The amplitude of self-oscillating mechanical resonators in cavity optomechanical systems is typically limited by nonlinearities arising from the cavity’s finite optical bandwidth. We propose and demonstrate a feedback technique for increasing this limit. By modulating the cavity input field with a signal derived from its output intensity, we increase the amplitude of a self-oscillating GHz frequency mechanical resonator by 22% (an increase in coherent phonon number of 50%), limited only by the achievable optomechanical cooperativity of the system. This technique will advance applications dependent on high dynamic mechanical stress, such as coherent spin-phonon coupling, as well as the implementation of sensors based on self-oscillating resonators.

Feedback is a key element of many classical and quantum technologies, ranging from the control of quantum systems1 to the implementation of mechanical sensors, such as atomic force microscopes2 and highly coherent optical sources.3,4 Cavity optomechanical systems5 commonly use various forms of feedback6–9 to dampen the mechanical motion of resonators interacting with an optical cavity and cool them to their quantum ground state. Feedback can also be used to amplify mechanical motion in these systems to probe the limits of nanomechanical sensors,10–14 generate frequency combs15 and rf and microwave oscillators,16–21 and study nonlinear dynamics.22 Self-oscillating phonon lasers have also played a role in optomechanically driving electronic spin qubits associated with defects in materials, such as diamonds.23 

By localizing optical fields near or within a mechanical resonator, cavity optomechanical systems provide coupling between optical and mechanical degrees of freedom, which is essential for using feedback to control resonator dynamics. The presence of a delay in the system’s optical response to resonator displacement—dynamical optomechanical backaction—allows the optomechanical coupling to modify the resonator’s effective mechanical dissipation.5 Both internal feedback from the cavity mode’s delayed optical response to changes in resonator geometry24–26 and delayed external feedback derived from the measured27,28 or re-circulated9 cavity field have been used to cool mechanical resonators. Similarly, the excitation of coherent resonator motion through these forms of feedback has been demonstrated.14,16,27 Cavity-less optomechanical systems can also harness feedback; for example, adaptive control has been employed for cooling levitated mechanical objects,29 and photothermal optomechanical coupling allows modification of resonator dynamics by microscope field gradients30 without the aid of a cavity.

Although phonon lasing typically increases the resonator amplitude by several orders of magnitude, its maximum displacement is clamped by nonlinear optomechanical effects arising when the mechanically induced change in cavity frequency exceeds the cavity linewidth,31 limiting its potential for many of the applications discussed above. Enhancing the dynamics of self-oscillating resonators by modulating the continuous wave field that parametrically excites their mechanical motion has been demonstrated in studies of frequency injection locking.32 Here, we examine whether modulation of the drive field allows the self-oscillation amplitude to be increased. We show that the amplitude of a self-oscillating optomechanical resonator can be enhanced with the aid of linear external feedback that complements the cavity’s internal backaction and compensates for the optomechanical nonlinearities limiting the oscillation amplitude. This external feedback is similar to the linear velocity feedback used in Ref. 29 for nanoparticle levitation and is applied here to control the self-oscillations of a GHz frequency diamond microdisk cavity optomechanical device. After being driven into phonon lasing, the measured mechanical motion of the resonator is amplified, phase shifted, and input into a phase modulator that modulates the input laser frequency so that it follows the optical cavity frequency set by the mechanical displacement. This process effectively reduces dynamical backaction and enables the self-oscillating resonator to accumulate a higher number of phonons for a given input power.

Modification of mechanical resonator dynamics through coupling to an optical cavity driven by a continuous wave laser arises when the displacement of the resonator by the cavity field modifies the cavity mode frequency, which in turn affects the field built up in the cavity. If the delay set by the cavity bandwidth is comparable to or longer than the mechanical oscillation period, this optomechanical backaction results in damping (Γom > 0) or anti-damping (Γom < 0) of the resonator motion, depending on whether the laser is red (Δ < 0) or blue (Δ > 0) detuned from the cavity mode. The mechanical resonator is driven into self-sustained oscillation when the energy transfer rate Γom between the cavity field and the resonator is equal to the resonator’s internal dissipation rate Γm,33 and in a linearized theory of optomechanics, this modifies the resonator’s phonon number from its thermal occupation nth to n = nthΓm/(Γm + Γom). This expression reveals the divergence of n and the onset of self-oscillation when Γom = −Γm, a regime that is most efficiently accessed when Δ = ωm so that the parametric process of scattering a laser photon into the lower frequency cavity mode while generating a phonon is resonant.

