Abstract

Noise usually has an unwelcome influence on system performance. For instance, noise inevitably affects the low-frequency mechanical freedom in optomechanical experiments. However, we investigate here the beneficial effects of thermal noise on a basic optomechanical system with parametric instability. In a regime near parametric instability, it is found that thermal noise in the mechanical freedom can sustain long-term quasi-coherent oscillations when the system would otherwise remain in the equilibrium state. In an overlapping regime of parametric instability and bistability, intermittent switching between a self-sustained oscillating state and an equilibrium can be induced by adding a certain amount of noise. When a subthreshold periodic signal is applied to the mechanics, the switching between the self-sustained oscillations and the equilibrium exhibits good periodicity at a rate that is synchronized to the signal frequency, resulting in a significant amplification of the signal. Our results deepen the understanding of the interplay between optomechanical nonlinearity and noise and provide theoretical guidance for experimental observation of noise-induced beneficial phenomena in optomechanics.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (1)

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

2017 (3)

2016 (1)

F. Monifi, J. Zhang, S. K. özdemir, B. Peng, Y. xi Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10, 399–405 (2016).
[Crossref]

2015 (3)

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

L. Kabiraj, R. Steinert, A. Saurabh, and C. O. Paschereit, “Coherence resonance in a thermoacoustic system,” Phys. Rev. E 92, 042909 (2015).
[Crossref]

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
[Crossref] [PubMed]

2014 (7)

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, 043839 (2014).
[Crossref]

S. Aldana, C. Bruder, and A. Nunnenkamp, “Detection of weak forces based on noise-activated switching in bistable optomechanical systems,” Phys. Rev. A 90, 063810 (2014).
[Crossref]

O. Suchoi, L. Ella, O. Shtempluk, and E. Buks, “Intermittency in an optomechanical cavity near a subcritical hopf bifurcation,” Phys. Rev. A 90, 033818 (2014).
[Crossref]

L. Zhang and H.-Y. Kong, “Self-sustained oscillation and harmonic generation in optomechanical systems with quadratic couplings,” Phys. Rev. A 89, 023847 (2014).
[Crossref]

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

O. Kyriienko, T. C. H. Liew, and I. A. Shelykh, “Optomechanics with cavity polaritons: Dissipative coupling and unconventional bistability,” Phys. Rev. Lett. 112, 076402 (2014).
[Crossref] [PubMed]

G. Cao, H. Liu, X. Li, N. Huang, and Q. Sun, “Reconstructing signals via stochastic resonance generated by photorefractive two-wave mixing bistability,” Opt. Express 22, 4214–4223 (2014).
[Crossref] [PubMed]

2013 (1)

A. Samuel, B. Christoph, and N. Andreas, “Equivalence between an optomechanical system and a kerr medium,” Phys. Rev. A 88, 043826 (2013).
[Crossref]

2012 (2)

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

S. Zaitsev, O. Gottlieb, and E. Buks, “Nonlinear dynamics of a microelectromechanical mirror in an optical resonance cavity,” Nonlinear Dyn. 69, 1589–1610 (2012).
[Crossref]

2010 (1)

C. Y. Lee, W. Choi, J.-H. Han, and M. S. Strano, “Coherence resonance in a single-walled carbon nanotube ion channel,” Science 329, 1320–1324 (2010).
[Crossref] [PubMed]

2008 (2)

F. Mueller, S. Heugel, and L. J. Wang, “Observation of optomechanical multistability in a high-q torsion balance oscillator,” Phys. Rev. A 77, 031802 (2008).
[Crossref]

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[Crossref] [PubMed]

2007 (1)

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Signal amplification in a nanomechanical duffing resonator via stochastic resonance,” Appl. Phys. Lett. 90, 013508 (2007).
[Crossref]

2006 (1)

F. Marquardt, J. G. E. Harris, and S. M. Girvin, “Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities,” Phys. Rev. Lett. 96, 103901 (2006).
[Crossref] [PubMed]

2005 (2)

