Pure dephasing in flux qubits due to flux noise with spectral density scaling as $1/f^{\alpha }$

Abstract

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 $S_{\Phi }(f)=A^2/(f/1Hz{)}^{\alpha }$, where $S_{\Phi }$ is the flux noise spectral density, $A$ is of the order of 1 $\mu {\Phi }_0{Hz}^{-1/2}$ and $0.61\le \alpha \le 0.95$; ${\Phi }_0$ 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 ${\tau }_{\varphi }$ of Ramsey and echo pulse sequences on $\alpha$ for fixed $A$ show that ${\tau }_{\varphi }$ decreases rapidly as $\alpha$ is reduced. We find that ${\tau }_{\varphi }$ is relatively insensitive to the noise bandwidth, $f_1\le f\le f_2$, for all $\alpha$ provided the ultraviolet cutoff frequency $f_2>1/{\tau }_{\varphi }$. We calculate the ratio ${\tau }_{\varphi ,E}/{\tau }_{\varphi ,R}$ 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" $f_0\ne 1$ Hz while $\alpha$ is varied, and find that the choice of $f_0$ can greatly influence the sensitivity of ${\tau }_{\varphi ,E}$ and ${\tau }_{\varphi ,R}$ to the value of $\alpha$. Finally, we present calculated values of ${\tau }_{\varphi }$ in a qubit corresponding to the values of $A$ and $\alpha$ measured in our SQUID.

7 pages, 8 figures, 1 table