Abstract

We present a theoretical scheme to realize high-sensitive mass detection in a dispersive optomechanical system (DOMS) via nonlinear sum-sideband. In this scheme, DOMS assisted by a degenerate parametric amplifier (DPA) provides a well-established optomechanical circumstance, where nonlinear optomechanical interaction between cavity mode and mechanical mode of dielectric membrane is expected for creating the frequency components at optical sum-sideband. Such a scheme for mass detection mainly relies on monitoring the conversion efficiency of generated sum-sideband after the added mass is absorbed on the dielectric membrane. Using experimentally achievable parameters, we find that the conversion efficiency of sum-sideband and the sensitivity of mass detection can be simultaneously improved when the nonlinear gain of DPA increases. Furthermore, our results also demonstrate that this mass detection of DOMS can reach femtogram (fg) level resolution, when the method of mass detection relies on a direct relationship between maximum efficiency of sum-sideband and mass-change of membrane.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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  4. J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
    [Crossref] [PubMed]
  5. Q. Lin, B. He, and M. Xiao, “Mass sensing by detecting the quadrature of a coupled light field,” Phys. Rev. A 96, 043812 (2017).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  26. J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
    [Crossref]
  27. S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
    [Crossref]
  28. M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
    [Crossref]
  29. S. Liu, W. X. Yang, T. Shui, Z. Zhu, and A. X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7, 17637 (2017).
    [Crossref] [PubMed]
  30. S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
    [Crossref]
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    [Crossref]
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    [Crossref]

2018 (3)

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

L. Li, W.-X. Yang, Y. Zhang, T. Shui, A.-X. Chen, and Z. Jiang, “Enhanced generation of charge-dependent second-order sideband and high-sensitivity charge sensors in a gain-cavity-assisted optomechanical system,” Phys. Rev. A 98, 063840 (2018).
[Crossref]

S. Liu, W.-X. Yang, Z. Zhu, T. Shui, and L. Li, “Quadrature squeezing of a higher-order sideband spectrum in cavity optomechanics,” Opt. Lett. 43, 9–12 (2018).
[Crossref] [PubMed]

2017 (3)

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

S. Liu, W. X. Yang, T. Shui, Z. Zhu, and A. X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7, 17637 (2017).
[Crossref] [PubMed]

Q. Lin, B. He, and M. Xiao, “Mass sensing by detecting the quadrature of a coupled light field,” Phys. Rev. A 96, 043812 (2017).
[Crossref]

2016 (2)

G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93, 043844 (2016).
[Crossref]

H. Xiong, L.-G. Si, X.-Y. Lü, and Y. Wu, “Optomechanically induced sum sideband generation,” Opt. Express 24, 5773–5783 (2016).
[Crossref] [PubMed]

2014 (2)

2013 (3)

J. J. Li and K. D. Zhu, “All-optical mass sensing with coupled mechanical resonator systems,” Phys. Rep. 525, 223–254 (2013).
[Crossref]

F. Liu and M. Hossein-Zadeh, “Mass sensing with optomechanical oscillation,” IEEE Sensors 13, 146–147 (2013).
[Crossref]

F. Liu, S. Alaie, Z. C. Leseman, and M. Hossein-Zadeh, “Sub-pg mass sensing and measurement with an optomechanical oscillator,” Opt. Express 21, 19555–19567 (2013).
[Crossref] [PubMed]

2012 (3)

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref] [PubMed]

J. J. Li and K. D. Zhu, “Nonlinear optical mass sensor with an optomechanical microresonator,” Appl. Phys. Lett. 101, 141905 (2012).
[Crossref]

M. Aspelmeyer, P. Meystre, and K. C. Schwab, “Quantum optomechanics,” Phys. Today 65, 29–35 (2012).
[Crossref]

2011 (3)

Z. Yie, M. A. Zielke, C. B. Burgner, and K. L. Turner, “Comparison of parametric and linear mass detection in the presence of detection noise,” J. Micromech. Microeng. 21, 025027 (2011).
[Crossref]

K. L. Turner, C. Burgner, Y. Zi, S. W. Shaw, and N. Miller, “Nonlinear dynamics of MEMS systems,” AIP Conference Proceedings 1339, 111–117 (2011).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
[Crossref]

2010 (3)

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

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

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]

2009 (2)

M. D. Dai, K. Eom, and C. W. Kim, “Nanomechanical mass detection using nonlinear oscillations,” Appl. Phys. Lett. 95, 203104 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[Crossref]

