Abstract

We investigate theoretically the generation and enhancement of sum sideband in a quadratically coupled optomechanical system with parametric interactions. It is shown that the generation of frequency components at the sum sideband stems from the nonlinear optomechanical interactions via two-phonon processes in the quadratically coupled optomechanical system, while an optical parametric amplifier (OPA) inside the system can considerably improve the sum sideband generation (SSG). The dependence of SSG on the system parameters, including the power of the control field, the frequency detuning of the probe fields and the nonlinear gain of OPA are analyzed in detail. Our analytic calculation indicates that the SSG can be obtained even under weak driven fields and greatly enhanced via meeting the matching conditions. The effect of SSG may have potential applications for achieving measurement of electric charge (or other weak forces) with higher precision and on-chip manipulation of light propagation.

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

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2019 (5)

B. Chen, L. Shang, X.-F. Wang, J.-B. Chen, H.-B. Xue, X. Liu, and J. Zhang, “Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling,” Phys. Rev. A 99(6), 063810 (2019).
[Crossref]

W.-A. Li and G.-Y. Huang, “Enhancement of optomechanically induced sum sideband using parametric interactions,” Phys. Rev. A 100(2), 023838 (2019).
[Crossref]

T.-X. Lu, Y.-F. Jiao, H.-L. Zhang, F. Saif, and H. Jing, “Selective and switchable optical amplification with mechanical driven oscillators,” Phys. Rev. A 100(1), 013813 (2019).
[Crossref]

S.-P Liu, B. Liu, and W.-X. Yang, “Highly sensitive mass detection based on nonlinear sum-sideband in a dispersive optomechanical system,” Opt. Express 27(4), 3909–3919 (2019).
[Crossref]

X.-Y. Wang, L.-G. Si, Z.-X. Liu, X.-H. Lu, and Y. Wu, “Tunable optical amplification arises from blue detuning in a quadratically coupled optomechanical system,” J. Opt. Soc. Am. B 36(5), 1355 (2019).
[Crossref]

2018 (8)

H. Zhang, F. Saif, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26(19), 25199–25210 (2018).
[Crossref]

H. Jing, H. Lü, S. K. özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5(11), 1424 (2018).
[Crossref]

L.-G. Si, L.-X. Guo, H. Xiong, and Y. Wu, “Tunable high-order-sideband generation and carrier-envelope-phase-dependent effects via microwave fields in hybrid electro-optomechanical systems,” Phys. Rev. A 97(2), 023805 (2018).
[Crossref]

C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. A. Clerk, F. Massel, M. J. Woolley, and M. A. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556(7702), 478–482 (2018).
[Crossref]

X.-Y. Lü, G.-L. Zhu, L.-L. Zheng, and Y. Wu, “Entanglement and quantum superposition induced by a single photon,” Phys. Rev. A 97(3), 033807 (2018).
[Crossref]

Z.-X. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Magnetic-field-dependent slow light in strontium atom-cavity system,” Appl. Phys. Lett. 112(11), 111109 (2018).
[Crossref]

Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a Kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
[Crossref]

S.-M. Huang and A.-X. Chen, “Improving the cooling of a mechanical oscillator in a dissipative optomechanical system with an optical parametric amplifier,” Phys. Rev. A 98(6), 063818 (2018).
[Crossref]

2017 (4)

Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-X. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95(1), 013843 (2017).
[Crossref]

L.-G. Si, H. Xiong, M. S. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95(3), 033803 (2017).
[Crossref]

S.-P. Liu, W.-X. Yang, T. Shui, Z.-H. Zhu, and A.-X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7(1), 17637 (2017).
[Crossref]

H. Xiong, Z.-X. Liu, and Y. Wu, “Highly sensitive optical sensor for precision measurement of electrical charges based on optomechanically induced difference-sideband generation,” Opt. Lett. 42(18), 3630 (2017).
[Crossref]

2016 (5)

H. Xiong, L.-G. Si, X.-Y. Lü, and Y. Wu, “Optomechanically induced sum sideband generation,” Opt. Express 24(6), 5773 (2016).
[Crossref]

G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93(4), 043844 (2016).
[Crossref]

