Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems

We study the spectral effect of the fluctuations of the vibration frequency. Such fluctuations play a major role in nanomechanical and other mesoscopic vibrational systems. We find that, for periodically driven systems, the interplay of the driving and frequency fluctuations results in specific spectral features. We present measurements on a carbon nanotube resonator and show that our theory allows not only the characterization of the frequency fluctuations but also the quantification of the decay rate without ring-down measurements. The results bear on identifying the decoherence of mesoscopic oscillators and on the general problem of resonance fluorescence and light scattering by oscillators.

Figure 1

Top: sketches of the power spectra of a driven linear oscillator $\mathrm{\Phi }(\omega )$. Panels (a) and (b) refer to large and small correlation time of the frequency noise $t_c$ compared to the oscillator relaxation time $t_r$, respectively, i.e., to narrow- and broadband frequency noise. The blue (lower) line shows the spectrum of thermal fluctuations in the absence of driving; it is centered at the oscillator eigenfrequency ${\omega }_0=⟨{\omega }_{\text{osc}}(t)⟩$. In the presence of driving there is added a $\delta$ peak at the driving frequency ${\omega }_F$. The green areas show the spectral features from the interplay of the driving and fluctuations of ${\omega }_{\text{osc}}(t)$. Bottom panels: ${\omega }_{\text{osc}}(t)$ for $t_c\gg t_r$ (a) and $t_c\ll t_r$ (b).

Figure 2

The power spectrum of the oscillator with a Gaussian frequency noise with the spectrum $\mathrm{\Xi }(\mathrm{\Omega })=2D{\lambda }^2/({\lambda }^2+{\mathrm{\Omega }}^2)$. The noise intensity is $D/\mathrm{\Gamma }=2$. Panels (a) and (b) show the full spectrum. The color coding is the same as in Fig. 1, $F^2/16{\mathrm{\Gamma }}^2=20k_{B}T$. Panel (c) shows the driving-induced term. The solid lines and dots show the analytic theory and simulations; the consecutive curves are shifted by 0.25 along the ordinate.

Figure 3

(a) The power spectrum of the fluctuating current $\delta I(t)$ through a driven carbon nanotube. The measurement bandwidth is 4.7 Hz. The eigenfrequency of the studied flexural mode is 6.3 MHz. The driving frequency is 100 Hz below the resonance frequency. The lower jagged line (blue) refers to the power spectrum without driving; the green area above it shows the driving-induced spectral change. This change is separated into the broad peak (darker filling), narrow peak (lighter filling), and a delta spike at the modulation frequency. This spike lies within three bins, within our experimental resolution, and is represented by the black vertical lines. The separation of the broad and narrow peaks is done by the straight line that interpolates the broad peak. Shown in the lower panels is the dependence of the lighter green area (b), the darker green area (c), and the area under the $\delta$ peak (d) on the squared amplitude of the modulating gate voltage; as expected from the theory, it is close to linear.

Figure 4

The driving-induced part of the spectrum of a nonlinear oscillator. The dots show the results of simulations. The red solid line shows the analytical results for ${\mathrm{\Phi }}_F(\omega )$ for small $\mathrm{\Delta }\omega /\mathrm{\Gamma }$ for the parameters of curve 1. The scaled values of the nonlinearity parameter, the detuning, and the driving strength on the curves 1 and 2 are, respectively, $\mathrm{\Delta }\omega /\mathrm{\Gamma }=0.125$ and 1.25, $({\omega }_F-{\omega }_0)/\mathrm{\Gamma }=0.5$ and 5, and $3\gamma F^2/32{\omega }_0^3({\omega }_F-{\omega }_0{)}^3=0.64$ and 0.01. The inset shows the full spectrum for the parameters of curve 2 (light green dots, simulations); the spectrum without driving for the same $\mathrm{\Delta }\omega /\mathrm{\Gamma }$ is shown by the solid line (analytical) and blue dots on top of it (simulations).