Optomechanical Platform with a Three-dimensional Waveguide Cavity

At low temperatures, microwave cavities are often preferred for the readout and control of a variety of systems. In this paper, we present design and measurements on an optomechanical device based on a three-dimensional rectangular waveguide cavity. We show that by suitably modifying the electromagnetic field corresponding to the fundamental mode of the cavity, the equivalent circuit capacitance can be reduced to 29 fF. By coupling a mechanical resonator to the modified electromagnetic mode of the cavity, we achieve a capacitance participation ratio of $43\mathrm{\%}$. We demonstrate an optomechanical cooperativity, $C\sim$ 40, characterized by performing measurements in the optomechanically induced absorption limit. In addition, due to a low-impedance environment between the two halves of the cavity, our design has the flexibility of incorporating a dc bias across the mechanical resonator, often a desired feature in tunable optomechanical devices.

Figure 1

Simulation and design: (a) An intensity colormap plot of the magnitude of charge current flowing on the surface of a rectangular waveguide cavity for the fundamental mode. White (black) represents the maximum (minimum) current. Arrows overlaid on the surface show the direction of charge current at some instance of time. (b) Schematic representation of the simulation model, showing a silicon chip placed inside the cavity. The chip contains two electrodes running from the center of the top and bottom surfaces of the cavity and a lumped port at the center. (c) The equivalent model for environmental admittance ${\tilde{Y}}_p(\omega )$ seen by a port placed between the two connecting electrodes. The additional capacitance $C_m$ represents the contribution from the mechanical resonator. On the right side, decomposition of this lumped-distributive mode into equivalent circuit elements, $C_p$ and $L_p$. (d) Plots of the imaginary part of admittance as seen by the mechanical capacitor and total admittance defined as $Y_t(\omega )=Y_p(\omega )+\omega C_m$. Each zero crossing with a positive slope corresponds to a resonant mode.

Figure 2

Device: (a) Image of a transmission cavity used to capacitively couple a mechanical resonator to the electromagnetic mode. Two M3 screws are shown to highlight the relative size. The right subpanel shows the complete assembly with SMA connectors. (b) False-color scanning electron microscope image of one of the drumhead resonators fabricated with aluminum on the intrinsic $\mathrm{Si}$ substrate. The scale bar corresponds to 10 $\mu \mathrm{m}$. The drumhead has a diameter of 22 $\mu \mathrm{m}$, and is suspended approximately $300$ nm above the bottom plate. (c) Measurement of the transmission parameter of the optomechanical cavity at 20 mK (blue curve), along with the fitted curve (red) yielding a total dissipation rate $\kappa =2\pi \times 481$ kHz as indicated by the arrows.

Figure 3

(a) Cavity transmission in the presence of a strong, red-detuned pump signal at ${\omega }_c-{\omega }_m$ and a weak probe signal at ${\omega }_c$. (b) The colormap indicates variation in linewidth of the absorption feature at ${\omega }_c$ with drive (pump) power. (c)–(e) Plots of cooperativity as a function of injected pump photons into the cavity. (c),(d) are measured on device 1 that consisted of two mechanical resonators coupled to a 7 $\times$ 4 $\times$ 7 ${\mathrm{mm}}^3$ cavity. (e) is obtained for a single mechanical resonator (device 2) placed inside a 28 $\times$ 6 $\times$ 28 ${\mathrm{mm}}^3$ cavity. The highest values of cooperativity measured on the two devices differ due to the dynamic range of the two cavities.

Figure 4

(a) Representative image showing the assembly of a dc-enabled optomechanical cavity, along with a schematic diagram. The two halves of the cavity are electrically isolated by a thin insulating spacer (thickness approximately 10 $\mu \mathrm{m}$). A bias tee allows a low-frequency rf signal to be added and a dc voltage to be applied across the mechanical resonator. (b) Response of the mechanical oscillator while directly driving it with a low-frequency rf signal. (c) Color plot of the demodulated signal with dc gate voltage and rf drive frequency, showing the tunability of the resonant frequency of the mechanical resonator.