J.-H. Li, R. Yu, C.-L. Ding, and Y. Wu, “$\mathcal {PT}$PT-symmetry-induced evolution of sharp asymmetric line shapes and high-sensitivity refractive index sensors in a three-cavity array,” Phys. Rev. A 93(2), 023814 (2016).
[Crossref]

Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
[Crossref]

G. A. Brawley, M. R. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. P. Bowen, “Nonlinear optomechanical measurement of mechanical motion,” Nat. Commun. 7(1), 10988 (2016).
[Crossref]

2015 (3)

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

X. Chen, Y.-C. Liu, P. Peng, Y.-Y. Zhi, and Y.-F. Xiao, “Cooling of macroscopic mechanical resonators in hybrid atom-optomechanical systems,” Phys. Rev. A 92(3), 033841 (2015).
[Crossref]

H. R. Hamedi, “Transient absorption and lasing without inversion in an artificial molecule via Josephson coupling energy,” Laser Phys. Lett. 12(3), 035201 (2015).
[Crossref]

2014 (5)

J. Ma, C. You, L.-G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Formation and manipulation of optomechanical chaos via a bichromatic driving,” Phys. Rev. A 90(4), 043839 (2014).
[Crossref]

A. Chen, “Coherent manipulation of spontaneous emission spectra in coupled semiconductor quantum well structures,” Opt. Express 22(22), 26991 (2014).
[Crossref]

B. Peng, S. K. Özdemir, F.-C. Lei, F. Monifi, M. Gianfreda, G.-L. Long, S.-H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

H. Wang, X. Gu, Y.-X. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically-induced transparency: photon transmission through an optomechanical device with a two-level system,” Phys. Rev. A 90(2), 023817 (2014).
[Crossref]

2013 (4)

C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in $\mathcal {PT}$PT-symmetric coupled oscillators,” Phys. Rev. A 88(6), 062111 (2013).
[Crossref]

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

J.-Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88(2), 023853 (2013).
[Crossref]

C. Jiang, H.-X. Liu, Y.-S. Cui, X.-W. Li, G.-B. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165 (2013).
[Crossref]

2012 (2)

H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

J.-Q. Liao, H. K. Cheung, and C. K. Law, “Spectrum of single-photon emission and scattering in cavity optomechanics,” Phys. Rev. A 85(2), 025803 (2012).
[Crossref]

2011 (4)

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83(4), 043826 (2011).
[Crossref]

D. E. McClelland, N. Mavalvala, Y. Chen, and R. Schnabel, “Advanced interferometry, quantum optics and optomechanics in gravitational wave detectors,” Laser Photonics Rev. 5, 677–696 (2011).
[Crossref]

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(7341), 69–73 (2011).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
[Crossref]

2010 (4)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

J. M. Dobrindt and T. J. Kippenberg, “Theoretical Analysis of Mechanical Displacement Measurement Using a Multiple Cavity Mode Transducer,” Phys. Rev. Lett. 104(3), 033901 (2010).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

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(9), 707–712 (2010).
[Crossref]

2009 (6)

S.-H. Ouyang, J.-Q. You, and F. Nori, “Cooling a mechanical resonator via coupling to a tunable double quantum dot,” Phys. Rev. B 79(7), 075304 (2009).
[Crossref]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref]

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5(7), 489–493 (2009).
[Crossref]

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3(4), 201–205 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79(1), 013821 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80(3), 033807 (2009).
[Crossref]

2008 (4)

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref]

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77(3), 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 452(7183), 72–75 (2008).
[Crossref]

A. Rai and G. S. Agarwal, “Quantum optical spring,” Phys. Rev. A 78(1), 013831 (2008).
[Crossref]

2007 (2)

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(9), 093902 (2007).
[Crossref]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and Probing Multipartite Macroscopic Entanglement with Light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

2003 (1)

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93(4), 043844 (2016).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83(4), 043826 (2011).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79(1), 013821 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80(3), 033807 (2009).
[Crossref]

A. Rai and G. S. Agarwal, “Quantum optical spring,” Phys. Rev. A 78(1), 013831 (2008).
[Crossref]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Asjad, M.

C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. A. Clerk, F. Massel, M. J. Woolley, and M. A. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556(7702), 478–482 (2018).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and Probing Multipartite Macroscopic Entanglement with Light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

Bawaj, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

Bender, C. M.

