Detecting and characterizing frequency fluctuations of vibrational modes

In recent years, there has been made significant progress in developing microand nanomechanical systems that display slowly decaying vibrations. For different types of such systems, the ratio of the vibration eigenfrequency to the decay rate, the quality factor Q, has reached 105.1–3 This has allowed studying new physics, including quantum phenomena,4,5 and opened a way for numerous applications, like highly-sensitive mass sensing6–8 and, potentially, high-accuracy nanomechanical clocks. In parallel, high-Q modes of superconducting cavities have been used for control and measurement of Josephson-junction based qubits.9 An important problem in the studies of nanomechanical vibrations and superconducting cavity modes is to understand the mechanisms of their decay and loss of coherence. Often one separates decay and fluctuations of the vibration amplitude and fluctuations of the vibration phase. Phase fluctuations are not only interesting on their own but are particularly important for applications, as they can impose limits on the sensitivity of a device. They can come from the thermal noise that accompanies vibration decay and is a consequence of coupling to a thermal reservoir. A more delicate and often more important source is fluctuations of the vibration frequency. They can have various origins, see Ref. 10 and papers cited therein, with recent examples being random attachment or detachment of molecules to a resonator that changes its mass,7,8,11–13 molecule diffusion along the resonator,14,15 coupling of the vibrational mode to two-state fluctuators,16 and, for nonlinear vibrations, frequency modulation by fluctuations of the vibration amplitude.17 In this paper, we suggest a simple way of separating and characterizing frequency fluctuations in vibrational systems. In two-level systems, frequency fluctuations lead to the difference between the T1 and T2 relaxation times and are routinely separated from decay using nonlinear response to an external field.18 In contrast, the response of linear vibrations is inherently linear, and the spectrum of the response remains a major source of information about the dynamics. If frequency fluctuations are the dominating factor, this spectrum reveals some of their features.10,11,13,19 However, in many cases of interest it does not provide enough information, and often does not allow one to even detect frequency fluctuations at all. For example, for broadband Gaussian frequency noise, the absorption spectrum is Lorentzian, as if there were no frequency noise, even though the overall width of the spectrum exceeds the width due to decay. We show below that frequency fluctuations can be studied by using, in a different way, essentially the same measurement as that used to find the absorption spectrum, i.e., by looking at the response of a resonantly modulated oscillator. This applies to both classical and quantum oscillators. The idea is to study such correlators of the quadrature and in-phase components of the oscillator displacement that are specifically sensitive to frequency fluctuations. As we show, these are correlators and moments of the complex vibration amplitude. They allow one not only to reveal frequency noise, but also to study its statistics, for both classical and quantum vibrations. The sensitivity to the noise statistics is illustrated for important examples of the noise.