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

1. INTRODUCTION

The ability to engineer strong interactions between high-quality electromagnetic cavities and mechanical resonators has led to a rich variety of results in the field of optomechanics, including ground-state cooling [1,2], quantum-limited measurement [3,4], and optomechanical entanglement [5]. Optomechanical systems have further proven to be a powerful tool for applications involving precision sensing [68]. Minute perturbations in the physical environment can be detected through changes in the resonance frequency of a nanomechanical resonator, making these systems widely used as mass [9,10] and gas [11] sensors. Furthermore, many optomechanical systems can be fabricated on-chip, allowing dense arrays of sensors to be employed for applications such as multiparticle sensing for detection of lung cancer [12]. One important feature of optomechanical systems is the ability of the optical field to amplify mechanical motion. This regenerative amplification has been shown to allow enhanced sensitivity [13] and presents an avenue for the exploration of synchronization phenomena in optomechanical systems.

Nonlinear effects can cause two resonators to become synchronized, where the phases of their individual oscillations lock with respect to each other. A network of coupled oscillators can spontaneously synchronize, as was first reported for pendulum clocks hung from a common frame [14] and further observed in numerous biological systems, including the flashing of fireflies and the chirping of crickets [15]. Alternatively, a single oscillator can synchronize to the phase of an externally applied drive through an effect known as injection locking, as has been studied extensively in the context of electrical tank circuits [1618], implemented in nonlinear mechanical resonators [19], and further observed for the effect of light on human circadian rhythms [20]. Both these forms of synchronization have been explored in optomechanical systems. Arrays of optomechanical systems can synchronize when coupled through an overlap of their optical modes [21,22], via coupling to a common optical waveguide [23], or through direct inertial coupling [24,25]. Injection locking of a single optomechanical oscillator has also been demonstrated [2629].

Conventionally, the injection signal is delivered to the optomechanical oscillator through the optical field, achieved experimentally via modulation of the input laser power [2628]. Alternatively, as pointed out in a recent theoretical study [30], the oscillator can be driven with a direct inertial force. By evading the cavity filtering inherent to optical driving, this introduces different synchronization dynamics. Moreover, inertial drives can be locally applied to individual oscillators with integrated electrodes, thus presenting a scalable approach for applications utilizing arrays of oscillators. While direct inertial forcing has been achieved in a bolometric system [29], it has not been demonstrated previously with a radiation-pressure optomechanical oscillator. The conservative nature of the radiation-pressure interaction, in contrast to bolometric systems, enables many of the most important applications of optomechanical systems in quantum science and technology [8].

In this work, we present the first demonstration of the locking of a radiation-pressure optomechanical oscillator to an inertial drive. We demonstrate the ability of the injection signal to suppress phase noise by over 55 dBc/Hz and to tune the oscillation frequency by more than 2 million times the oscillation linewidth, and explore previously unreported locking dynamics. With our approach, the inertial injection signal is the electrostatic force between two electrodes directly patterned onto the body of the resonator. The feed-forward stabilization achieved with this electrical injection locking may prove more scalable for sensor arrays or complex many-body quantum networks as compared with optical injection locking, or implementing feedback circuitry for individual free-running resonators [31]. For single-oscillator sensors, a potential application of the inertial drive is to employ anomalous cooling [32] to suppress undesired regenerative oscillation of certain mechanical modes, an issue encountered in precision optical sensors such as gravitational wave detectors [3335].

Regenerative oscillation, often referred to as self-sustained oscillation, is maintained via an intrinsic feedback loop introduced by the radiation-pressure interaction, which induces dynamical backaction between the light in the optical cavity and the motion of the mechanical element (see Fig. 1). The synchronization phenomenon studied in this work is enabled by the inherent nonlinearity in the optomechanical interaction. As discussed above, all previous injection-locked radiation-pressure optomechanical oscillators have used optical injection, which is described by the same feedback loop [Fig. 1(a)] as studied widely in the context of injection-locked tank circuits [17]. In contrast, inertial injection bypasses the optical system response to drive the mechanical oscillator directly, qualitatively modifying the dynamical behavior of the system. This provides an opportunity to explore different synchronization behavior. Furthermore, in contrast to optical injection, the inertial force can be made larger than the radiation-pressure force. Our measurements uncover two different regimes of injection locking which display qualitatively different behavior when the resonator is partially locked. This may stem from the unique way in which our injection signal is applied.

 

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.

