Pure dephasing in flux qubits due to flux noise with spectral density scaling as 1/ f^α

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_Φ(f) = A²/\(f/1 hbox{Hz}\)^α, where S_Φ is the flux noise spectral density, A is of the order of 1 μΦ₀ hbox{Hz}^-1/2 and 0.61 leq αleq 0.95; Φ₀ 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 τ_φ of Ramsey and echo pulse sequences on α for fixed A show that τ_φ decreases rapidly as α is reduced. We find that τ_φ is relatively insensitive to the noise bandwidth, f₁ leq f leq f₂, for all α provided the ultraviolet cutoff frequency f₂ > 1/τ_φ. We calculate the ratio τ_φ,E / τ_φ,R of the echo (E) and Ramsey (R) sequences, and the dependence of the decay function on α and f₂. We investigate the case in which S_Φ\(f₀\) is fixed at the "pivot frequency" f₀ neq 1 Hz while α is varied, and find that the choice of f₀ can greatly influence the sensitivity of τ_φ,E and τ_φ,R to the value of α. Finally, we present calculated values of τ_φ in a qubit corresponding to the values of A and α measured in our SQUID.