Saturation of two-level systems and charge noise in Josephson junction qubits

We study the charge noise S_Q in Josephson qubits produced by fluctuating two-level systems (TLSs) with electric-dipole moments in the substrate. The TLSs are driven by an alternating electric field of angular frequency Ω and electric field intensity I. It is not widely appreciated that TLS in small qubits can easily be strongly saturated if I⪢I_c, where I_c is the critical electric field intensity. To investigate the effect of saturation on the charge noise, we express the noise spectral density in terms of density-matrix elements. To determine the dependence of the density-matrix elements on the ratio I/I_c, we find the steady-state solution for the density matrix using the Bloch-Redfield differential equations. We then obtain an expression for the spectral density of charge fluctuations as a function of frequency f and the ratio I/I_c. We find 1/f charge noise at low frequencies, and that the charge noise is white (constant) at high frequencies. Using a flat density of states, we find that TLS saturation has no effect on the charge noise at either high or low frequencies.