In practice, the amplitude of a self-oscillating resonator is clamped when the instantaneous optomechanical frequency shift δom(t) = Gx(t) of the cavity mode frequency induced by the oscillations is comparable to the cavity linewidth, as determined by the optomechanical coupling coefficient G = −o/dx and the oscillation amplitude A. This is illustrated in Fig. 1(a), which shows how a laser nominally blue detuned Δ = ωm to maximize anti-damping is shifted away from this detuning by δom(t) when A is large. Previous numerical studies have explored how varying the cavity optomechanical system’s parameters affects A.31 Here, we explore an alternative concept: whether A can be enhanced using feedback to dynamically compensate for δom(t) by modulating the input laser frequency.

FIG. 1.

(a) and (b) Temporal evolution of the optical cavity (blue) and laser (red) frequencies for an oscillating cavity optomechanical system, with and without external feedback used to shift the laser frequency so that it follows the instantaneous optomechanical frequency shift. (c) Experimental system used to implement the laser frequency feedback scheme.

FIG. 1.

(a) and (b) Temporal evolution of the optical cavity (blue) and laser (red) frequencies for an oscillating cavity optomechanical system, with and without external feedback used to shift the laser frequency so that it follows the instantaneous optomechanical frequency shift. (c) Experimental system used to implement the laser frequency feedback scheme.

Close modal
This approach is illustrated in Fig. 1(b): given knowledge of x(t), the laser frequency ωl is constantly adjusted by δl(t) to reduce its deviation away from Δ = ωm. Our experimental implementation of this scheme is shown in Fig. 1(c). A weak laser input to an optical cavity through an optical waveguide provides a readout of x(t) via photodetection of its transmission. This electronic signal is fed-back to an electro-optic modulator that dynamically shifts the phase of a strong field α input to a second cavity mode that parametrically drives the cavity’s mechanical resonance into self-oscillation for sufficient optical power |α|2 and appropriate detuning Δ. In the experiment described below, a diamond microdisk whispering gallery mode cavity evanescently coupled to a fiber taper waveguide is used. This system is described by equations of motion,
α̇=κ2α+i(Δ+Gx)α+κexαine+ifGx(tτ)/ωm,
(1)
ẍ+Γmẋ+ωm2x=Gmeff|α|2,
(2)
with feedback captured by the time varying shift fGx(tτ)/ωm in the phase of the input field α, where τ is the delay of an electronic feedback line and f is the feedback strength, which is controlled by a variable phase shifter and a variable gain amplifier, respectively. Here, κex is the coupling rate between the cavity and the input waveguide, and meff is the resonator mode’s effective mass. It is worth noting that mechanical nonlinearities are safely disregarded in this analysis since the amplitude of mechanical oscillations is orders of magnitude smaller than the threshold for observing nonlinear effects, such as the Duffing effect.34 This assumption is consistent with the observed dynamics of the device described below, which can be explained without the inclusion of mechanical nonlinearity. However, such effects will ultimately limit the achievable oscillation amplitude for sufficiently strong feedback enhanced driving.
The nonlinear dynamics of this system can be analyzed by generalizing the theory of Marquardt et al.33 to include external feedback. In a steady state, the power scattered into the resonator from optical radiation pressure is equal to the resonator’s dissipated power, so that ẍẋ=0. This leads to
2ωmκCom1|z0|2j=Jj(β)Jj+1(β)β/2Im{zjzj+1*(1feiφ)},
(3)
where β=GA1+f22fcos(φ)/ωm is the effective optomechanical modulation index, zj=κexαin/[κ/2i(Δ+jωm)] is the cavity mode response at sideband frequency ωl + m, |z0|2 is the intracavity photon number before the onset of self-oscillations, and Com=4g02|z0|2/κΓm is the optomechanical cooperativity before the onset of self-oscillations. Physically, Eq. (3) compares Γm to the sum of the phonon generation rates associated with optomechanical scattering between the j + 1 and j optical sidebands.

