Abstract

We present the optical and mechanical design of a mechanically compliant quasi-two-dimensional photonic crystal cavity formed from thin-film silicon in which a pair of linear nanoscale slots are used to create two coupled high-Q optical resonances. The optical cavity supermodes, whose frequencies are designed to lie in the 1500 nm wavelength band, are shown to interact strongly with mechanical resonances of the structure whose frequencies range from a few MHz to a few GHz. Depending upon the symmetry of the mechanical modes and the symmetry of the slot sizes, we show that the optomechanical coupling between the optical supermodes can be either linear or quadratic in the mechanical displacement amplitude. Tuning of the nanoscale slot size is also shown to adjust the magnitude and sign of the cavity supermode splitting 2J, enabling near-resonant motional scattering between the two optical supermodes and greatly enhancing the x2-coupling strength. Specifically, for the fundamental flexural mode of the central nanobeam of the structure at 10 MHz the per-phonon linear cross-mode coupling rate is calculated to be g˜+/2π=1MHz, corresponding to a per-phonon x2-coupling rate of g˜/2π=1kHz for a mode splitting 2J/2π = 1 GHz which is greater than the radiation-limited supermode linewidths.

© 2016 Optical Society of America

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  36. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
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2016 (1)

Y. Yanay, J. C. Sankey, and A. A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).

2015 (8)

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

T. K. Paraïso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity,” Phys. Rev. X 5, 041024 (2015).

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 6232 (2015).

X. Chen, C. Zhao, S. Danilishin, L. Ju, D. Blair, H. Wang, S. P. Vyatchanin, C. Molinelli, A. Kuhn, S. Gras, T. Briant, P.-F. Cohadon, A. Heidmann, I. Roch-Jeune, R. Flaminio, C. Michel, and L. Pinard, “Observation of three-mode parametric instability,” Phys. Rev. A 91, 033832 (2015).

L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun. 6, 5850 (2015).

H. Flayac, D. Gerace, and V. Savona, “An all-silicon single-photon source by unconventional photon blockade,” Sci. Rep. 5, 11223 (2015).

N. Lörch and K. Hammerer, “Sub-Poissonian phonon lasing in three-mode optomechanics,” Phys. Rev. A 91, 061803 (2015).

H. Kaviani, C. Healey, M. Wu, R. Ghobadi, A. Hryciw, and P. E. Barclay, “Nonlinear optomechanical paddle nanocavities,” Optica 2, 271–274 (2015).

2014 (5)

M.-A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).

T. Ojanen and K. Børkje, “Ground-state cooling of mechanical motion in the unresolved sideband regime by use of optomechanically induced transparency,” Phys. Rev. A 90, 013824 (2014).

M. H. Matheny, M. Grau, L. G. Villanueva, R. B. Karabalin, M. C. Cross, and M. L. Roukes, “Phase synchronization of two anharmonic nanomechanical oscillators,” Phys. Rev. Lett. 112, 014101 (2014).

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

C. Doolin, B. D. Hauer, P. H. Kim, a. J. R. MacDonald, H. Ramp, and J. P. Davis, “Nonlinear optomechanics in the stationary regime,” Phys. Rev. A 89, 1–6 (2014).

2013 (2)

X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B At. Mol. Opt. Phys. 46, 035502 (2013).

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111, 213902 (2013).

2012 (6)

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).

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).

F. Massel, S. U. Cho, J.-M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photon. 6, 768–772 (2012).

M. Schmidt, M. Ludwig, and F. Marquardt, “Optomechanical circuits for nanomechanical continuous variable quantum state processing,” New Journal of Physics 14, 125005 (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).

2011 (5)

D. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 023003 (2011).

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

M. Winger, T. D. Blasius, T. P. M. Alegre, A. H. Safavi-Naeini, S. Meenehan, J. Cohen, S. Stobbe, and O. Painter, “A chip-scale integrated cavity-electro-optomechanics platform,” Opt. Express 19, 24905–24921 (2011).

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective dynamics in optomechanical arrays,” Phys. Rev. Lett. 107, 043603 (2011).

A. W. Rodriguez, A. P. McCauley, P.-C. Hui, D. Woolf, E. Iwase, F. Capasso, M. Loncar, and S. G. Johnson, “Bonding, antibonding and tunable optical forces in asymmetric membranes,” Opt. Express 19, 2225–2241 (2011).

2010 (4)

A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Quantum measurement of phonon shot noise,” Phys. Rev. Lett. 104, 213603 (2010).

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).