2008 (3)

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature (London) 452, 72–76 (2008).
[Crossref]

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
[Crossref] [PubMed]

2007 (1)

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Noise squeezing in a nanomechanical duffing resonator,” Phys. Rev. Lett. 98, 078103 (2007).
[Crossref] [PubMed]

2006 (2)

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

E. Buks and B. Yurke, “Mass detection with a nonlinear nanomechanical resonator,” Phys. Rev. E 74, 046619 (2006).
[Crossref]

2005 (1)

B. Ilic, Y. Yang, K. Aubin, R. Reichenbach, S. Krylov, and H. G. Craighead, “Enumeration of DNA molecules bound to a nanomechanical oscillator,” Nano Lett. 5, 925–929 (2005).
[Crossref] [PubMed]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93, 043844 (2016).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
[Crossref]

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]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[Crossref]

Alaie, S.

Almog, R.

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Noise squeezing in a nanomechanical duffing resonator,” Phys. Rev. Lett. 98, 078103 (2007).
[Crossref] [PubMed]

Arcizet, O.

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

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

M. Aspelmeyer, P. Meystre, and K. C. Schwab, “Quantum optomechanics,” Phys. Today 65, 29–35 (2012).
[Crossref]

Aubin, K.

B. Ilic, Y. Yang, K. Aubin, R. Reichenbach, S. Krylov, and H. G. Craighead, “Enumeration of DNA molecules bound to a nanomechanical oscillator,” Nano Lett. 5, 925–929 (2005).
[Crossref] [PubMed]

Bachtold, A.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref] [PubMed]

Bhattacharya, M.

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

Briant, T.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Buks, E.

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Noise squeezing in a nanomechanical duffing resonator,” Phys. Rev. Lett. 98, 078103 (2007).
[Crossref] [PubMed]

E. Buks and B. Yurke, “Mass detection with a nonlinear nanomechanical resonator,” Phys. Rev. E 74, 046619 (2006).
[Crossref]

Burgner, C.

K. L. Turner, C. Burgner, Y. Zi, S. W. Shaw, and N. Miller, “Nonlinear dynamics of MEMS systems,” AIP Conference Proceedings 1339, 111–117 (2011).
[Crossref]

Burgner, C. B.

Z. Yie, M. A. Zielke, C. B. Burgner, and K. L. Turner, “Comparison of parametric and linear mass detection in the presence of detection noise,” J. Micromech. Microeng. 21, 025027 (2011).
[Crossref]

Butsch, A.

Ceballos, G.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref] [PubMed]

Chaste, J.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref] [PubMed]

Chen, A. X.

S. Liu, W. X. Yang, T. Shui, Z. Zhu, and A. X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7, 17637 (2017).
[Crossref] [PubMed]

Chen, A.-X.

L. Li, W.-X. Yang, Y. Zhang, T. Shui, A.-X. Chen, and Z. Jiang, “Enhanced generation of charge-dependent second-order sideband and high-sensitivity charge sensors in a gain-cavity-assisted optomechanical system,” Phys. Rev. A 98, 063840 (2018).
[Crossref]

Cohadon, P.-F.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Craighead, H. G.

B. Ilic, Y. Yang, K. Aubin, R. Reichenbach, S. Krylov, and H. G. Craighead, “Enumeration of DNA molecules bound to a nanomechanical oscillator,” Nano Lett. 5, 925–929 (2005).
[Crossref] [PubMed]

Dai, M. D.

M. D. Dai, K. Eom, and C. W. Kim, “Nanomechanical mass detection using nonlinear oscillations,” Appl. Phys. Lett. 95, 203104 (2009).
[Crossref]

Deléglise, S.

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

Eichler, A.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref] [PubMed]

Eom, K.

M. D. Dai, K. Eom, and C. W. Kim, “Nanomechanical mass detection using nonlinear oscillations,” Appl. Phys. Lett. 95, 203104 (2009).
[Crossref]

Francais, O.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise(Springer-Verlag, 2000).
[Crossref]

Gavartin, E.

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

Girvin, S. M.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature (London) 452, 72–76 (2008).
[Crossref]

Harris, J. G. E.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature (London) 452, 72–76 (2008).
[Crossref]

He, B.

Q. Lin, B. He, and M. Xiao, “Mass sensing by detecting the quadrature of a coupled light field,” Phys. Rev. A 96, 043812 (2017).
[Crossref]

Heidmann, A.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Hossein-Zadeh, M.