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2. ELECTRO-OPTOMECHANICAL SYSTEM

The electro-optomechanical system used in this work consists of a microtoroidal optomechanical oscillator [36,37] with an integrated electrical interface that allows a radial force to be applied directly to the mechanical resonator. In a previous work, this force was used to demonstrate high-bandwidth tuning of the optical resonance frequency [38]. The optical mode is a whispering gallery mode (WGM) of the silica microtoroid of radius 100 μm, described by its ladder operators a and a, with a resonance frequency of ωc/2π194  THz and a linewidth of κ/2π100  MHz. Figure 2(a) shows a false-color electron micrograph of the microtoroid device, consisting of a reflown silica disk (blue) atop an etched silicon pedestal (gray). Apart from having a larger radius, this device is identical to that studied in Ref. [38], which can be referred to for details on device fabrication. Measurements are performed by bringing the tapered section of an optical fiber in contact with the microtoroid to couple to the WGM while reducing taper drift. Laser light of frequency ωL (wavelength of 1550  nm) and power Pin is delivered to one end of the fiber and the transmitted power through the taper, Pout, is measured by a photodetector, allowing the optical WGM to be probed. The optical measurement setup is shown in green in Fig. 2(a). All measurements are made in ambient conditions.

 

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.

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The optical mode couples to a mechanical radial breathing mode of the microtoroid, described by its ladder operators b and b. Figure 2(b) shows the result of a finite-element method (COMSOL) simulation of the mechanical mode with a frequency of ωm/2π8.9  MHz, effective mass of meff30  ng, and zero-point motion of xzp0.3  fm. With ωm/κ0.09, the optomechanical system operates in the unresolved sideband regime. The strength of the optomechanical coupling is parameterized by the single-photon coupling rate, g0=Gxzp. G/2π0.9  GHz/nm, estimated by COMSOL simulations, is the amount by which the radial mechanical motion shifts the optical resonance frequency [6]. Detuning the laser frequency appropriately allows probing of the mechanical motion through its modulation of the transmitted optical power. The Hamiltonian in the frame rotating with the frequency of the laser drive, ωL, is

Hom=ωmbbΔaag0aa(b+b)+AL(a+a).
Here, Δ=ωLωc is the detuning and AL=ηPinκex/ωL is the optical drive amplitude, where κex is the optical coupling rate between the taper and the WGM and η0.15 captures the power lost from the laser to the taper. In addition to allowing the measurement of the mechanical motion, when blue-detuned, the laser drive applies a radiation-pressure force that amplifies the mechanical motion. There exists a threshold for Pin beyond which the amplification of the mechanical motion provided by the light exceeds the intrinsic loss rate of the mechanical mode, resulting in regenerative mechanical oscillations [6,39].

Figure 3(a) shows the measured power spectrum of the radial breathing mode for varying input power Pin. Operating in the unresolved sideband regime, the laser frequency ωL is tuned at each power setting to maximize the mechanical modulation of Pout. For Pin=4  mW, thermal excitation of the mode dominates its motion and an intrinsic mechanical linewidth (full width at half-maximum, or FWHM) of Γ/2π15  kHz can be measured. As Pin increases, the mechanical mode narrows and is frequency-upshifted, consistent with optomechanical theory [6,8]. Once Pin surpasses the threshold for regenerative oscillation, the oscillation amplitude increases by several orders of magnitude and the linewidth narrows beyond the frequency resolution of the spectrum analyzer, as shown for Pin=10  mW. At this point, the amplitude is limited by the inherent nonlinearity in the optomechanical interaction which drives the oscillations and enables the synchronization effects studied in this work. This nonlinearity can be intuitively understood by considering the Lorentzian lineshape of the optical mode as a function of the detuning caused by mechanical displacement [6,8]. For small oscillation amplitudes, the power coupled into the cavity varies linearly with mechanical displacement. However, at larger amplitudes comparable to or greater than κ/G that probe the Lorentzian shape of the mode, this is no longer the case. Any mechanical nonlinearity can be neglected given the relatively small oscillation amplitudes studied here, estimated to be <100  pm, several orders of magnitude smaller than the smallest feature in the device. Kerr and Raman optical nonlinearities can also be neglected as the incident powers used are orders of magnitude below the threshold required to observe these effects [40].

 

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.

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The electrical interface of the device allows an inertial force—which is completely independent of the optics—to be applied to the microtoroid [31,38]. This is achieved through the integration of a circular capacitor [highlighted in yellow in Fig. 2(a)] of capacitance C7  fF, whose attractive force occurs in the radial direction and, hence, has a strong overlap with the radial breathing mode. As shown in the figure, a circular slot is etched through the disk, between the gold electrodes, to increase mechanical compliance. A voltage bias is applied across the capacitor via tungsten probe tips. Baker et al. [38] studied the ability of an applied voltage to tune the optical resonance frequency and did not operate in a regime where optical amplification of the mechanical motion was appreciable. By contrast, here we seek to enhance the optomechanical gain by increasing the laser power inside the cavity and operating with a blue-detuned laser. This allows us to reach the regime of regenerative oscillation and, by applying an RF voltage with frequency ωdωm, study the synchronization of the mechanical oscillations.

