  We have spectroscopically measured the energy level separation of a
superconducting charge qubit coupled non-resonantly to a single mode of the
electromagnetic field of a superconducting on-chip resonator. The strong
coupling leads to large shifts in the energy levels of both the qubit and the
resonator in this circuit quantum electrodynamics system. The dispersive shift
of the resonator frequency is used to non-destructively determine the qubit
state and to map out the dependence of its energy levels on the bias
parameters. The measurement induces an ac-Stark shift of 0.6 MHz per photon in
the qubit level separation. Fluctuations in the photon number (shot noise)
induce level fluctuations in the qubit leading to dephasing which is the
characteristic back-action of the measurement. A cross-over from lorentzian to
gaussian line shape with increasing measurement power is observed and
theoretically explained. For weak measurement a long intrinsic dephasing time
of T_2 > 200 ns of the qubit is found.