Huang, S.

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93, 043844 (2016).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
[Crossref]

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]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[Crossref]

Ilic, B.

B. Ilic, Y. Yang, K. Aubin, R. Reichenbach, S. Krylov, and H. G. Craighead, “Enumeration of DNA molecules bound to a nanomechanical oscillator,” Nano Lett. 5, 925–929 (2005).
[Crossref] [PubMed]

Jayich, A. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature (London) 452, 72–76 (2008).
[Crossref]

Jensen, K.

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
[Crossref] [PubMed]

Jiang, Z.

L. Li, W.-X. Yang, Y. Zhang, T. Shui, A.-X. Chen, and Z. Jiang, “Enhanced generation of charge-dependent second-order sideband and high-sensitivity charge sensors in a gain-cavity-assisted optomechanical system,” Phys. Rev. A 98, 063840 (2018).
[Crossref]

Kim, C. W.

M. D. Dai, K. Eom, and C. W. Kim, “Nanomechanical mass detection using nonlinear oscillations,” Appl. Phys. Lett. 95, 203104 (2009).
[Crossref]

Kim, K.

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
[Crossref] [PubMed]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

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

Koehler, J. R.

Krylov, S.

B. Ilic, Y. Yang, K. Aubin, R. Reichenbach, S. Krylov, and H. G. Craighead, “Enumeration of DNA molecules bound to a nanomechanical oscillator,” Nano Lett. 5, 925–929 (2005).
[Crossref] [PubMed]

Leseman, Z. C.

Li, J. J.

J. J. Li and K. D. Zhu, “All-optical mass sensing with coupled mechanical resonator systems,” Phys. Rep. 525, 223–254 (2013).
[Crossref]

J. J. Li and K. D. Zhu, “Nonlinear optical mass sensor with an optomechanical microresonator,” Appl. Phys. Lett. 101, 141905 (2012).
[Crossref]

Li, L.

L. Li, W.-X. Yang, Y. Zhang, T. Shui, A.-X. Chen, and Z. Jiang, “Enhanced generation of charge-dependent second-order sideband and high-sensitivity charge sensors in a gain-cavity-assisted optomechanical system,” Phys. Rev. A 98, 063840 (2018).
[Crossref]

S. Liu, W.-X. Yang, Z. Zhu, T. Shui, and L. Li, “Quadrature squeezing of a higher-order sideband spectrum in cavity optomechanics,” Opt. Lett. 43, 9–12 (2018).
[Crossref] [PubMed]

Lin, Q.

Q. Lin, B. He, and M. Xiao, “Mass sensing by detecting the quadrature of a coupled light field,” Phys. Rev. A 96, 043812 (2017).
[Crossref]

Liu, F.

Liu, S.

S. Liu, W.-X. Yang, Z. Zhu, T. Shui, and L. Li, “Quadrature squeezing of a higher-order sideband spectrum in cavity optomechanics,” Opt. Lett. 43, 9–12 (2018).
[Crossref] [PubMed]

S. Liu, W. X. Yang, T. Shui, Z. Zhu, and A. X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7, 17637 (2017).
[Crossref] [PubMed]

Lü, X.-Y.

Mackowski, J.-M.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature (London) 452, 72–76 (2008).
[Crossref]

Meystre, P.

M. Aspelmeyer, P. Meystre, and K. C. Schwab, “Quantum optomechanics,” Phys. Today 65, 29–35 (2012).
[Crossref]

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

Michel, C.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Miller, N.

K. L. Turner, C. Burgner, Y. Zi, S. W. Shaw, and N. Miller, “Nonlinear dynamics of MEMS systems,” AIP Conference Proceedings 1339, 111–117 (2011).
[Crossref]

Moser, J.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref] [PubMed]

Noskov, R. E.

Pinard, L.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Pinard, M.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Reichenbach, R.

B. Ilic, Y. Yang, K. Aubin, R. Reichenbach, S. Krylov, and H. G. Craighead, “Enumeration of DNA molecules bound to a nanomechanical oscillator,” Nano Lett. 5, 925–929 (2005).
[Crossref] [PubMed]

Rivière, R.

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

Rousseau, L.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Rurali, R.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref] [PubMed]

Russell, P. St.J.

Sankey, J. C.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

Schliesser, A.

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

Schwab, K. C.

