Abstract

We theoretically demonstrate the mechanically mediated electromagnetically induced transparency in a two-mode cavity optomechanical system, where two cavity modes are coupled to a common mechanical resonator. When the two cavity modes are driven on their respective red sidebands by two pump beams, a transparency window appears in the probe transmission spectrum due to destructive interference. Under this situation the transmitted probe beam can be delayed as much as 4 μs, which can be easily controlled by the power of the pump beams.

© 2013 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. F. Marquardt and S. M. Girvin, “Optomechanics,” Physics2,40 (2009).
    [CrossRef]
  4. M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today65,29–35 (2012).
    [CrossRef]
  5. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science322,235–238 (2008).
    [CrossRef] [PubMed]
  6. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett.107,063601 (2011).
    [CrossRef] [PubMed]
  7. A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett.107,063602 (2011).
    [CrossRef] [PubMed]
  8. J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London)471, 204–208 (2011).
    [CrossRef]
  9. E. Verhagen, S. Delglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London)482,63–67 (2012).
    [CrossRef]
  10. B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A85,063820 (2012).
    [CrossRef]
  11. J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature (London)475,359–363 (2011).
    [CrossRef]
  12. J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London)478,89–92 (2011).
    [CrossRef]
  13. J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett.101,263602 (2008).
    [CrossRef] [PubMed]
  14. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London)460,724–727 (2009).
    [CrossRef]
  15. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A81, 041803 (2010).
    [CrossRef]
  16. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520–1523 (2010).
    [CrossRef] [PubMed]
  17. 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 (London)472, 69–73 (2011).
    [CrossRef]
  18. M. Karuza, C. Biancofiore, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a room temperature membrane-in-the-middle setup,” arXiv:1209.1352 (2012).
  19. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77,633–673 (2005).
    [CrossRef]
  20. L.V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London)397,594–598 (1999).
    [CrossRef]
  21. D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys.13,023003 (2011).
    [CrossRef]
  22. K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66,2593–2596 (1991).
    [CrossRef] [PubMed]
  23. M. C. Phillips, H. Wang, I. Rumyantsev, N. H. Kwong, R. Takayama, and R. Binder, “Electromagnetically Induced Transparency in Semiconductors via Biexciton Coherence,” Phys. Rev. Lett.91,183602 (2003).
    [CrossRef] [PubMed]
  24. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater.8,758–762 (2009).
    [CrossRef] [PubMed]
  25. C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett.97,247401 (2006).
    [CrossRef]
  26. X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys.9,179–184 (2013).
    [CrossRef]
  27. F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature480, 351–354 (2011).
    [CrossRef] [PubMed]
  28. F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys.14,123037 (2012).
    [CrossRef]
  29. J. M. Dobrindt and T. J. Kippenberg, “Theoretical analysis of mechanical displacement measurement using a multiple cavity mode transducer,” Phys. Rev. Lett.104,033901 (2010).
    [CrossRef] [PubMed]
  30. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett.109,063601 (2012).
    [CrossRef] [PubMed]
  31. P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon n onlinearities in two-mode optomechanics,” Phys. Rev. A87,013839 (2013).
    [CrossRef]
  32. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett.109,013603 (2012).
    [CrossRef] [PubMed]
  33. K. Qu and G. S. Agarwal, “Optical memories and transduction of fields in double cavity optomechanical systems,” arXiv:1210.4067 (2012).
  34. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun.3,1196 (2012).
    [CrossRef]
  35. C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” arXiv:1205.2360 (2012).
  36. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
    [CrossRef]
  37. R. W. Boyd, Nonlinear Optics (San Diego, CA: Academic) (2008).
  38. C. W. Gardiner and P. Zoller, Quantum Noise (Springer) (2004).
  39. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett.99,093901 (2007).
    [CrossRef] [PubMed]

2013

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys.9,179–184 (2013).
[CrossRef]

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon n onlinearities in two-mode optomechanics,” Phys. Rev. A87,013839 (2013).
[CrossRef]

