Abstract

Advances in optomechanics have enabled significant achievements in precision sensing and control of matter, including detection of gravitational waves and cooling of mechanical systems to their quantum ground states. Recently, the inherent nonlinearity in the optomechanical interaction has been harnessed to explore synchronization effects, including the spontaneous locking of an oscillator to a reference injection signal delivered via the optical field. Here, we present, to the best of our knowledge, the first demonstration of a radiation-pressure-driven optomechanical system locking to an inertial drive, with actuation provided by an integrated electrical interface. We use the injection signal to suppress the drift in the optomechanical oscillation frequency, strongly reducing phase noise by over 55 dBc/Hz at 2 Hz offset. We further employ the injection tone to tune the oscillation frequency by more than 2 million times its narrowed linewidth. In addition, we uncover previously unreported synchronization dynamics, enabled by the independence of the inertial drive from the optical drive field. Finally, we show that our approach may enable control of the optomechanical gain competition between different mechanical modes of a single resonator. The electrical interface allows enhanced scalability for future applications involving arrays of injection-locked precision sensors.

© 2017 Optical Society of America

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References

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2017 (3)

M. J. Seitner, M. Abdi, A. Ridolfo, M. J. Hartmann, and E. M. Weig, “Parametric oscillation, frequency mixing, and injection locking of strongly coupled nanomechanical resonator modes,” Phys. Rev. Lett. 118, 254301 (2017).
[Crossref]

E. Gil-Santos, M. Labousse, C. Baker, A. Goetschy, W. Hease, C. Gomez, A. Lemaître, G. Leo, C. Ciuti, and I. Favero, “Light-mediated cascaded locking of multiple nano-optomechanical oscillators,” Phys. Rev. Lett. 118, 063605 (2017).
[Crossref]

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

2016 (1)

2015 (5)

A. G. Krause, J. T. Hill, M. Ludwig, A. H. Safavi-Naeini, J. Chan, F. Marquardt, and O. Painter, “Nonlinear radiation pressure dynamics in an optomechanical crystal,” Phys. Rev. Lett. 115, 233601 (2015).
[Crossref]

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref]

S. Y. Shah, M. Zhang, R. Rand, and M. Lipson, “Master–slave locking of optomechanical oscillators over a long distance,” Phys. Rev. Lett. 114, 113602 (2015).
[Crossref]

K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91, 032910 (2015).
[Crossref]

M. P. Fernandes, S. Venkatesh, and B. G. Sudarshan, “Early detection of lung cancer using nano-nose—a review,” Open Biomed. Eng. J. 9, 228–233 (2015).
[Crossref]

2014 (3)

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

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

U. Kemiktarak, M. Durand, M. Metcalfe, and J. Lawall, “Mode competition and anomalous cooling in a multimode phonon laser,” Phys. Rev. Lett. 113, 030802 (2014).
[Crossref]

2013 (3)

M. Ludwig and F. Marquardt, “Quantum many-body dynamics in optomechanical arrays,” Phys. Rev. Lett. 111, 073603 (2013).
[Crossref]

Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. 8, 475–490 (2013).
[Crossref]

T. Palomaki, J. Teufel, R. Simmonds, and K. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref]

2012 (5)

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref]

I. Bargatin, E. B. Myers, J. S. Aldridge, C. Marcoux, P. Brianceau, L. Duraffourg, E. Colinet, S. Hentz, P. Andreucci, and M. L. Roukes, “Large-scale integration of nanoelectromechanical systems for gas sensing applications,” Nano Lett. 12, 1269–1274 (2012).
[Crossref]

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).
[Crossref]

S. P. Vyatchanin and S. E. Strigin, “Parametric oscillatory instability in gravitational antennas wave laser detectors,” Phys. Usp. 18255, 1115–1123 (2012).
[Crossref]

M. A. Taylor, A. Szorkovszky, J. Knittel, K. H. Lee, T. G. McRae, and W. P. Bowen, “Cavity optoelectromechanical regenerative amplification,” Opt. Express 20, 12742–12751 (2012).
[Crossref]

2011 (5)

