Quadrature readout and generation of squeezed states of a harmonic oscillator using a qubit-based indirect measurement

We present a protocol for measuring the quadrature of a harmonic oscillator (HO). The HO is coupled to a qubit, with an interaction modulated by the qubit control and effectively proportional to the HO quadrature $I$. Repeated measurement of the qubit leads to gradually increasing information on the quadrature $I$, leading to squeezing. We derive an analytical formula for the quadrature variance, $(\Delta I)^2 = 1/(1+4\phi^2 s)$, with $\phi$ the product of interaction strength and interaction time and $s$ the number of repetitions of the measurement. We discuss the robustness of this scheme against decoherence. We find that this protocol could lead to significant squeezing in a realistic setup formed of a superconducting flux qubit used to measure an electrical or mechanical resonator.