Self-Sustained Oscillations of a Nonlinear Optomechanical System in the Low-Excitation Regime

Abstract

Manifesting across all time, mass and length scales, nonlinearities lie at the core of numerous physical phenomena. Next-generation quantum applications, such as quantum sensing, require the combination of nonlinearity with non-classical correlations. This necessitates the search for an experimental platform which enables a nonlinear response at ultra-low excitation levels in a system with practical sensing potential and quantum compatibility. Here, we report the observation and theoretical modeling of nonlinear dynamics in a mechanical system driven at the single-excitation level. We achieve this using a cavity-optomechanical platform with large single-photon coupling rates and a nonlinear microwave resonator. Specifically, the large Kerr nonlinearity of our superconducting microwave circuit reduces the threshold for the observation of nonlinear dynamics by four orders of magnitude, making this regime experimentally accessible at the few-photon level. The parameter-based quantitative predicative power of the theoretical description underlines our deep understanding of the physics involved and that this device concept paves the way for experiments with non-classical microwave drive schemes.

Figures (15)

• Figure 1: Schematic representation of the device featuring a superconducting flux-tunable $\lambda/4$ coplanar waveguide (CPW) resonator (orange), which is shunted to ground via a dc-SQUID. Signal input $s_\mathrm{in} \left( t \right)$ and output $s_\mathrm{out} \left( t \right)$ are achieved through a transmission line (blue). The dc-SQUID is partially suspended, forming two nanomechanical string resonators (only one of which is depicted here). The frequency of the microwave resonator is tunable via an out-of-plane magnetic field $B_\mathrm{oop}$. The application of an in-plane magnetic field $B_\mathrm{ip}$ induces an optomechanical coupling between the out-of-plane displacement of the nanostring and the CPW resonator. The sample is probed at a temperature of $70mK$ in a commercial dilution refrigerator. Details on the device fabrication and cryogenic wiring can be found in Sections \ref{['SI:sec:fab-layout']} and \ref{['SI:sec:mw-setup']} of the SI, respectively.
• Figure 2: Stability diagram for an optomechanical system as described by Eq. (\ref{['main:eq:hamilton']}) as a function of the normalized detuning of the probe frequency from the cavity resonance, $\Delta / \Omega_\mathrm{m}$, and the input photon flux $n_\mathrm{in}$ normalized with respect to the critical photon flux $n_{\mathrm{in,crit}}$ at which the optomechanical system becomes bistable. On the left, panels a and c show a linear system with $\mathcal{K}=0$, whereas in panels b, d, and e on the right, a Kerr nonlinearity of $\mathcal{K} / 2 \pi = 70kHz$ is assumed. For all panels with non-zero coupling, $g_0 = 4.69kHz$. $n_{\mathrm{in,crit}}$ is the same in all panels and calculated for the case of $\mathcal{K} = 70kHz$. It corresponds to an input power of $-120dBm$, where the cavity starts to bifurcate at an occupation of $\bar{n}_\mathrm{c,crit} = 19$. These parameter values correspond to parameter set III in the experiment (see table \ref{['tab:param-sets-main']}). The colors depict different stability regions with different amounts of (un)stable solutions. In region i, we find one unstable, in ii one stable, in iii one stable and two unstable, and in iv two stable and one unstable fixed point. Unstable regions are encircled by a dashed orange line. In b, the nonlinearity of the cavity causes a Kerr-enhanced instability, which extends the unstable region compared to a linear system (orange shaded area). Note that panel c extends a to vastly larger input powers. Only then one can observe multi-stability in a linear system, whereas our Kerr nonlinear cavity reduces the input power required to reach this regime by four orders of magnitude. Panel d is a zoom into the low power region of b, where the dashed horizontal lines represent the input powers displayed in Fig. \ref{['main:fig3:s21comp']}c,d. Panel e shows the phase diagram for a classical nonlinear duffing resonator with zero optomechanical interaction $g_0 = 0$.