We apply a drive voltage Vd(t)=VDC+VACcos(ωdt). The DC and 2ωd components of the resulting attractive force between the capacitor plates (proportional to Vd2 [38]) are off-resonant and can be omitted to obtain the drive force Fd(t)=(δC/δx)VDCVACcos(ωdt). The change in capacitance per unit mechanical displacement in our device is estimated to be δC/δx4×104  fF/nm from finite-element method simulations. The complete system Hamiltonian is then H=Hom+Hd, where

Hd=xzpFd(t)(b+b).

As discussed in the previous section, we highlight that this form of a drive is distinct from previous demonstrations of locking in optomechanical systems, which used optical radiation-pressure modulation driving [26,28]. Although the two implementations require similar instrumentation overhead for the synchronization of a single oscillator, our electrical approach should be more scalable for applications involving arrays of oscillators. A single electrical drive can be straightforwardly distributed to all resonators on a chip, whereas the optical driving scheme would generally require one optical modulator per resonator due to mismatches in optical resonance frequencies. The electrical approach also benefits from in situ tuning of the optical resonance frequency with a DC voltage [38].

3. INJECTION LOCKING

A. Locking and Stability

Figure 3(a) shows the power spectrum of the regeneratively oscillating mechanical mode. We see that, although the natural linewidth of the mechanical mode is 15 kHz, the optomechanical gain significantly narrows the linewidth such that it is no longer resolved by the spectrum analyzer. We thus use a phase-noise analyzer to capture the narrowed linewidth. Figure 3(b) shows the measured phase noise for the optomechanical oscillator with Pin=10  mW. The expected phase noise (units of dBc/Hz) for the oscillator is [41]

L(f)=10log10(1πfhw(fhw)2+(Δf)2),
where Δf is the frequency offset from the carrier and fhw is the half-width at half-maximum. The data in Fig. 3(b) are fit to the above equation, confirming the expected 1/(Δf)2 lineshape and yielding a linewidth of 30 mHz. This value is 5×105 times smaller than the intrinsic linewidth of the mechanical mode.

While the linewidth of the regenerative oscillations is extremely small, this particular measurement was performed with frequency offsets above 2 Hz and therefore does not capture the fact that the oscillation frequency drifts significantly in the timescale of minutes. This drift would be detrimental to the use of the oscillator for applications that require long-term frequency stability such as mass or temperature sensors. To highlight the broadening caused by this drift, we compare a single acquired power spectrum (light green) with the average of 50 traces acquired consecutively over a timescale of minutes (dark green) for Pin=20  mW in Fig. 4(a). The oscillation frequency drifts over a range of 100  Hz in this time, corresponding to more than 3000 times the reduced linewidth. This significant drift can be eliminated by applying the external drive described by Eq. (2) to which the mechanical oscillations can synchronize.

 

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.

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To demonstrate this effect, we apply a drive of VAC=0.5  V with VDC=50  V at a frequency of ωd/2π8.914  MHz. Figure 4(a) shows a single trace (light blue) and the average of 50 acquired traces (dark blue) of the power spectrum of the locked oscillations. We observe that the peak of the averaged trace overlaps exactly with that of the single acquisition, indicating that the oscillations are indeed locked and that the frequency drift is eliminated. In addition to the elimination of frequency drift, a comparison of the single acquisitions of the locked (light blue) and unlocked (light green) traces reveals that the locking significantly reduces the quasi-instantaneous linewidth. This effect of injection locking is well known in other types of synchronized oscillators [18], and has been demonstrated before in optomechanics with a radiation-pressure-modulated drive [26]. It is also used, for instance, in laser systems to reduce the phase noise of a noisy high-power laser by locking it to a phase-stable low-power laser [42].

To explore this further, we measure the phase noise for the locked oscillations for varying drive strengths. Figure 4(b) shows plots of phase noise for VAC=0.1  V, 0.5 V, and 1.5 V with Pin=10  mW. Also included in the figure is the phase-noise trace of the unlocked oscillations for comparison [reproduced from Fig. 3(b)], as well as the phase noise of the RF voltage source itself. We clearly observe a significant suppression of phase noise when the oscillator is locked, with a maximum suppression of over 55 dBc/Hz at 2 Hz. A stronger drive results in a greater phase-noise suppression, approaching the noise of the RF source itself [26]. As discussed in Ref. [18] in the context of electrical tank circuits, and as shown in Fig. 4(b), the phase noise is suppressed up to a critical frequency offset which depends on the drive strength. Above that frequency, the phase noise of the unlocked or free-running oscillator is recovered. As we will see in the next section, where injection locking is used to tune the frequency of regenerative oscillations, this critical frequency corresponds to the locking range for the particular drive strength.