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

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nature Physics 6, 707–712 (2010),

2009 (5)

M. Eichenfield, J. Chan, R. Camacho, K. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a "zipper" photonic crystal optomechanical cavity,” Opt. Express 17, 3802 (2009).

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).

C. Zhao, L. Ju, H. Miao, S. Gras, Y. Fan, and D. G. Blair, “Three-mode optoacoustic parametric amplifier: a tool for macroscopic quantum experiments,” Phys. Rev. Lett. 102, 243902 (2009).

2008 (2)

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New Journal of Physics 10, 095008 (2008).

J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).

2005 (1)

T. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005).

2002 (4)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in power recycled ligo interferometer,” Phys. Lett. A 305, 111–124 (2002).

V. Braginsky and S. Vyatchanin, “Low quantum noise tranquilizer for fabry–perot interferometer,” Phys. Lett. A 293, 228–234 (2002).

K. Srinivasan and O. Painter, “Momentum space design of high-q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002).

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).

2001 (2)

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).

V. Braginsky, S. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in fabry–perot interferometer,” Phys. Lett. A 287, 331–338 (2001).

1975 (1)

D. Biegelsen, “Frequency dependence of the photoelastic coefficients of silicon,” Phys. Rev. B 12, 2427 (1975).

Abbott, R.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Adams, C.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Adhikari, R. X.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Alegre, T. P. M.

Arai, K.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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).

Aspelmeyer, M.

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M. H. Matheny, M. Grau, L. G. Villanueva, R. B. Karabalin, M. C. Cross, and M. L. Roukes, “Phase synchronization of two anharmonic nanomechanical oscillators,” Phys. Rev. Lett. 112, 014101 (2014).

Grote, H.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Grudinin, I. S.

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).

Habraken, S. J. M.

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).

Hafezi, M.

D. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 023003 (2011).

Hakonen, P. J.

F. Massel, S. U. Cho, J.-M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

Hammerer, K.

N. Lörch and K. Hammerer, “Sub-Poissonian phonon lasing in three-mode optomechanics,” Phys. Rev. A 91, 061803 (2015).

Harris, J.

J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New Journal of Physics 10, 095008 (2008).

Harris, J. G.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nature Physics 6, 707–712 (2010),

Harris, J. G. E.

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 6232 (2015).

A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Quantum measurement of phonon shot noise,” Phys. Rev. Lett. 104, 213603 (2010).

Hauer, B. D.

C. Doolin, B. D. Hauer, P. H. Kim, a. J. R. MacDonald, H. Ramp, and J. P. Davis, “Nonlinear optomechanics in the stationary regime,” Phys. Rev. A 89, 1–6 (2014).

Healey, C.

Heidmann, A.

X. Chen, C. Zhao, S. Danilishin, L. Ju, D. Blair, H. Wang, S. P. Vyatchanin, C. Molinelli, A. Kuhn, S. Gras, T. Briant, P.-F. Cohadon, A. Heidmann, I. Roch-Jeune, R. Flaminio, C. Michel, and L. Pinard, “Observation of three-mode parametric instability,” Phys. Rev. A 91, 033832 (2015).

Heikkilä, T. T.

F. Massel, S. U. Cho, J.-M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

Heinrich, G.

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective dynamics in optomechanical arrays,” Phys. Rev. Lett. 107, 043603 (2011).

Heinze, M.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Hill, J. T.

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

Hoch, S. W.

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 6232 (2015).

Hryciw, A.

Hui, P.-C.

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).

Iwase, E.

Izumi, K.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Jayich, A.

J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New Journal of Physics 10, 095008 (2008).

Jayich, A. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nature Physics 6, 707–712 (2010),

Joannopoulos, J.

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).

Joannopoulos, J. D.

Johnson, S. G.

Ju, L.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

X. Chen, C. Zhao, S. Danilishin, L. Ju, D. Blair, H. Wang, S. P. Vyatchanin, C. Molinelli, A. Kuhn, S. Gras, T. Briant, P.-F. Cohadon, A. Heidmann, I. Roch-Jeune, R. Flaminio, C. Michel, and L. Pinard, “Observation of three-mode parametric instability,” Phys. Rev. A 91, 033832 (2015).

C. Zhao, L. Ju, H. Miao, S. Gras, Y. Fan, and D. G. Blair, “Three-mode optoacoustic parametric amplifier: a tool for macroscopic quantum experiments,” Phys. Rev. Lett. 102, 243902 (2009).

Kalaee, M.

T. K. Paraïso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity,” Phys. Rev. X 5, 041024 (2015).

Karabalin, R. B.

