Hyperfine interaction induced dephasing of coupled spin qubits in semiconductor double quantum dots

We investigate theoretically the hyperfine-induced dephasing of two-electron-spin states in a double quantum dot with a finite singlet-triplet splitting $J$. In particular, we derive an effective pure dephasing Hamiltonian, which is valid when the hyperfine-induced mixing is suppressed due to the relatively large $J$ and the external magnetic field. Using both a quantum theory based on resummation of ring diagrams and semiclassical methods, we identify the dominant dephasing processes in regimes defined by values of the external magnetic field, the singlet-triplet splitting, and inhomogeneity in the total effective magnetic field. We address both free induction and Hahn echo decay of superposition of singlet and unpolarized triplet states (both cases are relevant for singlet-triplet qubits realized in double quantum dots). We also study hyperfine-induced exchange gate errors for two single-spin qubits. Results for III-V semiconductors as well as silicon-based quantum dots are presented.

Figure 1

The virtual transitions that lead to the second-order effective interactions ${\widehat{V}}_{SS}$ and ${\widehat{V}}_{S{T}_0}$, as described in text.

Figure 2

The time scale $T_{BC}$ of $S$-$T_0$ coherence decay due to $J^z$-linear terms [see Eq. (45)] for (a) GaAs DQD with $N_L=N_R={10}^6$ at $B=200$ mT ($|{\mu }_{\text{T}}|\approx 5.0\mu$eV), and (b) DQD based on natural silicon with $N_L=N_R={10}^5$ at $B=30$ mT ($|{\mu }_{\text{T}}|\approx 3.5\mu$eV). The regions in which ${\sigma }_{\perp }\ll |J\pm {\mu }_{\text{T}}|,|{\mu }_{\text{T}}|$ is not fulfilled are hatched.

Figure 3

The time scale $T_A$ of $S$-$T_0$ coherence decay. (a) GaAs DQD with $N_L=N_R={10}^6$ at $B=200$ mT ($|{\mu }_{\text{T}}|\approx 5.0\mu$eV), and (b) natural Si DQD with $N_L=N_R={10}^5$ at $B=30$ mT ($|{\mu }_{\text{T}}|\approx 3.5\mu$eV). In both cases, the effective field gradient ${\theta }_{\mathrm{T}}=0$. The hatched regions are where ${\sigma }_{\perp }\ll |J\pm {\mu }_{\text{T}}|,|{\mu }_{\text{T}}|$ is not fulfilled. No narrowing is assumed ($n_F=1$).

Figure 4

The time scale $T_{SS}$ of $S$-$T_0$ coherence decay due to the ${\widehat{V}}_{SS}$ term for (a) GaAs DQD with $N_L=N_R={10}^6$ at $B=200$ mT ($|{\mu }_{\text{T}}|\approx 5.0\mu$eV), and (b) DQD based on natural silicon with $N_L=N_R={10}^5$ at $B=30$ mT ($|{\mu }_{\text{T}}|\approx 5.0\mu$eV). The diagonal hatching regions are where ${\sigma }_{\perp }\ll |J\pm {\mu }_{\text{T}}|,|{\mu }_{\text{T}}|$ is not fulfilled.

Figure 5

Lower bound for $T_{\delta \theta }$ as a function of $J$ for (a) GaAs with $N_L=N_R={10}^6$ at $B=200$ mT (${\mu }_{\text{T}}\approx 5.0\mu$eV) and (b) natural Si with $N_L=N_R={10}^6$ at $B=30$ mT ($|{\mu }_{\text{T}}|\approx 3.5\mu$eV). The diagonal hatching regions are where ${\sigma }_{\perp }\ll |J\pm {\mu }_{\text{T}}|,|{\mu }_{\text{T}}|$ is not fulfilled. No narrowing is assumed ($n_F=1$).

