Abstract

We study a double-cavity optomechanical system in which a movable mirror with perfect reflection is inserted between two fixed mirrors with partial transmission. This optomechanical system is driven from both fixed end mirrors in a symmetric scheme by two strong coupling fields and two weak probe fields. We find that three interesting phenomena: coherent perfect absorption (CPA), coherent perfect transmission (CPT), and coherent perfect synthesis (CPS) can be attained within different parameter regimes. That is, we can make two input probe fields totally absorbed by the movable mirror without yielding any energy output from either end mirror (CPA); make an input probe field transmitted from one end mirror to the other end mirror without suffering any energy loss in the two cavities (CPT); make two input probe fields synthesized into one output probe field after undergoing either a perfect transmission or a perfect reflection (CPS). These interesting phenomena originate from the efficient hybrid coupling of optical and mechanical modes and may be all-optically controlled to realize novel photonic devices in quantum information networks.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]

2013 (6)

S. Mahajan, T. Kumar, A. B. Bhattacherjee, and ManMohan, “Ground-state cooling of a mechanical oscillator and detection of a weak force using a Bose-Einstein condensate,” Phys. Rev. A 87, 013621 (2013).
[Crossref]

A. Dalafi, M. H. Naderi, M. Soltanolkotabi, and Sh. Barzanjeh, “Nonlinear effects of atomic collisions on the optomechanical properties of a Bose-Einstein condensate in an optical cavity,” Phys. Rev. A 87, 013417 (2013).
[Crossref]

Y.-D. Wang and A. A. Clerk., “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
[Crossref] [PubMed]

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

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Y. X. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110, 223603 (2013).
[Crossref] [PubMed]

2012 (3)

S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012).
[Crossref]

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A 86, 042323 (2012).
[Crossref]

A. Mari and J. Eisert, “Cooling by heating: Very hot thermal light can significantly cool quantum systems,” Phys. Rev. Lett. 108, 120602 (2012).
[Crossref] [PubMed]

2011 (6)

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]

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[Crossref] [PubMed]

S. I. Schmid, K. Y. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011).
[Crossref]

G. De Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A 83, 052324 (2011).
[Crossref]

S. K. Steinke and P. Meystre, “Role of quantum fluctuations in the optomechanical properties of a Bose-Einstein condensate in a ring cavity,” Phys. Rev. A 84, 023834 (2011).
[Crossref]

2010 (4)

M. Paternostro, G. De Chiara, and G. M. Palma, “Cold-atom-induced control of an optomechanical device,” Phys. Rev. Lett. 104, 243602 (2010).
[Crossref] [PubMed]

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

P. Verlot, A. Tavernarakis, T. Briant, P.-F. Cohadon, and A. Heidmann, “Backaction amplification and quantum limits in optomechanical measurements,” Phys. Rev. Lett. 104, 133602 (2010).
[Crossref] [PubMed]

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

2009 (3)

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

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

F. Marquardt and S. M. Girvin, “Trend: Optomechanics,” Physics 2, 40 (2009).
[Crossref]

2008 (2)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

2007 (3)

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
[Crossref] [PubMed]

M. Bhattacharya and P. Meystre, “Trapping and cooling a mirror to its quantum mechanical ground state,” Phys. Rev. Lett. 99, 073601 (2007).
[Crossref] [PubMed]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

2006 (2)

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature (London) 444, 75–78 (2006).
[Crossref]

2005 (1)

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

2003 (1)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421, 925–928 (2003).
[Crossref]

Agarwal, G. S.

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

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

G. S. Agarwal and Sumei Huang, “Coherent perfect absorption in cavity opto-mechanics,” arXiv:1304.7323.

Aksyuk, V.

Y. X. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110, 223603 (2013).
[Crossref] [PubMed]

Anetsberger, G.

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

Arcizet, O.

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

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421, 925–928 (2003).
[Crossref]

Aspelmeyer, M.

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

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

Barbour, R.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[Crossref] [PubMed]

Barzanjeh, Sh.

A. Dalafi, M. H. Naderi, M. Soltanolkotabi, and Sh. Barzanjeh, “Nonlinear effects of atomic collisions on the optomechanical properties of a Bose-Einstein condensate in an optical cavity,” Phys. Rev. A 87, 013417 (2013).
[Crossref]

Bäuerle, D.

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

Bawaj, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[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 nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Bhattacharya, M.

