Decoherence in Josephson-junction qubits due to critical-current fluctuations

We compute the decoherence caused by $1∕f$ fluctuations at low frequency $f$ in the critical current $I_0$ of Josephson junctions incorporated into flux, phase, charge, and hybrid flux-charge superconducting quantum bits (qubits). The dephasing time ${\tau }_{\varphi }$ scales as $I_0∕\Omega \Lambda S_{I_0}^{1∕2}\left(1\mathrm{Hz}\right)$, where $\Omega ∕2\pi$ is the energy-level splitting frequency, $S_{I_0}\left(1\mathrm{Hz}\right)$ is the spectral density of the critical-current noise at $1\mathrm{Hz}$, and $\Lambda \equiv \mid I_{0}d\Omega ∕\Omega d{I}_0\mid$ is a parameter computed for given parameters for each type of qubit that specifies the sensitivity of the level splitting to critical-current fluctuations. Computer simulations show that the envelope of the coherent oscillations of any qubit after time $t$ scales as $\exp (-t^2∕2{\tau }_{\varphi }^2)$ when the dephasing due to critical-current noise dominates the dephasing from all sources of dissipation. We compile published results for fluctuations in the critical current of Josephson tunnel junctions fabricated with different technologies and a wide range in $I_0$ and area $\mathcal{A}$, and show that their values of $S_{I_0}\left(1\mathrm{Hz}\right)$ scale to within a factor of 3 of $[144{(I_0∕\mu \mathrm{A})}^2∕(\mathcal{A}∕\mu {\mathrm{m}}^2)]{\left(\mathrm{pA}\right)}^2∕\mathrm{Hz}$ at $4.2\mathrm{K}$. We empirically extrapolate $S_{I_0}^{1∕2}\left(1\mathrm{Hz}\right)$ to lower temperatures using a scaling $T\left(\mathrm{K}\right)∕4.2$. Using this result, we find that the predicted values of ${\tau }_{\varphi }$ at $100\mathrm{mK}$ range from $0.8\text{to}12\mu \mathrm{s}$, and are usually substantially longer than values measured experimentally at lower temperatures.