M. Abdi and A. R. Bahrampour, “Effect of higher-order waves in parametric oscillatory instability in optical cavities,” Phys. Scripta 83, 045401 (2011).
[Crossref]

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
[Crossref]

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

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

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

2010 (1)

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett. 104, 123604 (2010).
[Crossref]

2009 (1)

J. F. Duffy and C. A. Czeisler, “Effect of light on human circadian physiology,” Sleep Med. Clin. 4, 165–177 (2009).
[Crossref]

2008 (3)

X. Feng, C. White, A. Hajimiri, and M. L. Roukes, “A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator,” Nat. Nanotechnol. 3, 342–346 (2008).
[Crossref]

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
[Crossref]

M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical RF oscillator,” Appl. Phys. Lett. 93, 191115 (2008).
[Crossref]

2006 (2)

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Français, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantum-limited optomechanical sensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

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

2005 (2)

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
[Crossref]

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).
[Crossref]

2004 (2)

T. Kippenberg, S. Spillane, and K. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[Crossref]

B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuits 39, 1415–1424 (2004).
[Crossref]

2003 (1)

M. Zalalutdinov, K. L. Aubin, M. Pandey, A. T. Zehnder, R. H. Rand, H. G. Craighead, J. M. Parpia, and B. H. Houston, “Frequency entrainment for micromechanical oscillator,” Appl. Phys. Lett. 83, 3281–3283 (2003).
[Crossref]

2002 (1)

V. B. Braginskii, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in signal recycled LIGO interferometer,” Phys. Lett. A 305, 111–124 (2002).
[Crossref]

2000 (1)

A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: a unifying theory and numerical methods for characterization,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 47, 655–674 (2000).
[Crossref]

1990 (1)

R. E. Mirollo and S. H. Strogatz, “Synchronization of pulse-coupled biological oscillators,” SIAM J. Appl. Math. 50, 1645–1662 (1990).
[Crossref]

1973 (1)

C. J. Buczek, R. J. Freiberg, and M. Skolnick, “Laser injection locking,” Proc. IEEE 61, 1411–1431 (1973).
[Crossref]

1965 (1)

L. J. Paciorek, “Injection locking of oscillators,” Proc. IEEE 53, 1723–1727 (1965).
[Crossref]

1946 (1)

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
[Crossref]

Abdi, M.

M. J. Seitner, M. Abdi, A. Ridolfo, M. J. Hartmann, and E. M. Weig, “Parametric oscillation, frequency mixing, and injection locking of strongly coupled nanomechanical resonator modes,” Phys. Rev. Lett. 118, 254301 (2017).
[Crossref]

M. Abdi and A. R. Bahrampour, “Effect of higher-order waves in parametric oscillatory instability in optical cavities,” Phys. Scripta 83, 045401 (2011).
[Crossref]

Adler, R.

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
[Crossref]

Aldridge, J. S.

I. Bargatin, E. B. Myers, J. S. Aldridge, C. Marcoux, P. Brianceau, L. Duraffourg, E. Colinet, S. Hentz, P. Andreucci, and M. L. Roukes, “Large-scale integration of nanoelectromechanical systems for gas sensing applications,” Nano Lett. 12, 1269–1274 (2012).
[Crossref]

Alegre, T. P. M.

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

Allman, M.

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

Amitai, E.

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

Andreucci, P.

I. Bargatin, E. B. Myers, J. S. Aldridge, C. Marcoux, P. Brianceau, L. Duraffourg, E. Colinet, S. Hentz, P. Andreucci, and M. L. Roukes, “Large-scale integration of nanoelectromechanical systems for gas sensing applications,” Nano Lett. 12, 1269–1274 (2012).
[Crossref]

Appel, J.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Arcizet, O.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Français, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantum-limited optomechanical sensor,” Phys. Rev. Lett. 97, 133601 (2006).
[Crossref]

Aspelmeyer, M.

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

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

Aubin, K. L.