O. V. Ushakov, H.-J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, and M. A. Zaks, “Coherence resonance near a hopf bifurcation,” Phys. Rev. Lett. 95, 123903 (2005).
[Crossref] [PubMed]

R. L. Badzey and P. Mohanty, “Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance,” Nature 437, 995–998 (2005).
[Crossref] [PubMed]

2001 (1)

Z. Liu and Y.-C. Lai, “Coherence resonance in coupled chaotic oscillators,” Phys. Rev. Lett. 86, 4737–4740 (2001).
[Crossref] [PubMed]

1998 (1)

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

1997 (1)

A. S. Pikovsky and J. Kurths, “Coherence resonance in a noise-driven excitable system,” Phys. Rev. Lett. 78, 775–778 (1997).
[Crossref]

1995 (1)

C. K. Law, “Interaction between a moving mirror and radiation pressure: A hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995).
[Crossref] [PubMed]

1994 (1)

W.-M. Liu, “Criterion of hopf bifurcations without using eigenvalues,” J. Math. Analysis Appl. 182, 250–256 (1994).
[Crossref]

1993 (1)

H. Gang, T. Ditzinger, C. Z. Ning, and H. Haken, “Stochastic resonance without external periodic force,” Phys. Rev. Lett. 71, 807–810 (1993).
[Crossref] [PubMed]

1982 (1)

R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, “Stochastic resonance in climatic change,” Tellus 34, 10–16 (1982).
[Crossref]

1981 (1)

R. Benzi, A. Sutera, and A. Vulpiani, “The mechanism of stochastic resonance,” J. Phys. A 14, L453–L457 (1981).
[Crossref]

Aldana, S.

S. Aldana, C. Bruder, and A. Nunnenkamp, “Detection of weak forces based on noise-activated switching in bistable optomechanical systems,” Phys. Rev. A 90, 063810 (2014).
[Crossref]

Almog, R.

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Signal amplification in a nanomechanical duffing resonator via stochastic resonance,” Appl. Phys. Lett. 90, 013508 (2007).
[Crossref]

Alvermann, A.

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
[Crossref] [PubMed]

Andreas, N.

A. Samuel, B. Christoph, and N. Andreas, “Equivalence between an optomechanical system and a kerr medium,” Phys. Rev. A 88, 043826 (2013).
[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. Schwab, “Quantum optomechanics,” Phys. Today 65, 29–35 (2012).
[Crossref]

Badzey, R. L.

R. L. Badzey and P. Mohanty, “Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance,” Nature 437, 995–998 (2005).
[Crossref] [PubMed]

Bakemeier, L.

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
[Crossref] [PubMed]

Benzi, R.

R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, “Stochastic resonance in climatic change,” Tellus 34, 10–16 (1982).
[Crossref]

R. Benzi, A. Sutera, and A. Vulpiani, “The mechanism of stochastic resonance,” J. Phys. A 14, L453–L457 (1981).
[Crossref]

Biermann, K.

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

Bo, F.

F. Monifi, J. Zhang, S. K. özdemir, B. Peng, Y. xi Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10, 399–405 (2016).
[Crossref]

Bonilla, L. L.

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

Brauer, F.

F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations: An Introduction(Dover Publications, 1989).

Bruder, C.

S. Aldana, C. Bruder, and A. Nunnenkamp, “Detection of weak forces based on noise-activated switching in bistable optomechanical systems,” Phys. Rev. A 90, 063810 (2014).
[Crossref]

Buks, E.

O. Suchoi, L. Ella, O. Shtempluk, and E. Buks, “Intermittency in an optomechanical cavity near a subcritical hopf bifurcation,” Phys. Rev. A 90, 033818 (2014).
[Crossref]

S. Zaitsev, O. Gottlieb, and E. Buks, “Nonlinear dynamics of a microelectromechanical mirror in an optical resonance cavity,” Nonlinear Dyn. 69, 1589–1610 (2012).
[Crossref]

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Signal amplification in a nanomechanical duffing resonator via stochastic resonance,” Appl. Phys. Lett. 90, 013508 (2007).
[Crossref]

Cao, G.

