Evidence for Quantized Displacement in Macroscopic Nanomechanical Oscillators

We report the observation of discrete displacement of nanomechanical oscillators with gigahertz-range resonance frequencies at millikelvin temperatures. The oscillators are nanomachined single-crystal structures of silicon, designed to provide two distinct sets of coupled elements with very low and very high frequencies. With this novel design, femtometer-level displacement of the frequency-determining element is amplified into collective motion of the entire micron-sized structure. The observed discrete response possibly results from energy quantization at the onset of the quantum regime in these macroscopic nanomechanical oscillators.

Figure 1

(a) Scanning electron micrograph of the suspended antenna oscillator. It consists of a central Si beam, $10.7\mu \mathrm{m}$ long and 400 nm wide, as well as two arrays of 500 nm long and 200 nm wide paddles on both sides. The total thickness of the structure is 245 nm, comprised of the device layer of silicon (185 nm) and the thermally evaporated gold electrode (60 nm), colorized in yellow. (b) Modal simulation of the antenna structure, showing the low frequency ($f\simeq 10\mathrm{M}\mathrm{H}\mathrm{z}$) fundamental resonance mode. (c) In the high order collective mode, the paddles vibrate at their own natural frequency ($f\ge 1\mathrm{G}\mathrm{H}\mathrm{z}$), and the induced in-phase strain drives the central beam, which acts as an amplifier of the paddle motion.

Figure 2

Low order resonance mode of the antenna structure at 21 MHz with $Q=11000$. (a) The induced voltage $V_{emf}$ is an inverted Lorentzian peak on top of a noise background. Inversion results from extra phase in the coaxial cable impedance. Three representative sweeps show the resonance peak at 5, 6, and 8 T. (b) As expected from the magnetomotive scheme, Eq. (1), the peak height varies as $B^2$ for constant $I_{\mathrm{d}\mathrm{r}}$. (c) Plot of the linear power dependence of the resonance peak at 2.5 T field, on a log-log scale. The corresponding displacement $x$ is linear with driving force $F_{\mathrm{d}\mathrm{r}}$ for forces below 1.3 pN. Nonlinearity ensues for larger $F_{\mathrm{d}\mathrm{r}}$. The effective spring constant $k_{\mathrm{e}\mathrm{f}\mathrm{f}}=m{\omega }_0^2$ measured on resonance $\omega \to {\omega }_0$ is $330\mathrm{N}/\mathrm{m}$.

Figure 3

Classical response of the antenna structure in the 1.48 GHz collective high order mode with $Q=150$. At $T=1000\mathrm{m}\mathrm{K}$ the occupation factor is $N_{\mathrm{t}\mathrm{h}}\simeq 14$ for this mode. (a) Four plots (A, B, C, and D) of the induced voltage $V_{emf}$ show the main peak at 1.479 GHz and a smaller secondary peak at 1.49 GHz, for various values of the magnetic field $B$. The data are normalized to the noise background at 0 T. (b) In agreement with the magnetomotive expression, the plot of the peak height versus magnetic field $B$ shows quadratic dependence, entirely analogous to the 21 MHz plot. (c) Linear dependence of the induced power $P_{\mathrm{i}\mathrm{n}}$ on driving power $P_{\mathrm{d}\mathrm{r}}$ shows direct evidence of classical Hooke’s law oscillator behavior. The effective spring constant is $k_{\mathrm{e}\mathrm{f}\mathrm{f}}=188\mathrm{N}/\mathrm{m}$.

Figure 4

Nonclassical response in the 1.49 GHz collective mode at $T=110\mathrm{m}\mathrm{K}$. (a) Four discrete plots showing rapid peak growth within the narrow $B$ field range from 6.0 to 6.87 T, which corresponds to 45 to 52 pN of force. We observe a 0.6% frequency blueshift with decreasing temperature. (b) The peak growth is nonmonotonic, as seen in this separate continuous field sweep at the center frequency $f_0=1.49\mathrm{G}\mathrm{H}\mathrm{z}$. The peak undergoes several transitions with amplitude 500 nV between two well-defined states. The labels A, B, C, and D indicate the approximate location of peak growth in plot (a). (c) Two sweeps of the magnetic field, up (black) and down (blue), reproducing the same transition, with occasional transitions to a possible intermediate state. (d) With all parameters held constant, the system spontaneously decays in time, with no further changes observed within a measurement period of 2 h.