B. Peng, S. K. Özdemir, F.-C. Lei, F. Monifi, M. Gianfreda, G.-L. Long, S.-H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in $\mathcal {PT}$PT-symmetric coupled oscillators,” Phys. Rev. A 88(6), 062111 (2013).
[Crossref]

Bhattacharya, M.

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77(3), 033819 (2008).
[Crossref]

Biancofiore, C.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

Boisen, A.

G. A. Brawley, M. R. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. P. Bowen, “Nonlinear optomechanical measurement of mechanical motion,” Nat. Commun. 7(1), 10988 (2016).
[Crossref]

Bowen, W. P.

G. A. Brawley, M. R. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. P. Bowen, “Nonlinear optomechanical measurement of mechanical motion,” Nat. Commun. 7(1), 10988 (2016).
[Crossref]

Boyd, R. W.

R. W. Boyd, “Nonlinear Optics,” 3rd ed., (Academic Press, Burlington, MA, 2008).

Brawley, G. A.

G. A. Brawley, M. R. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. P. Bowen, “Nonlinear optomechanical measurement of mechanical motion,” Nat. Commun. 7(1), 10988 (2016).
[Crossref]

Brukner, C.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and Probing Multipartite Macroscopic Entanglement with Light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

Carmon, T.

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(7341), 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(7341), 69–73 (2011).
[Crossref]

Chen, A.

Chen, A.-X.

S.-M. Huang and A.-X. Chen, “Improving the cooling of a mechanical oscillator in a dissipative optomechanical system with an optical parametric amplifier,” Phys. Rev. A 98(6), 063818 (2018).
[Crossref]

S.-P. Liu, W.-X. Yang, T. Shui, Z.-H. Zhu, and A.-X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7(1), 17637 (2017).
[Crossref]

Chen, B.

B. Chen, L. Shang, X.-F. Wang, J.-B. Chen, H.-B. Xue, X. Liu, and J. Zhang, “Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling,” Phys. Rev. A 99(6), 063810 (2019).
[Crossref]

C. Jiang, H.-X. Liu, Y.-S. Cui, X.-W. Li, G.-B. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165 (2013).
[Crossref]

Chen, G.-B.

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(9), 093902 (2007).
[Crossref]

Chen, J.-B.

B. Chen, L. Shang, X.-F. Wang, J.-B. Chen, H.-B. Xue, X. Liu, and J. Zhang, “Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling,” Phys. Rev. A 99(6), 063810 (2019).
[Crossref]

Chen, X.

X. Chen, Y.-C. Liu, P. Peng, Y.-Y. Zhi, and Y.-F. Xiao, “Cooling of macroscopic mechanical resonators in hybrid atom-optomechanical systems,” Phys. Rev. A 92(3), 033841 (2015).
[Crossref]

Chen, Y.

D. E. McClelland, N. Mavalvala, Y. Chen, and R. Schnabel, “Advanced interferometry, quantum optics and optomechanics in gravitational wave detectors,” Laser Photonics Rev. 5, 677–696 (2011).
[Crossref]

Cheung, H. K.

J.-Q. Liao, H. K. Cheung, and C. K. Law, “Spectrum of single-photon emission and scattering in cavity optomechanics,” Phys. Rev. A 85(2), 025803 (2012).
[Crossref]

Clerk, A. A.

C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. A. Clerk, F. Massel, M. J. Woolley, and M. A. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556(7702), 478–482 (2018).
[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(9), 093902 (2007).
[Crossref]

Cui, Y.-S.

Damskägg, E.

C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. A. Clerk, F. Massel, M. J. Woolley, and M. A. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556(7702), 478–482 (2018).
[Crossref]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Ding, C.-L.

J.-H. Li, R. Yu, C.-L. Ding, and Y. Wu, “$\mathcal {PT}$PT-symmetry-induced evolution of sharp asymmetric line shapes and high-sensitivity refractive index sensors in a three-cavity array,” Phys. Rev. A 93(2), 023814 (2016).
[Crossref]

Dobrindt, J. M.

J. M. Dobrindt and T. J. Kippenberg, “Theoretical Analysis of Mechanical Displacement Measurement Using a Multiple Cavity Mode Transducer,” Phys. Rev. Lett. 104(3), 033901 (2010).
[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(7341), 69–73 (2011).
[Crossref]

Eisert, J.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and Probing Multipartite Macroscopic Entanglement with Light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

Fan, S.-H.

