Generation of non-classical states of a harmonic oscillator by measurement using a two-level system

Generation of non-classical states of a harmonic oscillator is a topic of considerable fundamental interest and has practical implications to the development of sensors. We discuss the indirect measurement of a harmonic oscillator using a two-level system, or qubit. We analyze in detail a protocol which leads to the preparation of states with a reduced uncertainty in one of the quadratures. We also analyze relevant imperfections in this protocol, including oscillator energy relaxation, qubit dephasing, qubit measurement errors, and finite temperatures. We discuss the prospects for practical implementation of this protocol with mechanical or electrical resonators measured using superconducting qubits.