M. H. Matheny, M. Grau, L. G. Villanueva, R. B. Karabalin, M. C. Cross, and M. L. Roukes, “Phase synchronization of two anharmonic nanomechanical oscillators,” Phys. Rev. Lett. 112, 014101 (2014).

Kaviani, H.

Kawabe, K.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Kells, B.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Kim, P. H.

C. Doolin, B. D. Hauer, P. H. Kim, a. J. R. MacDonald, H. Ramp, and J. P. Davis, “Nonlinear optomechanics in the stationary regime,” Phys. Rev. A 89, 1–6 (2014).

King, E.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Kippenberg, T.

T. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005).

Kippenberg, T. J.

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

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

Kokeyama, K.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Komar, P.

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).

Krause, A. G.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photon. 6, 768–772 (2012).

Kubala, B.

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective dynamics in optomechanical arrays,” Phys. Rev. Lett. 107, 043603 (2011).

Kuhn, A.

X. Chen, C. Zhao, S. Danilishin, L. Ju, D. Blair, H. Wang, S. P. Vyatchanin, C. Molinelli, A. Kuhn, S. Gras, T. Briant, P.-F. Cohadon, A. Heidmann, I. Roch-Jeune, R. Flaminio, C. Michel, and L. Pinard, “Observation of three-mode parametric instability,” Phys. Rev. A 91, 033832 (2015).

Lee, D.

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 6232 (2015).

Lee, H.

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).

Lemonde, M.-A.

M.-A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).

Li, Y.-J.

X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B At. Mol. Opt. Phys. 46, 035502 (2013).

Lin, Q.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photon. 6, 768–772 (2012).

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

Lipson, M.

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).

Loncar, M.

Lörch, N.

N. Lörch and K. Hammerer, “Sub-Poissonian phonon lasing in three-mode optomechanics,” Phys. Rev. A 91, 061803 (2015).

Ludwig, M.

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).

M. Schmidt, M. Ludwig, and F. Marquardt, “Optomechanical circuits for nanomechanical continuous variable quantum state processing,” New Journal of Physics 14, 125005 (2012).

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective dynamics in optomechanical arrays,” Phys. Rev. Lett. 107, 043603 (2011).

Lukin, M. D.

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).

MacDonald, a. J. R.

C. Doolin, B. D. Hauer, P. H. Kim, a. J. R. MacDonald, H. Ramp, and J. P. Davis, “Nonlinear optomechanics in the stationary regime,” Phys. Rev. A 89, 1–6 (2014).

Manipatruni, S.

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).

Marquardt, F.

T. K. Paraïso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity,” Phys. Rev. X 5, 041024 (2015).

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

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111, 213902 (2013).

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).

M. Schmidt, M. Ludwig, and F. Marquardt, “Optomechanical circuits for nanomechanical continuous variable quantum state processing,” New Journal of Physics 14, 125005 (2012).

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective dynamics in optomechanical arrays,” Phys. Rev. Lett. 107, 043603 (2011).

A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Quantum measurement of phonon shot noise,” Phys. Rev. Lett. 104, 213603 (2010).

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New Journal of Physics 10, 095008 (2008).

J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).

Martynov, D.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Mason, D.

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 6232 (2015).

Massel, F.

F. Massel, S. U. Cho, J.-M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

Massinger, T. J.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Matheny, M. H.

M. H. Matheny, M. Grau, L. G. Villanueva, R. B. Karabalin, M. C. Cross, and M. L. Roukes, “Phase synchronization of two anharmonic nanomechanical oscillators,” Phys. Rev. Lett. 112, 014101 (2014).

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 472, 69–73 (2011).

McCauley, A. P.

McEuen, P.

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).

Meenehan, S.

Miao, H.

C. Zhao, L. Ju, H. Miao, S. Gras, Y. Fan, and D. G. Blair, “Three-mode optoacoustic parametric amplifier: a tool for macroscopic quantum experiments,” Phys. Rev. Lett. 102, 243902 (2009).

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).

Michel, C.

X. Chen, C. Zhao, S. Danilishin, L. Ju, D. Blair, H. Wang, S. P. Vyatchanin, C. Molinelli, A. Kuhn, S. Gras, T. Briant, P.-F. Cohadon, A. Heidmann, I. Roch-Jeune, R. Flaminio, C. Michel, and L. Pinard, “Observation of three-mode parametric instability,” Phys. Rev. A 91, 033832 (2015).

Miller, J.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Molinelli, C.