2012

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett.109,013603 (2012).
[CrossRef] [PubMed]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun.3,1196 (2012).
[CrossRef]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett.109,063601 (2012).
[CrossRef] [PubMed]

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys.14,123037 (2012).
[CrossRef]

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today65,29–35 (2012).
[CrossRef]

E. Verhagen, S. Delglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London)482,63–67 (2012).
[CrossRef]

B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A85,063820 (2012).
[CrossRef]

2011

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

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London)478,89–92 (2011).
[CrossRef]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett.107,063601 (2011).
[CrossRef] [PubMed]

A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett.107,063602 (2011).
[CrossRef] [PubMed]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London)471, 204–208 (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 (London)472, 69–73 (2011).
[CrossRef]

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature480, 351–354 (2011).
[CrossRef] [PubMed]

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys.13,023003 (2011).
[CrossRef]

2010

J. M. Dobrindt and T. J. Kippenberg, “Theoretical analysis of mechanical displacement measurement using a multiple cavity mode transducer,” Phys. Rev. Lett.104,033901 (2010).
[CrossRef] [PubMed]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A81, 041803 (2010).
[CrossRef]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520–1523 (2010).
[CrossRef] [PubMed]

2009

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics2,40 (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 (London)460,724–727 (2009).
[CrossRef]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater.8,758–762 (2009).
[CrossRef] [PubMed]

2008

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
[CrossRef]

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett.101,263602 (2008).
[CrossRef] [PubMed]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science322,235–238 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science321,1172–1176 (2008).
[CrossRef] [PubMed]

2007

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express15,17172–17205 (2007).
[CrossRef] [PubMed]

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

2006

C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett.97,247401 (2006).
[CrossRef]

2005

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77,633–673 (2005).
[CrossRef]

2003

M. C. Phillips, H. Wang, I. Rumyantsev, N. H. Kwong, R. Takayama, and R. Binder, “Electromagnetically Induced Transparency in Semiconductors via Biexciton Coherence,” Phys. Rev. Lett.91,183602 (2003).
[CrossRef] [PubMed]

1999

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

1991

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

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A81, 041803 (2010).
[CrossRef]

K. Qu and G. S. Agarwal, “Optical memories and transduction of fields in double cavity optomechanical systems,” arXiv:1210.4067 (2012).

Alegre, T. P.

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London)478,89–92 (2011).
[CrossRef]

Allman, M. S.

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

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

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520–1523 (2010).
[CrossRef] [PubMed]

Aspelmeyer, M.

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today65,29–35 (2012).
[CrossRef]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London)478,89–92 (2011).
[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 (London)460,724–727 (2009).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
[CrossRef]

Beausoleil, R. G.

C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett.97,247401 (2006).
[CrossRef]

Behroozi, C. H.

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

Bennett, S. D.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon n onlinearities in two-mode optomechanics,” Phys. Rev. A87,013839 (2013).
[CrossRef]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett.109,013603 (2012).
[CrossRef] [PubMed]

Biancofiore, C.

M. Karuza, C. Biancofiore, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a room temperature membrane-in-the-middle setup,” arXiv:1209.1352 (2012).

Binder, R.

M. C. Phillips, H. Wang, I. Rumyantsev, N. H. Kwong, R. Takayama, and R. Binder, “Electromagnetically Induced Transparency in Semiconductors via Biexciton Coherence,” Phys. Rev. Lett.91,183602 (2003).
[CrossRef] [PubMed]

Boller, K.-J.

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

Borkje, K.

A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett.107,063602 (2011).
[CrossRef] [PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (San Diego, CA: Academic) (2008).

Brennecke, F.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science322,235–238 (2008).
[CrossRef] [PubMed]

Chan, J.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun.3,1196 (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, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London)472, 69–73 (2011).
[CrossRef]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London)478,89–92 (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 (London)472, 69–73 (2011).
[CrossRef]

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys.13,023003 (2011).
[CrossRef]

Cho, S. U.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature480, 351–354 (2011).
[CrossRef] [PubMed]

Cicak, K.

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

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

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520–1523 (2010).
[CrossRef] [PubMed]

Delglise, S.

