Capacitive Coupling of Two Transmission Line Resonators Mediated by the Phonon Number of a Nanoelectromechanical Oscillator

Detection of quantum features in mechanical systems at the nanoscale constitutes a challenging task, given the weak interaction with other elements and the available technics. Here we describe how the interaction between two monomodal transmission-line resonators (TLRs) mediated by vibrations of a nano-electromechanical oscillator can be described. This scheme is then employed for quantum non-demolition detection of the number of phonons in the nano-electromechanical oscillator through a direct current measurement in the output of one of the TLRs. For that to be possible an undepleted field inside one of the TLR works as a amplifier for the interaction between the mechanical resonator and the remaining TLR. We also show how how the non-classical nature of this system can be used for generation of tripartite entanglement and conditioned mechanical coherent superposition states, which may be further explored for detection processes.