M. Zalalutdinov, K. L. Aubin, M. Pandey, A. T. Zehnder, R. H. Rand, H. G. Craighead, J. M. Parpia, and B. H. Houston, “Frequency entrainment for micromechanical oscillator,” Appl. Phys. Lett. 83, 3281–3283 (2003).
[Crossref]

Bachtold, A.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref]

Bagci, T.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Bagheri, M.

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
[Crossref]

Bahrampour, A. R.

M. Abdi and A. R. Bahrampour, “Effect of higher-order waves in parametric oscillatory instability in optical cavities,” Phys. Scripta 83, 045401 (2011).
[Crossref]

Baker, C.

E. Gil-Santos, M. Labousse, C. Baker, A. Goetschy, W. Hease, C. Gomez, A. Lemaître, G. Leo, C. Ciuti, and I. Favero, “Light-mediated cascaded locking of multiple nano-optomechanical oscillators,” Phys. Rev. Lett. 118, 063605 (2017).
[Crossref]

Baker, C. G.

Bargatin, I.

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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groeblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
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O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Français, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantum-limited optomechanical sensor,” Phys. Rev. Lett. 97, 133601 (2006).
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J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
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A. G. Krause, J. T. Hill, M. Ludwig, A. H. Safavi-Naeini, J. Chan, F. Marquardt, and O. Painter, “Nonlinear radiation pressure dynamics in an optomechanical crystal,” Phys. Rev. Lett. 115, 233601 (2015).
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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groeblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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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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K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91, 032910 (2015).
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J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. D. Whittaker, K. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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S. P. Vyatchanin and S. E. Strigin, “Parametric oscillatory instability in gravitational antennas wave laser detectors,” Phys. Usp. 18255, 1115–1123 (2012).
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V. B. Braginskii, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in signal recycled LIGO interferometer,” Phys. Lett. A 305, 111–124 (2002).
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R. E. Mirollo and S. H. Strogatz, “Synchronization of pulse-coupled biological oscillators,” SIAM J. Appl. Math. 50, 1645–1662 (1990).
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K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91, 032910 (2015).
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Tang, H. X.

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
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T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
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T. Palomaki, J. Teufel, R. Simmonds, and K. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
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J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. D. Whittaker, K. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
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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).
[Crossref]

T. Kippenberg, S. Spillane, and K. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[Crossref]

Vahala, K. J.

M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical RF oscillator,” Appl. Phys. Lett. 93, 191115 (2008).
[Crossref]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
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M. P. Fernandes, S. Venkatesh, and B. G. Sudarshan, “Early detection of lung cancer using nano-nose—a review,” Open Biomed. Eng. J. 9, 228–233 (2015).
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T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
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S. P. Vyatchanin and S. E. Strigin, “Parametric oscillatory instability in gravitational antennas wave laser detectors,” Phys. Usp. 18255, 1115–1123 (2012).
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V. B. Braginskii, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in signal recycled LIGO interferometer,” Phys. Lett. A 305, 111–124 (2002).
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E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
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M. J. Seitner, M. Abdi, A. Ridolfo, M. J. Hartmann, and E. M. Weig, “Parametric oscillation, frequency mixing, and injection locking of strongly coupled nanomechanical resonator modes,” Phys. Rev. Lett. 118, 254301 (2017).
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X. Feng, C. White, A. Hajimiri, and M. L. Roukes, “A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator,” Nat. Nanotechnol. 3, 342–346 (2008).
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J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. D. Whittaker, K. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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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).
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M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’s clocks,” in Proceedings: Mathematics, Physical and Engineering Sciences (2002), pp. 563–579.

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K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91, 032910 (2015).
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Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. 8, 475–490 (2013).
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T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
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K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91, 032910 (2015).
[Crossref]

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M. Zalalutdinov, K. L. Aubin, M. Pandey, A. T. Zehnder, R. H. Rand, H. G. Craighead, J. M. Parpia, and B. H. Houston, “Frequency entrainment for micromechanical oscillator,” Appl. Phys. Lett. 83, 3281–3283 (2003).
[Crossref]

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M. Zalalutdinov, K. L. Aubin, M. Pandey, A. T. Zehnder, R. H. Rand, H. G. Craighead, J. M. Parpia, and B. H. Houston, “Frequency entrainment for micromechanical oscillator,” Appl. Phys. Lett. 83, 3281–3283 (2003).
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K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
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T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Zhang, M.

