Abstract

We employ a decoupled Heisenberg–Langevin equation for the observation and physical interpretation of mechanical-mode splitting (MMS) of the movable mirror in a generic optomechanical system. Then we identify some observable and significant features of MMS in a two-mode cavity. That is, the second control field coupled to another optical mode is input to the system to modify the mechanical mode, leading to the suppression of transmission, the appearance of the doublet in the spectrum of the anti-Stokes field, and the emergence of optomechanically induced transparency in corresponding new mechanical modes. Furthermore, we open two transparent windows in virtue of MMS and find the second splitting of the mechanical mode in this two-mode optomechanical system.

© 2014 Optical Society of America

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  19. A. Ashkin, J. Dziedzic, J. Bjorkholm, and S. Chu, Opt. Lett. 11, 288 (1986).
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  28. M. Bhattacharya and P. Meystre, Phys. Rev. Lett. 99, 073601 (2007).
    [CrossRef]
  29. H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, Phys. Rev. A 86, 013815 (2012).
    [CrossRef]

2014

P. C. Chaumet and A. Rahmani, Nat. Nanotechnol. 9, 252 (2014).
[CrossRef]

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nat. Photonics 8, 234 (2014).
[CrossRef]

2013

H. Xiong, L.-G. Si, X.-Y. Lü, X. Yang, and Y. Wu, Opt. Lett. 38, 353 (2013).
[CrossRef]

A. Kronwald and F. Marquardt, Phys. Rev. Lett. 111, 133601 (2013).
[CrossRef]

K. Brøkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, Phys. Rev. Lett. 111, 053603 (2013).
[CrossRef]

X.-Y. Lü, W.-M. Zhang, S. Ashhab, Y. Wu, and F. Nori, Sci. Rep. 3, 2943 (2013).

2012

S. Basiri-Esfahani, U. Akram, and G. J. Milburn, New J. Phys. 14, 085017 (2012).
[CrossRef]

A. Mari and J. Eisert, Phys. Rev. Lett. 108, 120602 (2012).
[CrossRef]

A. Bonakdar, J. Kohoutek, D. Dey, and H. Mohseni, Opt. Lett. 37, 3258 (2012).
[CrossRef]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rab, Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, Phys. Rev. Lett. 109, 063601 (2012).
[CrossRef]

H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, Phys. Rev. A 86, 013815 (2012).
[CrossRef]

2011

2010

G. S. Agarwal and S. Huang, Phys. Rev. A 81, 041803 (2010).
[CrossRef]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
[CrossRef]

W. He, J.-J. Li, and K.-D. Zhu, Opt. Lett. 35, 339 (2010).
[CrossRef]

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, Nat. Phys. 6, 707 (2010).
[CrossRef]

2009

M. Tomes and T. Carmon, Phys. Rev. Lett. 102, 113601 (2009).
[CrossRef]

2008

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, Phys. Rev. Lett. 101, 263602 (2008).
[CrossRef]

2007

Y. Wu and X. Yang, Phys. Rev. A 76, 013832 (2007).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, Opt. Express 15, 17172 (2007).
[CrossRef]

M. Bhattacharya and P. Meystre, Phys. Rev. Lett. 99, 073601 (2007).
[CrossRef]

2005

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

1986

1975

T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

1970

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).
[CrossRef]

Abram, I.

Agarwal, G. S.

G. S. Agarwal and S. Huang, Phys. Rev. A 81, 041803 (2010).
[CrossRef]

Akram, U.

S. Basiri-Esfahani, U. Akram, and G. J. Milburn, New J. Phys. 14, 085017 (2012).
[CrossRef]

Antoni, T.

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
[CrossRef]

Ashhab, S.

X.-Y. Lü, W.-M. Zhang, S. Ashhab, Y. Wu, and F. Nori, Sci. Rep. 3, 2943 (2013).

Ashkin, A.

Basiri-Esfahani, S.

S. Basiri-Esfahani, U. Akram, and G. J. Milburn, New J. Phys. 14, 085017 (2012).
[CrossRef]

Bennett, S. D.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rab, Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

Bhattacharya, M.

M. Bhattacharya and P. Meystre, Phys. Rev. Lett. 99, 073601 (2007).
[CrossRef]

Bjorkholm, J.

Bonakdar, A.

Braive, R.

Briant, T.

Brøkje, K.

K. Brøkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, Phys. Rev. Lett. 111, 053603 (2013).
[CrossRef]

Carmon, T.

X. Zhang, M. Tomes, and T. Carmon, Opt. Express 19, 9066 (2011).
[CrossRef]

M. Tomes and T. Carmon, Phys. Rev. Lett. 102, 113601 (2009).
[CrossRef]

Chan, J.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, Nature 472, 69 (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, Nature 472, 69 (2011).
[CrossRef]

Chaumet, P. C.

