Low temperature 1/f noise in microwave dielectric constant of amorphous dielectrics in Josephson qubits

The accurate analytical solution for the low temperature $1/f$ noise in a microwave dielectric constant of amorphous films containing tunneling two-level systems (TLSs) is derived within the standard tunneling model including the weak dipolar or elastic TLS-TLS interactions. The results are consistent with the recent experimental investigations of $1/f$ noise in Josephson junction qubits including the power law increase of the noise amplitude with decreasing temperature at low temperatures $T<0.1$K. The long time correlations needed for $1/f$ noise are provided by the logarithmic broadening of TLS absorption resonances with time due to their interaction with neighboring TLSs. The noise behavior at higher temperatures $T>0.1$K and its possible sensitivity to quasi-particle excitations are discussed.