B. Peng, S. K. Özdemir, F.-C. Lei, F. Monifi, M. Gianfreda, G.-L. Long, S.-H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Favero, I.

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3(4), 201–205 (2009).
[Crossref]

Galassi, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Geng, Z.

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

Gianfreda, M.

B. Peng, S. K. Özdemir, F.-C. Lei, F. Monifi, M. Gianfreda, G.-L. Long, S.-H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in $\mathcal {PT}$PT-symmetric coupled oscillators,” Phys. Rev. A 88(6), 062111 (2013).
[Crossref]

Gigan, S.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and Probing Multipartite Macroscopic Entanglement with Light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

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 452(7183), 72–75 (2008).
[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(9), 093902 (2007).
[Crossref]

Giuseppe, G. D.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

Gröblacher, S.

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref]

Gu, X.

H. Wang, X. Gu, Y.-X. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically-induced transparency: photon transmission through an optomechanical device with a two-level system,” Phys. Rev. A 90(2), 023817 (2014).
[Crossref]

Guo, L.-X.

L.-G. Si, L.-X. Guo, H. Xiong, and Y. Wu, “Tunable high-order-sideband generation and carrier-envelope-phase-dependent effects via microwave fields in hybrid electro-optomechanical systems,” Phys. Rev. A 97(2), 023805 (2018).
[Crossref]

Hamedi, H. R.

H. R. Hamedi, “Transient absorption and lasing without inversion in an artificial molecule via Josephson coupling energy,” Laser Phys. Lett. 12(3), 035201 (2015).
[Crossref]

Hammerer, K.

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[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(9), 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 452(7183), 72–75 (2008).
[Crossref]

Hill, J. T.

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(7341), 69–73 (2011).
[Crossref]

Huang, G.-Y.

W.-A. Li and G.-Y. Huang, “Enhancement of optomechanically induced sum sideband using parametric interactions,” Phys. Rev. A 100(2), 023838 (2019).
[Crossref]

Huang, S.

G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93(4), 043844 (2016).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83(4), 043826 (2011).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79(1), 013821 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80(3), 033807 (2009).
[Crossref]

Huang, S.-M.

S.-M. Huang and A.-X. Chen, “Improving the cooling of a mechanical oscillator in a dissipative optomechanical system with an optical parametric amplifier,” Phys. Rev. A 98(6), 063818 (2018).
[Crossref]

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(9), 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 452(7183), 72–75 (2008).
[Crossref]

Jiang, C.

Jiao, Y.

H. Zhang, F. Saif, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26(19), 25199–25210 (2018).
[Crossref]

Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
[Crossref]

Jiao, Y.-F.

T.-X. Lu, Y.-F. Jiao, H.-L. Zhang, F. Saif, and H. Jing, “Selective and switchable optical amplification with mechanical driven oscillators,” Phys. Rev. A 100(1), 013813 (2019).
[Crossref]

Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a Kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
[Crossref]

Jing, H.

T.-X. Lu, Y.-F. Jiao, H.-L. Zhang, F. Saif, and H. Jing, “Selective and switchable optical amplification with mechanical driven oscillators,” Phys. Rev. A 100(1), 013813 (2019).
[Crossref]

H. Zhang, F. Saif, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26(19), 25199–25210 (2018).
[Crossref]

H. Jing, H. Lü, S. K. özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5(11), 1424 (2018).
[Crossref]

Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a Kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
[Crossref]

Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
[Crossref]

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

Karrai, K.

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3(4), 201–205 (2009).
[Crossref]

Karuza, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

Kim, M. S.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and Probing Multipartite Macroscopic Entanglement with Light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 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(6010), 1520–1523 (2010).
[Crossref]

J. M. Dobrindt and T. J. Kippenberg, “Theoretical Analysis of Mechanical Displacement Measurement Using a Multiple Cavity Mode Transducer,” Phys. Rev. Lett. 104(3), 033901 (2010).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[Crossref]

Kong, C.

Z.-X. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Magnetic-field-dependent slow light in strontium atom-cavity system,” Appl. Phys. Lett. 112(11), 111109 (2018).
[Crossref]

Larsen, P. E.

