  We report a direct measurement of the low-frequency noise spectrum in a
superconducting flux qubit. Our method uses the noise sensitivity of a
free-induction Ramsey interference experiment, comprising free evolution in the
presence of noise for a fixed period of time followed by single-shot
qubit-state measurement. Repeating this procedure enables Fourier-transform
noise spectroscopy with access to frequencies up to the achievable repetition
rate, a regime relevant to dephasing in ensemble-averaged time-domain
measurements such as Ramsey interferometry. Rotating the qubit's quantization
axis allows us to measure two types of noise: effective flux noise and
effective critical-current or charge noise. For both noise sources, we observe
that the very same 1/f-type power laws measured at considerably higher
frequencies (0.2-20 MHz) are consistent with the noise in the 0.01-100-Hz range
measured here. We find no evidence of temperature dependence of the noises over
65-200 mK, and also no evidence of time-domain correlations between the two
noises. These methods and results are pertinent to the dephasing of all
superconducting qubits.
