Resonant Optomechanics with a Vibrating Carbon Nanotube and a Radio-Frequency Cavity

In an optomechanical setup, the coupling between cavity and resonator can be increased by tuning them to the same frequency. We study this interaction between a carbon nanotube resonator and a radio-frequency tank circuit acting as a cavity. In this resonant regime, the vacuum optomechanical coupling is enhanced by the dc voltage coupling the cavity and the mechanical resonator. Using the cavity to detect the nanotube’s motion, we observe and simulate interference between mechanical and electrical oscillations. We measure the mechanical ring down and show that further improvements to the system could enable the measurement of mechanical motion at the quantum limit.

Figure 1

(a) Experimental setup. A gated and contacted carbon nanotube resonator is connected to a radio-frequency tank circuit acting as a readout cavity. Nanotube motion is excited via a gate drive; the cavity can independently be probed directly via a directional coupler. The cavity response is detected via a cryogenic amplifier. The combined setup acts as a three-port rf device, with transmission measurements possible from ports 1 or 2 to port 3. Dotted capacitors indicate coupling between gates 4 and 5 and between these gates and the nanotube. dc voltages ${\overline{V_G}}_1$ to ${\overline{V_G}}_3$ are applied to gates 1–3. Bias tees (not shown) allow dc voltages ${\overline{V_G}}_4$ and ${\overline{V_G}}_5$ to be added to the rf signal on gates 4 and 5, respectively. (b) Transmission as a function of frequency with cavity tuning voltage $V_S=9\mathrm{V}$ (solid line) and $V_S=2\mathrm{V}$ (dotted line) for both direct and gate drive, showing a tuneable resonance. The different insertion losses are mainly due to different fixed attenuators in the two paths. (c) Numerically differentiated $dI/d{f}_d$ as a function of $f_d$ and $\overline{V_G}$. The gate drive power was $P=-33\mathrm{dBm}$. Inset shows $I$ over the same $\overline{V_G}$ range with $V_{\mathrm{sd}}=10\mathrm{mV}$ and with no rf excitation.

Figure 2

(a) Schematic pulsed excitation and measurement scheme showing gating signals $G_{\mathrm{in}}$ and $G_{\text{out}}$ together with simulated drive $\delta V_G$, displacement $u$, and cavity output voltage $V_{\text{out}}$. Durations of drive ($t_d$), wait ($t_w$), and acquisition ($t_r$) are indicated. Inset: Room-temperature gating scheme using rf switches. (b),(c) Time-averaged $\mathrm{\Delta }|S_{31}{|}^2$ measured using the scheme in (a), as a function of $\overline{V_G}$ and $f_d$ for two different values of $t_w$. Each column of data is averaged for 5 s. Measurement parameters were $V_S=9\mathrm{V}$, $V_{\mathrm{sd}}=2\mathrm{mV}$, $P=-38\mathrm{dBm}$, and $t_d=t_r=300\mathrm{ns}$. The jump at $\overline{V_G}\approx -6900\mathrm{mV}$ indicates an electrostatic switcher in the device. (d),(e) Simulated $\mathrm{\Delta }|S_{31}{|}^2$ (see text) reproducing panels (b) and (c), respectively.

Figure 3

(a) Numerically differentiated $dI/d{f}_d$ as a function of $f_d$ and $\overline{V_{\mathrm{TB}}}$ for $P=-38\mathrm{dBm}$ and $V_{\mathrm{sd}}=10\mathrm{mV}$. The inset shows $I$ for the same range of $\overline{V_{\mathrm{TB}}}$ and same $V_{\mathrm{sd}}$ with no microwaves applied. As $\overline{V_{\mathrm{TB}}}$ increases, the tunnel barriers are pinched off so that $I$ cannot be detected. (b) Measured $\mathrm{\Delta }|S_{31}{|}^2$ following the scheme in Fig. 2 and plotted as a function of $f_d$ and $\overline{V_{\mathrm{TB}}}$. The resonance is seen even with transport pinched off. The measurement parameters were $V_S=0.2\mathrm{V}$, $t_d=t_r=300\mathrm{ns}$, and $t_w\sim 15\mathrm{ns}$.

Figure 4

(a) Gating scheme using a rf switch to gate the drive and a digital sampling oscilloscope (DSO), triggered by the gating signal $G_{\mathrm{in}}$, to capture $V_{\text{out}}$. (b) Schematic of $G_{\mathrm{in}}$ and $V_{\text{out}}$. Durations of drive ($t_d$) and acquisition ($t_{\mathrm{\Delta }}$) are indicated. (c),(d) Averaged PSD for two fixed gate voltages marked in (e) and (f), respectively. Solid (dashed) curves correspond to $t_{\mathrm{\Delta }}=3.32\mu \mathrm{s}$ ($t_{\mathrm{\Delta }}=1.82\mu \mathrm{s}$). The measurement parameters were $V_{\mathrm{sd}}=10\mathrm{mV}$, $P=-38\mathrm{dBm}$, $t_d=330\mathrm{ns}$, and the time between drive pulses was 3660 ns. For (c), $f_d=315\mathrm{MHz}$ and $V_S=9\mathrm{V}$, while for (d), $f_d=273\mathrm{MHz}$ and $V_S=0.2\mathrm{V}$. Insets show averaged $V_{\text{out}}$ traces. The drive burst ends $\sim 50\mathrm{ns}$ after the beginning of these traces and acquisition begins $\sim 80\mathrm{ns}$ later (dashed lines). (e),(f) Measured PSD as a function of frequency and gate voltage for the two different device configurations studied, averaged over 65 500 repetitions at each gate voltage. A background due to purely electrical resonances was subtracted. The signal-to-noise ratio is better in (e) than in (f), presumably reflecting larger electrical coupling when the nanotube is in a more conducting state.