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Temperature Measurement and Phonon Number Statistics of a Nanoelectromechanical Resonator

Measuring thermodynamic quantities can be easy or not, depending on the system that is being studied. For a macroscopic object, measuring temperatures can be as simple as measuring how much a column of mercury rises when in contact with the object. At the small scale of quantum electromechanical systems, such simple methods are not available and invariably detection processes disturb the system state. Here we propose a method for measuring the temperature on a suspended semiconductor membrane clamped at both ends. In this method, the membrane is mediating a capacitive coupling between two transmission line resonators (TLR). The first TLR has a strong dispersion, that is, its decaying rate is larger than its drive, and its role is to pump in a pulsed way the interaction between the membrane and the second TLR. By averaging the pulsed measurements of the quadrature of the second TLR we show how the temperature of the membrane can be determined. Moreover the statistical description of the state of the membrane, which is directly accessed in this approach is significantly improved by the addition of a Josephson Junction coupled to the second TLR.