X. Chen, C. Zhao, S. Danilishin, L. Ju, D. Blair, H. Wang, S. P. Vyatchanin, C. Molinelli, A. Kuhn, S. Gras, T. Briant, P.-F. Cohadon, A. Heidmann, I. Roch-Jeune, R. Flaminio, C. Michel, and L. Pinard, “Observation of three-mode parametric instability,” Phys. Rev. A 91, 033832 (2015).

Mueller, C.

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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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).

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photon. 6, 768–772 (2012).

D. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 023003 (2011).

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

M. Winger, T. D. Blasius, T. P. M. Alegre, A. H. Safavi-Naeini, S. Meenehan, J. Cohen, S. Stobbe, and O. Painter, “A chip-scale integrated cavity-electro-optomechanics platform,” Opt. Express 19, 24905–24921 (2011).

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).

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J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a "zipper" photonic crystal optomechanical cavity,” Opt. Express 17, 3802 (2009).

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M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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X. Chen, C. Zhao, S. Danilishin, L. Ju, D. Blair, H. Wang, S. P. Vyatchanin, C. Molinelli, A. Kuhn, S. Gras, T. Briant, P.-F. Cohadon, A. Heidmann, I. Roch-Jeune, R. Flaminio, C. Michel, and L. Pinard, “Observation of three-mode parametric instability,” Phys. Rev. A 91, 033832 (2015).

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F. Massel, S. U. Cho, J.-M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

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L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun. 6, 5850 (2015).

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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).

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X. Chen, C. Zhao, S. Danilishin, L. Ju, D. Blair, H. Wang, S. P. Vyatchanin, C. Molinelli, A. Kuhn, S. Gras, T. Briant, P.-F. Cohadon, A. Heidmann, I. Roch-Jeune, R. Flaminio, C. Michel, and L. Pinard, “Observation of three-mode parametric instability,” Phys. Rev. A 91, 033832 (2015).

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T. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005).

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M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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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).

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D. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 023003 (2011).

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

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M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun. 6, 5850 (2015).

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M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New Journal of Physics 10, 095008 (2008).

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M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).

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M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

Yanay, Y.

Y. Yanay, J. C. Sankey, and A. A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).

Yang, C.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nature Physics 6, 707–712 (2010),

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New Journal of Physics 10, 095008 (2008).

Zang, L.

T. K. Paraïso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity,” Phys. Rev. X 5, 041024 (2015).

Zhang, M.

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).

Zhao, C.

X. Chen, C. Zhao, S. Danilishin, L. Ju, D. Blair, H. Wang, S. P. Vyatchanin, C. Molinelli, A. Kuhn, S. Gras, T. Briant, P.-F. Cohadon, A. Heidmann, I. Roch-Jeune, R. Flaminio, C. Michel, and L. Pinard, “Observation of three-mode parametric instability,” Phys. Rev. A 91, 033832 (2015).

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

C. Zhao, L. Ju, H. Miao, S. Gras, Y. Fan, and D. G. Blair, “Three-mode optoacoustic parametric amplifier: a tool for macroscopic quantum experiments,” Phys. Rev. Lett. 102, 243902 (2009).

Zoller, P.

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).

Zwickl, B.

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New Journal of Physics 10, 095008 (2008).

J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).

Zwickl, B. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nature Physics 6, 707–712 (2010),

J. Phys. B At. Mol. Opt. Phys. (1)

X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B At. Mol. Opt. Phys. 46, 035502 (2013).

Nat. Commun. (3)

L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun. 6, 5850 (2015).

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 6232 (2015).

F. Massel, S. U. Cho, J.-M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

Nat. Photon. (1)

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photon. 6, 768–772 (2012).

Nature (4)

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

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).

J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).

M. Eichenfield, J. Chan, R. Camacho, K. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).

Nature Physics (1)

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nature Physics 6, 707–712 (2010),

New Journal of Physics (3)

D. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 023003 (2011).

M. Schmidt, M. Ludwig, and F. Marquardt, “Optomechanical circuits for nanomechanical continuous variable quantum state processing,” New Journal of Physics 14, 125005 (2012).

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New Journal of Physics 10, 095008 (2008).

Opt. Express (5)

Optica (1)

Phys. Lett. A (3)

V. Braginsky, S. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in fabry–perot interferometer,” Phys. Lett. A 287, 331–338 (2001).

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in power recycled ligo interferometer,” Phys. Lett. A 305, 111–124 (2002).

V. Braginsky and S. Vyatchanin, “Low quantum noise tranquilizer for fabry–perot interferometer,” Phys. Lett. A 293, 228–234 (2002).