B. End-of-Lock-Range Dynamics

As well as stabilizing the mechanical oscillations, the injection signal can also be used to tune the oscillation frequency. The range of frequencies over which the mechanical mode can be locked has been studied extensively in the context of electrical tank circuits [16,18] and, more recently, optomechanical systems [30]. By definition, the oscillator is locked to the drive when the phase difference between it and the external drive, Δϕ, is stationary in time. When locked, Δϕ is zero at the natural mechanical resonance frequency, and grows with detuning to ±π/2 at the edges of the lock range, beyond which the oscillator fails to lock [16]. This change in phase with mechanical resonance frequency allows an injection-locked oscillator to be used for sensing applications where shifts in the mechanical resonance frequency are to be detected. This may offer benefits for sensor arrays compared with other approaches, such as using free-running regenerative oscillators or phase locking of driven oscillators, as stability is achieved without the need for individual feedback control circuitry to each sensor.

Here, we explore the dependence of the lock range on the two independent variables in our system, namely, (i) drive strength and (ii) optical power, and show good agreement with the recent theoretical study performed by Amitai et al. [30]. In addition to this, we find that previously studied end-of-lock-range dynamics occur only for sufficiently high optical power, with a new class of behavior evident at lower optical powers.

We begin by setting the system to regeneratively oscillate with Pin=23  mW and applying an AC voltage of VAC=0.5  V at a frequency close to the mechanical resonance. With these settings, the amplitude of the mechanical oscillations due to the radiation-pressure force dominates the inertial drive by at least an order of magnitude. The oscillations successfully lock to the drive and ωd is then increased (decreased) to find the upper (lower) end of the lock range. The end of the lock range is unambiguously signaled on the spectrum analyzer, marked by a characteristic spectrum that is representative of quasi-locking [18]. As discussed in Ref. [18], this regime of quasi-locking is characterized by the oscillator slipping in and out of lock. This manifests as Δϕ alternating between periods of being held near ±π/2 and cycling through 2π radians before being locked near ±π/2 again (c.f., Supplement 1). The fraction of the time for which Δϕ is held near ±π/2 decreases as the drive frequency moves further out of the locking range [18], until the oscillator is no longer locked at all. Supplement 1 covers the phase dynamics in greater detail. Here, it suffices to consider that this results in a phase difference which oscillates in time. This translates to a periodic frequency modulation, resulting in the sidebands that make up the triangular single-sided spectrum characteristic of quasi-locking [18], as shown in Fig. 5(a). The top (bottom) trace corresponds to the upper (lower) end of the lock range, whereas the two spectra in between are within the lock range.

 

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).

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For injection frequencies outside the lock range (not shown), we observe, as expected [18], injection pulling of the optomechanical oscillator. This term comes from the fact that, in this range of detuning, the resonance peak appears to be pulled toward the injection signal [18].

The locking phenomena discussed above are qualitatively the same as those observed for injection locking of electrical tank circuits [18], as well as a previous demonstration with an optomechanical system [26]. However, when these parameters are varied to obtain a larger locking range, we find that the end-of-range dynamics are strikingly different.

Decreasing Pin to 10 mW with the same drive strength (VAC=0.5  V) to obtain a larger locking range [30], we repeat the search for the ends of the lock range. With these settings, the amplitude of the mechanical oscillations due to the radiation-pressure force and due to the inertial drive are of the same order. We find that the end of the lock range is no longer marked by the distinct quasi-lock spectrum. Figure 5(c) shows the spectra near the lower end of the lock range, where the drive frequency is decreased from the top to bottom traces. The end of the lock range is now instead marked by an increase in power at the natural resonance frequency, rather than the quasi-lock spectrum. Outside the lock range here, injection pulling is not observed. We refer to this apparently distinct regime of partial locking as the “continuous-suppression” regime, as the peak at the mechanical resonance frequency appears to be suppressed as the injection signal moves into the locking range. The transition from the quasi-lock regime to the continuous-suppression regime occurs when the driven oscillation amplitude approaches or even exceeds the amplitude due to optical pumping. The cross-over is not entirely well defined, however, as some settings result in one end of the lock range being marked by quasi-locking while the other being marked by continuous suppression of the natural peak.

In order to understand these two seemingly distinct locking regimes, we modeled the system using classical equations of motion for the coupled optical mode (characterized by its light amplitude α) and mechanical mode (characterized by its position x), which can be derived from the Hamiltonian given earlier:

α˙=κ2α+i(Δ+Gx)α+AL,
meff[x¨+Γx˙+ωm2x]=G|α|2+Fd(t).

The mechanical mode, with a decay rate of Γ/2π=15  kHz, is subject to both a radiation-pressure force and an electrical drive. We omit in our simulations thermo-optic effects, which affect the refractive index of the material [23], as well as the thermal Langevin force [6]. We numerically solve the coupled differential equations to simulate the dynamics of the system in the locking regimes explored experimentally in Fig. 5(a) and 5(c), and plot the results in Figs. 5(b) and 5(d), respectively. We find excellent qualitative agreement between the simulation and experiment, highlighting that these simple classical equations of motion capture all the experimentally observed locking dynamics, including the cross-over between the quasi-lock and continuous-suppression regimes.