Feedback manifests in the term (1 − fe) in Eq. (3) and a modification of β, affecting the optomechanical scattering rates between sidebands. As written, Eq. (3) can be conveniently solved numerically for mechanical amplitude A as a function of Com, allowing the influence of f and φ to be studied for a given κ/ωm. Figure 2(a) shows the result of this calculation for κ/ωm = 0.8 and Com = 2.73, chosen since they correspond to the operating conditions of the cavity optomechanical system studied experimentally below. We see that the oscillation amplitude, expressed in terms of the phonon occupation of the mechanical resonator mode n(GA/ωm)2, possesses a maximum that is 1.5× larger than the amplitude n0 in the absence of feedback (f = 0). The dependence of n/n0 on f and φ is consistent with the behavior of optomechanical damping: dynamical back-action is responsible for phonon lasing5 but also limits the oscillation amplitude.31 It is worth noting that the points where the ratio is zero indicate that the system is not self-oscillating and the number of coherent phonons is zero. Note that for all of these calculations, Δ = ωm, and that the input field strength α and coupling rate κ do not need to be specified in this analysis.

FIG. 2.

(a) Theoretically predicted phonon occupation of the mechanical resonance as a function of feedback parameters, normalized by the number of phonons during pure self-oscillation (f = 0) for the device parameters used in this work (κ/ωm = 0.8) and an optimally blue detuned drive field with intensity corresponding to Com = 2.73. The dashed lines correspond to parameter ranges studied experimentally below. (b) Predicted normalized phonon occupation for varying Com with optimized feedback parameters (red) and feedback turned off (blue). (c) and (d) Repeat part (a) for κ/ωm = 0.08 and Com = 2.73, and κ/ωm = 0.8 and Com = 20, respectively.

FIG. 2.

(a) Theoretically predicted phonon occupation of the mechanical resonance as a function of feedback parameters, normalized by the number of phonons during pure self-oscillation (f = 0) for the device parameters used in this work (κ/ωm = 0.8) and an optimally blue detuned drive field with intensity corresponding to Com = 2.73. The dashed lines correspond to parameter ranges studied experimentally below. (b) Predicted normalized phonon occupation for varying Com with optimized feedback parameters (red) and feedback turned off (blue). (c) and (d) Repeat part (a) for κ/ωm = 0.08 and Com = 2.73, and κ/ωm = 0.8 and Com = 20, respectively.

Close modal

To further explore the feedback’s behavior, we solve Eq. (3) for varying cooperativity, optimizing the feedback parameters at each Com to maximize n. Figure 2(b) shows the result. This clearly indicates that in the absence of feedback (f = 0), as Com increases, which for a given set of cavity parameters corresponds to increasing the number of intracavity drive photons, ncav, the resonator amplitude saturates. Conversely, when feedback is used, n grows as Com increases. One can show that at the optimal point, A is linearly proportional to the cooperativity GA/ωmCom, with the proportionality constant to be maximized over f and φ. This is shown in Fig. 2(b), where GA/ωm2 scales quadratically with Com. This shows that adding feedback allows the phonon number to be enhanced by orders of magnitude, which is limited, in practice, by the experimentally achievable Com. We also note that self-oscillation is possible using feedback for Com < 1, as shown in the inset in Fig. 2(b). Figures 2(c) and 2(d) illustrate the impact of device parameters on the feedback enhancement. For κ/ωm = 0.08 [Fig. 2(c)], corresponding to increasing Qo or ωm by an order of magnitude, the maximum phonon occupation is not significantly affected if Com is kept constant. However, note that a higher Qo allows for a higher Com for a given ncav. When Com is increased to 20 while κ/ωm remains constant [Fig. 2(d)], the maximum phonon occupation increases by nearly an order of magnitude.

