Feedback enhanced phonon lasing of a microwave frequency resonator Open Access

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 feedback^6–9 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 geometry^24–26 and delayed external feedback derived from the measured^27,28 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 (Γ_om > 0) or anti-damping (Γ_om < 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 Γ_om between the cavity field and the resonator is equal to the resonator’s internal dissipation rate Γ_m,³³ and in a linearized theory of optomechanics, this modifies the resonator’s phonon number from its thermal occupation n_th to n = n_thΓ_m/(Γ_m + Γ_om). This expression reveals the divergence of n and the onset of self-oscillation when Γ_om = −Γ_m, a regime that is most efficiently accessed when Δ = ω_m 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 δ_om(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 = −dω_o/dx and the oscillation amplitude A. This is illustrated in Fig. 1(a), which shows how a laser nominally blue detuned Δ = ω_m to maximize anti-damping is shifted away from this detuning by δ_om(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 δ_om(t) by modulating the input laser frequency.

Feedback manifests in the term (1 − fe^iφ) 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 C_om, allowing the influence of f and φ to be studied for a given κ/ω_m. Figure 2(a) shows the result of this calculation for κ/ω_m = 0.8 and C_om = 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 = ω_m, and that the input field strength α and coupling rate κ do not need to be specified in this analysis.

To further explore the feedback’s behavior, we solve Eq. (3) for varying cooperativity, optimizing the feedback parameters at each C_om to maximize n. Figure 2(b) shows the result. This clearly indicates that in the absence of feedback (f = 0), as C_om increases, which for a given set of cavity parameters corresponds to increasing the number of intracavity drive photons, n_cav, the resonator amplitude saturates. Conversely, when feedback is used, n grows as C_om increases. One can show that at the optimal point, A is linearly proportional to the cooperativity GA/ω_m ∝ C_om, with the proportionality constant to be maximized over f and φ. This is shown in Fig. 2(b), where $GA/ωm2$ scales quadratically with C_om. 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 C_om. We also note that self-oscillation is possible using feedback for C_om < 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/ω_m = 0.08 [Fig. 2(c)], corresponding to increasing Q_o or ω_m by an order of magnitude, the maximum phonon occupation is not significantly affected if C_om is kept constant. However, note that a higher Q_o allows for a higher C_om for a given n_cav. When C_om is increased to 20 while κ/ω_m 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 coupling^35,36 and spin–optomechanics.²³ This device supports a ω_m/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 λ_d = 1563.4 nm) has a quality factor Q_o,d = 1.1 × 10⁵ sufficiently high to place it near the sideband resolved regime, while the probe mode (λ_p = 1509.5 nm) has a lower Q_o,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 Q_m = ω_m/Γ_m = 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 n_cav = 0.9 × 10⁶ photons (1.4 mW dropped power) is used to realize photon-enhanced C_om ≈ 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 ω_m (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 C_om ≈ 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 C_om = 1). Note that for this value of C_om, self-oscillation is possible for 0 ≤ f < 0.75.

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 C_om, in contrast to its behavior in conventional phonon lasing, where n saturates due to dynamical back-action. Increasing C_om 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, C_om 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 C_om in Eq. (3) are not affected by Γ_m. For example, increasing Q_m 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 Q_m > 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.