S. Y. Shah, M. Zhang, R. Rand, and M. Lipson, “Master–slave locking of optomechanical oscillators over a long distance,” Phys. Rev. Lett. 114, 113602 (2015).
[Crossref]

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref]

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).
[Crossref]

Appl. Phys. Lett. (2)

M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical RF oscillator,” Appl. Phys. Lett. 93, 191115 (2008).
[Crossref]

M. Zalalutdinov, K. L. Aubin, M. Pandey, A. T. Zehnder, R. H. Rand, H. G. Craighead, J. M. Parpia, and B. H. Houston, “Frequency entrainment for micromechanical oscillator,” Appl. Phys. Lett. 83, 3281–3283 (2003).
[Crossref]

Front. Phys. (1)

Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. 8, 475–490 (2013).
[Crossref]

IEEE J. Solid-State Circuits (1)

B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuits 39, 1415–1424 (2004).
[Crossref]

IEEE Trans. Circuits Syst. I Fundam. Theory Appl. (1)

A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: a unifying theory and numerical methods for characterization,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 47, 655–674 (2000).
[Crossref]

Nano Lett. (1)

I. Bargatin, E. B. Myers, J. S. Aldridge, C. Marcoux, P. Brianceau, L. Duraffourg, E. Colinet, S. Hentz, P. Andreucci, and M. L. Roukes, “Large-scale integration of nanoelectromechanical systems for gas sensing applications,” Nano Lett. 12, 1269–1274 (2012).
[Crossref]

Nat. Nanotechnol. (4)

X. Feng, C. White, A. Hajimiri, and M. L. Roukes, “A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator,” Nat. Nanotechnol. 3, 342–346 (2008).
[Crossref]

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
[Crossref]

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref]

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
[Crossref]

Nature (3)

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

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

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Open Biomed. Eng. J. (1)

M. P. Fernandes, S. Venkatesh, and B. G. Sudarshan, “Early detection of lung cancer using nano-nose—a review,” Open Biomed. Eng. J. 9, 228–233 (2015).
[Crossref]

Opt. Express (2)

Phys. Lett. A (1)

V. B. Braginskii, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in signal recycled LIGO interferometer,” Phys. Lett. A 305, 111–124 (2002).
[Crossref]

Phys. Rev. A (1)

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

Phys. Rev. E (1)

K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91, 032910 (2015).
[Crossref]

Phys. Rev. Lett. (15)

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
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K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett. 104, 123604 (2010).
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U. Kemiktarak, M. Durand, M. Metcalfe, and J. Lawall, “Mode competition and anomalous cooling in a multimode phonon laser,” Phys. Rev. Lett. 113, 030802 (2014).
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Supplementary Material (1)

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» Supplement 1       Supplementary information

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Figures (7)