We next experimentally demonstrate feedback enhanced optomechanical self-oscillation. The cavity optomechanical system used here consisted of a diamond microdisk previously used for coherent photon–phonon coupling35,36 and spin–optomechanics.23 This device supports a ωm/2π = 2.1 GHz radial breathing mode mechanical resonance that interacts strongly with the optical whispering gallery modes of the device. A fiber taper waveguide evanescently couples to optical modes used for driving and probing the mechanical motion of the microdisk. The drive mode (wavelength λd = 1563.4 nm) has a quality factor Qo,d = 1.1 × 105 sufficiently high to place it near the sideband resolved regime, while the probe mode (λp = 1509.5 nm) has a lower Qo,p = 1.1 × 104, allowing its field to instantaneously respond to the motion of the mechanical resonator. Coherent photon–phonon coupling between the drive mode and the radial breathing mode can be achieved thanks to their high optomechanical coupling rate g0/2π = 25 kHz, the mechanical resonance’s high quality factor Qm = ωmm = 4300, and diamond’s ability to support high intensity fields before the cavity exhibits heating instability and nonlinear absorption. In the measurements presented below, a drive field of approximately ncav = 0.9 × 106 photons (1.4 mW dropped power) is used to realize photon-enhanced Com ≈ 2.73. In all measurements, the probe laser power is sufficiently low for it to have no effect on the mechanical resonance dynamics.

To implement the feedback scheme, the RF component of the photodetected probe field signal is amplified, delayed, and input to an electro-optic modulator (EOM) that shifts the phase of the drive field. At the fiber taper input, the probe and drive fields are combined using a 90:10 fiber coupler, and at the output, they are separated using a wavelength division multiplexer (Montclair Fiber MFT-MC-51-30-AFC/AFC-1). The transmitted probe field is monitored using a fast photoreceiver (Thorlabs RXM25AA) whose output is filtered near ωm (passband 2.0–2.3 GHz, Mini Circuits VBFZ-2130-S+) and measured using an RF power detector (Mini Circuits ZX47-50LN-S+) to generate the signal input to the feedback circuit. Two amplifiers (Mini Circuits ZKL-33ULN-S+ and ZX60-83LN-S+) and a variable attenuator (Mini Circuits ZX73-2500-S+) realize a variable gain amplifier that varies f and a phase shifter (Mini Circuits JSPHS-2484+) that varies τ. The resulting signal is used to drive the EOM (EOSPACE PM-5S5-20-PFA-PFA-UV-UL).

Figure 3 shows the measured power spectral density with and without feedback when the device is excited into self-oscillation by the blue detuned drive laser. The area under the peak is a measure of the mechanical power, corresponding phonon occupation, and oscillation amplitude. An increase in mechanical power corresponding to n/n0 ≈ 1.5 is obtained when feedback is on and the feedback parameters are optimized. Figure 3(b) compares the measured phase noise for the self-oscillation when the feedback is on and off. As indicated, the phase noise is barely different in the two cases since the injected signal to the phase modulator is obtained from the oscillator’s displacement itself, which completes the feedback loop. No mechanism of phase locking is introduced in our system, and therefore, the phase noise in both cases is similar. Figures 3(c) and 3(d) show the dependence of n/n0 on these parameters, revealing a clear maxima at f ≈ 0.5 and φ ≈ 20°. A cooperativity of Com ≈ 2.73 is estimated for these measurements by referencing the drive field input power to its value at the onset of self-oscillations (corresponding to Com = 1). Note that for this value of Com, self-oscillation is possible for 0 ≤ f < 0.75.

FIG. 3.

Experimental demonstration of feedback enhanced optomechanical self-oscillation. (a) Power spectral density of the photodetected signal showing the self-oscillation of the microdisk radial breathing mode with and without electronic feedback turned on. The signal is measured from the probe laser transmission through the fiber taper waveguide when it is tuned near resonance with the probe cavity mode, and the drive laser is detuned by ωm from the drive mode. (b) Phase noise of the oscillator with and without feedback for frequency offsets between 1 kHz and 1 MHz. In (c) and (d), the dependence of the area under the self-oscillation resonance is plotted for varying f and φ, respectively, and compared with theoretical predictions. In each plot, the non-varying feedback parameter is fixed at its optimal value.

FIG. 3.