Chen, J.

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

Choi, W.

C. Y. Lee, W. Choi, J.-H. Han, and M. S. Strano, “Coherence resonance in a single-walled carbon nanotube ion channel,” Science 329, 1320–1324 (2010).
[Crossref] [PubMed]

Christoph, B.

A. Samuel, B. Christoph, and N. Andreas, “Equivalence between an optomechanical system and a kerr medium,” Phys. Rev. A 88, 043826 (2013).
[Crossref]

Ditzinger, T.

H. Gang, T. Ditzinger, C. Z. Ning, and H. Haken, “Stochastic resonance without external periodic force,” Phys. Rev. Lett. 71, 807–810 (1993).
[Crossref] [PubMed]

Ella, L.

O. Suchoi, L. Ella, O. Shtempluk, and E. Buks, “Intermittency in an optomechanical cavity near a subcritical hopf bifurcation,” Phys. Rev. A 90, 033818 (2014).
[Crossref]

Fan, B.

B. Fan and M. Xie, “Stochastic resonance in a tristable optomechanical system,” Phys. Rev. A 95, 023808 (2017).
[Crossref]

Favero, I.

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[Crossref] [PubMed]

Fehske, H.

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
[Crossref] [PubMed]

Gammaitoni, L.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

Gang, H.

H. Gang, T. Ditzinger, C. Z. Ning, and H. Haken, “Stochastic resonance without external periodic force,” Phys. Rev. Lett. 71, 807–810 (1993).
[Crossref] [PubMed]

Girvin, S. M.

F. Marquardt, J. G. E. Harris, and S. M. Girvin, “Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities,” Phys. Rev. Lett. 96, 103901 (2006).
[Crossref] [PubMed]

Gottlieb, O.

S. Zaitsev, O. Gottlieb, and E. Buks, “Nonlinear dynamics of a microelectromechanical mirror in an optical resonance cavity,” Nonlinear Dyn. 69, 1589–1610 (2012).
[Crossref]

Grahn, H. T.

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

Haken, H.

H. Gang, T. Ditzinger, C. Z. Ning, and H. Haken, “Stochastic resonance without external periodic force,” Phys. Rev. Lett. 71, 807–810 (1993).
[Crossref] [PubMed]

Han, J.

Han, J.-H.

C. Y. Lee, W. Choi, J.-H. Han, and M. S. Strano, “Coherence resonance in a single-walled carbon nanotube ion channel,” Science 329, 1320–1324 (2010).
[Crossref] [PubMed]

Han, S.

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

Hänggi, P.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

Harris, J. G. E.

F. Marquardt, J. G. E. Harris, and S. M. Girvin, “Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities,” Phys. Rev. Lett. 96, 103901 (2006).
[Crossref] [PubMed]

Henneberger, F.

O. V. Ushakov, H.-J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, and M. A. Zaks, “Coherence resonance near a hopf bifurcation,” Phys. Rev. Lett. 95, 123903 (2005).
[Crossref] [PubMed]

Heugel, S.

F. Mueller, S. Heugel, and L. J. Wang, “Observation of optomechanical multistability in a high-q torsion balance oscillator,” Phys. Rev. A 77, 031802 (2008).
[Crossref]

Huang, N.

Jung, P.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

Kabiraj, L.

L. Kabiraj, R. Steinert, A. Saurabh, and C. O. Paschereit, “Coherence resonance in a thermoacoustic system,” Phys. Rev. E 92, 042909 (2015).
[Crossref]

Kang, L.

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

Karrai, K.

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[Crossref] [PubMed]

Khovanov, I. A.

O. V. Ushakov, H.-J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, and M. A. Zaks, “Coherence resonance near a hopf bifurcation,” Phys. Rev. Lett. 95, 123903 (2005).
[Crossref] [PubMed]

Kippenberg, T. J.