M. Aspelmeyer, P. Meystre, and K. C. Schwab, “Quantum optomechanics,” Phys. Today 65, 29–35 (2012).
[Crossref]

Shaw, S. W.

K. L. Turner, C. Burgner, Y. Zi, S. W. Shaw, and N. Miller, “Nonlinear dynamics of MEMS systems,” AIP Conference Proceedings 1339, 111–117 (2011).
[Crossref]

Shtempluck, O.

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Noise squeezing in a nanomechanical duffing resonator,” Phys. Rev. Lett. 98, 078103 (2007).
[Crossref] [PubMed]

Shui, T.

L. Li, W.-X. Yang, Y. Zhang, T. Shui, A.-X. Chen, and Z. Jiang, “Enhanced generation of charge-dependent second-order sideband and high-sensitivity charge sensors in a gain-cavity-assisted optomechanical system,” Phys. Rev. A 98, 063840 (2018).
[Crossref]

S. Liu, W.-X. Yang, Z. Zhu, T. Shui, and L. Li, “Quadrature squeezing of a higher-order sideband spectrum in cavity optomechanics,” Opt. Lett. 43, 9–12 (2018).
[Crossref] [PubMed]

S. Liu, W. X. Yang, T. Shui, Z. Zhu, and A. X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7, 17637 (2017).
[Crossref] [PubMed]

Si, L.-G.

Thompson, J. D.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature (London) 452, 72–76 (2008).
[Crossref]

Turner, K. L.

Z. Yie, M. A. Zielke, C. B. Burgner, and K. L. Turner, “Comparison of parametric and linear mass detection in the presence of detection noise,” J. Micromech. Microeng. 21, 025027 (2011).
[Crossref]

K. L. Turner, C. Burgner, Y. Zi, S. W. Shaw, and N. Miller, “Nonlinear dynamics of MEMS systems,” AIP Conference Proceedings 1339, 111–117 (2011).
[Crossref]

Uys, H.

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

Weis, S.

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

Wu, Y.

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

H. Xiong, L.-G. Si, X.-Y. Lü, and Y. Wu, “Optomechanically induced sum sideband generation,” Opt. Express 24, 5773–5783 (2016).
[Crossref] [PubMed]

Xiao, M.

Q. Lin, B. He, and M. Xiao, “Mass sensing by detecting the quadrature of a coupled light field,” Phys. Rev. A 96, 043812 (2017).
[Crossref]

Xiong, H.

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

H. Xiong, L.-G. Si, X.-Y. Lü, and Y. Wu, “Optomechanically induced sum sideband generation,” Opt. Express 24, 5773–5783 (2016).
[Crossref] [PubMed]

Yang, C.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

Yang, W. X.

S. Liu, W. X. Yang, T. Shui, Z. Zhu, and A. X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7, 17637 (2017).
[Crossref] [PubMed]

Yang, W.-X.

S. Liu, W.-X. Yang, Z. Zhu, T. Shui, and L. Li, “Quadrature squeezing of a higher-order sideband spectrum in cavity optomechanics,” Opt. Lett. 43, 9–12 (2018).
[Crossref] [PubMed]

L. Li, W.-X. Yang, Y. Zhang, T. Shui, A.-X. Chen, and Z. Jiang, “Enhanced generation of charge-dependent second-order sideband and high-sensitivity charge sensors in a gain-cavity-assisted optomechanical system,” Phys. Rev. A 98, 063840 (2018).
[Crossref]

Yang, Y.

B. Ilic, Y. Yang, K. Aubin, R. Reichenbach, S. Krylov, and H. G. Craighead, “Enumeration of DNA molecules bound to a nanomechanical oscillator,” Nano Lett. 5, 925–929 (2005).
[Crossref] [PubMed]

Yie, Z.

Z. Yie, M. A. Zielke, C. B. Burgner, and K. L. Turner, “Comparison of parametric and linear mass detection in the presence of detection noise,” J. Micromech. Microeng. 21, 025027 (2011).
[Crossref]

Yurke, B.

E. Buks and B. Yurke, “Mass detection with a nonlinear nanomechanical resonator,” Phys. Rev. E 74, 046619 (2006).
[Crossref]

Zaitsev, S.

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Noise squeezing in a nanomechanical duffing resonator,” Phys. Rev. Lett. 98, 078103 (2007).
[Crossref] [PubMed]

Zettl, A.

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
[Crossref] [PubMed]

Zhang, Y.