Figure 6

Dephasing times induced by ${\widehat{H}}_A$ (solid red line), ${\widehat{V}}_{SS}$ (dashed blue line), and ${\widehat{V}}_{\delta \theta }$ (dotted black line) as a function of $J/{\mu }_{\text{T}}$ for GaAs at (a) $B=1$ T ($|{\mu }_{\text{T}}|\approx 25\mu$eV) and (b) $B=200$ mT ($|{\mu }_{\text{T}}|\approx 5.0\mu$eV), both with $N_L=N_R={10}^6$, resulting in ${\sigma }_z={\sigma }_{\perp }\approx 0.1$ $\mu$eV with no narrowing ($n_F=1$).

Figure 7

Dephasing times induced by ${\widehat{H}}_A$ (solid red line), ${\widehat{V}}_{SS}$ (dashed blue line), and ${\widehat{V}}_{\delta \theta }$ (dotted black line) as a function of $J/{\mu }_{\text{T}}$ for Si at $B=30$ mT ($|{\mu }_{\text{T}}|\approx 3.5\mu$eV), both with $N_L=N_R={10}^5$, resulting in ${\sigma }_z={\sigma }_{\perp }\approx 1.5$ neV. No narrowing is assumed ($n_F=1$).

Figure 8

The decay of the decoherence function, calculated separately for each of the mechanisms, for In${}_{0.5}$Ga${}_{0.5}$As DQD with two electrons at $B=200$ mT (assuming the $g$ factors of the electrons in both dots to be $\approx$$0.5$), with $J=100$ $\mu$eV and for $N_L=N_R={10}^5$. Here, $|{\mu }_{\text{T}}|\approx 5.8\mu$eV and $n_F=1$.

Figure 9

The dephasing time (a) $T_{2,{\theta }_{\mathrm{T}}}^{*}$ due to the $\delta \widehat{\theta }$-linear terms and (b) $T_A^{\prime }$ from the second canonical transformation in GaAs as a function of ${\theta }_{\mathrm{T}}$ at two fixed values of $J$. The rms of the Overhauser field difference between the dots is ${\sigma }_z=0.1$ $\mu$eV. For ${\theta }_{\mathrm{T}}\gg J$, $T_{2,{\theta }_{\mathrm{T}}}^{*}$ is saturated at $\sim$10 ns.

Figure 10

The dephasing times $T_{SS}^{\prime }$ from Eq. (81) and $T_H$ from Eq. (83), due to the terms given in Eqs. (80) and (82), respectively, for GaAs DQD with $N_L=N_R={10}^6$ at $B=200$ mT ($|{\mu }_{\text{T}}|\approx 5.0\mu$eV).

Figure 11

(a) $W_{SS}^{\mathrm{HE}}\left(t\right)$ and (b) $W_{\theta }^{\mathrm{HE}}\left(t\right)$ for GaAs DQD with $N_L=N_R={10}^6$ at $B=200$ mT and ${\theta }_{\mathrm{T}}=40$ mT with various values of $J$. Here, $|{\mu }_{\text{T}}|\approx 5.0\mu$eV.

Figure 12

(a) $W_{SS}^{\mathrm{HE}}\left(t\right)$ and (b) $W_{\theta }^{\mathrm{HE}}\left(t\right)$ for GaAs DQD with $N_L=N_R={10}^6$ at $B=700$ mT and ${\theta }_{\mathrm{T}}=2.5$ mT with various values of $J$. The parameters here are chosen to correspond closely to the ones employed in recent experimental work from Ref. 42.

Figure 13

The SWAP gate error in GaAs at $B=200$ mT for two input states: (a) $|{\psi }_{\text{in}}\rangle =|\psi \rangle =|\uparrow \downarrow \rangle$ (blue lines) and (b) $|{\psi }_{\text{in}}\rangle =|\varphi \rangle =\frac{1}{\sqrt{2}}\left(\right|\uparrow \rangle +|\downarrow \rangle )\otimes \frac{1}{\sqrt{2}}\left(\right|\uparrow \rangle -|\downarrow \rangle )$ (green lines). The blue lines have steeper slopes than the green ones. Solid lines show the gate errors with the Overhauser fluctuations ${\sigma }_z\approx 0.1$ $\mu$eV, while dashed lines are with ${\sigma }_z\approx 0.05$ $\mu$eV. Here, ${\sigma }_z=n_F^{\prime }{\sigma }_{\perp }$, as discussed in the main text below Eq. (104).