Dephasing of Solid-State Qubits at Optimal Points

Motivated by recent experiments with Josephson-junction circuits, we analyze the influence of various noise sources on the dynamics of two-level systems at optimal operation points where the linear coupling to low-frequency fluctuations is suppressed. We study the decoherence due to nonlinear (quadratic) coupling, focusing on the experimentally relevant $1/f$ and Ohmic noise power spectra. For $1/f$ noise strong higher-order effects influence the evolution.

Figure 1

Line shape ${\chi }_{{\sigma }_x}^{\prime \prime }(\omega )$, in units of $1/({\Gamma }_f{L}_{\Gamma })$ and as a function of $(\omega -\epsilon )/({\Gamma }_f{L}_{\Gamma })$. Equation (8) sets a singular universal shape, $\propto (\omega -\epsilon {)}^{-1/2}$. The high-frequency contribution (9, 12) washes out this singularity on scale ${\Gamma }_f$ ($L_{\Gamma }^{-1}$ in these units) and shifts the peak. The dashed line shows the result of the Gaussian approximation $F_2$ combined with the phase shift $F_1$.

Figure 2

(a) Linked-cluster expansion. The factor ${\delta }_t(\Delta \omega )=\sin (\Delta \omega /2)/(\pi \Delta \omega )$ at each vertex violates the frequency conservation. (b) High- and low frequency regions dominating the integration in the cluster diagrams.