L. Li, W.-X. Yang, Y. Zhang, T. Shui, A.-X. Chen, and Z. Jiang, “Enhanced generation of charge-dependent second-order sideband and high-sensitivity charge sensors in a gain-cavity-assisted optomechanical system,” Phys. Rev. A 98, 063840 (2018).
[Crossref]

Zhu, K. D.

J. J. Li and K. D. Zhu, “All-optical mass sensing with coupled mechanical resonator systems,” Phys. Rep. 525, 223–254 (2013).
[Crossref]

J. J. Li and K. D. Zhu, “Nonlinear optical mass sensor with an optomechanical microresonator,” Appl. Phys. Lett. 101, 141905 (2012).
[Crossref]

Zhu, Z.

S. Liu, W.-X. Yang, Z. Zhu, T. Shui, and L. Li, “Quadrature squeezing of a higher-order sideband spectrum in cavity optomechanics,” Opt. Lett. 43, 9–12 (2018).
[Crossref] [PubMed]

S. Liu, W. X. Yang, T. Shui, Z. Zhu, and A. X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7, 17637 (2017).
[Crossref] [PubMed]

Zi, Y.

K. L. Turner, C. Burgner, Y. Zi, S. W. Shaw, and N. Miller, “Nonlinear dynamics of MEMS systems,” AIP Conference Proceedings 1339, 111–117 (2011).
[Crossref]

Zielke, M. A.

Z. Yie, M. A. Zielke, C. B. Burgner, and K. L. Turner, “Comparison of parametric and linear mass detection in the presence of detection noise,” J. Micromech. Microeng. 21, 025027 (2011).
[Crossref]

Zoller, P.

C. W. Gardiner and P. Zoller, Quantum Noise(Springer-Verlag, 2000).
[Crossref]

Zwickl, B. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature (London) 452, 72–76 (2008).
[Crossref]

AIP Conference Proceedings (1)

K. L. Turner, C. Burgner, Y. Zi, S. W. Shaw, and N. Miller, “Nonlinear dynamics of MEMS systems,” AIP Conference Proceedings 1339, 111–117 (2011).
[Crossref]

Appl. Phys. Lett. (2)

J. J. Li and K. D. Zhu, “Nonlinear optical mass sensor with an optomechanical microresonator,” Appl. Phys. Lett. 101, 141905 (2012).
[Crossref]

M. D. Dai, K. Eom, and C. W. Kim, “Nanomechanical mass detection using nonlinear oscillations,” Appl. Phys. Lett. 95, 203104 (2009).
[Crossref]

Appl. Phys. Rev. (1)

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

IEEE Sensors (1)

F. Liu and M. Hossein-Zadeh, “Mass sensing with optomechanical oscillation,” IEEE Sensors 13, 146–147 (2013).
[Crossref]

J. Micromech. Microeng. (1)

Z. Yie, M. A. Zielke, C. B. Burgner, and K. L. Turner, “Comparison of parametric and linear mass detection in the presence of detection noise,” J. Micromech. Microeng. 21, 025027 (2011).
[Crossref]

Nano Lett. (1)

B. Ilic, Y. Yang, K. Aubin, R. Reichenbach, S. Krylov, and H. G. Craighead, “Enumeration of DNA molecules bound to a nanomechanical oscillator,” Nano Lett. 5, 925–929 (2005).
[Crossref] [PubMed]

Nat. Nanotechnol. (2)

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
[Crossref] [PubMed]

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref] [PubMed]

Nat. Phys. (1)

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

Nature (London) (1)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature (London) 452, 72–76 (2008).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Optica (1)

Phys. Rep. (1)

J. J. Li and K. D. Zhu, “All-optical mass sensing with coupled mechanical resonator systems,” Phys. Rep. 525, 223–254 (2013).
[Crossref]

Phys. Rev. A (8)

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93, 043844 (2016).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
[Crossref]

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

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]

L. Li, W.-X. Yang, Y. Zhang, T. Shui, A.-X. Chen, and Z. Jiang, “Enhanced generation of charge-dependent second-order sideband and high-sensitivity charge sensors in a gain-cavity-assisted optomechanical system,” Phys. Rev. A 98, 063840 (2018).
[Crossref]

Q. Lin, B. He, and M. Xiao, “Mass sensing by detecting the quadrature of a coupled light field,” Phys. Rev. A 96, 043812 (2017).
[Crossref]

Phys. Rev. E (1)

