We present measurements of 1/f frequency noise in both linear andJosephson-junction-embedded superconducting aluminum resonators in the lowpower, low temperature regime - typical operating conditions forsuperconducting qubits. The addition of the Josephson junction does not resultin additional frequency noise, thereby placing an upper limit for fractionalcritical current fluctuations of $10^{-8}$ (Hz$^{-1/2}$) at 1 Hz forsub-micron, shadow evaporated junctions. These values imply a minimum dephasingtime for a superconducting qubit due to critical current noise of 40 -- 1400$\mu$s depending on qubit architecture. Occasionally, at temperatures above 50mK, we observe the activation of individual fluctuators which increase thelevel of noise significantly and exhibit Lorentzian spectra.
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