  In the dispersive regime of qubit-cavity coupling, classical cavity drive
populates the cavity, but leaves the qubit state unaffected. However, the
dispersive Hamiltonian is derived after both a frame transformation and an
approximation. Therefore, to connect to external experimental devices, the
inverse frame transformation from the dispersive frame back to the lab frame is
necessary. In this work, we show that in the lab frame the system is best
described by an entangled state known as the dressed coherent state, and thus
even in the dispersive regime, entanglement is generated between the qubit and
the cavity. Also, we show that further qubit evolution depends on both the
amplitude and phase of the dressed coherent state, and use the dressed coherent
state to calculate the measurement contrast of a recently developed dispersive
readout protocol.
