Geometrical Effects in Noise Spectra of Superconducting Flux Qubits

We present theoretical study of geometrical effects related to spin diffusion in superconducting flux qubits. We adopt a model of a long superconducting wire surrounded by a thin oxide layer with spins distributed uniformly over cross-sectional area of the oxide layer. Using a continuous transformation from a round cylinder to a flat wire strip, we demonstrate that the noise spectral density tends to a power law $S(\omega)\propto (\omega/\Gamma)^{-s}$ with $s\agt 3/4$, approaching $s=3/4$ for very thin wires. The $\omega^{-s}$ dependence is valid in a broad frequency range above $\omega\agt\Gamma$ stretching up to four orders of magnitude in units of characteristic diffusion decay rate $\Gamma \sim 1- 10^2$ Hz. The effect is highly sensitive to a cross-sectional aspect ratio of a thin wire thus revealing its geometrical origin. We substantiate our findings by detailed comparison with available experimental data and conclude that $3/4$ power law distinguishes spin diffusion flux noise from generic ``$1/f$" family.