M. Bhattacharya and P. Meystre, “Trapping and cooling a mirror to its quantum mechanical ground state,” Phys. Rev. Lett. 99, 073601 (2007).
[Crossref] [PubMed]

Bhattacherjee, A. B.

S. Mahajan, T. Kumar, A. B. Bhattacherjee, and ManMohan, “Ground-state cooling of a mechanical oscillator and detection of a weak force using a Bose-Einstein condensate,” Phys. Rev. A 87, 013621 (2013).
[Crossref]

Biancofiore, C.

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

Blaser, F.

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

Böhm, H.

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

Bouwmeester, D.

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature (London) 444, 75–78 (2006).
[Crossref]

Briant, T.

P. Verlot, A. Tavernarakis, T. Briant, P.-F. Cohadon, and A. Heidmann, “Backaction amplification and quantum limits in optomechanical measurements,” Phys. Rev. Lett. 104, 133602 (2010).
[Crossref] [PubMed]

Brukner, C.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[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, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472, 69–73 (2011).
[Crossref]

Chang, D. E.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (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]

Clerk., A. A.

Y.-D. Wang and A. A. Clerk., “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
[Crossref] [PubMed]

Cohadon, P.-F.

P. Verlot, A. Tavernarakis, T. Briant, P.-F. Cohadon, and A. Heidmann, “Backaction amplification and quantum limits in optomechanical measurements,” Phys. Rev. Lett. 104, 133602 (2010).
[Crossref] [PubMed]

Dalafi, A.

A. Dalafi, M. H. Naderi, M. Soltanolkotabi, and Sh. Barzanjeh, “Nonlinear effects of atomic collisions on the optomechanical properties of a Bose-Einstein condensate in an optical cavity,” Phys. Rev. A 87, 013417 (2013).
[Crossref]

Davanco, M.

Y. X. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110, 223603 (2013).
[Crossref] [PubMed]

De Chiara, G.

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A 86, 042323 (2012).
[Crossref]

G. De Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A 83, 052324 (2011).
[Crossref]

M. Paternostro, G. De Chiara, and G. M. Palma, “Cold-atom-induced control of an optomechanical device,” Phys. Rev. Lett. 104, 243602 (2010).
[Crossref] [PubMed]

Deléglise, S.

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

Di Giuseppe, G.

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

Eichenfield, M.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472, 69–73 (2011).
[Crossref]

Eisert, J.

A. Mari and J. Eisert, “Cooling by heating: Very hot thermal light can significantly cool quantum systems,” Phys. Rev. Lett. 108, 120602 (2012).
[Crossref] [PubMed]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Evers, J.

S. I. Schmid, K. Y. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011).
[Crossref]

Fiore, V.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[Crossref] [PubMed]

Fleischhauer, M.

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

Galassi, M.

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

Gavartin, E.

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

Gigan, S.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

Girvin, S. M.

F. Marquardt and S. M. Girvin, “Trend: Optomechanics,” Physics 2, 40 (2009).
[Crossref]

Gröblacher, S.

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

Habraken, S. J. M.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Hafezi, M.

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]

Hammerer, K.

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

Heidmann, A.

P. Verlot, A. Tavernarakis, T. Briant, P.-F. Cohadon, and A. Heidmann, “Backaction amplification and quantum limits in optomechanical measurements,” Phys. Rev. Lett. 104, 133602 (2010).
[Crossref] [PubMed]

Hertzberg, J.

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

Hill, J. T.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472, 69–73 (2011).
[Crossref]

Huang, S.

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

Huang, Sumei

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

G. S. Agarwal and Sumei Huang, “Coherent perfect absorption in cavity opto-mechanics,” arXiv:1304.7323.

Imamoglu, A.

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

Jing, H.

S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012).
[Crossref]

Karuza, M.

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

Kim, M. S.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Kippenberg, T. J.

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

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
[Crossref] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421, 925–928 (2003).
[Crossref]

Kleckner, D.

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature (London) 444, 75–78 (2006).
[Crossref]

Kómár, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Kronwald, A.

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the single-photon strong coupling regime,” arXiv:1034.5230.

Kumar, T.

S. Mahajan, T. Kumar, A. B. Bhattacherjee, and ManMohan, “Ground-state cooling of a mechanical oscillator and detection of a weak force using a Bose-Einstein condensate,” Phys. Rev. A 87, 013621 (2013).
[Crossref]

Kuzyk, M. C.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[Crossref] [PubMed]

Langer, G.

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

Lin, Q.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472, 69–73 (2011).
[Crossref]

Liu, Y. X.