Phys. Rev. A (6)

T. Ojanen and K. Børkje, “Ground-state cooling of mechanical motion in the unresolved sideband regime by use of optomechanically induced transparency,” Phys. Rev. A 90, 013824 (2014).

M.-A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).

N. Lörch and K. Hammerer, “Sub-Poissonian phonon lasing in three-mode optomechanics,” Phys. Rev. A 91, 061803 (2015).

C. Doolin, B. D. Hauer, P. H. Kim, a. J. R. MacDonald, H. Ramp, and J. P. Davis, “Nonlinear optomechanics in the stationary regime,” Phys. Rev. A 89, 1–6 (2014).

X. Chen, C. Zhao, S. Danilishin, L. Ju, D. Blair, H. Wang, S. P. Vyatchanin, C. Molinelli, A. Kuhn, S. Gras, T. Briant, P.-F. Cohadon, A. Heidmann, I. Roch-Jeune, R. Flaminio, C. Michel, and L. Pinard, “Observation of three-mode parametric instability,” Phys. Rev. A 91, 033832 (2015).

Y. Yanay, J. C. Sankey, and A. A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).

Phys. Rev. B (1)

D. Biegelsen, “Frequency dependence of the photoelastic coefficients of silicon,” Phys. Rev. B 12, 2427 (1975).

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).

Phys. Rev. Lett. (12)

C. Zhao, L. Ju, H. Miao, S. Gras, Y. Fan, and D. G. Blair, “Three-mode optoacoustic parametric amplifier: a tool for macroscopic quantum experiments,” Phys. Rev. Lett. 102, 243902 (2009).

A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Quantum measurement of phonon shot noise,” Phys. Rev. Lett. 104, 213603 (2010).

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, K. Arai, R. Bork, B. Kells, J. Rollins, N. Smith-Lefebvre, G. Vajente, H. Yamamoto, C. Adams, S. Aston, J. Betzweiser, V. Frolov, A. Mullavey, A. Pele, J. Romie, M. Thomas, K. Thorne, S. Dwyer, K. Izumi, K. Kawabe, D. Sigg, R. Derosa, A. Effler, K. Kokeyama, S. Ballmer, T. J. Massinger, A. Staley, M. Heinze, C. Mueller, H. Grote, R. Ward, E. King, D. Blair, L. Ju, and C. Zhao, “Observation of Parametric Instability in Advanced LIGO,” Phys. Rev. Lett. 114, 161102 (2015).

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).

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G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective dynamics in optomechanical arrays,” Phys. Rev. Lett. 107, 043603 (2011).

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111, 213902 (2013).

M. H. Matheny, M. Grau, L. G. Villanueva, R. B. Karabalin, M. C. Cross, and M. L. Roukes, “Phase synchronization of two anharmonic nanomechanical oscillators,” Phys. Rev. Lett. 112, 014101 (2014).

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).

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).

Phys. Rev. X (1)

T. K. Paraïso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity,” Phys. Rev. X 5, 041024 (2015).

Rev. Mod. Phys. (1)

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

Sci. Rep. (1)

H. Flayac, D. Gerace, and V. Savona, “An all-silicon single-photon source by unconventional photon blockade,” Sci. Rep. 5, 11223 (2015).

Science (1)

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

Other (2)

COMSOL Multiphysics, version 3.5a.

A. H. Safavi-Naeini, “Quantum Optomechanics with Silicon Nanostructures,” Ph.D. thesis, California Institute of Technology (2013).

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Figures (10)

Fig. 1
Fig. 1

(a) Schematic of a multimode membrane-in-the-middle optomechanical system consisting of a central movable membrane (b3) and two movable end mirrors (b1, b2). Due to tunneling (rate J) of light through the partially transmitting central membrane, the left and right individual optical cavity modes (a1, a2; not shown) hybridize into the supermodes a+ and a. (b) Schematic of a photonic crystal implementation of a similar multimode optomechanical system. The structure consists of a pair of top and bottom photonic crystal slabs which are separated from a central photonic crystal slab by nanoscale air slots. A pair of optical waveguide modes localize around each nanoscale slot, propagating along the axial direction (x) of the structure. By varying the photonic crystal unit cell along the length of the structure, one can form optical cavity modes which are localized to a central “defect” region of the structure. Light in the cavity modes surrounding the top and bottom nanoscale air slots (a1, a2) tunnel across the central photonic crystal slab, forming hybridized supermodes (a+, a). Mechanical motion of the structure includes in-plane flexural motion of the central nanobeam (b3), the top photonic crystal slab (b1), and the bottom photonic crystal slab (b2).