E. Verhagen, S. Delglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London)482,63–67 (2012).
[CrossRef]

Di Giuseppe, G.

M. Karuza, C. Biancofiore, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a room temperature membrane-in-the-middle setup,” arXiv:1209.1352 (2012).

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.104,033901 (2010).
[CrossRef] [PubMed]

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett.101,263602 (2008).
[CrossRef] [PubMed]

Dong, C.

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” arXiv:1205.2360 (2012).

Donner, T.

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

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science322,235–238 (2008).
[CrossRef] [PubMed]

Dutton, Z.

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

Eichenfield, M.

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D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys.13,023003 (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 (London)472, 69–73 (2011).
[CrossRef]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London)478,89–92 (2011).
[CrossRef]

Saloniemi, H.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature480, 351–354 (2011).
[CrossRef] [PubMed]

Santori, C.

C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett.97,247401 (2006).
[CrossRef]

Schliesser, A.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys.9,179–184 (2013).
[CrossRef]

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys.14,123037 (2012).
[CrossRef]

E. Verhagen, S. Delglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London)482,63–67 (2012).
[CrossRef]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520–1523 (2010).
[CrossRef] [PubMed]

Schwab, K.

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today65,29–35 (2012).
[CrossRef]

Sillanpää, M. A.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature480, 351–354 (2011).
[CrossRef] [PubMed]

Simmonds, R. W.

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

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

Sirois, A. J.

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

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

Stannigel, K.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon n onlinearities in two-mode optomechanics,” Phys. Rev. A87,013839 (2013).
[CrossRef]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett.109,013603 (2012).
[CrossRef] [PubMed]

Takayama, R.

M. C. Phillips, H. Wang, I. Rumyantsev, N. H. Kwong, R. Takayama, and R. Binder, “Electromagnetically Induced Transparency in Semiconductors via Biexciton Coherence,” Phys. Rev. Lett.91,183602 (2003).
[CrossRef] [PubMed]

Tamarat, P.

C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett.97,247401 (2006).
[CrossRef]

Teufel, J. D.

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

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

Tian, L.

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” arXiv:1205.2360 (2012).

Tombesi, P.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
[CrossRef]

M. Karuza, C. Biancofiore, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a room temperature membrane-in-the-middle setup,” arXiv:1209.1352 (2012).

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science321,1172–1176 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express15,17172–17205 (2007).
[CrossRef] [PubMed]

Vanner, M. R.

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

Verhagen, E.

E. Verhagen, S. Delglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London)482,63–67 (2012).
[CrossRef]

Vitali, D.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
[CrossRef]

M. Karuza, C. Biancofiore, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a room temperature membrane-in-the-middle setup,” arXiv:1209.1352 (2012).

Wang, H.

M. C. Phillips, H. Wang, I. Rumyantsev, N. H. Kwong, R. Takayama, and R. Binder, “Electromagnetically Induced Transparency in Semiconductors via Biexciton Coherence,” Phys. Rev. Lett.91,183602 (2003).
[CrossRef] [PubMed]

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” arXiv:1205.2360 (2012).

Weis, S.

E. Verhagen, S. Delglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London)482,63–67 (2012).
[CrossRef]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520–1523 (2010).
[CrossRef] [PubMed]

Weiss, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater.8,758–762 (2009).
[CrossRef] [PubMed]

Whittaker, J. D.

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

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

Wilson-Rae, I.

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett.101,263602 (2008).
[CrossRef] [PubMed]

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

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, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London)472, 69–73 (2011).
[CrossRef]

Wrachtrup, J.

C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett.97,247401 (2006).
[CrossRef]

Zhou, X.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys.9,179–184 (2013).
[CrossRef]

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys.14,123037 (2012).
[CrossRef]

Zoller, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon n onlinearities in two-mode optomechanics,” Phys. Rev. A87,013839 (2013).
[CrossRef]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett.109,013603 (2012).
[CrossRef] [PubMed]

C. W. Gardiner and P. Zoller, Quantum Noise (Springer) (2004).

Zwerger, W.

