Abstract

We theoretically investigate nonlinear effects such as four-wave mixing (FWM) and bistability in a cavity electromechanical system based on the electromagnetically induced transparency effect. We show that the FWM can be resonantly enhanced under conditions of reduced linear absorption. We also demonstrate that bistable behavior of the mean intracavity photon number can appear in this system. The system may have potential applications in communication networks for frequency conversion.

© 2012 Optical Society of America

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    [CrossRef]
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  12. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
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    [CrossRef]
  26. S. Huang and G. S. Agarwal, “Normal-mode splitting and antibunching in Stokes and anti-Stokes processes in cavity optomechanics: radiation-pressure-induced four-wave-mixing cavity optomechanics,” Phys. Rev. A 81, 033830 (2010).
    [CrossRef]
  27. J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
    [CrossRef]
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    [CrossRef]
  29. B. I. Greence, J. F. Mueller, J. Orenstein, D. H. Rapkine, S. S. Rink, and M. Thakur, “Phonon-mediated optical nonlinearity in polydiacetylene,” Phys. Rev. Lett. 61, 325–328 (1988).
    [CrossRef]
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    [CrossRef]
  32. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
    [CrossRef]
  33. R. Kanamoto and P. Meystre, “Optomechanics of a quantum-degenerate Fermi gas,” Phys. Rev. Lett. 104, 063601 (2010).
    [CrossRef]
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    [CrossRef]
  36. A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
    [CrossRef]
  37. B. P. Venkatesh, J. Larson, and D. H. J. O’Dell, “Band-structure loops and multistabilityin cavity QED,” Phys. Rev. A 83, 063606 (2011).
    [CrossRef]
  38. R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
    [CrossRef]
  39. C. Jiang, B. Chen, and K. D. Zhu, “Tunable pulse delay and advancement device based on a cavity electromechanical system,” Europhys. Lett. 94, 38002 (2011).
    [CrossRef]
  40. M. D. Reid and D. F. Walls, “Generation of squeezed states via degenerate four-wave mixing,” Phys. Rev. A 31, 1622–1635 (1985).
    [CrossRef]
  41. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
    [CrossRef]

2011 (7)

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[CrossRef]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[CrossRef]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

B. P. Venkatesh, J. Larson, and D. H. J. O’Dell, “Band-structure loops and multistabilityin cavity QED,” Phys. Rev. A 83, 063606 (2011).
[CrossRef]

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[CrossRef]

C. Jiang, B. Chen, and K. D. Zhu, “Tunable pulse delay and advancement device based on a cavity electromechanical system,” Europhys. Lett. 94, 38002 (2011).
[CrossRef]

J. Li, L. O’Faolain, I. H. Rey, and T. F. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express 19, 4458–4463 (2011).
[CrossRef]

2010 (5)

S. Huang and G. S. Agarwal, “Normal-mode splitting and antibunching in Stokes and anti-Stokes processes in cavity optomechanics: radiation-pressure-induced four-wave-mixing cavity optomechanics,” Phys. Rev. A 81, 033830 (2010).
[CrossRef]

R. Kanamoto and P. Meystre, “Optomechanics of a quantum-degenerate Fermi gas,” Phys. Rev. Lett. 104, 063601 (2010).
[CrossRef]

T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature 463, 72–75 (2010).
[CrossRef]

G. S. Agarwal and S. M. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[CrossRef]

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

2009 (1)

S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

2008 (7)

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008).
[CrossRef]

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Phys. 4, 555–560 (2008).
[CrossRef]

J. D. Teufel, J. W. Harlow, C. A. Regal, and K. W. Lehnert, “Dynamical backaction of microwave fields on a nanomechanical oscillator,” Phys. Rev. Lett. 101, 197203 (2008).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[CrossRef]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[CrossRef]

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

2007 (3)

S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
[CrossRef]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[CrossRef]

2006 (2)

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

A. Wiberg, P. P. Millán, M. V. Andrés, and P. O. Hedekvist, “Microwave-photonic frequency multiplication utilizing optical four-wave mixing and fiber Bragg gratings,” J. Lightwave Technol. 24, 329–334 (2006).
[CrossRef]

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

1997 (3)

S. Diez, C. Schmidt, R. Ludwig, H. G. Weber, K. Obermann, S. Kindt, I. Koltchanov, and K. Petermann, “Four-wave mixing in semiconductor optical amplifiers for frequency conversion and fast optical switching,” IEEE J. Sel. Top. Quantum Electron. 3, 1131–1145 (1997).
[CrossRef]

K. Kitayama, “Highly stabilized millimeter-wave generation by using fiber-optic frequency-tunable comb generator,” J. Lightwave Technol. 15, 883–893 (1997).
[CrossRef]

B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Enhanced nondegenerate four-wave mixing owing to electromagnetically induced transparency in a spectral hole-burning crystal,” Opt. Lett. 22, 1138–1140 (1997).
[CrossRef]

1996 (2)

1994 (1)

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[CrossRef]

1991 (1)

K.-J. Boller, A. Imamoǧlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef]

1990 (1)

S. E. Harris, J. E. Field, and A. Imamoǧlu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef]

1988 (1)

B. I. Greence, J. F. Mueller, J. Orenstein, D. H. Rapkine, S. S. Rink, and M. Thakur, “Phonon-mediated optical nonlinearity in polydiacetylene,” Phys. Rev. Lett. 61, 325–328 (1988).
[CrossRef]

1985 (2)

M. D. Reid and D. F. Walls, “Generation of squeezed states via degenerate four-wave mixing,” Phys. Rev. A 31, 1622–1635 (1985).
[CrossRef]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef]

1983 (1)

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550–1553 (1983).
[CrossRef]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Normal-mode splitting and antibunching in Stokes and anti-Stokes processes in cavity optomechanics: radiation-pressure-induced four-wave-mixing cavity optomechanics,” Phys. Rev. A 81, 033830 (2010).
[CrossRef]

G. S. Agarwal and S. M. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[CrossRef]

Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[CrossRef]

Andrés, M. V.

