Quantum back-action and entanglement in microwave optomechanics*

Quantum mechanics sets a limit for the precision of continuous measurement of the position of an oscillator. Mechanical oscillators affected by radiation pressure forces allow to explore such quantum limits in measurement and amplification. An interesting setup for the purpose consists of superconducting microwave cavities coupled to a micromechanical vibrating membranes. We show how it is possible to measure an oscillator without quantum back-action of the measurement by constructing one effective oscillator from two physical oscillators. We realize such a quantum mechanics-free subsystem using two micromechanical oscillators, and show the measurements of two collective quadratures while evading the quantum back-action by 8 decibels on both of them, obtaining a total noise within a factor of 2 of the full quantum limit. By perturbing the measurement slightly, such measurements could be used to generate stabilized entanglement between two macroscopic mechanical oscillators. This prepares a canonical entangled state known as the two-mode squeezed state. It corresponds to the variances of collective position and momentum quadratures being reduced below the quantum zero-point fluctuations level. Moreover, our approach allows for full tomographic characterization of the prepared entangled state. We carry out this measurement, and verify the existence of entanglement in the steady state by direct access to fluctuations in all the collective motional quadratures.