Experimental demonstration of feedback enhanced optomechanical self-oscillation. (a) Power spectral density of the photodetected signal showing the self-oscillation of the microdisk radial breathing mode with and without electronic feedback turned on. The signal is measured from the probe laser transmission through the fiber taper waveguide when it is tuned near resonance with the probe cavity mode, and the drive laser is detuned by ωm from the drive mode. (b) Phase noise of the oscillator with and without feedback for frequency offsets between 1 kHz and 1 MHz. In (c) and (d), the dependence of the area under the self-oscillation resonance is plotted for varying f and φ, respectively, and compared with theoretical predictions. In each plot, the non-varying feedback parameter is fixed at its optimal value.

Close modal

In addition, shown in Figs. 3(c) and 3(d), is the theoretically predicted dependence of n/n0 on the feedback parameters. Each of these plots is sliced through the parameter space shown in Fig. 2(a), with the non-varying feedback parameter fixed at its optimal setting. When the gain is varied in Fig. 1(b), we see that, as discussed in Sec. II, the number of phonons initially increases with increasing f, until reaching a maximum at f = 0.5. For larger f, the phonon number decreases, and for f > 0.75, we find that the number of coherently generated phonons becomes zero, meaning that there is no self-oscillation and that the effective cooperativity becomes less than one. When the feedback’s phase shift is varied, as shown in Fig. 1(c), we see that the enhancement decreases gradually on either side of its maxima. Comparing both sets of measurements with the theoretically predicted values of n/n0, we find excellent agreement, including the observed asymmetry of n/n0 with respect to φ, which is found to be due to the non-zero cavity linewidth κ.

The enhancement in phonon occupation demonstrated here is not limited by fundamental mechanisms. Rather, as discussed above, n is predicted to increase quadratically with Com, in contrast to its behavior in conventional phonon lasing, where n saturates due to dynamical back-action. Increasing Com beyond the value demonstrated here can be achieved most directly through operating at higher laser input power. However, in practice, this is limited by local heating of the diamond device, which can lead to thermal instability of the optical mode.37 Heating could potentially be reduced by using higher purity diamond samples, for example, by employing element six “quantum grade” material in place of the “optical grade” material used for the devices studied here. High input power can also damage the optical fiber taper waveguide; this limitation could be avoided by fabricating on-chip waveguides. Alternatively, Com can be increased by improving the device parameters. Reducing its mechanical dissipation is particularly desirable since the scattering processes governing the dependence of n on Com in Eq. (3) are not affected by Γm. For example, increasing Qm to 9000, as observed in Ref. 38 would immediately offer a further 2× increase in phonon occupation. Implementing the feedback scheme using optomechanical crystals,39 which have been demonstrated with Qm > 1010,40 would offer orders of magnitude higher enhancement.

The increase in mechanical self-oscillation amplitude accessible using this feedback scheme will enable more efficient optomechanical driving23 of electronic spin systems such as diamond nitrogen vacancy (NV) and silicon vacancy (SiV) color centers, enabling spin–optomechanical control to enter the regime of coherent spin-phonon coupling.41 Enhanced self-oscillation amplitude will also provide access to rich nonlinear optomechanical dynamics22 and assist in observing nonlinear nanomechanical effects42 of interest for applications, such as generating squeezed states.43 Implementing this scheme using high signal to noise homodyne detection may allow the excitation of thermal states into self-oscillation.

We wish to acknowledge the support from Alberta Innovates (Strategic Research Project), the Government of Alberta Major Innovation Fund, the Canadian Foundation for Innovation, the National Research Council (NanoInitiative Program), and the Natural Sciences and Engineering Research Council of Canada (Discovery Grant and Strategic Partnership Grant programs).

The authors have no conflicts to disclose.

Peyman Parsa: Conceptualization (supporting); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (supporting). Prasoon Kumar Shandilya: Conceptualization (supporting); Investigation (supporting); Project administration (equal); Software (supporting); Validation (equal); Visualization (supporting). David P. Lake: Conceptualization (lead); Resources (equal). Matthew E. Mitchell: Conceptualization (supporting); Resources (equal). Paul E. Barclay: Funding acquisition (lead); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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