Dynamical decoupling and noise spectroscopy with a superconducting flux qubit

We demonstrate dynamical decoupling in a superconducting flux qubit with a long energy-relaxation time, $T_1 = 12\,\mu$s. Low-frequency noise acts to dephase the qubit, reducing its transverse coherence time $T_2$. At the noise-optimal bias point we observe a free-induction decay time $T_2^* = 2.5\,\mu$s and $T_1$-limited spin-echo decay, $T_{2E} = 2\,T_1$. Biased away from this point, the increased sensitivity to flux noise leads to increased echo and free-induction decay rates. We moderate the dephasing effects of this noise by applying dynamical-decoupling sequences with up to 200 $\pi$-pulses. Using the CPMG sequence, we achieve a more than 50-fold enhanced decay time over $T_2^*$, and Gaussian pure-dephasing times $T_\varphi > 100\,\mu$s. We use the filtering property of this pulse sequence to facilitate spectroscopy of the environmental noise and reconstruct its $1/f$ power spectral density, which we independently confirm by a Rabi-spectroscopy approach. We characterize the noise sources coupling to the energy-bias and tunnel-coupling terms of the Hamiltonian.