Anetsberger, G.

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

Arcizet, O.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

Aspelmeyer, M.

S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

Bahrampour, A. R.

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Boller, K.-J.

K.-J. Boller, A. Imamoǧlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef]

Bourzeix, S.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008), pp. 297–304.

Brennecke, F.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[CrossRef]

Briant, T.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

Chan, J.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[CrossRef]

Chang, D. E.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[CrossRef]

Chen, B.

C. Jiang, B. Chen, and K. D. Zhu, “Tunable pulse delay and advancement device based on a cavity electromechanical system,” Europhys. Lett. 94, 38002 (2011).
[CrossRef]

Chen, J. P.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[CrossRef]

Cicak, K.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[CrossRef]

Clerk, A. A.

T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature 463, 72–75 (2010).
[CrossRef]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[CrossRef]

Cohadon, P.-F.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

Delèglise, S.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Diez, S.

S. Diez, C. Schmidt, R. Ludwig, H. G. Weber, K. Obermann, S. Kindt, I. Koltchanov, and K. Petermann, “Four-wave mixing in semiconductor optical amplifiers for frequency conversion and fast optical switching,” IEEE J. Sel. Top. Quantum Electron. 3, 1131–1145 (1997).
[CrossRef]

Doherty, A. C.

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008).
[CrossRef]

Donner, T.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[CrossRef]

Dorsel, A.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550–1553 (1983).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Eichenfield, M.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[CrossRef]

Esslinger, T.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[CrossRef]

Fabre, C.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[CrossRef]

Field, J. E.

S. E. Harris, J. E. Field, and A. Imamoǧlu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Schematic of a nanomechanical resonator capacitively coupled to a microwave cavity denoted by equivalent inductance L and equivalent capacitance C in the presence of a strong pump field ωp and a weak probe field ωr; (b) equivalent circuit.

Fig. 2.
Fig. 2.

Steady-state intracavity photon number as a function of (a) cavity–pump detuning Δp for Pp=5, 50, 100, and 120 nW (from bottom to top) and (b) pump power for Δp=ωn. Other parameters used are ωc=2π×7.5GHz, ωn=2π×6.3MHz, κ=2π×600kHz, λ=250Hz, and γn=40Hz.

Fig. 3.
Fig. 3.

(a) FWM intensity as a function of cavity–pump detuning with probe–cavity detuning Δr=0, (b) the normalized magnitude of the cavity transmission as a function of the probe–cavity detuning when the cavity is driven on the red sideband. The effect of electromagnetically induced transparency (EIT) appears in this case. Other parameters are Pp=50nW, ωc=2π×7.5GHz, ωn=2π×6.3MHz, κ=2π×600kHz, λ=250Hz, and γn=40Hz.

Fig. 4.
Fig. 4.

FWM intensity versus probe–cavity detuning with Δp=ωn for (a) Pp=4, 10, and 40 pW (from bottom to top) and (b) Pp=50, 120, and 160 nW (from bottom to top). Other parameters used are ωc=2π×7.5GHz, ωn=2π×6.3MHz, κ=2π×600kHz, λ=250Hz, and γn=40Hz.

Fig. 5.
Fig. 5.

FWM intensity as a function of pump power Pp when Δp=ωn and Δr=0. Other parameters used are ωc=2π×7.5GHz, ωn=2π×6.3MHz, κ=2π×600kHz, λ=250Hz, and γn=40Hz.

Fig. 6.
Fig. 6.

FWM conversion efficiency as a function of probe–cavity detuning when Δp=ωn and Pp=100pW. Other parameters used are ωc=2π×7.5GHz, ωn=2π×6.3MHz, κ=2π×600kHz, λ=250Hz, and γn=40Hz.

Equations (10)

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H=Δpaa+ωnbbλaa(b+b)+i(Epa+Ep*a)+i(Eraeiδt+Er*aeiδt),
dadt=(iΔp+κ)a+iλaQ+Ep+Ereiδt,
d2Qdt2+γndQdt+ωn2Q=2ωnλaa,
a+=κ+θi(δ+Δp)(κiδ)2(θiΔp)2βEr,
a=iαη*ωn(κ+iΔpiαωnnp)2·Ep2Er(κ+iδ)2(θ*+iΔp)2β*,
np[κ2+(Δpωnαnp)2]=|Ep|2.
aout(t)=(Ep2κa0)e-iωpt+(Er2κa+)ei(δ+ωp)t2κaei(δωp)t=(Ep-2κa0)e-iωpt+(Er2κa+)eiωrt2κaei(2ωpωr)t.
T=Er2κa+Er=1-2κκ+θi(δ+Δp)(κiδ)2(θiΔp)2β.
FWM=|2κaEr|2=|2κiαη*ωn(κ+iΔpiαωnnp)2·Ep2(κ+iδ)2(θ*+iΔp)2β*|2.
ηFWM=PiPr=(2ωpωr)|2κa|2Pr=|2κaEr|2,

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