Figure 1

(Color online) Dispersion of the SWAP and $Z$ splittings for ${\omega }_c/\Omega =0.01$. Left: $\delta {\omega }_{21}/\Omega$ from numerical diagonalization of ${\mathcal{H}}_0+{\mathcal{H}}_{\text{I}}$ (top). Zoom around the origin highlights the interplay of second and fourth order terms, barrier height $\propto {\omega }_c^3$ (bottom). Right: comparative behavior of dispersions in the two subspaces. Top: SWAP exact splitting blue (gray) line, expansion (1) for $x_2=0$, $y_i=0$ dashed blue (dashed gray), second order expansion (dotted), $Z$ splitting (thick gray) $\delta {\omega }_{30}\simeq -\frac{\text{cos}\phi }{2}\left\{\left(y_1+y_2\right)+\left[1+\frac{1}{2}{\left(\frac{{\omega }_c}{\Omega }\right)}^2\right]\frac{x_1^2+x_2^2}{\Omega }\right\}$, and single-qubit dispersion (circles). Bottom: longitudinal dispersions. The $Z$ subspace (light gray) is much more sensitive both to transverse and longitudinal variations.Reuse & Permissions

Figure 2

(Color online) Envelope of the concurrence in the SPA, $\Omega ={10}^{11}\text{rad}/\text{s}$. To elucidate the significance of the optimal coupling scheme, we consider $1/f$ amplitudes larger than expected from single-qubit measurements (Ref. 19). Left: effect of transverse noise with ${\Sigma }_x/\Omega =0.08$ and ${\omega }_c/\Omega =0.01,0.02,0.06,0.08,0.1$ (dotted, dot-dashed, triangles, thick (gray) red, dashed). Light gray is the single-qubit coherence, $\left|{\rho }_{+-}\left(t\right)\right|={\left[1+\left({\Sigma }_x^{2}t/{\Omega }^2\right)\right]}^{-1/4}$ (Ref. 8). Inset: $C\left(t\right)$ and its envelope for ${\omega }_c/\Omega =0.02,0.08$. For optimal coupling ${\tilde{\omega }}_c\approx {\Sigma }_x$, at 3 ns, already 8-SWAP cycles occurred. Right: effect of transverse plus longitudinal noise on qubit 2, ${\Sigma }_{z_2}/\Omega =2.5\times {10}^{-3}$. Inset: effect of longitudinal noise, ${\omega }_c$ values as in left panel.Reuse & Permissions

Figure 3

(Color online) Concurrence for ${\omega }_c/\Omega =0.01$ blue (gray) line and for optimal coupling ${\tilde{\omega }}_c/\Omega =0.08$ (light gray). (a) Effect of high-frequency noise, $S_{x_i}\left(\omega \right)\approx 8\times {10}^5{\text{s}}^{-1}$, $S_{z_2}\left(\omega \right)\approx 4\times {10}^7{\text{s}}^{-1}⪢S_{z_1}\left({\omega }_c=0.08\Omega \right)\approx 6\times {10}^3{\text{s}}^{-1}$ (Ref. 23). Inset: at short times $C\left(t\right)\approx 2\left|\text{Im}\left\{{\rho }_{12}\left(t\right)\right\}\right|$ (diamonds). (b) Effect of $1/f$ noise (parameters as in Fig. 2) and white quantum noise. Inset: asymptotic behavior. Results are minimally modified considering the dynamics of fluctuators generating $1/f$ transverse (longitudinal) noise in ${\gamma }_m=1{\text{s}}^{-1}$, ${\gamma }_M={10}^6\left({10}^{10}\right){\text{s}}^{-1}$ (numerical solution of the stochastic Schrödinger equation).Reuse & Permissions

Figure 4

(Color online) Qubit 1 switching probability to $|-⟩$, $P^{\left(1\right)}\left(t\right)$, and probability $P^{\left(2\right)}\left(t\right)$ to find qubit 2 in the initial state $|-⟩$ in the presence of $1/f$ and white noise for ${\omega }_c/\Omega =0.01$ blue (gray) line and for optimal coupling ${\tilde{\omega }}_c={\Sigma }_x=0.08\Omega$ (light gray). (a) $P^{\left(1\right)}$: exponential short-time limit at ${\tilde{\omega }}_c$, algebraic otherwise. (b) $P^{\left(1\right)}$ and $P^{\left(2\right)}$ [dashed (gray) red] antiphase oscillations for ${\tilde{\omega }}_c$ (main), ${\omega }_c/\Omega =0.01$ (inset).Reuse & Permissions