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

Kong, H.-Y.

L. Zhang and H.-Y. Kong, “Self-sustained oscillation and harmonic generation in optomechanical systems with quadratic couplings,” Phys. Rev. A 89, 023847 (2014).
[Crossref]

Kurths, J.

A. S. Pikovsky and J. Kurths, “Coherence resonance in a noise-driven excitable system,” Phys. Rev. Lett. 78, 775–778 (1997).
[Crossref]

Kyriienko, O.

O. Kyriienko, T. C. H. Liew, and I. A. Shelykh, “Optomechanics with cavity polaritons: Dissipative coupling and unconventional bistability,” Phys. Rev. Lett. 112, 076402 (2014).
[Crossref] [PubMed]

Lai, Y.-C.

Z. Liu and Y.-C. Lai, “Coherence resonance in coupled chaotic oscillators,” Phys. Rev. Lett. 86, 4737–4740 (2001).
[Crossref] [PubMed]

Law, C. K.

C. K. Law, “Interaction between a moving mirror and radiation pressure: A hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995).
[Crossref] [PubMed]

Lee, C. Y.

C. Y. Lee, W. Choi, J.-H. Han, and M. S. Strano, “Coherence resonance in a single-walled carbon nanotube ion channel,” Science 329, 1320–1324 (2010).
[Crossref] [PubMed]

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, 043839 (2014).
[Crossref]

Li, X.

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

G. Cao, H. Liu, X. Li, N. Huang, and Q. Sun, “Reconstructing signals via stochastic resonance generated by photorefractive two-wave mixing bistability,” Opt. Express 22, 4214–4223 (2014).
[Crossref] [PubMed]

Liew, T. C. H.

O. Kyriienko, T. C. H. Liew, and I. A. Shelykh, “Optomechanics with cavity polaritons: Dissipative coupling and unconventional bistability,” Phys. Rev. Lett. 112, 076402 (2014).
[Crossref] [PubMed]

Liu, H.

Liu, W.

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

Liu, W.-M.

W.-M. Liu, “Criterion of hopf bifurcations without using eigenvalues,” J. Math. Analysis Appl. 182, 250–256 (1994).
[Crossref]

Liu, Z.

Z. Liu and Y.-C. Lai, “Coherence resonance in coupled chaotic oscillators,” Phys. Rev. Lett. 86, 4737–4740 (2001).
[Crossref] [PubMed]

Ludwig, M.

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[Crossref] [PubMed]

Ma, 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, 043839 (2014).
[Crossref]

Marchesoni, F.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

Marquardt, F.

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

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[Crossref] [PubMed]

F. Marquardt, J. G. E. Harris, and S. M. Girvin, “Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities,” Phys. Rev. Lett. 96, 103901 (2006).
[Crossref] [PubMed]

Metzger, C.

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[Crossref] [PubMed]

Meystre, P.

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

Mohanty, P.

R. L. Badzey and P. Mohanty, “Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance,” Nature 437, 995–998 (2005).
[Crossref] [PubMed]

Monifi, F.

F. Monifi, J. Zhang, S. K. özdemir, B. Peng, Y. xi Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10, 399–405 (2016).
[Crossref]

Mueller, F.

F. Mueller, S. Heugel, and L. J. Wang, “Observation of optomechanical multistability in a high-q torsion balance oscillator,” Phys. Rev. A 77, 031802 (2008).
[Crossref]

Neuenhahn, C.

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[Crossref] [PubMed]

Ning, C. Z.

H. Gang, T. Ditzinger, C. Z. Ning, and H. Haken, “Stochastic resonance without external periodic force,” Phys. Rev. Lett. 71, 807–810 (1993).
[Crossref] [PubMed]

Nohel, J. A.

F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations: An Introduction(Dover Publications, 1989).

Nori, F.

F. Monifi, J. Zhang, S. K. özdemir, B. Peng, Y. xi Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10, 399–405 (2016).
[Crossref]

Nunnenkamp, A.

