Dephasing of a superconducting flux qubit

In order to gain a better understanding of the origin of decoherence in superconducting flux qubits, we have measured the magnetic field dependence of the characteristic energy relaxation time ($T_1$) and echo phase relaxation time ($T_2^{\rm echo}$) near the optimal operating point of a flux qubit. We have measured $T_2^{\rm echo}$ by means of the phase cycling method. At the optimal point, we found the relation $T_2^{\rm echo}\approx 2T_1$. This means that the echo decay time is {\it limited by the energy relaxation} ($T_1$ process). Moving away from the optimal point, we observe a {\it linear} increase of the phase relaxation rate ($1/T_{2}^{\rm echo}$) with the applied external magnetic flux. This behavior can be well explained by the influence of magnetic flux noise with a $1/f$ spectrum on the qubit.