Y. X. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110, 223603 (2013).
[Crossref] [PubMed]

Lukin, M. D.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Mahajan, S.

S. Mahajan, T. Kumar, A. B. Bhattacherjee, and ManMohan, “Ground-state cooling of a mechanical oscillator and detection of a weak force using a Bose-Einstein condensate,” Phys. Rev. A 87, 013621 (2013).
[Crossref]

ManMohan,

S. Mahajan, T. Kumar, A. B. Bhattacherjee, and ManMohan, “Ground-state cooling of a mechanical oscillator and detection of a weak force using a Bose-Einstein condensate,” Phys. Rev. A 87, 013621 (2013).
[Crossref]

Marangos, J. P.

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

Mari, A.

A. Mari and J. Eisert, “Cooling by heating: Very hot thermal light can significantly cool quantum systems,” Phys. Rev. Lett. 108, 120602 (2012).
[Crossref] [PubMed]

Marquardt, F.

F. Marquardt and S. M. Girvin, “Trend: Optomechanics,” Physics 2, 40 (2009).
[Crossref]

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the single-photon strong coupling regime,” arXiv:1034.5230.

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

Meystre, P.

S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012).
[Crossref]

S. K. Steinke and P. Meystre, “Role of quantum fluctuations in the optomechanical properties of a Bose-Einstein condensate in a ring cavity,” Phys. Rev. A 84, 023834 (2011).
[Crossref]

M. Bhattacharya and P. Meystre, “Trapping and cooling a mirror to its quantum mechanical ground state,” Phys. Rev. Lett. 99, 073601 (2007).
[Crossref] [PubMed]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).

Molinelli, C.

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

Naderi, M. H.

A. Dalafi, M. H. Naderi, M. Soltanolkotabi, and Sh. Barzanjeh, “Nonlinear effects of atomic collisions on the optomechanical properties of a Bose-Einstein condensate in an optical cavity,” Phys. Rev. A 87, 013417 (2013).
[Crossref]

Natali, R.

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

Painter, O.

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]

Palma, G. M.

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A 86, 042323 (2012).
[Crossref]

G. De Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A 83, 052324 (2011).
[Crossref]

M. Paternostro, G. De Chiara, and G. M. Palma, “Cold-atom-induced control of an optomechanical device,” Phys. Rev. Lett. 104, 243602 (2010).
[Crossref] [PubMed]

Paternostro, M.

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A 86, 042323 (2012).
[Crossref]

G. De Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A 83, 052324 (2011).
[Crossref]

M. Paternostro, G. De Chiara, and G. M. Palma, “Cold-atom-induced control of an optomechanical device,” Phys. Rev. Lett. 104, 243602 (2010).
[Crossref] [PubMed]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

Rabl, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Rivière, R.

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

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

Rogers, B.

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A 86, 042323 (2012).
[Crossref]

Safavi-Naeini, A. H.

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]

Schliesser, A.

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

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

Schmid, S. I.

S. I. Schmid, K. Y. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011).
[Crossref]

Schwab, K.

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

Singh, S.

S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012).
[Crossref]

Soltanolkotabi, M.

A. Dalafi, M. H. Naderi, M. Soltanolkotabi, and Sh. Barzanjeh, “Nonlinear effects of atomic collisions on the optomechanical properties of a Bose-Einstein condensate in an optical cavity,” Phys. Rev. A 87, 013417 (2013).
[Crossref]

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421, 925–928 (2003).
[Crossref]

Srinivasan, K.

Y. X. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110, 223603 (2013).
[Crossref] [PubMed]

Stannigel, K.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Steinke, S. K.

S. K. Steinke and P. Meystre, “Role of quantum fluctuations in the optomechanical properties of a Bose-Einstein condensate in a ring cavity,” Phys. Rev. A 84, 023834 (2011).
[Crossref]

Tavernarakis, A.

P. Verlot, A. Tavernarakis, T. Briant, P.-F. Cohadon, and A. Heidmann, “Backaction amplification and quantum limits in optomechanical measurements,” Phys. Rev. Lett. 104, 133602 (2010).
[Crossref] [PubMed]

Tian, L.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[Crossref] [PubMed]

Tombesi, P.

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

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
[Crossref] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421, 925–928 (2003).
[Crossref]

Vanner, M.

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

Verlot, P.

P. Verlot, A. Tavernarakis, T. Briant, P.-F. Cohadon, and A. Heidmann, “Backaction amplification and quantum limits in optomechanical measurements,” Phys. Rev. Lett. 104, 133602 (2010).
[Crossref] [PubMed]

Vitali, D.