C. Quantifying the Locking Range

In this section, we characterize the dependence of the lock range on the optical power and drive strength and compare our data with the predictions of Amitai et al. [30]. We first determine the largest lock range achievable with our device, by maximizing the output of the RF source to VAC=5  V and lowering the optical power to Pin=11  mW. Figure 6 shows plots of traces of the power spectrum for varying drive frequencies, spanning the lock range for these settings, marked by green dots. The lock range achieved is 71 kHz, over 2×106 times the linewidth of the unlocked regenerative oscillations. While this range can be easily increased by increasing VAC or VDC, this value already represents a tuning percentage of 1%. For applications which require arrays of optomechanical systems oscillating regeneratively at the same frequency, it is important to compare this tuning percentage to the expected variations in natural mechanical resonance frequencies. For the case of a circular optomechanical resonator, the resonance frequency of a radial breathing mode is inversely proportional to the radius, ωm1/R. Given that a 100 μm disk can be fabricated to well within 1 μm precision with standard lithographic techniques, the tuning range demonstrated here is already sufficient to overcome the <1% variations expected in ωm.

 

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.

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Referring back to the locked traces in Fig. 6, we find that setting the injection frequency of the strong drive (VAC=5  V) close to the mechanical resonance frequency causes the optical cavity to be shifted out of resonance such that regenerative oscillations cease. This is shown by the absence of traces marked by green dots between 10  kHz and 10 kHz. We suspect that this is due to thermal effects in the optomechanical system, as explained in Supplement 1. Here, we emphasize that this effect does not preclude locking over the entire range, as VAC can be reduced to lock near the natural mechanical resonance frequency. This is demonstrated by the power spectra marked by orange squares in Fig. 6(c), where VAC is reduced to 0.5 V to successfully lock near the center of the range.

Amitai et al. [30] show theoretically that the locking range 2ωr is directly proportional to the drive strength and inversely proportional to the amplitude of mechanical oscillations, ro (units of m). Adapting their expression to our parameters, we have

ωr=xzp2δCδxVACVDCro.
The amplitude of regenerative oscillations, ro, is a function of the optomechanical parameters of the system as well as Pin, which can be controlled in the experiment. We first determine the lock range as a function of Pin. As explained previously, two regimes of end-of-range dynamics are observed in the experiment. In one, ωr can be unambiguously quantified owing to the appearance of the quasi-lock spectrum. While some hysteresis was observed depending on the direction of the frequency sweep [29], its effect on quantifying ωr was minor and thus neglected. In the continuous-suppression regime, however, the end of the lock range is marked by a relatively gradual re-emergence of the peak at the natural frequency ωm. For consistency, we choose to define ωr in this regime to be the frequency offset at which the peak at ωm reaches 30 dB below the peak at ωd. We note that this choice of threshold does not have a significant effect on the results as the peak at ωm rises from being completely suppressed (below the noise floor) back to its original amplitude when ωd is varied over a frequency offset range corresponding to 5% of the locking range.

Figure 7(a) shows plots of the measured locking range 2ωr as a function of Pin for VAC=2.5  V (green squares) and 5 V (yellow circles). While the data include measurements in both the quasi-lock and continuous-suppression regimes, the data points nevertheless follow clear trends. In order to compare these results with locking ranges predicted by Eq. (6), we require the regenerative oscillation amplitude ro, which is not precisely determined in the experiment. We thus use the equations of motion [Eq. (5)] to calculate ro in the absence of an inertial drive. Figure 7(b) shows plots of ro as a function of optical power Pin for detunings of Δ/2π=42  MHz, 44 MHz, and 50 MHz, illustrating the sensitivity of ro to both these parameters. We find that, when inserted into Eq. (6), ro(Pin,Δ) for Δ/2π=44  MHz (solid line) produces the best fit to the measured locking ranges. The fit is plotted as a solid curve in Fig. 7(a), demonstrating good agreement between theory and experiment. Note that a free scaling parameter β=0.65 is added to Eq. (6), to account for uncertainties in δC/δx and xzp. The fit could be further improved by taking into account the modification of the effective optical detuning with Pin. Increasing Pin modifies the effective cavity resonance due to the steady-state radiation-pressure-induced cavity enlargement. This dependency is naturally corrected for in the experiment by always tuning the laser frequency to maximize the regenerative oscillation signal. However, for simplicity, the calculations for Fig. 7(b) were performed with a fixed laser frequency. Adding this correction, which we estimate to result in a shift of a few MHz in detuning over the range of laser powers used in our experiments, would reduce the steepness of the calculated curves at lower laser power, resulting in a closer match to the experimental data. As confirmation of this mechanism, we include the calculated locking ranges for Δ/2π=42  MHz and 50 MHz to Fig. 7(a), which enclose all the experimental data.

 

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).

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In contrast, the measurement of locking range as a function of VAC does not involve changes in the optical detuning and can thus be more straightforwardly compared to the theory. Figure 7(c) shows plots for this for Pin=12  mW (red squares) and 16 mW (blue circles), confirming the linear dependence predicted by Eq. (6).

