Nanomechanical Measurement of the Thermal Fluctuation Spectrum of a Liquid

It is challenging to measure the fluctuating Langevin force, which gives rise to thermal fluctuations in a system. Here, we explore a novel approach for measuring the spectrum of thermal fluctuations in a liquid using a nanomechanical resonator. First, the nanomechanical resonator is coherently driven in the liquid at an amplitude above its thermal noise within the linear response regime. The drive force is provided by an electrothermal actuator, while the displacement is measured using optical interferometry. Separately, the position fluctuations of the resonator are measured. The spring constant, and hence the linear response function, of the resonator is determined from the equipartition theorem and the linear response of the resonator. Since the Langevin force excites the thermal fluctuations of the resonator in proportion to the square of the linear response function, it is then possible to extract the spectrum of the thermal force. We compare our experimental measurements with a theoretical description that assumes a long and thin beam fluctuating in a viscous fluid with a frequency dependent thermal driving force. We observe that the thermal force spectrum is a monotonically increasing function of frequency.