G. A. Brawley, M. R. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. P. Bowen, “Nonlinear optomechanical measurement of mechanical motion,” Nat. Commun. 7(1), 10988 (2016).
[Crossref]

Law, C. K.

J.-Q. Liao, H. K. Cheung, and C. K. Law, “Spectrum of single-photon emission and scattering in cavity optomechanics,” Phys. Rev. A 85(2), 025803 (2012).
[Crossref]

Lei, F.-C.

B. Peng, S. K. Özdemir, F.-C. Lei, F. Monifi, M. Gianfreda, G.-L. Long, S.-H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Li, J.

J. Ma, C. You, L.-G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Formation and manipulation of optomechanical chaos via a bichromatic driving,” Phys. Rev. A 90(4), 043839 (2014).
[Crossref]

Li, J.-H.

J.-H. Li, R. Yu, C.-L. Ding, and Y. Wu, “$\mathcal {PT}$PT-symmetry-induced evolution of sharp asymmetric line shapes and high-sensitivity refractive index sensors in a three-cavity array,” Phys. Rev. A 93(2), 023814 (2016).
[Crossref]

Li, W.-A.

W.-A. Li and G.-Y. Huang, “Enhancement of optomechanically induced sum sideband using parametric interactions,” Phys. Rev. A 100(2), 023838 (2019).
[Crossref]

Li, X.-W.

Li, Y.

Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
[Crossref]

Liao, J.-Q.

J.-Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88(2), 023853 (2013).
[Crossref]

J.-Q. Liao, H. K. Cheung, and C. K. Law, “Spectrum of single-photon emission and scattering in cavity optomechanics,” Phys. Rev. A 85(2), 025803 (2012).
[Crossref]

Lin, Q.

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(7341), 69–73 (2011).
[Crossref]

Liu, B.

Liu, H.-X.

Liu, S.-P

Liu, S.-P.

S.-P. Liu, W.-X. Yang, T. Shui, Z.-H. Zhu, and A.-X. Chen, “Tunable two-phonon higher-order sideband amplification in a quadratically coupled optomechanical system,” Sci. Rep. 7(1), 17637 (2017).
[Crossref]

Liu, X.

B. Chen, L. Shang, X.-F. Wang, J.-B. Chen, H.-B. Xue, X. Liu, and J. Zhang, “Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling,” Phys. Rev. A 99(6), 063810 (2019).
[Crossref]

Liu, Y.-C.

X. Chen, Y.-C. Liu, P. Peng, Y.-Y. Zhi, and Y.-F. Xiao, “Cooling of macroscopic mechanical resonators in hybrid atom-optomechanical systems,” Phys. Rev. A 92(3), 033841 (2015).
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H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
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Figures (4)

Fig. 1.
Fig. 1. (a) Schematic diagram of a quadratic optomechanical system consisting of an OPA and a MR inserted between two fixed mirrors. The system is excited by two relatively weak probe fields and a strong control field. The two probe fields are represented by probe field 1 with frequency $\omega _{1}$ and probe field 2 with frequency $\omega _{2}$, respectively. And $\omega _{c}$ is the frequency of the control field. The MR and cavity field are quadratically coupled via the radiation pressure. (b) Frequency spectrogram of sum sideband generation in the quadratically coupled optomechanical system with a control field and two probe fields driven.
Fig. 2.
Fig. 2. (a) The efficiency ${\log _{10}\eta _{s}^+}$ (in logarithmic form) of a USSG and (b) the efficiency ${\log _{10}\eta _{s}^-}$ (in logarithmic form) of a LSSG as a function of the control field power $P_{c}$ and the probe field 1 detuning $\Omega _{1}$ for $\Omega _{2}=0.02\times 2\Omega _{m}$. The specific parameters are as follows: $m=100$ pg, $\Omega _{m}=2\pi \times 0.1$ MHz, $R$=0.8, $\kappa =2\pi \times 75$ KHz, $T=50$ K, $Q=\Omega _{m}/\Gamma _{m}=3.14\times 10^4$, $L$=67 mm, $\lambda _{c}=2\pi c/\omega _{c}$=532 nm, $P_{1}=P_{2}$=0.1 nW, $G=0$.
Fig. 3.
Fig. 3. (a) The efficiency ${\log _{10}\eta _{s}^+}$ (in logarithmic form) of a USSG and (b) the efficiency ${\log _{10}\eta _{s}^-}$ (in logarithmic form) of a LSSG as a function of the probe field 1 detuning $\Omega _{1}$ for different values of $G$ with $P_c$=1 nW, $\Omega _{2}=-0.01\times 2\Omega _{m}$ . Other parameters are the same as those in Fig. 2.
Fig. 4.
Fig. 4. (a), (c) The efficiency ${\log _{10}\eta _{s}^+}$ (in logarithmic form) of a USSG and (b), (d) the efficiency ${\log _{10}\eta _{s}^-}$ (in logarithmic form) of a LSSG as a function of the probe field 1 detuning $\Omega _{1}$ and the probe field 2 detuning $\Omega _{2}$ for $P_{c}$=1 nW. For the cases of (a), (b) we use $G=0$, and (c), (d) we use $G=0.6\kappa$. Other parameters are the same as those in Fig. 2.

