Thermodynamics of a single mesoscopic phononic mode

In recent decades, the laws of thermodynamics have been pushed down to smaller and smaller scales, within the theoretical field of stochastic thermodynamics and state-of-the-art experiments performed on microfabricated mesoscopic systems. These measurements concern thermal properties of electrons, photons, and mesoscopic mechanical objects. Here we report on the measurements of thermal fluctuations of a single mechanical mode in equilibrium with a heat reservoir. The device under study is a nanomechanical beam with a first flexural mode resonating at 3. 80. 16em{0ex}MHz, cooled down to temperatures in the range from 100 to 4000. 16em{0ex}mK. The technique is constructed around a microwave optomechanical setup using a cryogenic high electron mobility transistor, and is based on two parametric amplifications implemented in series: an in-built optomechanical ``blue detuned'' pumping plus a traveling wave parametric amplifier stage. We demonstrate our ability to resolve energy fluctuations of the mechanical mode in real time up to the fastest relevant speed given by the mechanical relaxation rate. The energy probability distribution is then exponential, matching the expected Boltzmann distribution. The variance of fluctuations is found to be ({k₁T) }^2 with no free parameters. Our microwave detection floor is about three times the standard quantum limit at 60. 16em{0ex}GHz; the resolution of our fastest acquisition tracks reached about 1000. 16em{0ex}phonons, and is directly related to the rather poor optomechanical coupling of the device (g₀/20. 50. 16em{0ex}Hz). This result is deeply in the classical regime, but shall be extended to the quantum case in the future with systems presenting a much larger g₀ (up to 22500. 16em{0ex}Hz), potentially reaching the resolution of a single mechanical quantum. We believe that it will open an experimental field, phonon-based quantum stochastic thermodynamics, with fundamental implications for quantum heat transport and macroscopic mechanical quantum coherence.