Fig. 2
Fig. 2

Bandstructure of a triangular lattice of air holes in a silicon slab with two line defects (air slots). We plot the TE-like bands with even (vector) symmetry about the slab mid-plane. The blue line and gray shaded region delimit the light cone of the air cladding surrounding the silicon slab. We focus on the fundamental waveguide modes (solid lines) inside the pseudo-bandgap of the triangular lattice. The y-polarization of the electric field, Ey, of the odd and even modes at the X-point are shown in the insets to the right of the bandstructure. Red and blue correspond to the positive and negative normalized amplitude of the y-polarized electric field. The bandstructure here is computed for a silicon slab with the following set of parameters: refractive index n = 3.42, thickness t = 220 nm, lattice constant a = 480 nm, hole radius r = 0.3a and slot widths s1 = s2 = 100 nm.

Fig. 3
Fig. 3

Simulated bandstructures of the coupled linear waveguide modes for three different separations of the line defects: (a) one, (b) three and (c) five rows of holes in the central nanobeam. The waveguide modes of interest are shown as solid red (odd modes) and black (even modes) lines. The figure shows the decrease in frequency splitting between the targeted optical modes at the X-point as we increase the number of rows in the central nanobeam. The band diagrams are calculated for the same geometrical parameter as in Fig. 2.

Fig. 4
Fig. 4

Influence of the hole ellipticity and lattice constant on the X-point bandedge frequency of the even and odd waveguide supermodes in the case of a single row of holes in the central nanobeam. (a) Shift of the X-point frequencies due to a change of hole ellipticity in the central nanobeam. The odd mode is unaffected by the change. The three insets illustrate the shape of holes for three different aspect ratios η = 0.55, 1 and 1.7 with (a, r, s1, s2) = (480 nm, 0.3a, 100 nm, 100 nm). (b) Increase of the optical waveguide supermodes’ X-point frequency with a decrease of the lattice constant from a = 480 nm to aD = 468 nm = 0.975a calculated with (r, η, s1, s2) = (0.3a, 0.55, 100 nm, 100 nm).

Fig. 5
Fig. 5

(a) Left: optical bandstructure of the mirror section of the cavity structure. Right: waveguide supermode frequencies at the X-point as the waveguide unit cell transitions from the outer mirror section to the center of the defect region. As described in the main text, this transition involves a change in the lattice constant (a) and in the nanobeam hole ellipticity (η). In the mirror cell, we use a lattice constant a = 480 nm and elliptical holes with major axis of ry = 194 nm and minor axis of rx = 107 nm (η = 0.55). In the defect cell, we use a = 0.975a and η = 1 (circular holes). The slot widths are s1 = s2=100 nm. The cavity is designed with total of Nx = 42 waveguide unit cells along the x-axis, with a central defect region consisting of ND = 7 defect cells. The outer slabs contain Ny = 9 rows of holes in the transverse y-direction. (b) Plot of the FEM-simulated amplitude of the y-polarization of the electric field Ey(r) of the even cavity mode. (c) Plot of the FEM-simulated odd cavity mode. In (b) and (c), red and blue correspond to positive and negative Ey field amplitudes, respectively.

Fig. 6
Fig. 6

(a) Cavity eigenfrequencies for a slot width varying from s = 90 nm to s = 100 nm. The frequency of the odd mode is more influenced by a change of the slot width than the even mode, leading to a change in the spectral ordering of the odd and even eigenmodes. (b) Extracted frequency splitting Δω = 2J as function of the slot width. Arbitrarily small splittings are achieved around s = 95 nm. The parameters of the structure are identical to Fig. 5.

Fig. 7
Fig. 7

Simulated anti-crossing curves obtained by varying s2 while s1 is kept constant at (a) 90 nm (b) 95 nm and (c) 100 nm. The splitting between the odd (red circles) and even (black squares) cavity supermodes is inverted between (a) and (c). In (b), the splitting is reduced from 2J/2π > 100 GHz to 2J/2π = 17 GHz. (d) Plot of the Q-factor of the even and odd supermode branches versus the second slot width s2 for fixed slot width s1 = 95 nm. The parameters of the structure, save the slot width, are the same as in Fig. 5 for all simulations in (a–d).

Fig. 8
Fig. 8

Normalized displacement profile of (a) the in-plane slab modes and (b) the nanobeam first and higher order in-plane flexural modes. The inset on top of (b) shows the profiles of |Ey |2 along the waveguides for both the odd and even symmetry optical supermodes. The deformations are exaggerated for clarity. The photonic crystal parameters are the same as in Fig. 5. The central nanobeam is 731 nm wide and 24 μm long. The outer slabs are suspended by tethers of length lt = 2.5 μm and width wt = 150 nm.