4. CONCLUSION

We have reported the first observation of locking of the radiation-pressure-driven regenerative oscillations of an optomechanical system to a direct inertial drive. We demonstrate a suppression of over 55 dBc/Hz of the phase noise of the regenerative oscillations at 2 Hz and a locking range of 71 kHz, more than 2 million times the 30 mHz oscillation linewidth. This tuning range is sufficient to overcome variations in natural mechanical resonance frequencies due to limits in fabrication precision. Applications requiring distributed optomechanical systems or arrays of oscillators within a single chip may benefit from injection locking. Although phase-locked loops can also be used to stabilize the mechanical resonance frequency, they require feedback circuitry, which may prove impractical for large arrays of devices. The feed-forward nature of injection locking allows stabilization to be achieved without feedback circuitry. Moreover, the electrical drive presented here can be implemented for arrays more efficiently than optical injection, which requires either additional lasers or optical power modulators. This is especially true in the case where multiple oscillation frequencies are desired.

At a system level, the direct inertial drive we implement is distinct from that used in prior explorations of injection locking in optomechanical systems as well as on other physical platforms. To our knowledge, the only previously known difference between the two drive forms is the absence of harmonic locking effects for the direct drive [29]. We present previously unreported locking dynamics as described by the continuous-suppression regime, enabled by the ability to increase the inertial injection signal to approach and even exceed that of the radiation-pressure force. Further research is required to explain the underlying mechanisms responsible for the cross-over between the two regimes with distinct locking dynamics. Recently, Tóth et al. [43] also demonstrated locking with an injection signal which bypasses the nonlinearity in a radiation-pressure electromechanical system. This experiment explored the reverse regime to that studied here, with the mechanical oscillator driving regenerative oscillations of a microwave field, allowing injection locking of the microwave resonance.

Beyond applications in sensing, the electrical drive technique introduced in this work may enable the possibility to engineer optomechanical gain competition. For instance, we observed that for some devices, non-ideal circularity split the radially symmetric mechanical mode into two quasi-degenerate modes separated by <1  kHz, as visible in Fig. 5(c). Upon application of the drive to the resonance frequency of either mode, the system could be made to oscillate regeneratively on the targeted mode. Indeed, much like for the optical modes of a laser, different mechanical modes compete for gain in an optomechanical resonator [32]. We provide a detailed numerical analysis of this phenomenon in Supplement 1 and confirm that the electrical drive can reorient the oscillator along a new stable trajectory. The switch of optomechanical gain from one mode to the other persists even once the drive has been turned off, potentially serving as a form of “non-volatile” optomechanical memory [44] or enabling the exploration of normally inaccessible stable dynamical attractors of the system [45,46].

Funding

Australian Research Council (ARC) (CE110001013, FT140100650, LP140100595); University of Queensland (UQ) (UQFEL1719237).

Acknowledgment

This research was primarily funded by the Australian Research Council and the Lockheed Martin Corporation through an Australian Research Council Linkage Grant. Support was also provided by a Lockheed Martin Corporation seed grant and the Australian Research Council Centre of Excellence for Engineered Quantum Systems. W. P. B. and R. K. acknowledge fellowships from the Australian Research Council and the University of Queensland, respectively. This work was performed in part at the Queensland node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nano- and microfabrication facilities for Australia’s researchers.

 

See Supplement 1 for supporting content.

REFERENCES

1. 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]  

2. 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]  

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]  

4. 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]  

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

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

7. 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]  

8. W. P. Bowen and G. J. Milburn, Quantum Optomechanics (CRC Press, 2015).

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

10. 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]  

11. 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]  

12. 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]  

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

14. M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’s clocks,” in Proceedings: Mathematics, Physical and Engineering Sciences (2002), pp. 563–579.

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

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

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

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

19. 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]  

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

21. 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]  

22. 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]  

23. 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]  

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

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

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

27. 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]  

28. 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]  

29. 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]  

30. 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]  

31. 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]  

32. 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]  

33. 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]  

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

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

36. 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]  

37. 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]  

38. C. G. Baker, C. Bekker, D. L. McAuslan, E. Sheridan, and W. P. Bowen, “High bandwidth on-chip capacitive tuning of microtoroid resonators,” Opt. Express 24, 20400–20412 (2016). [CrossRef]  

39. 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]  

40. 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]  

41. 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]  

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

43. L. Tóth, N. Bernier, A. Feofanov, and T. Kippenberg, “A maser based on dynamical backaction on microwave light,” Phys. Lett. A (to be published) (2017). [CrossRef]  

44. 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]  

45. 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]  

46. 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]  

