Qubit dephasing due to quasiparticle tunneling

We study dephasing of a superconducting qubit due to quasiparticle tunneling through a Josephson junction. While qubit decay due to tunneling processes is well understood within a golden rule approximation, pure dephasing due to BCS quasiparticles gives rise to a divergent golden rule rate. We calculate qubit dephasing due to quasiparticle tunneling beyond lowest-order approximation in coupling between qubit and quasiparticles. Summing up a certain class of diagrams, we show that qubit dephasing due to purely longitudinal coupling to quasiparticles leads to dephasing $\sim exp[-x(t\left)\right]$ where $x\left(t\right)\propto t^{3/2}$ for short-time scales and $x\left(t\right)\propto tln\left(t\right)$ for long-time scales.

Figure 1

Here we sketch the different time scales discussed in the text. We plot the weighting function (39), $g(x,t)={\text{sinc}}^2(\omega t/2)$, for long, intermediate, and short times in comparison with an effective Boltzmann distribution (dark brown line). For short times (blue line), the weighting function remains almost constant for all frequencies with significant quasiparticle density. Hence, all quasiparticles present in the system contribute with equal weight to dephasing. On the other hand, for long times the weighting function decays in a narrow region and we can approximate the distribution function with a constant in this region.

Figure 2

Normalized non-Markovian dephasing function over time $[x_2\left(t\right)/t]$ versus $t$ for different interference angles $\vartheta =0$ (blue, increasing with time) and $0.2\pi$ (red, decreasing with time). The function is calculated using (a) an effective Boltzmann distribution with $k_{B}T=0.1\Delta \approx 230\text{mK}$ and (b) an effective Fermi distribution with $k_{B}T={10}^{-3}\Delta \approx 2.3\text{mK}$ and $\mu =0.992\Delta$. Both distribution functions yield a quasiparticle density $x_{qp}\sim \mathcal{O}\left({10}^{-5}\right)$. Thin horizontal lines are the corresponding self-consistent rates ${\Gamma }_{2^{*}}\Delta$, while dashed lines are obtained with the analytical approximation (46) for short times. Dashed dotted lines show analytical long-time behavior from Eq. (50) and are in good agreement with the numerical long-time behavior. $\vartheta =0$ describes the special case without divergence and the long-time behavior follows the self-consistent rate ${\Gamma }_{2^{*},0}$. For the narrow Fermi distribution, the short-time approximation shows good agreement with the numerical data for a large time range and the long-time behavior sets in rather late. On the other hand, for the Boltzmann distribution the short-time behavior shows discrepancy between analytical and numerical results for the times plotted, while the long-time behavior sets in early.

Figure 3

Normalized dephasing function $x\left(t\right)/t$ for increasing Dynes parameters [${\gamma }_D/\Delta =2$ (red line) to ${\gamma }_D/\Delta =8$ (green line) for two different interference angles, $\vartheta =0$ (dashed lines) and $\vartheta =0.2\pi$ (solid lines)]. Time in units of Dynes parameter. Numerical data show that for ${\gamma }_{D}t\gtrsim 1$ a crossover from $x\left(t\right)\sim tlnt$ behavior to linear dephasing $x\left(t\right)\sim t$ occurs.

Figure 4

Fourth-order diagrams (a) two-vertex loop type and (b) different kind of diagram.