E. Buks and B. Yurke, “Mass detection with a nonlinear nanomechanical resonator,” Phys. Rev. E 74, 046619 (2006).
[Crossref]

Phys. Rev. Lett. (2)

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantumlimited optomechanicalsensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Noise squeezing in a nanomechanical duffing resonator,” Phys. Rev. Lett. 98, 078103 (2007).
[Crossref] [PubMed]

Phys. Today (1)

M. Aspelmeyer, P. Meystre, and K. C. Schwab, “Quantum optomechanics,” Phys. Today 65, 29–35 (2012).
[Crossref]

Rev. Mod. Phys. (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Sci. Rep. (1)

S. Liu, W. X. Yang, T. Shui, Z. Zhu, and A. X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7, 17637 (2017).
[Crossref] [PubMed]

Science (1)

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

Other (1)

C. W. Gardiner and P. Zoller, Quantum Noise(Springer-Verlag, 2000).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 (a) Schematic diagram of all-optical mass detection based on DOMS. A DPA is embedded in the optomechanical cavity, while a thin dielectric membrane is located at an antinode of the cavity field. When DOMS is driven by a strong control field (with frequency ωd) and a relatively weak probe pulse (with frequency ωp), the added mass deposited on the dielectric membrane can be measured by monitoring the conversion efficiency of sum-sideband. (b) Frequency spectrogram of nonlinear sum sideband in DOMS. The output fields at sum-sideband (with frequency ± Ω + = ± ( Δ p + Δ G ) in a frame rotating at ωd) are used to weigh the added mass deposited on the dielectric membrane.
Fig. 2
Fig. 2 (a) The conversion efficiency ηs of upper sum-sideband and (b) the sensitivity S (in units of pg) of mass detection as a function of the probe-pulsed detuning Δp for different signal field detuning of DPA. Other parameters are m m = 100 p g, ω m = 2 π × 0.1 M H z, Q = ω m / Γ m = π × 10 4, L = 67 m m, T = 50 K, κ = 0.2 ω m, Δ c = 2 ω m, η L = 0.499, ε p = 0.05 ε d, P d = 230 μ W, δ m = 0 p g and G a = 0.01 κ, respectively.
Fig. 3
Fig. 3 Contour maps of (a) the conversion efficiency ηs of upper sum-sideband and (b) the sensitivity S (in units of pg) of mass detection as a function of the probe-pulsed detuning Δp and nonlinear gain Ga of DPA. Other parameters are the same as in Fig. 2 except for Δ G = 0.2 ω m.
Fig. 4
Fig. 4 Contour maps of (a) the conversion efficiency ηs of upper sum-sideband and (b) the sensitivity S (in units of pg) of mass detection as a function of the probe-pulsed detuning Δp and the added mass δm. Other parameters are the same as in Fig. 2 except for G a = 1 κ and Δ G = 0.2 ω m.
Fig. 5
Fig. 5 (a) The maximum efficiency η s m a x of upper sum-sideband and (b) the maximum sensitivity S m a x (in units of pg) of mass detection versus the added mass δm. Other parameters are the same as in Fig. 2 except for G a = 1 κ and Δ G = 0.2 ω m.
Fig. 6
Fig. 6 (a) The conversion efficiency ηs of upper sum-sideband and (b) the sensitivity S (in units of pg) of mass detection varying with the probe-pulsed detuning Δp for different materials-induced loss Γ m m. Other parameters are the same as in Fig. 2 except for Γ m d = 20 H z, G a = 1 κ, Δ G = 0.2 ω m and δ m = 20 p g.

Equations (26)