• Figure 3: a, Measurement protocol in the time domain: A probe signal at a given frequency $\omega_{\mathrm{p},1}$ is applied sufficiently long for the mechanical oscillator to ring up and reach its steady state. A waiting period follows, which allows the excitation to ring down completely. We repeat this for different frequencies $\omega_{\mathrm{p},2}$, $\omega_{\mathrm{p},3}$, ... of the probe signal. The detuning $\Delta$ is calculated for each probe frequency with respect to the resonance frequency $\omega_\mathrm{c}$ of the cavity in the linear regime, i.e. $\Delta = \omega_{\mathrm{p},i} - \omega_\mathrm{c}$ for $i = 1,2,3,...$. b, Measurement protocol in the frequency domain: A single microwave probe/pump tone with frequency $\omega_\mathrm{p}$ is swept around the cavity resonance $\omega_\mathrm{c}$. We measure its complex scattering parameter $S_\mathrm{21} \propto s_\mathrm{out} / s_\mathrm{in}$. c and d, Scattering response $|S_{21}|$ observed in the experiment (purple dots), calculated with our analytical model (orange dots) and obtained from numerical simulation (gray lines) for parameter set III and an input power of $n_\mathrm{in} / n_{\mathrm{in,crit}} = 0.008$ and $1.10$ corresponding to $P_\mathrm{d} = -139.6dBm$ and $-118.6dBm$, respectively. The different stability regions are highlighted in the background by the color code introduced in Fig. \ref{['main:fig2:stability-dia']}. We see that both the analytical model and the simulation capture the nonlinear features, including the self-sustained oscillations observed as an additional absorption dip around $\Delta = 1$ in d.
• Figure 4: Power and parameter dependence of the cavity scattering response. The panels are labeled with capital Roman numbers according to the parameter sets, so that, e.g. the leftmost two columns II.a,b,c,d,e refer to parameter set II in table \ref{['tab:param-sets-main']}. The two color plots II.d,e compare simulation and experiment for a wide set of input powers. The linecuts in II.a,b,c show the experimental data (points) and numerical simulation (lines) for a single input power increasing from top to bottom. The powers are indicated by the dashed vertical lines in II.d,e, and correspond to $n_\mathrm{in} / n_{\mathrm{in,crit}} = 0.003$, $1.63$, and $4.09$, translating to $P_\mathrm{d} = -139.6dBm$, $-111.6dBm$, and $-107.6dBm$. For III.a,b,c, where both $g_0$ and $\mathcal{K}$ are increased, we show $n_\mathrm{in} / n_{\mathrm{in,crit}} = 0.008$, $1.10$, and $2.77$ ($P_\mathrm{d} = -139.6dBm$, $-118.6dBm$, and $-114.6dBm$). Since IV.a,b,c on the right has the largest single-photon coupling $g_0$ and Kerr nonlinearity $\mathcal{K}$, the lowest absolute input powers are required for the additional absorption dips to appear on the blue sideband. Input powers of $n_\mathrm{in} / n_{\mathrm{in,crit}} = 0.04$, $1.45$, and $2.88$ are depicted, which correspond to $P_\mathrm{d} = -139.6dBm$, $-123.6dBm$, and $-120.6dBm$. For all linecuts, the background is shaded according to the color code introduced with the stability diagram in Fig. \ref{['main:fig2:stability-dia']}.
• Figure 5: Cavity parameter determination.a, Flux tuning curve measured at $B_\mathrm{ip} = 30mT$. The four selected working points corresponding to parameter sets I to IV, which are used in the presented measurements, are marked with orange stars. Besides the flux-tunable microwave resonator, the resonance of a fixed frequency resonator coupled to the same feedline can be seen at around $7.274GHz$. We avoid this frequency range to ensure that this parasitic resonance does not affect the results. b, Power sweep performed at the top of the tuning curve, where $\omega_\mathrm{c}$ is maximal and $g_0 \approx 0$. In this regime, only the Kerr-induced shift of the cavity resonance towards lower frequencies is observed with increasing power. No additional absorption dips appear on the blue sideband of the cavity, as the optomechanical interaction is effectively turned off. We fit the power-induced frequency shift of the cavity resonance to obtain the Kerr nonlinearity $\mathcal{K}$. Note that this measurement was conducted at $B_\mathrm{ip} = 10mT$. c, Analytical calculation of the power-dependent scattering response based on the fit results from b. The apparent excellent agreement between theory and experiment validates the value of the Kerr nonlinearity $\mathcal{K}$ obtained by the fit.