Intermittency in an optomechanical cavity near a subcritical Hopf bifurcation

We experimentally study an optomechanical cavity consisting of an oscillating mechanical resonator embedded in a superconducting microwave transmission line cavity. Tunable optomechanical coupling between the mechanical resonator and the microwave cavity is introduced by positioning a niobium-coated single-mode optical fiber above the mechanical resonator. The capacitance between the mechanical resonator and the coated fiber gives rise to optomechanical coupling, which can be controlled by varying the fiber-resonator distance. We study radiation-pressure-induced self-excited oscillation as a function of microwave driving parameters (frequency and power). Intermittency between limit-cycle and steady-state behaviors is observed with blue-detuned driving frequency. The experimental results are accounted for by a model that takes into account the Duffing-like nonlinearity of the microwave cavity. A stability analysis reveals a subcritical Hopf bifurcation near the region where intermittency is observed.

Figure 1

(a) Experimental setup. The microwave cavity is a microstrip made of aluminum over high-resistivity silicon wafer coated with a 100-nm-thick SiN layer. The mechanical resonator at the end of the microstrip is a suspended trampoline supported by four beams. (b) Electron micrograph of the trampoline. The optomechanical coupling is generated using a niobium-coated optical fiber that is positioned at submicron distance from the trampoline. (c) Several windows are opened in the niobium layer on the fiber tip using FIB. The optical setup (seen above the sample) allows using the optical fiber for displacement detection, whereas the microwave setup (seen below the sample) is employed for measuring the cavity response. The coated fiber is galvanically connected to both ac and dc voltage sources with a bias-T (not seen in the sketch), which can be used to externally actuate the mechanical resonator. The sample is mounted inside a closed copper package, which is internally coated with niobium. Measurements are performed in a dilution refrigerator operating at a temperature of $0.28\mathrm{K}$ and in vacuum. (d) The measured (solid) and calculated (dashed) cavity resonance frequency $f_0$ vs fiber-trampoline distance $\Delta z$.

Figure 2

Self-excited oscillation. The reflected power at angular frequency ${\omega }_{\mathrm{p}}-{\omega }_b$ for (a) backward and (b) forward frequency sweeps. The pump critical power is $P_{\mathrm{c}}=-19.5$ dB m, and the pump critical detuning is ${\Delta }_{\mathrm{c}}=-2\pi \times 0.7\mathrm{MHz}$. (c) The reflected power at angular frequency ${\omega }_{\mathrm{p}}$. The solid black, dotted cyan, and dotted green lines represent, respectively, supercritical Hopf (labeled $H_{-}$), subcritical Hopf (labeled $H_{+}$), and limit point of cycle bifurcations (labeled $\mathrm{LPC}$). The following cavity parameters are employed for the bifurcation calculation: $K_a/{\omega }_a=-1.44\times {10}^{-15}$ and ${\gamma }_{a3}/K_a =0.04$.

Figure 3

Intermittency. (a) The measurement of the reflected signal using a spectrum analyzer. (b) and (c) The time traces measured using an oscilloscope for two values of $f_{\mathrm{p}}$, which are indicated in (a) by the two dashed lines. The solid white line in (a) shows the reflected power at the pump frequency $f_{\mathrm{p}}.$ Noise-induced transitions between a limit cycle and a fixed point are seen in (b). (d) The projection of the limit cycle (red line) and the fixed point (blue cross) on the complex $A_a$ plane.