Fig. 9
Fig. 9

(a,b) Simulated phononic bandstructure of the nanobeam and defect mode drawn from the Γ-point. The breathing mode band is specified by the solid red line. The even (red lines) and odd (black lines) symmetry acoustic modes are defined with respect to σy mirror operator. (a) Shift of the Γ-point frequency of the breathing mode band for a defect formed by both variations in lattice constant and ellipticity of holes. The defect parameters are the same as in Fig. 5. (c) and (d) show the shift of the breathing mode frequency at the Γ point due to variations of the lattice constant and of the holes aspect ratio, respectively. (e) Mechanical beam and (f) normalized displacement field Q(r) of the localized breathing mode. The color scale indicates the magnitude of Q(r). (g) Exaggerated deformation of the structure due to the breathing mode. All acoustic mode simulations were performed using COMSOL [38].

Fig. 10
Fig. 10

Optomechanical coupling rates as a function of static central nanobeam displacement. (a) Linear self-mode optomechanical coupling, (b) linear cross-mode optomechanical coupling, and (c) x2-coupling of the ω + ( x ¯ 3 ) supermode branch to the fundamental in-plane mechanical resonances of the nanobeam (b3, red curve) and outer slabs (b1, blue curve; b2, green curve). The asterisk (∗) correspond to numerical FEM simulations of the coupling rates using the perturbation theory with different in-plane static displacement x ¯ 3 of the nanobeam from its symmetric equilibrium position. The geometrical parameters of the simulated double-slotted structure are: (a, r, s1, s2) = (480 nm, 0.3a, 90 nm + x ¯ 3, 90 nm x ¯ 3). The solid lines are theoretical fits based on Eqs. (2224).

Tables (3)

Tables Icon

Table 1 In-plane flexural modes of the nanobeam. We consider the modes with symmetric displacement with respect to σx. The geometric parameters of the nanobeam are the same as in Fig. 5.

Tables Icon

Table 2 Strength of the linear optomechanical coupling of the optical supermodes to the fundamental in-plane flexural modes of the outer slabs for three different slot widths. The second and third columns display the linear coupling strengths calculated numerically using the perturbation theory. The fourth column gives the values of the optomechanical coupling constant obtained by fitting the anti-crossing curves shown in Fig. 7. The geometric parameters of the nanobeam are the same as in Fig. 8.

Tables Icon

Table 3 Linear cross-mode optomechanical (vacuum) coupling rates g ˜ + of the optical supermodes to the nanobeam’s in-plane fundamental and higher order flexural modes. The x2-coupling rate g ˜ + is inferred using Eq. (9) for a minimum splitting of 2J/2π = 1 GHz. The geometric parameters of the nanobeam are the same as in Fig. 8.

Equations (24)

