Fast Amplification and Rephasing of Entangled Cat States in a Qubit-Oscillator System

We study a qubit-oscillator system, with a time-dependent coupling coefficient, and present a scheme for generating entangled Schr\"odinger-cat states with large mean photon numbers and also a scheme that protects the cat states against dephasing caused by the nonlinearity in the system. We focus on the case where the qubit frequency is small compared to the oscillator frequency. We first present the exact quantum state evolution in the limit of infinitesimal qubit frequency. We then analyze the first-order effect of the nonzero qubit frequency. Our scheme works for a wide range of coupling strength values, including the recently achieved deep-strong-coupling regime.