Equations (10)

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H = Δ a ^ a ^ + ( p ^ 2 2 m + 1 2 m Ω m 2 x ^ 2 ) + g a ^ a ^ x ^ 2 + i G ( a ^ 2 e i θ a ^ 2 e i θ ) +                 i η κ [ ( ε c + ε 1 e i Ω 1 t + ε 2 e i Ω 2 t ) a ^ H . c . ] ,
t x ^ = p ^ m , t p ^ = [ m Ω m 2 + 2 g a ^ a ^ ] x ^ Γ m p ^ + F ^ t h , t a ^ = [ κ 2 + i ( Δ + g x ^ 2 ) ] a ^ + 2 G e i θ a ^ + η κ ( ε c + ε 1 e i Ω 1 t + ε 2 e i Ω 2 t ) + a ^ i n ,
t X = Q m , t P = [ m Ω m 2 + 2 g a a ] Q + Γ m ( 1 + 2 n ) m Ω m 2 Γ m P , t Q = [ 4 g a a + 2 m Ω m 2 ] X Γ m Q + 2 P m , t a = [ κ 2 + i ( Δ + g X ) ] a + 2 G e i θ a ^ + η κ ( ε c + ε 1 e i Ω 1 t + ε 2 e i Ω 2 t ) ,
δ a = A + 1 e i Ω 1 t + A 1 e i Ω 1 t + A + 2 e i Ω 2 t + A 2 e i Ω 2 t + A + s e i Ω + t + A s e i Ω + t , δ a = A 1 e i Ω 1 t + A + 1 e i Ω 1 t + A 2 e i Ω 2 t + A + 2 e i Ω 2 t + A s e i Ω + t + A + s e i Ω + t , δ Q = Q 1 e i Ω 1 t + Q 1 e i Ω 1 t + Q 2 e i Ω 2 t + Q 2 e i Ω 2 t + Q s e i Ω + t + Q s e i Ω + t , δ X = X 1 e i Ω 1 t + X 1 e i Ω 1 t + X 2 e i Ω 2 t + X 2 e i Ω 2 t + X s e i Ω + t + X s e i Ω + t , δ P = P 1 e i Ω 1 t + P 1 e i Ω 1 t + P 2 e i Ω 2 t + P 2 e i Ω 2 t + P s e i Ω + t + P s e i Ω + t ,
Q ¯ = 0 , a ¯ = ( β + 2 G e i θ ) β β + 4 G 2 η κ ε c , X ¯ = P ¯ m 2 Ω m 2 ( 1 + 2 α ) , P ¯ = ( 1 + 2 n ) m Ω m 2 ,
( β i Ω j ) A + j = i g a ¯ X j + 2 G e i θ A j + η κ ε j ( β + i Ω j ) A j = i g a ¯ X j + 2 G e i θ A + j ( 2 Γ m i Ω j ) P j = ( m Ω m 2 + 2 g | a ¯ | 2 ) Q j ( Γ m i Ω j ) Q j = 4 g [ a ¯ A + j + a ¯ A j ] X ¯ 2 m Ω m 2 ( 1 + 2 α ) X j + 2 P j m Q j + i Ω 1 m X j = 0.                 ( j = 1 , 2 )
( β i Ω + ) A + s = i g ( a ¯ X s + X 1 A + 2 + X 2 A + 1 ) + 2 G e i θ A s ( β + i Ω + ) A s = i g ( a ¯ X s + X 1 A 2 + X 2 A 1 ) + 2 G e i θ A + s ( 2 Γ m i Ω + ) P s = m Ω m 2 ( 1 + 2 α ) Q s 2 g [ a ¯ ( A + 1 Q 2 + A + 2 Q 1 ) + a ¯ ( A 1 Q 2 + A 2 Q 1 ) ] ( Γ m i Ω + ) Q s = 4 g [ a ¯ A + s + a ¯ A s + A 1 A + 2 + A 2 A + 1 ] X ¯ 4 g [ a ¯ ( A + 1 X 2 + A + 2 X 1 )                                                       + a ¯ ( A 1 X 2 + A 2 X 1 ) ] 2 m Ω m 2 ( 1 + 2 α ) X s + 2 P s m Q s + i Ω + m X s = 0.
