Detecting and characterizing frequency fluctuations of vibrational modes

We show how frequency fluctuations of a vibrational mode can be separated from other sources of phase noise. The method is based on the analysis of the time dependence of the complex amplitude of forced vibrations. The moments of the complex amplitude sensitively depend on the frequency noise statistics and its power spectrum. The analysis applies to classical and to quantum vibrations.

Figure 1

The scaled second moment of the complex amplitude of forced vibrations $u$ as a function of the frequency of the driving field, $\delta \omega ={\omega }_F-{\omega }_0$. The solid, short-dash and long-dash curves show the results for the Gaussian, Poisson, and telegraph noises, respectively. The left and right panels refer to the relative Gaussian noise intensities $D/\Gamma =0.1$ and 1; note the different scales. For the Poisson noise on these panels, we took $g=1$ and the same intensities as the Gaussian noise, $\nu g^2/2=D$. For the telegraph noise, we chose $W_{12}=W_{21}=D$ and the variance ${\xi }_c^2=2D^2$.

Figure 2

The real (upper panel) and imaginary (lower panel) parts of the normalized moments of the complex amplitude of forced vibrations. The squares, circles, and triangles show the results for the Gaussian, Poisson, and telegraph noises, respectively; the lines are guides for the eye. The data refer to $\delta \omega /\Gamma =-1$ and the same noise parameters as in the right panel of Fig. 1. Inset: dependence of the scaled complex amplitude Im $\langle u_F^{*}\rangle =(4{\omega }_F/F)\mathrm{Im}\langle u^{*}\rangle$ on the driving field frequency for the different types of frequency noise; the plotted quantity gives the oscillator absorption spectrum. The coding of the curves and the noise parameters are the same as in the main figure.

Figure 3

The real part and the absolute value of the normalized moments of the complex amplitude of forced vibrations at resonance, $\delta \omega =0$. The coding of the curves and the values of the noise parameters are the same as in Fig. 2.