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

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).

Wang, H.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[Crossref] [PubMed]

Wang, Y.-D.

Y.-D. Wang and A. A. Clerk., “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
[Crossref] [PubMed]

Weis, S.

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

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]

Wright, E. M.

S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012).
[Crossref]

Xia, K. Y.

S. I. Schmid, K. Y. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011).
[Crossref]

Yang, Y.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[Crossref] [PubMed]

Zeilinger, A.

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

Zoller, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Nature (London) (5)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421, 925–928 (2003).
[Crossref]

S. Gigan, H. Böhm, M. Paternostro, F. Blaser, G. Langer, J. Hertzberg, K. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature (London) 444, 67–70 (2006).
[Crossref]

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature (London) 444, 75–78 (2006).
[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]

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

Nature Phys. (1)

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

New J. Phys. (1)

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]

Opt. Express (1)

Phys. Rev. A (11)

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

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

S. I. Schmid, K. Y. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011).
[Crossref]

S. Mahajan, T. Kumar, A. B. Bhattacherjee, and ManMohan, “Ground-state cooling of a mechanical oscillator and detection of a weak force using a Bose-Einstein condensate,” Phys. Rev. A 87, 013621 (2013).
[Crossref]

G. De Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A 83, 052324 (2011).
[Crossref]

S. K. Steinke and P. Meystre, “Role of quantum fluctuations in the optomechanical properties of a Bose-Einstein condensate in a ring cavity,” Phys. Rev. A 84, 023834 (2011).
[Crossref]

S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012).
[Crossref]

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A 86, 042323 (2012).
[Crossref]

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Phys. Rev. Lett. (8)

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Physics (1)

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[Crossref]

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Other (3)

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the single-photon strong coupling regime,” arXiv:1034.5230.

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Figures (7)

Fig. 1
Fig. 1

A double-cavity optomechanical system with a movable mirror inserted between two fixed mirrors. The two cavities have identical cavity lengths L and mode frequencies ω0 in the absence of radiation pressure. Two coupling (probe) fields with identical frequencies ωc (ωp) but different amplitudes εcL,cR (εL,R) act upon opposite sides of the movable mirror after entering either left or right cavities. The two input probe fields have a fixed relative phase θ while the two output probe fields are denoted by εoutL and εoutR, respectively.

Fig. 2
Fig. 2

Normalized output probe field energy |εoutL+|2/|εL|2 = |εoutR+|2/|εR|2 vs. normalized input probe field detuning x/κ for G = 0.5κ (red-dotted), G = κ / 2 (black-solid), and G = 1.5κ (blue-dashed) with n = 1. Other parameters are given at the beginning of section 3 for a realistic experimental setup as restricted by Eq.s (10).

Fig. 3
Fig. 3

Normalized output probe field energy |εoutL+|2/|εL|2 = |εoutR+|2/|εR|2 vs. normalized input probe field detuning x/κ for n = 0 (red-dotted), n = 1 (black-solid), and n = 1.5 (blue-dashed) with G = 2κ. Other parameters are given at the beginning of section 3 for a realistic experimental setup as restricted by Eq.s (10).

Fig. 4
Fig. 4

Normalized output probe field energy |εoutL+/εL|2 vs. normalized input probe field detuning x/κ for G = 0.2κ (black-solid), G = 1.2κ (red-dotted), and G = 1.6κ (blue-dashed) with n = 1. Other parameters are given at the beginning of section 3 for a realistic experimental setup as restricted by Eq.s (11).

Fig. 5
Fig. 5

Normalized output probe field energy |εoutR+/εL|2 vs. normalized input probe field detuning x/κ for G = 0.2κ (black-solid), G = 1.2κ (red-dotted), and G = 1.6κ (blue-dashed) with n = 1. Other parameters are given at the beginning of section 3 for a realistic experimental setup as restricted by Eq.s (11).

Fig. 6
Fig. 6

Normalized output probe field energy |εoutL+/εL|2 vs. normalized input probe field detuning x/κ for θ = 0 (black-solid), θ = 0.5π (red-dotted), θ = π (green-dashed), and θ = 1.5π (blue-dash-dotted) with G = 0.5κ. Other parameters are given at the beginning of section 3 for a realistic experimental setup as restricted by Eq. (12).

