  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.
