Frequency Comb from a Single Driven Nonlinear Nanomechanical Mode

Phononic frequency combs have attracted increasing attention both as a qualitatively new type of nonlinear phenomena in vibrational systems and from the point of view of applications. It is commonly believed that at least two modes must be involved in generating a comb. We demonstrate that a comb can be generated by a single nanomechanical mode driven by a resonant monochromatic drive. The comb emerges where the drive is still weak, so the anharmonic part of the mode potential energy remains small. We relate the experimental observation to a negative nonlinear friction induced by the resonant drive, which makes the vibrations at the drive frequency unstable. We directly map the measured trajectories of the emerging oscillations in the rotating frame and show how these oscillations lead to the frequency comb in the laboratory frame. The results go beyond nanomechanics and suggest a qualitatively new approach to generating tunable frequency combs in single-mode vibrational systems. They demonstrate new sides of the interplay of conservative and dissipative nonlinearities in driven systems.

Popular Summary

Optical frequency combs—laser light consisting of a series of narrow and equidistant spectral peaks—have revolutionized the field of metrology, ranging from unprecedentedly accurate timekeeping to molecular sensing and distance measurements. The same spectral pattern also is generated by nano- and micromechanical oscillators and is referred to as a phononic frequency comb. Irrespective of their implementation, all frequency combs to date emerge from two or more modes that are coupled by a nonlinear interaction, and it is commonly believed that at least two modes or a two-tone driving are required to generate a comb. Here, we demonstrate that a frequency comb can be generated by a single resonantly driven nanomechanical mode.

Our experiment shows that the comb results from the low-frequency oscillations of the amplitude and phase of the resonantly driven mode. Such oscillations are strongly nonlinear, with multiple overtones. These overtones modulate the forced vibrations and are seen as a frequency comb in the laboratory frame. The onset of the oscillations is a consequence of a negative nonlinear friction force induced by the driving, which overcomes the positive friction once the driving becomes sufficiently strong. The theory that accounts for this force and the experiment are in quantitative agreement.

The occurrence of a frequency comb in a single-mode system provides new insight into the nature of the combs and opens a qualitatively new route to generating them. The phenomenon is not limited to nanomechanics. It deserves further exploration and can lead to new technological applications.

Figure 1

Nanomechanical string resonator in the single-mode regime. (a) Scanning electron micrograph of the doubly clamped silicon nitride string resonator (green) and two control electrodes (yellow). (b) Linear response for a drive power of $-56\mathrm{dBm}$ along with a Lorentzian fit (solid red line). A constant noise background has been subtracted from the data ($1.5\times {10}^{-6}\mathrm{V}$). (c) Amplitude of forced vibrations for a moderately strong drive power of $-24\mathrm{dBm}$ as a function of drive frequency along with a fit (solid red line, see Appendix pp2). (d) Vibration spectra for a resonant drive applied at the OOP mode in the linear regime ($-46\mathrm{dBm}$, light blue), and for a stronger drive (5 dBm, purple) that exhibits an overtone at twice the drive frequency. Even under strong driving, no other modes are excited. A noise background has been subtracted from the data.

Figure 2

Frequency comb induced by resonant drive. (a) Power spectra of the mode measured for increasing drive power at $f_d=f_0$. The transition from the regime of two thermal-noise-induced satellite peaks to the frequency comb consisting of a series of equidistantly spaced, multiple satellite peaks is clearly seen. It occurs at ${\mathcal{P}}_{\mathrm{th}}=-2.5\mathrm{dBm}$ (black vertical line). The red dots show the theoretical model for ${\mathrm{H}}_1$. The inset shows the magnified region inside the dashed box. Arrows indicate drive power of the line cuts depicted in Fig. 3. (b) Extracted linewidth of the two thermal noise-induced satellite peaks ${\mathrm{H}}_1$ (black) and ${\mathrm{L}}_1$ (gray) as a function of the drive power. The dashed line represents the prediction based on the theory that disregards driving-induced friction, whereas the solid line includes it [see Eqs. (4) and (5)]. (c) Amplitude of forced vibrations at $f_d=f_0$ as a function of the drive power. The dashed line represents the prediction based on the theory that disregards driving-induced friction, whereas the solid line includes it [see Eqs. (4) and (5)].