S. Aldana, C. Bruder, and A. Nunnenkamp, “Detection of weak forces based on noise-activated switching in bistable optomechanical systems,” Phys. Rev. A 90, 063810 (2014).
[Crossref]

Ortlieb, A.

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[Crossref] [PubMed]

özdemir, S. K.

F. Monifi, J. Zhang, S. K. özdemir, B. Peng, Y. xi Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10, 399–405 (2016).
[Crossref]

Parisi, G.

R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, “Stochastic resonance in climatic change,” Tellus 34, 10–16 (1982).
[Crossref]

Paschereit, C. O.

L. Kabiraj, R. Steinert, A. Saurabh, and C. O. Paschereit, “Coherence resonance in a thermoacoustic system,” Phys. Rev. E 92, 042909 (2015).
[Crossref]

Peng, B.

F. Monifi, J. Zhang, S. K. özdemir, B. Peng, Y. xi Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10, 399–405 (2016).
[Crossref]

Pikovsky, A. S.

A. S. Pikovsky and J. Kurths, “Coherence resonance in a noise-driven excitable system,” Phys. Rev. Lett. 78, 775–778 (1997).
[Crossref]

Samuel, A.

A. Samuel, B. Christoph, and N. Andreas, “Equivalence between an optomechanical system and a kerr medium,” Phys. Rev. A 88, 043826 (2013).
[Crossref]

Saurabh, A.

L. Kabiraj, R. Steinert, A. Saurabh, and C. O. Paschereit, “Coherence resonance in a thermoacoustic system,” Phys. Rev. E 92, 042909 (2015).
[Crossref]

Schimansky-Geier, L.

O. V. Ushakov, H.-J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, and M. A. Zaks, “Coherence resonance near a hopf bifurcation,” Phys. Rev. Lett. 95, 123903 (2005).
[Crossref] [PubMed]

Schwab, K.

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

Shao, Z.

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

Shelykh, I. A.

O. Kyriienko, T. C. H. Liew, and I. A. Shelykh, “Optomechanics with cavity polaritons: Dissipative coupling and unconventional bistability,” Phys. Rev. Lett. 112, 076402 (2014).
[Crossref] [PubMed]

Shtempluck, O.

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Signal amplification in a nanomechanical duffing resonator via stochastic resonance,” Appl. Phys. Lett. 90, 013508 (2007).
[Crossref]

Shtempluk, O.

O. Suchoi, L. Ella, O. Shtempluk, and E. Buks, “Intermittency in an optomechanical cavity near a subcritical hopf bifurcation,” Phys. Rev. A 90, 033818 (2014).
[Crossref]

Si, L.-G.

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, 043839 (2014).
[Crossref]

Song, H.

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

Steinert, R.

L. Kabiraj, R. Steinert, A. Saurabh, and C. O. Paschereit, “Coherence resonance in a thermoacoustic system,” Phys. Rev. E 92, 042909 (2015).
[Crossref]

Strano, M. S.

C. Y. Lee, W. Choi, J.-H. Han, and M. S. Strano, “Coherence resonance in a single-walled carbon nanotube ion channel,” Science 329, 1320–1324 (2010).
[Crossref] [PubMed]

Suchoi, O.

O. Suchoi, L. Ella, O. Shtempluk, and E. Buks, “Intermittency in an optomechanical cavity near a subcritical hopf bifurcation,” Phys. Rev. A 90, 033818 (2014).
[Crossref]

Sun, G.

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

Sun, Q.

Sutera, A.

R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, “Stochastic resonance in climatic change,” Tellus 34, 10–16 (1982).
[Crossref]

R. Benzi, A. Sutera, and A. Vulpiani, “The mechanism of stochastic resonance,” J. Phys. A 14, L453–L457 (1981).
[Crossref]

Ushakov, O. V.

O. V. Ushakov, H.-J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, and M. A. Zaks, “Coherence resonance near a hopf bifurcation,” Phys. Rev. Lett. 95, 123903 (2005).
[Crossref] [PubMed]

Vulpiani, A.

