Pure dephasing in flux qubits due to flux noise with spectral density scaling as 1/fα

For many types of superconducting qubits, magnetic flux noise is a source of pure dephasing. Measurements on a representative dc superconducting quantum interference device (SQUID) over a range of temperatures show that S-Phi(f) = A(2)/(f/1 Hz)(alpha), where S-Phi is the flux noise spectral density, A is of the order of 1 mu Phi(0) Hz(-1/2), 0.61 <= alpha <= 0.95, and Phi(0) is the flux quantum. For a qubit with an energy level splitting linearly coupled to the applied flux, calculations of the dependence of the pure dephasing time tau(phi) of Ramsey and echo pulse sequences on alpha for fixed A show that tau(phi) decreases rapidly as alpha is reduced. We find that tau(phi) is relatively insensitive to the noise bandwidth, f(1) <= f <= f(2), for all alpha provided the ultraviolet cutoff frequency f(2) > 1/tau(phi). We calculate the ratio tau(phi,E)/tau(phi,R) of the echo (E) and Ramsey (R) sequences and the dependence of the decay function on alpha and f(2). We investigate the case in which S-Phi(f(0)) is fixed at the "pivot frequency" f(0) not equal 1 Hz while alpha is varied and find that the choice of f(0) can greatly influence the sensitivity of tau(phi,E) and tau(phi,R) to the value of alpha. Finally, we present calculated values of tau(phi) in a qubit corresponding to the values of A and alpha measured in our SQUID.