Mechanically Modulated Sideband and Squeezing Effects of Membrane Resonators

We investigate the sideband spectra of a driven nonlinear mode with its eigenfrequency being modulated at a low frequency ($<1\mathrm{kHz}$). This additional parametric modulation leads to prominent antiresonance line shapes in the sideband spectra, which can be controlled through the vibration state of the driven mode. We also establish a direct connection between the antiresonance frequency and the squeezing of thermal fluctuation in the system. Our Letter not only provides a simple and robust method for squeezing characterization, but also opens a new possibility toward sideband applications.

Figure 1

(a) Schematic drawing of the Si-N membrane resonator structures (cross section shown as inset) and the on-chip electromagnetic detection scheme. (b) The frequency response curve of the fundamental (1,1) mode (solid blue line) measured at $V_{\mathrm{exc}}=0.1\mathrm{V}$ fitted by the Duffing model (red dashed line). (c) The measured power spectrum of a two-tone experiment with $f_d=251\mathrm{kHz}$ and $V_{\mathrm{exc}}=0.5\mathrm{V}$, $f_p=380\mathrm{Hz}$ and $V_p=0.3\mathrm{V}$. (d) Schematic drawing of the measurement scheme and the frequency relation between the drive and probe tone and their response.

Figure 2

(a) Overlaid power spectra with a sweeping probe frequencies $f_p$ increasing from 10 to 1000 Hz in steps of 10 Hz and $V_p=0.5\mathrm{V}$ under a given drive $V_{\mathrm{exc}}=0.5\mathrm{V}$ and $f_d=251\mathrm{kHz}$. The color coding indicates different probe frequencies. The blue dashed line shows a parameter-free sideband calculation using Eq. (1). The coordinate $M(f_{\mathrm{ar}},P_{\mathrm{ar}})$ marks the calculated minimum point of the silent region. (b) The phases of the sideband captured under the same condition in (a) and the calculated phase are shown in blue dots and red-dashed line, respectively. (a) and (b) share the same frequency axis.

Figure 3

Theoretically calculated antiresonance frequencies $f_{\mathrm{ar}}$ and their power $P_{\mathrm{ar}}$ under different drive frequency $f_d$ and power $V_{\mathrm{exc}}$. The white spheres are the experimentally extracted $f_{\mathrm{ar}}$ at drive strength (0.5 and 1.0 V) and different fixed $f_d$ with an error bar of $\pm 10\mathrm{Hz}$. The lower panel shows line cuts for two fixed voltages comparing the model (lines) and the experimental results (dot).

Figure 4

(a) Theoretical sideband amplitudes driven by broadband noise (black) and a probe tone (red) with the same drive tone. (b) Calculated squeezing parameter $\varphi$ using $f_{\mathrm{ar}}$ under different drive conditions. (c) Detuning-dependent sidebands measured with $V_{\mathrm{exc}}=0.1\mathrm{V}$. (d) Drive-dependent sidebands measured at $f_d=252.5\mathrm{kHz}$. (e) The intensity area ratio between the left and the right sideband $I_{-}/I_{+}$ of (c). (f) The $I_{-}/I_{+}$ of (d). In both (c) and (d), the noise bandwidth is 300 kHz and $V_{\text{noise}}=0.4\mathrm{V}$. The theoretical results from our model are shown as red solid lines and the experiments are shown as blue dots, and they share the same $y$ axis. (c),(e) and (d),(f) pairwise share the same $x$ axis.