P. C. Chaumet and A. Rahmani, Nat. Nanotechnol. 9, 252 (2014).
[CrossRef]

Chu, S.

Cohadon, P.-F.

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
[CrossRef]

Dey, D.

Dobrindt, J. M.

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, Phys. Rev. Lett. 101, 263602 (2008).
[CrossRef]

Dziedzic, J.

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, Nature 472, 69 (2011).
[CrossRef]

Eisert, J.

A. Mari and J. Eisert, Phys. Rev. Lett. 108, 120602 (2012).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
[CrossRef]

Giacobino, E.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nat. Photonics 8, 234 (2014).
[CrossRef]

Giner, L.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nat. Photonics 8, 234 (2014).
[CrossRef]

Girvin, S. M.

K. Brøkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, Phys. Rev. Lett. 111, 053603 (2013).
[CrossRef]

Grudinin, I. S.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” arXiv:0805.0803 (2008).

Habraken, S. J. M.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rab, Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

Hansch, T. W.

T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

Harris, J. G. E.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, Nat. Phys. 6, 707 (2010).
[CrossRef]

He, W.

Heidmann, A.

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, Nature 472, 69 (2011).
[CrossRef]

Huang, S.

G. S. Agarwal and S. Huang, Phys. Rev. A 81, 041803 (2010).
[CrossRef]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Jayich, A. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, Nat. Phys. 6, 707 (2010).
[CrossRef]

Kippenberg, T. J.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
[CrossRef]

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, Phys. Rev. Lett. 101, 263602 (2008).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, Opt. Express 15, 17172 (2007).
[CrossRef]

Kohoutek, J.

Komar, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rab, Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

Kronwald, A.

A. Kronwald and F. Marquardt, Phys. Rev. Lett. 111, 133601 (2013).
[CrossRef]

Kuhn, A. G.

Laurat, J.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nat. Photonics 8, 234 (2014).
[CrossRef]

Le Gratiet, L.

Li, J.-J.

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, Nature 472, 69 (2011).
[CrossRef]

Lü, X.-Y.

X.-Y. Lü, W.-M. Zhang, S. Ashhab, Y. Wu, and F. Nori, Sci. Rep. 3, 2943 (2013).

H. Xiong, L.-G. Si, X.-Y. Lü, X. Yang, and Y. Wu, Opt. Lett. 38, 353 (2013).
[CrossRef]

Ludwig, M.

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, Phys. Rev. Lett. 109, 063601 (2012).
[CrossRef]

Lukin, M. D.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rab, Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

Maleki, L.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” arXiv:0805.0803 (2008).

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Mari, A.

A. Mari and J. Eisert, Phys. Rev. Lett. 108, 120602 (2012).
[CrossRef]

Marquardt, F.

A. Kronwald and F. Marquardt, Phys. Rev. Lett. 111, 133601 (2013).
[CrossRef]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, Phys. Rev. Lett. 109, 063601 (2012).
[CrossRef]

Matsko, A. B.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” arXiv:0805.0803 (2008).

Maxein, D.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nat. Photonics 8, 234 (2014).
[CrossRef]

Mayer Alegre, T. P.

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, Nature 472, 69 (2011).
[CrossRef]

Meystre, P.

M. Bhattacharya and P. Meystre, Phys. Rev. Lett. 99, 073601 (2007).
[CrossRef]

Milburn, G. J.

S. Basiri-Esfahani, U. Akram, and G. J. Milburn, New J. Phys. 14, 085017 (2012).
[CrossRef]

Mohseni, H.

Nicolas, A.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nat. Photonics 8, 234 (2014).
[CrossRef]

Nori, F.

X.-Y. Lü, W.-M. Zhang, S. Ashhab, Y. Wu, and F. Nori, Sci. Rep. 3, 2943 (2013).

Nunnenkamp, A.

K. Brøkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, Phys. Rev. Lett. 111, 053603 (2013).
[CrossRef]

Painter, O.

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, Phys. Rev. Lett. 109, 063601 (2012).
[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, Nature 472, 69 (2011).
[CrossRef]

Philip, I. R.

Rab, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rab, Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

Rahmani, A.

P. C. Chaumet and A. Rahmani, Nat. Nanotechnol. 9, 252 (2014).
[CrossRef]

Rivière, R.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
[CrossRef]

Safavi-Naeini, A. H.

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, Phys. Rev. Lett. 109, 063601 (2012).
[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, Nature 472, 69 (2011).
[CrossRef]

Sagnes, I.

Sankey, J. C.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, Nat. Phys. 6, 707 (2010).
[CrossRef]

Schawlow, A. L.

T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

Schliesser, A.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
[CrossRef]

Si, L.-G.

H. Xiong, L.-G. Si, X.-Y. Lü, X. Yang, and Y. Wu, Opt. Lett. 38, 353 (2013).
[CrossRef]

H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, Phys. Rev. A 86, 013815 (2012).
[CrossRef]

Stannigel, K.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rab, Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

Teufel, J. D.

K. Brøkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, Phys. Rev. Lett. 111, 053603 (2013).
[CrossRef]

Tomes, M.

X. Zhang, M. Tomes, and T. Carmon, Opt. Express 19, 9066 (2011).
[CrossRef]

M. Tomes and T. Carmon, Phys. Rev. Lett. 102, 113601 (2009).
[CrossRef]

Vahala, K. J.

Veissier, L.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nat. Photonics 8, 234 (2014).
[CrossRef]

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
[CrossRef]

Wilson-Rae, I.

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, Phys. Rev. Lett. 101, 263602 (2008).
[CrossRef]

Winger, 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, Nature 472, 69 (2011).
[CrossRef]

Wu, Y.

X.-Y. Lü, W.-M. Zhang, S. Ashhab, Y. Wu, and F. Nori, Sci. Rep. 3, 2943 (2013).

H. Xiong, L.-G. Si, X.-Y. Lü, X. Yang, and Y. Wu, Opt. Lett. 38, 353 (2013).
[CrossRef]

H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, Phys. Rev. A 86, 013815 (2012).
[CrossRef]

Y. Wu and X. Yang, Phys. Rev. A 76, 013832 (2007).
[CrossRef]

Xiong, H.

H. Xiong, L.-G. Si, X.-Y. Lü, X. Yang, and Y. Wu, Opt. Lett. 38, 353 (2013).
[CrossRef]

H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, Phys. Rev. A 86, 013815 (2012).
[CrossRef]

Yang, C.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, Nat. Phys. 6, 707 (2010).
[CrossRef]

Yang, X.

H. Xiong, L.-G. Si, X.-Y. Lü, X. Yang, and Y. Wu, Opt. Lett. 38, 353 (2013).
[CrossRef]

H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, Phys. Rev. A 86, 013815 (2012).
[CrossRef]

Y. Wu and X. Yang, Phys. Rev. A 76, 013832 (2007).
[CrossRef]

Zhang, W.-M.

X.-Y. Lü, W.-M. Zhang, S. Ashhab, Y. Wu, and F. Nori, Sci. Rep. 3, 2943 (2013).

Zhang, X.

Zheng, A.-S.

H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, Phys. Rev. A 86, 013815 (2012).
[CrossRef]

Zhu, K.-D.

Zoller, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rab, Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

Zwickl, B. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, Nat. Phys. 6, 707 (2010).
[CrossRef]

Nat. Nanotechnol.

P. C. Chaumet and A. Rahmani, Nat. Nanotechnol. 9, 252 (2014).
[CrossRef]

Nat. Photonics

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nat. Photonics 8, 234 (2014).
[CrossRef]

Nat. Phys.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, Nat. Phys. 6, 707 (2010).
[CrossRef]

Nature

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, Nature 472, 69 (2011).
[CrossRef]

New J. Phys.

S. Basiri-Esfahani, U. Akram, and G. J. Milburn, New J. Phys. 14, 085017 (2012).
[CrossRef]

Opt. Commun.

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Figures (3)

Fig. 1.
Fig. 1.

(a) Calculation result of |Q| varies with Ω in different optical power Pl of the control field with frequency ωl. (b) The dependence of the peak and dip position of |Q| on Pl. We use m=20ng, ωm=2π×51.8MHz, γm=2π×41.0kHz, G=2π×12GHz/nm, κ=2π×15.0MHz, Δ=ωm throughout the work, which are taken from the experiments in [9]. And we fix εp=εl/100.

Fig. 2.
Fig. 2.

(a) Suppression of OMIT signal in the case of Δ2=ωm. (b) Spectrum of anti-Stokes field from the scattering of light from the control field with frequency ωl1. (c) OMIT signals by considering corresponding new mechanical modes. (d)  OMIT signals by considering left mechanical mode (top figure) and right mechanical mode (bottom figure)in cases of different Pl2.

Fig. 3.
Fig. 3.

(a) Transmission of the probe field in corresponding Δ2. (b) Two transparent windows induced by modulating Δ2 for a strong Pl2. (c) The second splitting of the mechanical mode of the movable mirror. (d) The spectrum of anti-Stokes field excited from the control field with frequency ωl1.

Equations (4)

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md2dt2δq+mγmddtδq+mωm2δq=2|f(Ω)|cos(Ωt+ϕf)
|Q|=|f(Ω)Ω2iγmΩ+ωm2|.
H=(p^22m+mωm2q^22)Δ1c^1c^1Δ2c^2c^2+Gq^c^1c^1+Gq^c^2c^2+εin,
C2=[1+if(Ω)]ηcκεp2(Δ2Gq0)f(Ω)iΩ+β2,Q=(iΩ+β2)C2ηcκεpiGc20,C1=iGc10QiΩ+β1,

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