Fig. 1.
Fig. 1. Block diagrams of optomechanical systems where an injection signal Ainj is delivered (a) via the optical mode or (b) as an inertial drive directly to the mechanical resonator. Note that the optical pumping of the cavity is omitted in the schematic.
Fig. 2.
Fig. 2. (a) Scanning electron micrograph of the silica (blue) microtoroid optomechanical cavity. A circular slot is etched through the device to increase compliance for radial motion [38]. Circular capacitor electrodes (yellow) are patterned on either side of the slot. The components of the optical and electrical measurement setup are shown in green and yellow, respectively. VDC and VAC are applied to the capacitor via a bias-tee and probe tips controlled by micromanipulators [38]. Laser light is coupled into one end of an optical fiber, which feeds into a fiber polarization controller (FPC). The tapered section of the fiber is coupled to the silica WGM using a micropositioning stage; the transmitted light is then collected on a high-speed photodetector (PD). (b) Result of COMSOL simulation showing the “radial breathing mode”-like excitation of such a microtoroid supported by a single spoke.
Fig. 3.
Fig. 3. (a) Power spectra of the mechanical mode measured via its modulation of the transmitted optical power Pout for Pin=4  mW, 6 mW, 8 mW, and 10 mW. For each setting of Pin, the laser detuning (in the range of κ/2) is modified to maximize the mechanical modulation of Pout. At low input powers Pin, the mechanical motion is dominated by thermal excitation. At 10 mW, regenerative oscillation is observed, marked by a significant increase in oscillation amplitude and linewidth narrowing. (b) Phase-noise measurement of the mechanical oscillations with Pin=10  mW. A fit to the data shows that the linewidth of the regenerative oscillations is 30 mHz. Note that we add to our fit a Lorentzian peak at an offset of 220 Hz and the noise floor at 120  dBc/Hz.
Fig. 4.
Fig. 4. (a) Comparison of the power spectra of unlocked (green) and locked (blue) regenerative oscillations with Pin=20  mW. The light and dark traces correspond, respectively, to single acquisitions and averages of 50 consecutive acquisitions, presented to highlight the effect of drift in the mechanical oscillation frequency. (b) Comparison of phase noise traces for varying VAC with Pin=10  mW and the direct phase noise of the RF signal generator. Higher drive strengths result in a greater suppression of mechanical phase noise over a wider range of frequency offsets.
Fig. 5.
Fig. 5. Comparison of end-of-lock-range dynamics for the two regimes observed. (a) Mechanical power spectra of regenerative oscillations at Pin=23  mW with VDC=50  V and an injection signal of VAC=0.5  V for varying ωd. Each consecutive trace corresponds to a shift of 500  Hz in ωd and is offset in the graph by 100  dBm for clarity. The edges of the lock range (top and bottom traces) are clearly marked by the characteristic quasi-lock spectrum. (b) Result of numerical solutions of the equations of motion for the electro-optomechanical system with parameters approximating those used in the measurements for sub-figure (a). Good qualitative agreement is shown between experiment and theory of the end-of-lock-range dynamics in the quasi-locking regime. (c) Measured power spectra where Pin is reduced to 10 mW to achieve a larger lock range. Each consecutive trace corresponds to a shift of 500  Hz in ωd, spanning over the lower end of the lock range, which is here marked by the emergence of the peak at the natural resonance frequency. (d) Result of numerical solutions with parameters approximating those used in measurements for sub-figure (c).
Fig. 6.
Fig. 6. Demonstration of the large lock range achieved in the experiment: 71 kHz as compared with the 30 mHz linewidth with Pin=11  mW, with traces offset for clarity. The traces marked by green dots correspond to locking with VAC=5  V where ωd is varied. This large drive cannot be used close to ωm due to thermo-optic effects as explained in Supplement 1. Nevertheless, decreasing VAC allows the oscillations to be locked over the entire lock range, as demonstrated by the traces marked by orange squares where VAC is reduced to 0.5 V.
Fig. 7.
Fig. 7. (a) Locking range as a function of Pin for VAC=2.5  V (green squares) and 5 V (yellow circles). The fits to the measured locking ranges use the results of the calculated mechanical oscillation amplitude ro as a function of Pin, plotted in (b) for Δ/2π=42  MHz (dashed–dotted), 44 MHz (solid), and 50 MHz (dashed). Using Eq. (6), the resulting calculated locking ranges are correspondingly plotted in (a), with the best fit provided by ro(Pin,Δ) with Δ/2π=44  MHz, and upper and lower bounds with 42 MHz and 50 MHz, respectively. The data lie within the shaded regions defined by the bounds, indicating good agreement between experiment and theory. (c) Locking range as a function of VAC for Pin=12  mW (red squares) and 16 mW (blue circles), following the linear dependence described by Eq. (6).

Equations (6)

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Hom=ωmbbΔaag0aa(b+b)+AL(a+a).
Hd=xzpFd(t)(b+b).
L(f)=10log10(1πfhw(fhw)2+(Δf)2),
α˙=κ2α+i(Δ+Gx)α+AL,
meff[x¨+Γx˙+ωm2x]=G|α|2+Fd(t).
ωr=xzp2δCδxVACVDCro.

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