References

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  1. 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]
  2. 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]
  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]
  4. 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]
  5. T. Palomaki, J. Teufel, R. Simmonds, and K. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
    [Crossref]
  6. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  7. 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]
  8. W. P. Bowen and G. J. Milburn, Quantum Optomechanics (CRC Press, 2015).
  9. K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
    [Crossref]
  10. 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]
  11. 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]
  12. 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]
  13. X. Feng, C. White, A. Hajimiri, and M. L. Roukes, “A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator,” Nat. Nanotechnol. 3, 342–346 (2008).
    [Crossref]
  14. M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’s clocks,” in Proceedings: Mathematics, Physical and Engineering Sciences (2002), pp. 563–579.
  15. R. E. Mirollo and S. H. Strogatz, “Synchronization of pulse-coupled biological oscillators,” SIAM J. Appl. Math. 50, 1645–1662 (1990).
    [Crossref]
  16. R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
    [Crossref]
  17. L. J. Paciorek, “Injection locking of oscillators,” Proc. IEEE 53, 1723–1727 (1965).
    [Crossref]
  18. B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuits 39, 1415–1424 (2004).
    [Crossref]
  19. 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]
  20. J. F. Duffy and C. A. Czeisler, “Effect of light on human circadian physiology,” Sleep Med. Clin. 4, 165–177 (2009).
    [Crossref]
  21. 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]
  22. 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]
  23. 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]
  24. G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective dynamics in optomechanical arrays,” Phys. Rev. Lett. 107, 043603 (2011).
    [Crossref]
  25. M. Ludwig and F. Marquardt, “Quantum many-body dynamics in optomechanical arrays,” Phys. Rev. Lett. 111, 073603 (2013).
    [Crossref]
  26. M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical RF oscillator,” Appl. Phys. Lett. 93, 191115 (2008).
    [Crossref]
  27. 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]
  28. 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]
  29. 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]
  30. 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]
  31. 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]
  32. 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]
  33. 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]
  34. M. Abdi and A. R. Bahrampour, “Effect of higher-order waves in parametric oscillatory instability in optical cavities,” Phys. Scripta 83, 045401 (2011).
    [Crossref]
  35. S. P. Vyatchanin and S. E. Strigin, “Parametric oscillatory instability in gravitational antennas wave laser detectors,” Phys. Usp. 18255, 1115–1123 (2012).
    [Crossref]
  36. 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]
  37. 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]
  38. C. G. Baker, C. Bekker, D. L. McAuslan, E. Sheridan, and W. P. Bowen, “High bandwidth on-chip capacitive tuning of microtoroid resonators,” Opt. Express 24, 20400–20412 (2016).
    [Crossref]
  39. 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]
  40. 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]
  41. 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]
  42. C. J. Buczek, R. J. Freiberg, and M. Skolnick, “Laser injection locking,” Proc. IEEE 61, 1411–1431 (1973).
    [Crossref]
  43. L. Tóth, N. Bernier, A. Feofanov, and T. Kippenberg, “A maser based on dynamical backaction on microwave light,” Phys. Lett. A (to be published) (2017).
    [Crossref]
  44. 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]
  45. 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]
  46. 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]

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. 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]

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

2004 (2)

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

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.

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]

Barnard, A.

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]

Baskin, I.

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]

Bekker, C.

Bennett, M.

M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’s clocks,” in Proceedings: Mathematics, Physical and Engineering Sciences (2002), pp. 563–579.

Bernier, N.

L. Tóth, N. Bernier, A. Feofanov, and T. Kippenberg, “A maser based on dynamical backaction on microwave light,” Phys. Lett. A (to be published) (2017).
[Crossref]

Bowen, W. P.

Braginskii, V. B.

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]

Brianceau, 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]

Briant, T.

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]

Bruder, C.

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]

Buczek, C. J.

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

Buks, E.

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]

Cardenas, J.

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]

Carmon, 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).
[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).
[Crossref]

Ceballos, G.

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]

Chan, J.

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]

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]

Chaste, J.

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]

Cicak, K.

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]

Ciuti, 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]

Cohadon, P.-F.

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]

Colinet, E.

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]

Craighead, H. G.

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]

Czeisler, C. A.

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

Demir, A.

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]

Donner, T.

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]

Duffy, J. F.

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

Duraffourg, L.

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]

Durand, M.

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]

Eichler, 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]

Favero, I.

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]

Feng, X.

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

Feofanov, A.

L. Tóth, N. Bernier, A. Feofanov, and T. Kippenberg, “A maser based on dynamical backaction on microwave light,” Phys. Lett. A (to be published) (2017).
[Crossref]

Fernandes, M. P.

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]

Français, 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]

Freiberg, R. J.

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

Gil-Santos, E.

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]

Girvin, S. M.

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]

Goetschy, A.

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]

Gomez, 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]

Gong, Q.

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]

Groeblacher, S.

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]

Hajimiri, A.

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

Harlow, J.

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]

Harris, G. I.

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]

Harris, J. G. E.

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]

Hartmann, M. J.

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]

Hease, W.

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]

Heidmann, A.

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]

Heinrich, G.

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

Hentz, 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]

Hill, J. T.