R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, “Stochastic resonance in climatic change,” Tellus 34, 10–16 (1982).
[Crossref]

R. Benzi, A. Sutera, and A. Vulpiani, “The mechanism of stochastic resonance,” J. Phys. A 14, L453–L457 (1981).
[Crossref]

Wang, L. J.

F. Mueller, S. Heugel, and L. J. Wang, “Observation of optomechanical multistability in a high-q torsion balance oscillator,” Phys. Rev. A 77, 031802 (2008).
[Crossref]

Wang, N.

Wang, Z.

Wen, X.

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

Wu, P.

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

Wu, Y.

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, 043839 (2014).
[Crossref]

Wünsche, H.-J.

O. V. Ushakov, H.-J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, and M. A. Zaks, “Coherence resonance near a hopf bifurcation,” Phys. Rev. Lett. 95, 123903 (2005).
[Crossref] [PubMed]

xi Liu, Y.

F. Monifi, J. Zhang, S. K. özdemir, B. Peng, Y. xi Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10, 399–405 (2016).
[Crossref]

Xie, M.

B. Fan and M. Xie, “Stochastic resonance in a tristable optomechanical system,” Phys. Rev. A 95, 023808 (2017).
[Crossref]

Xiong, H.

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, 043839 (2014).
[Crossref]

Xu, W.

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

Yang, L.

F. Monifi, J. Zhang, S. K. özdemir, B. Peng, Y. xi Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10, 399–405 (2016).
[Crossref]

Yang, X.

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, 043839 (2014).
[Crossref]

Yin, Z.

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

You, C.

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, 043839 (2014).
[Crossref]

Yu, Y.

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

Yu, Z.

Zaitsev, S.

S. Zaitsev, O. Gottlieb, and E. Buks, “Nonlinear dynamics of a microelectromechanical mirror in an optical resonance cavity,” Nonlinear Dyn. 69, 1589–1610 (2012).
[Crossref]

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Signal amplification in a nanomechanical duffing resonator via stochastic resonance,” Appl. Phys. Lett. 90, 013508 (2007).
[Crossref]

Zaks, M. A.

O. V. Ushakov, H.-J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, and M. A. Zaks, “Coherence resonance near a hopf bifurcation,” Phys. Rev. Lett. 95, 123903 (2005).
[Crossref] [PubMed]

Zhai, J.

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

Zhang, J.

F. Monifi, J. Zhang, S. K. özdemir, B. Peng, Y. xi Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10, 399–405 (2016).
[Crossref]

Zhang, L.

L. Zhang and H.-Y. Kong, “Self-sustained oscillation and harmonic generation in optomechanical systems with quadratic couplings,” Phys. Rev. A 89, 023847 (2014).
[Crossref]

Zhang, Y.

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

Zheng, B.

Zheng, H.

Zhu, J.

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

Appl. Phys. Lett. (2)

G. Sun, J. Zhai, X. Wen, Y. Yu, L. Kang, W. Xu, J. Chen, P. Wu, and S. Han, “Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device,” Appl. Phys. Lett. 106, 172602 (2015).
[Crossref]

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Signal amplification in a nanomechanical duffing resonator via stochastic resonance,” Appl. Phys. Lett. 90, 013508 (2007).
[Crossref]

J. Math. Analysis Appl. (1)

W.-M. Liu, “Criterion of hopf bifurcations without using eigenvalues,” J. Math. Analysis Appl. 182, 250–256 (1994).
[Crossref]

J. Phys. A (1)

R. Benzi, A. Sutera, and A. Vulpiani, “The mechanism of stochastic resonance,” J. Phys. A 14, L453–L457 (1981).
[Crossref]

Nat. Photon. (1)

F. Monifi, J. Zhang, S. K. özdemir, B. Peng, Y. xi Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10, 399–405 (2016).
[Crossref]

Nature (1)

