal microcavity modes to a ridge waveguide," Appl. Phys. Lett. 90, 193106 (2007).
[CrossRef]

M.W. McCutcheon, G.W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, and R. L. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221110 (2005).

C. R. Physique

J. A. Gaj,  et al., "Semiconductor heterostructures for spintronics and quantum information," C. R. Physique 8, 243-252 (2007).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

T. Asano, Bong-Shik Song, Y. Akahane, and S. Noda, "Ultrahigh-Q Nanocavities in Two-Dimensional Photonic Crystal Slabs," IEEE J. Sel. Top. Quantum Electron. 12, 1123-1134 (2006).
[CrossRef]

Int. J. Mod. Phys. B

G. Weihs, "Parametric down-conversion in photonic crystal waveguides," Int. J. Mod. Phys. B 20, 1543-1550 (2006).
[CrossRef]

J. Chem. Phys.

S. Nordholm and S. A. Rice, "A quantum ergodic theory approach to unimolecular fragmentation," J. Chem. Phys. 62, 157-168 (1975).
[CrossRef]

J. Opt. Soc. Am. B

S. Reynaud, C. Fabre, and E. Giacobino, " Quantum fluctuations in a 2-mode parametric oscillator," J. Opt. Soc. Am. B 4, 152-1524 (1987).
[CrossRef]

Nature

E. Knill, L. Laflamme, and G. J. Milburn, "Efficient linear optics quantum computation," Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Nature Photon.

A. J. Shields, "Semiconductor quantum light sources," Nature Photon. 1, 215-223 (2007).
[CrossRef]

Nature Phys.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, "A single-photon server with just one atom," Nature Phys. 3, 253-255 (2007).
[CrossRef]

G. Burkard, "Spin qubits: Connect the dots," Nature Phys. 2, 807-808 (2006).
[CrossRef]

Opt. Commun.

L. Xiao, Y. Wang, W. Zhang, Y. Huang, and J. Peng, "A 2-D photonic crystal based source of polarization entangled photon pairs with high nonlinear conversion efficiency and without walk-off compensation," Opt. Commun. 272, 525-528 (2007).
[CrossRef]

Opt. Express

Phys. Rep.

M. Shapiro and P. Brumer, "Quantum control of bound and continuum state dynamics," Phys. Rep. 425, 195-264 (2006).
[CrossRef]

Phys. Rev. A

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Theory of spontaneous parametric downconversion from photonic crystals," Phys. Rev. A 70, 043810 (2004).
[CrossRef]

M. D. Reid and P. D. Drummond, "Correlations in nondegenerate parametric oscillation II, below threshold results," Phys. Rev. A 41, 3930-3949 (1990).
[CrossRef] [PubMed]

M. Hillery and L. Mlodinow, "Quantized fields in a nonlinear dielectric medium: a microscopic approach," Phys. Rev. A 55, 678-689 (1997).
[CrossRef]

C. Viviescas and G. Hackenbroich, "Field quantization for open optical cavities," Phys. Rev. A 67, 013805 (2003).
[CrossRef]

Phys. Rev. Lett

M. D. Reid and P.D. Drummond, "Quantum correlations of phase in nondegenerate parametric oscillation," Phys. Rev. Lett,  60, 2731-2733 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, "Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002).
[PubMed]

Quantum Opt.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, "Squeezing in detuned degenerate optical parametric oscillators," Quantum Opt. 2, 159-187 (1990).
[CrossRef]

Rev. Mod. Phys.

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77513-577 (2005).
[CrossRef]

Science

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
[CrossRef] [PubMed]

Other

M. Dusek, N. Lutkenhaus, and M. Hendrych, "Quantum Cryptography," Progress in Optics, E. Wolf, ed., (Elsevier, 2006), Vol. 49.

S. L. Braunstein and A. K. Pati, Quantum information with continuous variables (Kluwer Academic Publishers, 2003).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2005).

J. P. Karr, A. Bass, R. Houdre, and E. Giacobino, "Squeezing in semiconductor microcavities in the strongcoupling regime," Phys. Rev. A 69,031802(R) (2004).

H. J. Kimble, Fundamental Systems in Quantum Optics (Elsevier Science Publishing, 1992) Chap. 10.

H. Carmichael, An open systems approach to quantum optics (Springer-Verlag, 1993).

M. Hillery, Quantum Squeezing (Springer-Verlag, 2004), edited by P. D. Drummond, and Z. Ficek, chap. 2.

S. Adachi, GaAs and Related Materials: Bulk Semiconducting and Superlattice Properties (World Scientific Publishing Company, 1994).
[CrossRef]

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

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 (6)

Fig. 1.
Fig. 1.

2D photonic crystal microcavities, a) an isolated cavity and b) adding a 1D waveguide channel to the cavity structure.

Fig. 2.
Fig. 2.

The model cavity that communicates with several output channels.

Fig. 3.
Fig. 3.

a) Spectrum at threshold of squeezing for the Y quadrature in a degenerate down-conversion process. b) Squeezing versus g factor at Ω=0. The solid lines are for the case when the cavity couples to a single channel (γx =0) and the dashed lines are when γx =γ 1.

Fig. 4.
Fig. 4.

a) Shift of the holes next to the cavity in order to increase its Q and also its coupling efficiency to the waveguide, b) the cavity is tilted with respect to the waveguide to boost their coupling efficiencies.

Fig. 5.
Fig. 5.

Intensity profile of a) X component and b) Y component of the electric field associated with the cavity in Fig.(4), and c) total intensity of the pump beam in the vicinity of the cavity. Center of the cavity is located at (x=0,y=0).

