Effect of Two Level System Saturation on Charge Noise in Josephson Junction Qubits

We show that charge noise $S_Q$ in Josephson qubits can be produced by fluctuating two level systems (TLS) with electric dipole moments in the substrate using a flat density of states. At high frequencies the frequency and temperature dependence of the charge noise depends on the ratio $J\mathrm{/}{J}_c$ of the electromagnetic flux $J$ to the critical flux $J_c$. It is not widely appreciated that TLS in small qubits can easily be strongly saturated with $J\mathrm{/}{J}_c\gg 1$. Our results are consistent with experimental conclusions that $S_Q\sim 1\mathrm{/}f$ at low frequencies and $S_Q\sim f$ at high frequencies.