R. L. Badzey and P. Mohanty, “Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance,” Nature 437, 995–998 (2005).
[Crossref] [PubMed]

Nonlinear Dyn. (1)

S. Zaitsev, O. Gottlieb, and E. Buks, “Nonlinear dynamics of a microelectromechanical mirror in an optical resonance cavity,” Nonlinear Dyn. 69, 1589–1610 (2012).
[Crossref]

Opt. Express (3)

Phys. Rev. A (8)

C. K. Law, “Interaction between a moving mirror and radiation pressure: A hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995).
[Crossref] [PubMed]

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, 043839 (2014).
[Crossref]

S. Aldana, C. Bruder, and A. Nunnenkamp, “Detection of weak forces based on noise-activated switching in bistable optomechanical systems,” Phys. Rev. A 90, 063810 (2014).
[Crossref]

O. Suchoi, L. Ella, O. Shtempluk, and E. Buks, “Intermittency in an optomechanical cavity near a subcritical hopf bifurcation,” Phys. Rev. A 90, 033818 (2014).
[Crossref]

A. Samuel, B. Christoph, and N. Andreas, “Equivalence between an optomechanical system and a kerr medium,” Phys. Rev. A 88, 043826 (2013).
[Crossref]

F. Mueller, S. Heugel, and L. J. Wang, “Observation of optomechanical multistability in a high-q torsion balance oscillator,” Phys. Rev. A 77, 031802 (2008).
[Crossref]

B. Fan and M. Xie, “Stochastic resonance in a tristable optomechanical system,” Phys. Rev. A 95, 023808 (2017).
[Crossref]

L. Zhang and H.-Y. Kong, “Self-sustained oscillation and harmonic generation in optomechanical systems with quadratic couplings,” Phys. Rev. A 89, 023847 (2014).
[Crossref]

Phys. Rev. E (1)

L. Kabiraj, R. Steinert, A. Saurabh, and C. O. Paschereit, “Coherence resonance in a thermoacoustic system,” Phys. Rev. E 92, 042909 (2015).
[Crossref]

Phys. Rev. Lett. (9)

Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable GaAs/al0.45ga0.55As superlattice,” Phys. Rev. Lett. 121, 086806 (2018).
[Crossref]

Z. Liu and Y.-C. Lai, “Coherence resonance in coupled chaotic oscillators,” Phys. Rev. Lett. 86, 4737–4740 (2001).
[Crossref] [PubMed]

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[Crossref] [PubMed]

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
[Crossref] [PubMed]

O. Kyriienko, T. C. H. Liew, and I. A. Shelykh, “Optomechanics with cavity polaritons: Dissipative coupling and unconventional bistability,” Phys. Rev. Lett. 112, 076402 (2014).
[Crossref] [PubMed]

F. Marquardt, J. G. E. Harris, and S. M. Girvin, “Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities,” Phys. Rev. Lett. 96, 103901 (2006).
[Crossref] [PubMed]

H. Gang, T. Ditzinger, C. Z. Ning, and H. Haken, “Stochastic resonance without external periodic force,” Phys. Rev. Lett. 71, 807–810 (1993).
[Crossref] [PubMed]

A. S. Pikovsky and J. Kurths, “Coherence resonance in a noise-driven excitable system,” Phys. Rev. Lett. 78, 775–778 (1997).
[Crossref]

O. V. Ushakov, H.-J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, and M. A. Zaks, “Coherence resonance near a hopf bifurcation,” Phys. Rev. Lett. 95, 123903 (2005).
[Crossref] [PubMed]

Phys. Today (1)

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

Rev. Mod. Phys. (2)

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

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

Science (1)

C. Y. Lee, W. Choi, J.-H. Han, and M. S. Strano, “Coherence resonance in a single-walled carbon nanotube ion channel,” Science 329, 1320–1324 (2010).
[Crossref] [PubMed]

Tellus (1)

R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, “Stochastic resonance in climatic change,” Tellus 34, 10–16 (1982).
[Crossref]

Other (2)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, eds., Cavity optomechanics : Nano-and Micromechanical Resonators Interacting with Light (Springer-Verlag, 2014).