Figure 3

Spectral vs homodyne measurement. (a) Frequency combs at drive power $-3\mathrm{dBm}$ (top), $-1\mathrm{dBm}$ (center), and 8 dBm (bottom). Data are taken from line cuts of Fig. 2 indicated by small arrows. (b) Trajectories in the rotating frame for the same drive powers as in (a). Data have been rescaled as described in Appendix pp2. Solid lines depict the theoretical model. (c) Amplitude of the satellites ${\mathrm{L}}_1$, ${\mathrm{H}}_1$, ${\mathrm{H}}_2$ as well as ${\mathrm{H}}_3$ and ${\mathrm{H}}_4$ (magnified in the inset) from Fig. 2 (black dots), compared to the Fourier components obtained from the theoretical model (red lines).

Figure 4

Frequency comb for detuned drive. (a) Power spectra measured for increasing detuning $f_d-f_0$ at a fixed drive power of 0 dBm. The dotted line represents the theoretical model for ${\mathrm{H}}_1$. The inset magnifies the region indicated by the dashed box. Arrows indicate drive power of the trajectories depicted in Fig. 5. (b) Frequency comb in line cut of diagram (a) taken at a detuning of 840 Hz (dark green arrow). (c) Amplitude of forced vibrations at $f_d$ as a function of detuning extracted from diagram (a). Black data points refer to the large-amplitude solution, whereas light purple indicates the small-amplitude solution for which there is no instability. The dashed line represents the prediction based on the theory that disregards driving-induced friction, whereas the solid line includes it; see Eqs. (4) and (5).

Figure 5

Nonsinusoidal trajectories for detuned drive. (a) Trajectories in the rotating frame for a detuning of 0 Hz (yellow), 250 Hz (olive), and 850 Hz (dark green) at a drive power of 0 dBm. Data have been rescaled as described in Appendix pp2. (b) Amplitude of the satellites ${\mathrm{H}}_1$, ${\mathrm{H}}_2$ (top) as well as ${\mathrm{L}}_1$, ${\mathrm{L}}_2$ (bottom) as a function of detuning from Fig. 4 (black dots), compared to the Fourier components obtained from the theory (red lines).

Figure 6

Mode spectra of the investigated sample hosting six working nanostring resonators. The applied drive powers are $-20\mathrm{dBm}$, $-10\mathrm{dBm}$, and $-4\mathrm{dBm}$, from top to bottom; the lowest two panels are both measured for a drive power of $-4\mathrm{dBm}$. The lowest-frequency mode at approximately 6.5 MHz is the OOP mode of the longest string discussed in this work. Identified modes are labeled by the string number (${\mathrm{S}}^I$ to ${\mathrm{S}}^{VI}$, from the longest to the shortest working nanostring), the mode polarization (OOP or IP for out-of-plane and in-plane modes), and the harmonic (1 to 4 from the fundamental up to the fourth harmonic). The label ${\mathrm{S}}_O^I$ refers to the second overtone at exactly twice the eigenfrequency of the OOP mode at 6.5 MHz. It does not correspond to an eigenmode of the system. The spurious mode at 19.56 MHz in the third panel remains unlabeled. Other modes in this panel are third harmonics, which, however, could not be unambiguously assigned to the eigenmodes of the six-string system.

Figure 7

Frequency response measurement in the RIFF regime (drive power 0 dBm), sweeping the drive frequency through the eigenfrequency of the OOP mode. The four lines show the response of the sample at the drive frequency, as well as at 2, 3, and 4 times its value.