  We present a systematic study of the phase-coherent dynamics of a
superconducting three-Josephson-junction flux qubit. The qubit state is
detected with the integrated-pulse method, which is a variant of the pulsed
switching DC SQUID method. In this scheme the DC SQUID bias current pulse is
applied via a capacitor instead of a resistor, giving rise to a narrow
band-pass instead of a pure low-pass filter configuration of the
electromagnetic environment. Measuring one and the same qubit with both setups
allows a direct comparison. With the capacitive method about four times faster
switching pulses and an increased visibility are achieved. Furthermore, the
deliberate engineering of the electromagnetic environment, which minimizes the
noise due to the bias circuit, is facilitated. Right at the degeneracy point
the qubit coherence is limited by energy relaxation. We find two main noise
contributions. White noise is limiting the energy relaxation and contributing
to the dephasing far from the degeneracy point. 1/f-noise is the dominant
source of dephasing in the direct vicinity of the optimal point. The influence
of 1/f-noise is also supported by non-random beatings in the Ramsey and spin
echo decay traces. Numeric simulations of a coupled qubit-oscillator system
indicate that these beatings are due to the resonant interaction of the qubit
with at least one point-like fluctuator, coupled especially strongly to the
qubit.
