  The quantum state of a superconducting qubit nonresonantly coupled to a
transmission line resonator can be determined by measuring the quadrature
amplitudes of an electromagnetic field transmitted through the resonator. We
present experiments in which we analyze in detail the dynamics of the
transmitted field as a function of the measurement frequency for both weak
continuous and pulsed measurements. We find excellent agreement between our
data and calculations based on a set of Bloch-type differential equations for
the cavity field derived from the dispersive Jaynes-Cummings Hamiltonian
including dissipation. We show that the measured system response can be used to
construct a measurement operator from which the qubit population can be
inferred accurately. Such a measurement operator can be used in tomographic
methods to reconstruct single and multiqubit states in ensemble-averaged
measurements.