A + j = τ ( Ω j ) X j + η κ ε j ( β i Ω j ) s ( Ω j ) , X j = η κ ε j f ( Ω j ) F ( Ω j ) + d ( Ω j ) + σ ( Ω j ) + D ( Ω j ) , A j = i g a ¯ X j + 2 G e i θ A + j β + i Ω j ,                     ( j = 1 , 2 ) A + s = τ ( Ω + ) X s + W s ( Ω + ) , A s = i g a ¯ X s i g ( X 1 A 2 + X 2 A 1 ) + 2 G e i θ A + s β + i Ω + , X s = f ( Ω + ) W + [ ξ X 2 M ( Ω 2 ) X 1 M ( Ω 1 ) ] ( β i Ω + ) [ F ( Ω + ) + σ ( Ω + ) ] ( β i Ω + ) τ ( Ω + ) f ( Ω + ) ,
s ( Ω y ) = ( β i Ω y ) ( β i Ω y ) 4 G 2 , σ ( Ω y ) = 4 i g 2 | a ¯ | 2 X ¯ ( 2 Γ m i Ω y ) s ( Ω y ) , τ ( Ω y ) = i g [ a ¯ ( β i Ω y ) 2 G a ¯ e i θ ] , d ( Ω y ) = 4 i g 2 | a ¯ | 2 X ¯ ( 2 Γ m i Ω y ) [ ( β i Ω y ) 2 + 4 G 2 ] , D ( Ω y ) = 8 i g 2 G X ¯ ( 2 Γ m i Ω y ) ( β i Ω y ) [ a ¯ 2 e i θ a ¯ 2 e i θ ] , f ( Ω y ) = 4 g X ¯ ( 2 Γ m i Ω y ) ( β i Ω y ) [ a ¯ ( β i Ω y ) + 2 G a ¯ e i θ ] , F ( Ω y ) = ( β i Ω y ) s ( Ω y ) ( Γ m i Ω y ) [ ( 2 Γ m i Ω y ) ( i Ω y m ) + 4 m Ω m 2 ( 1 + 2 α ) ] , M ( Ω 1 ) = 4 g ( β i Ω + ) s ( Ω + ) ( 2 Γ m i Ω + i Ω 1 ) ( a ¯ A + 2 + a ¯ A 2 ) , M ( Ω 2 ) = 4 g ( β i Ω + ) s ( Ω + ) ( 2 Γ m i Ω + i Ω 2 ) ( a ¯ A + 1 + a ¯ A 1 ) , ξ = 4 g X ¯ ( 2 Γ m i Ω + ) s ( Ω + ) [ i g a ¯ ( X 1 A 2 + X 2 A 1 ) + ( β i Ω + ) ( A 1 A + 2 + A 2 A + 1 ) ] , W = i g [ ( β i Ω + ) ( X 1 A + 2 + X 2 A + 1 ) + 2 G e i θ ( X 1 A 2 + X 2 A 1 ) ] ,
s t o u t ( t ) = η κ a ( t ) s i n = ( η κ a ¯ ε c ) + ( η κ A + 1 ε 1 ) e i Ω 1 t + η κ A 1 e i Ω 1 t +                                       ( η κ A + 2 ε 2 ) e i Ω 2 t + η κ A 2 e i Ω 2 t + η κ A + s e i Ω + t + η κ A s e i Ω + t .

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