Fig. 6.
Fig. 6.

The spectrum of squeezing for the Y quadrature of the sample in Fig. (4) for a crystal oriented along the [111] direction pumped with 500 ps pulses of 80 MHz repetition rate and 10 mW average power.

Equations (30)

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

f m ( r , ω ) = λ α λ m ( ω ) U λ ( r ) + n d ω β n m ( ω , ω ) V n ( r , ω ) .
H ̂ = λ h ̅ ω λ a ̂ λ a ̂ λ + m d ω h ̅ ω r ̂ m ( ω ) r ̂ m ( ω )
+ h ̅ λ m d ω [ W λ m ( ω ) a ̂ λ r ̂ m ( ω ) + W λ m * ( ω ) a ̂ λ r ̂ m ( ω )
+ T λ m ( ω ) a ̂ λ r ̂ m ( ω ) + T λ m * ( ω ) a ̂ λ r ̂ m ( ω ) ] ,
H ̂ I = 2 ε 0 3 d 3 r E ̂ ( r , t ) . χ ( 2 ) ( r ) : E ̂ ( r , t ) E ̂ ( r , t ) ,
E ̂ c ( r , t ) = i h ̅ ω 1 2 ε 0 [ a ̂ 1 ( t ) U 1 ( r ) a ̂ 1 ( t ) U 1 * ( r ) ] .
E p ( r , t ) = i A p ( t ) [ U p ( r ) e 2 i ω 0 t U p * ( r ) e + 2 i ω 0 t ] ,
H ̂ I = i h ̅ [ g a ̂ 1 ( t ) a ̂ 1 ( t ) e 2 i ω 0 t g * a ̂ 1 ( t ) a ̂ 1 ( t ) e 2 i ω 0 t ] ,
g = ω 1 3 A p ( t ) d 3 r [ U 1 . χ ( 2 ) : U 1 U p + U 1 . χ ( 2 ) : U p U 1 + U p . χ ( 2 ) : U 1 U 1 ] .
a ˜ 1 ( Ω ) = 2 g 2 γ 1 r ˜ 1 in ( Ω ) + 2 g m = 2 2 γ m r ˜ m in ( Ω ) [ Γ i ( Δ + Ω ) ] [ Γ + i ( Δ Ω ) ] 4 g 2
+ [ Γ + i ( Δ Ω ) ] [ 2 γ 1 r ˜ 1 in ( Ω ) + m = 2 2 γ m r ˜ m in ( Ω ) ] [ Γ i ( Δ + Ω ) ] [ Γ + i ( Δ Ω ) ] 4 g 2 ,
S X ( Ω ) = < X out ( Ω ) , X out ( Ω ) > 1 ,
S Y ( Ω ) = < Y out ( Ω ) , Y out ( Ω ) > 1 ,
X out ( Ω ) = r ˜ 1 out ( Ω ) + r ˜ 1 out ( Ω ) ,
Y out ( Ω ) = i [ r ˜ 1 out ( Ω ) r ˜ 1 out ( Ω ) ] .
r ˜ 1 out ( Ω ) r ˜ 1 in ( Ω ) = 2 γ 1 a ˜ 1 ( Ω ) ,
S X ( Ω ) = 8 g γ 1 ( γ 1 2 g ) 2 + Ω 2 ,
S Y ( Ω ) = 8 g γ 1 ( γ 1 + 2 g ) 2 + Ω 2 .
r ˜ m out ( Ω ) r ˜ m in ( Ω ) = 2 γ 1 a ˜ 1 ( Ω ) ,
S X ( Ω ) = 8 g γ 1 [ ( γ 1 + γ x ) 2 g ] 2 + Ω 2 ,
S Y ( Ω ) = 8 g γ 1 [ ( γ 1 + γ x ) + 2 g ] 2 + Ω 2 ,
γ x = m = 2 γ m ,
S Y ( Ω = 0 ) = γ 1 γ 1 + γ x ,
χ [ 001 ] ( 2 ) ( ω 1 ; 2 ω 0 , ω 1 ) = ( χ xxx ( 2 ) χ xxy ( 2 ) χ xxz ( 2 ) χ xyx ( 2 ) χ xyy ( 2 ) χ xyz ( 2 ) χ xzx ( 2 ) χ xzy ( 2 ) χ xzz ( 2 ) χ yxx ( 2 ) χ yxy ( 2 ) χ yxz ( 2 ) χ yyx ( 2 ) χ yyy ( 2 ) χ yyz ( 2 ) χ yzx ( 2 ) χ yzy ( 2 ) χ yzz ( 2 ) χ zxx ( 2 ) χ zxy ( 2 ) χ zxz ( 2 ) χ zyx ( 2 ) χ zyy ( 2 ) χ zyz ( 2 ) χ zzx ( 2 ) χ zzy ( 2 ) χ zzz ( 2 ) )
= β ( 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 ) ,
χ [ 111 ] ( 2 ) ( ω 1 ; 2 ω 0 , ω 1 ) = β ( 3 3 0 2 2 3 0 1 3 0 2 2 3 0 1 2 2 3 0 1 3 0 1 3 0 2 3 0 2 3 0 2 2 3 0 1 2 2 3 0 2 3 0 1 2 2 3 0 3 2 3 )
1 Q T = 1 Q i + 1 Q wg ,
γ 1 = ω 1 Q wg = 46 GHz , γ x = ω 1 Q i = 16 GHz .
g [ 111 ] = 0.839 GHz .
g [ 111 ] = 4.199 GHz .

Metrics