Oscillator state reconstruction via tunable qubit coupling in Markovian environments

We show that a parametrically coupled qubit can be used to fully reconstruct the quantum state of a harmonic oscillator even when both systems are subject to decoherence. By controlling the coupling strength of the qubit over time, the characteristic function of the oscillator at any phase-space point can be directly measured by combining the expectation values of two Pauli operators. The effect of decoherence can be filtered out from the measured data, provided a sufficient number of experimental runs are performed. In situations where full state reconstruction is not practical or not necessary, the method can still be used to estimate low-order moments of the mechanical quadratures. We also show that in the same framework it is possible to prepare superposition states of the oscillator. The model is very general but particularly appropriate for nanomechanical systems.