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 $S_Φ(f) = A^2/(f/1 \hbox{Hz})^α$, where $S_Φ$ is the flux noise spectral density, $A$ is of the order of 1 $μΦ_0 \, \hbox{Hz}^{-1/2}$ and $0.61 \leq α\leq 0.95$; $Φ_{0}$ 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 $τ_ϕ$ 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, $f_1 \leq f \leq f_2$, for all $α$ provided the ultraviolet cutoff frequency $f_2 > 1/τ_ϕ$. 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 $f_2$. We investigate the case in which $S_Φ(f_0)$ is fixed at the "pivot frequency" $f_0 \neq 1$ Hz while $α$ is varied, and find that the choice of $f_0$ 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.

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