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

Nat. Commun.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun.3,1196 (2012).
[CrossRef]

Nat. Mater.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater.8,758–762 (2009).
[CrossRef] [PubMed]

Nat. Phys.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys.9,179–184 (2013).
[CrossRef]

Nature

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature480, 351–354 (2011).
[CrossRef] [PubMed]

Nature (London)

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

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

E. Verhagen, S. Delglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London)482,63–67 (2012).
[CrossRef]

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

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London)478,89–92 (2011).
[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 (London)460,724–727 (2009).
[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 (London)472, 69–73 (2011).
[CrossRef]

New J. Phys.

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys.13,023003 (2011).
[CrossRef]

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys.14,123037 (2012).
[CrossRef]

Opt. Express

Phys. Rev. A

B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A85,063820 (2012).
[CrossRef]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A81, 041803 (2010).
[CrossRef]

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon n onlinearities in two-mode optomechanics,” Phys. Rev. A87,013839 (2013).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
[CrossRef]

Phys. Rev. Lett.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett.109,013603 (2012).
[CrossRef] [PubMed]

C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett.97,247401 (2006).
[CrossRef]

J. M. Dobrindt and T. J. Kippenberg, “Theoretical analysis of mechanical displacement measurement using a multiple cavity mode transducer,” Phys. Rev. Lett.104,033901 (2010).
[CrossRef] [PubMed]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett.109,063601 (2012).
[CrossRef] [PubMed]

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

M. C. Phillips, H. Wang, I. Rumyantsev, N. H. Kwong, R. Takayama, and R. Binder, “Electromagnetically Induced Transparency in Semiconductors via Biexciton Coherence,” Phys. Rev. Lett.91,183602 (2003).
[CrossRef] [PubMed]

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett.101,263602 (2008).
[CrossRef] [PubMed]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett.107,063601 (2011).
[CrossRef] [PubMed]

A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett.107,063602 (2011).
[CrossRef] [PubMed]

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

Phys. Today

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today65,29–35 (2012).
[CrossRef]

Physics

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics2,40 (2009).
[CrossRef]

Rev. Mod. Phys.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77,633–673 (2005).
[CrossRef]

Science

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520–1523 (2010).
[CrossRef] [PubMed]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science322,235–238 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science321,1172–1176 (2008).
[CrossRef] [PubMed]

Other

M. Karuza, C. Biancofiore, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a room temperature membrane-in-the-middle setup,” arXiv:1209.1352 (2012).

K. Qu and G. S. Agarwal, “Optical memories and transduction of fields in double cavity optomechanical systems,” arXiv:1210.4067 (2012).

R. W. Boyd, Nonlinear Optics (San Diego, CA: Academic) (2008).

C. W. Gardiner and P. Zoller, Quantum Noise (Springer) (2004).

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” arXiv:1205.2360 (2012).

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Figures (4)

Fig. 1
Fig. 1

Schematic of a two-mode optomechanical system where two optical cavity modes, a1 and a2, are coupled to the same mechanical mode b. The left cavity is driven by a strong pump beam EL in the simultaneous presence of a weak probe beam Ep while the right cavity is only driven by a pump beam ER.

Fig. 2
Fig. 2

Probe transmission as a function of the probe-cavity detuning Δp = ωpω1 for left pump power PL equals to 0, 0.1, 1, and 10 μW, respectively. The right pump power is kept equal to 0.1 μW. Both the cavities are pumped on their respective red sidebands, i.e., Δ1 = ωm and Δ2 = ωm. Other parameters used are ω1 = 2π × 205.3 THz, ω2 = 2π × 194.1 THz, κ1 = 2π × 520 MHz, κ2 = 1.73 GHz, κe,1 = 0.2κ1, κe,2 = 0.42κ2, g1 = 2π × 960 kHz, g2 = 2π × 430 kHz, ωm = 2π × 4 GHz, Qm = 87 × 103.

Fig. 3
Fig. 3

(a) Magnitude and (b) phase of the transmitted probe beam versus probe-cavity detuning Δp for PL = 10 μW and PR = 0.1 μW. Other parameters are the same with figure 2.