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]

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]

Hossein-Zadeh, M.

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

Houston, B. H.

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]

Hu, Y. W.

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]

Jensen, K.

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

Kemiktarak, U.

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]

Kim, K.

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

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).
[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]

L. Tóth, N. Bernier, A. Feofanov, and T. Kippenberg, “A maser based on dynamical backaction on microwave light,” Phys. Lett. A (to be published) (2017).
[Crossref]

Kippenberg, T. J.

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

Knittel, J.

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]

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]

Krause, A.

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]

Krause, A. G.

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]

Kubala, B.

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

Labousse, M.

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]

Lawall, J.

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]

Lee, K. H.

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]

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]

Lehnert, K.

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

Lemaître, A.

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]

Leo, G.

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]

Li, D.

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]

Li, 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]

Lipson, M.

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]

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]

Liu, Y. C.

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]

Lörch, N.

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]

Ludwig, M.

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

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

Mackowski, J.-M.

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]

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).
[Crossref]

Marcoux, C.

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]

Marquardt, F.

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

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

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

McAuslan, D. L.

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).
[Crossref]

McRae, T. G.

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]

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]

Mehrotra, A.

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]

Metcalfe, M.

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]

Michel, C.

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]

Milburn, G. J.

W. P. Bowen and G. J. Milburn, Quantum Optomechanics (CRC Press, 2015).

Mirollo, R. E.

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

Moser, J.

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]

Myers, E. B.

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]

Nunnenkamp, A.

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]

Paciorek, L. J.

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

Painter, O.

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]

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]

Palomaki, T.

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

Pandey, M.

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]

Parpia, J. M.

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]

Pernice, W. P. H.

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]

Pinard, L.

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]

Pinard, M.

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]

Polzik, E. S.

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]

Poot, 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]

Qian, J.

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

Rand, R.

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]

Rand, R. H.

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]

Razavi, B.

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

Ridolfo, A.

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]

Rockwood, H.

M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’s clocks,” in Proceedings: Mathematics, Physical and Engineering Sciences (2002), pp. 563–579.

Rokhsari, H.

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. 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]

Roukes, M. L.

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]

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

Rousseau, L.

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]

Roychowdhury, J.

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]

Rurali, R.

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]

Safavi-Naeini, A. H.

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]

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]

Schatz, M. F.

M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’s clocks,” in Proceedings: Mathematics, Physical and Engineering Sciences (2002), pp. 563–579.

Scherer, A.

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]

Schliesser, A.

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]

Schmid, S.

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]

Seitner, M. J.

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]

Shah, S.

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]

Shah, S. Y.

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]

Sheridan, E.

Shlomi, K.

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]

Simmonds, R.

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

Simmonds, R. W.

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]

Simonsen, A.

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]

Sirois, A.

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]

Skolnick, M.

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

Sørensen, A.

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]

Spillane, S.

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]

Strigin, S. E.

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

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]

Strogatz, S. H.

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

Suchoi, O.

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]

Sudarshan, B. G.

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]

Szorkovszky, A.

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).
[Crossref]

Taylor, J. M.

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]

Taylor, M. A.

Teufel, J.

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

Tóth, L.

L. Tóth, N. Bernier, A. Feofanov, and T. Kippenberg, “A maser based on dynamical backaction on microwave light,” Phys. Lett. A (to be published) (2017).
[Crossref]

Usami, K.

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]

Vahala, K.

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).
[Crossref]

Venkatesh, S.

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]

Villanueva, L. G.

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]

Vyatchanin, S. P.

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

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]

Walter, S.

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]

Weig, E. 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]

White, C.

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

Whittaker, J. D.

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]

Wiederhecker, G. 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).
[Crossref]

Wiesenfeld, K.

M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’s clocks,” in Proceedings: Mathematics, Physical and Engineering Sciences (2002), pp. 563–579.

Winik, R.

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]

Xiao, Y. F.

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]

Yang, L.

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]

Yuvaraj, D.

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]

Zalalutdinov, M.

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]

Zehnder, A. T.

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]

Zettl, A.

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

Zeuthen, E.

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).
[Crossref]

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]

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]

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]

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, 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]

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]

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]

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

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

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]

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]

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]

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]

Phys. Scripta (1)

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

Phys. Usp. (1)

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

Proc. IEEE (2)

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

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

Proc. IRE (1)

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

Rev. Mod. Phys. (1)

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

Science (1)

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

SIAM J. Appl. Math. (1)

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

Sleep Med. Clin. (1)

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

Other (3)

M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’s clocks,” in Proceedings: Mathematics, Physical and Engineering Sciences (2002), pp. 563–579.

W. P. Bowen and G. J. Milburn, Quantum Optomechanics (CRC Press, 2015).

L. Tóth, N. Bernier, A. Feofanov, and T. Kippenberg, “A maser based on dynamical backaction on microwave light,” Phys. Lett. A (to be published) (2017).
[Crossref]

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)

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

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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