Quantitative calibration of a traveling-wave parametric amplifier applied to an optomechanical platform

In the past decade, the microwave quantum electronics toolbox has been enriched with quantum limited detection devices such as traveling-wave parametric amplifiers (TWPAs). The extreme sensitivity that they provide is not only mandatory for some physics applications within quantum information processing, but is also the key element that will determine the detection limit of quantum sensing setups. In the framework of microwave optomechanical systems, an unprecedented range of small motions and forces is accessible, for which a specific quantitative calibration becomes necessary. We report on near quantum limited measurements performed with an aluminum drumhead mechanical device within the temperature range 4--400 mK. The whole setup is carefully calibrated, especially taking into account the power dependence of microwave absorption in the superconducting optomechanical cavity. This effect is commonly attributed to two-level systems (TLSs) present in the metal oxide. We demonstrate that a similar feature exists in the TWPA, and can be phenomenologically fitted with adapted expressions. If not taken into account, the error on the signal strength can be as large as a factor of about 2, which is unacceptable for quantitative experiments. The power and temperature dependence is studied over the full parameter range, leading to an absolute definition of the phonon population (i.e., Brownian-motion amplitude), with an uncertainty $\ifmmode\pm\else\textpm\fi{}20\mathrm{%}$ limited by sources of noise internal to the optomechanical element.