Fig. 4
Fig. 4

Group delay τg of the (a)–(c) transmitted (d) reflected probe beam as a function of the left pump power PL with Δ1 = ωm and Δ2 = ωm considering the effects of κe,1 and PR. Other parameters are ω1 = 2π × 205.3 THz, ω2 = 2π × 194.1 THz, κ1 = 2π × 520 MHz, κ2 = 1.73 GHz, κe,2 = 0.42κ2, g1 = 2π × 960 kHz, g2 = 2π × 430 kHz, ωm = 2π × 4 GHz, Qm = 87 × 103.

Equations (17)

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

H = k = 1 , 2 h ¯ Δ k a k a k + h ¯ ω m b b k = 1 , 2 h ¯ g k a k a k ( b + b ) + i h ¯ κ e , 1 E L ( a 1 a 1 ) + i h ¯ κ e , 2 E R ( a 2 a 2 ) + i h ¯ κ e , 1 E p ( a 1 e i δ t a 1 e i δ t ) .
a ˙ 1 = i ( Δ 1 g 1 Q ) a 1 κ 1 a 1 + κ e , 1 ( E L + E p e i δ t ) + 2 κ 1 a in , 1 ,
a ˙ 2 = i ( Δ 2 g 2 Q ) a 2 κ 2 a 2 + κ e , 2 E R + 2 κ 2 a in , 2 ,
Q ¨ + γ m Q ˙ + ω m 2 Q = 2 g 1 ω m a 1 a 1 + 2 g 2 ω m a 2 a 2 + ξ ,
a s , 1 = κ e , 1 E L κ 1 + i Δ 1 , a s , 2 = κ e , 2 E R κ 2 + i Δ 2 , Q s = 2 ω m ( g 1 | a s , 1 | 2 + g 2 | a s , 2 | 2 ) ,
a 1 = a s , 1 + δ a 1 , a 2 = a s , 2 + δ a 2 , Q = Q s + δ Q .
δ a ˙ 1 = ( κ 1 + i Δ 1 ) δ a 1 + i g 1 Q s δ a 1 + i g 1 a s , 1 δ Q + κ e , 1 E p e i δ t ,
δ a ˙ 2 = ( κ 2 + i Δ 2 ) δ a 2 + i g 2 Q s δ a 2 + i g 2 a s , 2 δ Q ,
δ Q ¨ + γ m δ Q ˙ + ω m 2 δ Q = 2 ω m g 1 a s , 1 ( δ a 1 + δ a 1 ) + 2 ω m g 2 a s , 2 ( δ a 2 + δ a 2 ) .
a 1 + = κ e , 1 E p κ 1 + i Δ 1 i δ 1 d ( δ ) i g 1 2 n 1 κ e , 1 E p ( κ 1 + i Δ 1 i δ ) 2 ,
d ( δ ) = k = 1 , 2 2 Δ k g k 2 n k ( κ k i δ ) 2 + Δ k 2 ω m 2 δ 2 i δ γ m ω m ,
n 1 = κ e , 1 E L 2 κ 1 2 + [ Δ 1 2 g 1 / ω m ( g 1 n 1 g 2 n 2 ) ] 2 ,
n 2 = κ e , 2 E R 2 κ 2 2 + [ Δ 2 2 g 2 / ω m ( g 1 n 1 g 2 n 2 ) ] 2 .
a out ( t ) = ( E L κ e , 1 a s , 1 ) e i ω L t + ( E p κ e , 1 a 1 + ) e i ( δ + ω L ) t κ e , 1 a 1 e i ( δ ω L ) t
t ( ω p ) = E p κ e , 1 a 1 + E p = 1 [ κ e , 1 κ 1 + i Δ 1 i δ 1 d ( δ ) i g 1 2 n 1 κ e , 1 ( κ 1 + i Δ 1 i δ ) 2 ] .
τ g = d ϕ d ω p | ω p = ω 1 .
a 1 + κ e , 1 E p i x + κ 1 + g 1 2 n 1 / 2 i x + γ m / 2 + g 2 2 n 2 / 2 i x + κ 2 ,

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