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^ tot = ^ opt + ^ mec ^ int , ^ opt = ω 1 a ^ 1 a ^ 1 + ω 2 a ^ 2 a ^ 2 + J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) , ^ mec = k ω m , k b ^ k b ^ k , ^ int = i , j , k g i j , k a ^ i a ^ j ( b ^ k + b ^ k ) x zpf , k ,
^ tot = ^ 0 + ^ int , ^ 0 = ω + ( 0 ) a ^ + a ^ + + ω ( 0 ) a ^ a ^ + k ω m , k b ^ k b ^ k , ^ int = k x zpf , k ( b ^ k + b ^ k ) [ g 1 , k + g 2 , k 2 ( a ^ + a ^ + + a ^ a ^ ) + g 1 , k + g 2 , k 2 ( a ^ + a ^ + a ^ a ^ + ) ] ,
g + , k = g , k = g 1 , k + g 2 , k 2 ,
g + , k = g 1 , k + g 2 , k 2 .
ω ± ( { x k } ) = ω 0 + k g ± , k x k ± J 2 + ( k g + , k x k ) 2 ,
ω ± ( { x k , eq + x k } ) = ω ± ( { x k , eq } ) + δ ω ± , k ( 1 ) ( { x k , eq } ) x k + δ ω ± , k ( 2 ) ( { x k , eq } ) x k 2 + ,
δ ω ± , k ( 1 ) ( { x k , eq } ) = ω ± x k | { x k , eq } g ± , k ( { x k , eq } )
δ ω ± , k ( 2 ) ( { x k , eq } ) = 1 2 2 ω ± , k x k 2 | { x k , eq } g ± , k ( { x k , eq } ) .
g ± , k ( { 0 } ) g ± , k = ± ( g + , k ) 2 / 2 J .
× × | E = ( ω 2 c ) ϵ | E ,
δ ω ( 1 ) = ω ( 0 ) 2 E ( 0 ) | δ α | E ( 0 ) E ( 0 ) | ϵ ( 0 ) | E ( 0 ) ,
δ ω MB ( 1 ) = ω ( 0 ) 2 A d 2 r ( q n ) [ Δ ϵ | E ( 0 ) | 2 Δ ϵ 1 | D ( 0 ) | 2 ] E ( 0 ) | ϵ ( 0 ) | E ( 0 ) ,
m eff = V d 3 r ρ ( r ) | q ( r ) | 2 ,
δ ϵ = ϵ ( 0 ) pS ϵ 0 ϵ ( 0 ) ,
δ ϵ i j = ϵ 0 n 4 p i j k l S k l ,
δ ϵ = ϵ 0 n 4 × [ p 11 S x x + p 12 ( S y y + S z z ) p 44 S x y p 44 S x z p 44 S x y p 11 S y y + p 12 ( S x x + S z z ) p 44 S y z p 44 S x z p 44 S y z p 11 S z z + p 12 ( S x x + S y y ) ] .
δ ω PE ( 1 ) = ω ( 0 ) ϵ 0 n 4 2 E ( 0 ) | ϵ ( 0 ) | E ( 0 ) V d 3 r 2 [ Re ( ( E x ( 0 ) ) * E y ( 0 ) ) p 44 S x y + Re ( ( E x ( 0 ) ) * E z ( 0 ) ) p 44 S x z + Re ( ( E y ( 0 ) ) * E z ( 0 ) ) p 44 S y z + | E x ( 0 ) | 2 ( p 11 S x x + p 12 ( S y y + S z z ) ) + | E y ( 0 ) | 2 ( p 11 S y y + p 12 ( S x x + S z z ) ) + | E z ( 0 ) | 2 ( p 11 S z z + p 12 ( S y y + S x x ) ) ] .
g i j , k = ω i ( 0 ) ω j ( 0 ) 2 E i ( 0 ) | δ α k | E j ( 0 ) ( E i ( 0 ) | ϵ ( 0 ) | E i ( 0 ) ) 1 / 2 ( E j ( 0 ) | ϵ ( 0 ) | E j ( 0 ) ) 1 / 2 ,
g + , k = ω + ( 0 ) ω ( 0 ) 2 A d 2 r ( q k n ) [ Δ ϵ ( E , + ( 0 ) ) * E , ( 0 ) Δ ϵ 1 ( D , + ( 0 ) ) * D , ( 0 ) ] ( E + ( 0 ) | ϵ ( 0 ) | E + ( 0 ) ) 1 / 2 ( E ( 0 ) | ϵ ( 0 ) | E ( 0 ) ) 1 / 2 .
δ ω i ( 2 ) = 3 8 ω ( 0 ) | E i ( 0 ) | δ α | E i ( 0 ) E i ( 0 ) | ϵ ( 0 ) | E i ( 0 ) | 2 1 2 j i ω i ( 0 ) 3 ω j ( 0 ) 2 ω i ( 0 ) 2 | E j ( 0 ) | δ α | E i ( 0 ) | 2 E j ( 0 ) | ϵ ( 0 ) | E j ( 0 ) E i ( 0 ) | ϵ ( 0 ) | E i ( 0 ) .
δ ω + ( 2 ) ( { 0 } ) g + ω + ( 0 ) ( ω + ( 0 ) + ω ( 0 ) ) ( ω ( 0 ) ω + ( 0 ) ) ( ω + ( 0 ) ) 2 | E ( 0 ) | δ α | E + ( 0 ) | 2 E ( 0 ) | ϵ ( 0 ) | E ( 0 ) E + ( 0 ) | ϵ ( 0 ) | E + ( 0 ) , g + 2 2 J ,
g ± , b 3 ( x 3 , eq ) g 1 , b 3 + g 2 , b 3 2 ± ( g 1 , b 3 g 2 , b 3 2 ) Z 1 + Z 2 ,
g + , b 3 ( x 3 , eq ) ( g 1 , b 3 g 2 , b 3 2 ) 1 1 + Z 2 ,
g ± , b 3 ( x 3 , eq ) ± g + , b 3 2 2 [ 1 1 + Z 2 ] 3 ,

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