  We analyze the dispersive readout of superconducting charge and flux qubits
as a quantum measurement process. The measurement oscillator frequency is
considered much lower than the qubit frequency. This regime is interesting
because large detuning allows for strong coupling between the measurement
oscillator and the signal transmission line, thus allowing for fast readout.
Due to the large detuning we may not use the rotating wave approximation in the
oscillator-qubit coupling. Instead we start from an approximation where the
qubit follows the oscillator adiabatically, and show that non-adiabatic
corrections are small. We find analytic expressions for the measurement time,
as well as for the back-action, both while measuring and in the off-state. The
quantum efficiency is found to be unity within our approximation, both for
charge and flux qubit readout.