Fig. 7
Fig. 7

Normalized output probe field energy |εoutR+/εR|2 vs. normalized input probe field detuning x/κ for for θ = 0 (black-solid), θ = 0.5π (red-dotted), θ = π (green-dashed), and θ = 1.5π (blue-dash-dotted) with G = 0.5κ. Other parameters are given at the beginning of section 3 for a realistic experimental setup as restricted by Eq. (12).

Equations (12)

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H = h ¯ Δ c ( c 1 c 1 + c 2 c 2 ) + h ¯ g 0 ( c 2 c 2 c 1 c 1 ) ( b + b ) + h ¯ ω m b b + i h ¯ ε c L ( c 1 c 1 ) + i h ¯ ( c 1 ε L e i δ t c 1 ε L * e i δ t ) + i h ¯ ε c R ( c 2 c 2 ) + i h ¯ ( c 2 ε R e i θ e i δ t c 2 ε R * e i θ e i δ t )
b ˙ = i ω m b i g 0 ( c 2 c 2 c 1 c 1 ) γ m 2 b + γ m b in , c ˙ 1 = [ κ + i Δ c i g 0 ( b + b ) ] c 1 + ε c L + ε L e i δ t + 2 κ c 1 in , c ˙ 2 = [ κ + i Δ c + i g 0 ( b + b ) ] c 2 + ε c R + ε R e i θ e i δ t + 2 κ c 2 in
b = b s = i g 0 ( | c 2 s | 2 | c 1 s | 2 ) γ m 2 + i ω m , c 1 = c 1 s = ε c L κ + i Δ 1 , c 2 = c 2 s = ε c R κ + i Δ 2
δ b ˙ = i g 0 ( c 2 s * δ c 2 e i ( Δ 2 ω m ) t c 1 s * δ c 1 e i ( Δ 1 ω m ) t ) i g 0 ( c 2 s δ c 2 e i ( Δ 2 + ω m ) t c 1 s δ c 1 e i ( Δ 1 + ω m ) t ) γ m 2 δ b + γ m b in , δ c ˙ 1 = κ δ c 1 + i g 0 c 1 s ( δ b e i ( ω m Δ 1 t ) + δ b e i ( ω m + Δ 1 ) t ) + ε L e i ( δ Δ 1 ) t + 2 κ c 1 in , δ c ˙ 2 = κ δ c 2 i g 0 c 2 s ( δ b e i ( ω m Δ 2 ) t + δ b e i ( ω m + Δ 2 ) t ) + ε R e i θ e i ( δ Δ 2 t ) + 2 κ c 2 in .
δ b ˙ = i g 0 ( c 2 s * δ c 2 c 1 s * δ c 1 ) γ m 2 δ b + γ m b in , δ c ˙ 1 = κ δ c 1 + i g 0 c 1 s δ b + ε L e i x t + 2 κ c 1 in , δ c ˙ 2 = κ δ c 2 i g 0 c 2 s δ b + ε R e i θ e i x t + 2 κ c 2 in
δ b ˙ = i g 0 ( c 2 s * δ c 2 c 1 s * δ c 1 ) γ m 2 δ b , δ c ˙ 1 = κ δ c 1 + i g 0 c 1 s δ b + ε L e i x t , δ c ˙ 2 = κ δ c 2 i g 0 c 2 s δ b + ε R e i θ e i x t
δ b + = i G ( n ε R e i θ ε L ) ( κ i x ) ( γ m 2 i x ) + G 2 ( n 2 + 1 ) , δ c 1 + = ( n ε R e i θ + n 2 ε L ) G 2 + ε L ( κ i x ) ( γ m 2 i x ) ( κ i x ) 2 ( γ m 2 i x ) + G 2 ( n 2 + 1 ) ( κ i x ) , δ c 2 + = ( n ε L + ε R e i θ ) G 2 + ε R e i θ ( κ i x ) ( γ m 2 i x ) ( κ i x ) 2 ( γ m 2 i x ) + G 2 ( n 2 + 1 ) ( κ i x )
ε out L + ε L e i x t = 2 κ δ c 1 , ε out R + ε R e i θ e i x t = 2 κ δ c 2
ε out L + = 2 κ δ c 1 + ε L , ε out R + = 2 κ δ c 2 + ε R e i θ
θ = π , ε R = n ε L , γ m = 2 κ , x ± = ± ( n 2 + 1 ) G 2 κ 2
x 0 = 0 , x ± = ± 2 G 2 κ 2
x ± = 2 × 3 1 3 × G 2 3 1 3 × κ 2 + ρ 2 3 3 2 3 × ρ 1 3

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