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

H i n t = p ^ 2 2 m + 1 2 m ω m 2 x ^ 2 + Δ c a ^ a ^ + G a ^ a ^ x ^ 2 + i η L κ ε d ( a ^ a ^ ) + i η L κ ε p ( a ^ e i Δ p t a ^ e i Δ p t ) + i G a ( a ^ 2 e i Δ G t a ^ 2 e i Δ G t ) .
t x ^ = p ^ m ,
t p ^ = m ω m 2 x ^ Γ m p ^ 2 G a ^ a ^ x ^ + F ^ t h ,
t a ^ = [ κ 2 + i ( Δ c + G X ^ ) ] a ^ + 2 G a a ^ e i Δ G t + η L κ ( ε c + ε p e i Δ p t ) + a ^ i n ,
t X ^ = Q ^ m ,
t P ^ = ( 2 G a ^ a ^ + m ω m 2 ) Q ^ 2 Γ m P ^ + F ^ '   t h ,
t Q ^ = ( 4 G a ^ a ^ + 2 m ω m 2 ) X ^ + 2 P ^ m Γ m Q ^ ,
t δ a = ( κ 2 + i Δ ¯ c ) δ a i G a δ X i G δ a δ X + 2 G a a 0   * e i Δ G t + η L κ ( ε p e i Δ p t ) ,
t δ X = δ Q m ,
t δ P = m ω m 2 δ Q 2 Γ m δ P 2 G ( | a 0 | 2 δ Q + δ a * a 0 δ Q + a 0 * δ a δ Q + δ a * δ a δ Q ) ,
t δ Q = 2 m ω m 2 δ X Γ m δ Q + 2 δ P m 4 G ( | a 0 | 2 δ X + δ a * a 0 X 0 + a 0 * δ a X 0 + δ a * δ a X 0 + δ a * a 0 δ X + a 0 * δ a δ X + δ a * δ a δ X ) .
δ a = a Δ p + e i Δ p t + a Δ p e i Δ p t + a Δ G + e i Δ G t + a Δ G e i Δ G t + a s u m + e i Ω + t + a s u m e i Ω + t ,
δ O = O Δ p e i Δ p t + O Δ p * e i Δ p t + O Δ G e i Δ G t + O Δ G * e i Δ G t + O s u m e i Ω + t + O s u m * e i Ω + t ,
a Δ p + = i | a 0 | 2 β 1 ( Δ p ) + β 2 ( Δ p ) D ( Δ p ) η L κ ε p , a Δ p = i a 0 2 β 1 * ( Δ p ) D * ( Δ p ) η L κ ε p ,
a Δ G + = i | a 0 | 2 β 1 ( Δ G ) + β 2 ( Δ G ) D ( Δ G ) ( 2 a 0 * G a ) , a Δ G = i a 0 2 β 1 * ( Δ G ) D * ( Δ G ) ( 2 a 0 G a ) ,
X Δ p = a 0 * f 2 ( Δ p ) β 1 ( Δ p ) G D ( Δ p ) η L κ ε p , X Δ G = a 0 * f 2 ( Δ G ) β 1 ( Δ G ) G D ( Δ G ) ( 2 a 0 * G a ) ,
a s u m + = i G D ( Δ p ) { y s + i a 0 2 β 1 ( Ω + ) y c [ i | a 0 | 2 β 1 ( Ω + ) + β 2 ( Ω + ) ] y d } ,
f 1 ( z ) = κ 2 + i Δ ¯ c i z , f 2 ( z ) = κ 2 i Δ ¯ c i z , f 3 ( z ) = 2 Γ m i z ,
f 4 ( z ) = ( 4 α + 2 ) ω m 2 i z ( Γ m i z ) , β 1 ( z ) = 4 G 2 X 0 f 3 ( z ) m f 2 ( z ) ,
β 2 ( z ) = f 3 ( z ) f 4 ( z ) i ( 4 α + 2 ) z ω m 2 , D ( z ) = 2 Δ ¯ | a 0 | 2 β 1 ( z ) + f 1 ( z ) β 2 ( z ) ,
y a = i 2 G ( a 0 * a Δ G + Δ p X Δ p + a 0 * a Δ p + Δ G X Δ G + a Δ G * a 0 Δ p X Δ p + a Δ p * a 0 Δ G X Δ G ) ,
y b = 4 G m ( a Δ G * a Δ p + X 0 + a Δ p * a Δ G + X 0 + a 0 * a Δ G + X Δ p + a 0 * a Δ p + X Δ G + a Δ G * a 0 X Δ p + a Δ p * a 0 X Δ G ) ,
y s = 2 a 0 y a m + a 0 f 3 ( Ω + ) y b , y c = a Δ p * X Δ G + a Δ G * X Δ p , y d = a Δ p + X Δ G + a Δ G + X Δ p 2 G a a Δ p * / ( i G ) .
S o u t = η c κ a 0 ε c + ( η c κ a Δ p + ε p ) e i Δ p t + η c κ a Δ p e i Δ p t + η c κ a Δ G + e i Δ G t + η c κ a Δ G e i Δ G t + η c κ a s u m + e i Ω + t + η c κ a s u m e i Ω + t ,
η s = | η c κ a s u m + ε p | .
S = | d η s d δ m | .

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