F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations: An Introduction(Dover Publications, 1989).

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Figures (4)

Fig. 1
Fig. 1 (a) Optomechanical system driven by a coherent optical field Ed and a periodic force signal Fs. (b) System stability diagram. The parameter plane is divided into four regions: (i) one stable fixed point (grey); (ii) three fixed points (red); (iii) parametric instability (blue); (iv) overlap region (purple). The white dashed line indicates the detuning (Δ = −1.38ωm) used in the following numerical calculations. The system parameters are (g, κ, γm) = (0.21, 1, 0.25)× ωm.
Fig. 2
Fig. 2 Noise-sustained quasi-coherent oscillations. (a–c) Single trajectories of system dynamics (the mechanical position x) with different noise strengths. (d–f) Peak height Hω, peak width Δω, and coherence factor β as functions of noise strength Dm. The system parameters are (g, κ, γ, Ed, Δ) = (0.21, 1, 0.25, 2.85, −1.38) × ωm.
Fig. 3
Fig. 3 Time series of system variable (x) and their power spectra with (a,b) no thermal noise (Dm = 0) and (c,d) thermal noise (Dm = 0.55). The system parameters are the same as in Fig. 2 except for Ed = 3.11ωm.
Fig. 4
Fig. 4 Stochastic resonance (SR) with self-sustained oscillations. (a) Signal-to-noise ratio (SNR) versus noise strength Dm. The red squares are the original data for the SNR calculated from trajectories and the blue curve is a fit to the SNR data. Points A, B, and C are representative of the low-noise, optimal, and high-noise regimes, respectively. (b) Spectra at noise levels A (yellow), B (blue), and C (red). (c) Time series of mechanical position x with no noise. (d–f) Typical trajectories of mechanical position x at noise levels A–C, respectively. The parameters are (g, κ, γ, Ed, Δ, Fs, fs) = (0.21, 1, 0.25, 3.11, −1.38, 1.5, 0.05) × ωm.

Equations (17)

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H ^ = 1 2 ω m ( x ^ 2 + p ^ 2 ) Δ a ^ a ^ g a ^ a ^ x ^ + i E d ( a ^ a ^ ) + F s x ^ cos ( ω f t ) ,
d a ^ d t = ( i Δ + κ / 2 i g x ^ ) a ^ E d ,
d a ^ d t = ( i Δ + κ / 2 + i g x ^ ) a ^ E d ,
d p ^ d t = γ m p ^ ω m x ^ + g a ^ a ^ F s cos ( ω f t ) + ξ ,
d x ^ d t = ω m p ^ .
ω m ( κ 2 / 4 + ( Δ + g x s ) 2 ) x s g E d 2 = 0 .
y ˙ = J y ^ + ξ 0 ,
J = ( i ( Δ + g x s ) κ 2 0 0 i g α s 0 i ( Δ + g x s ) κ 2 0 i g α s * g α s * g α s γ m ω m 0 0 ω m 0 ) .
c 1 = γ m + κ ,
c 2 = ( g x s + Δ ) 2 + 1 4 κ 2 + γ m κ + ω m 2 ,
c 3 = γ m ( g x s + Δ ) 2 + 1 4 γ m κ 2 + κ ω m 2 ,
c 4 = ( ( g x s + Δ ) 2 + 2 g x s ( Δ + x s g ) + 1 4 κ 2 ) ω m 2 .
D 1 = c 1 ,
D 2 = | c 1 c 3 1 c 2 | ,
D 3 = | c 1 c 3 0 1 c 2 c 4 0 c 1 c 3 | ,
D 4 = | c 1 c 3 0 0 1 c 2 c 4 0 0 c 1 c 3 0 0 1 c 2 c 4 | = c 4 D 3 .
D 3 = 0 , D 1 > 0 , D 2 > 0 , c 4 > 0 , and dD 3 d λ > 0 .

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