Error rate of a charge qubit coupled to an acoustic phonon reservoir

We analyze decoherence of an electron in a double dot due to the interaction with acoustic phonons. For large tunneling rates between the quantum dots, the main contribution to decoherence comes from the phonon emission relaxation processes, while for small tunneling rates, the virtual-phonon, dephasing processes dominate. Our results show that in common semiconductors, such as Si and GaAs, the latter mechanism determines the upper limit for the double-dot charge qubit performance measure.

Figure 1

Sketch of the qubit: single electron within double-well potential.

Figure 2

Error rate estimate per cycle due to electron-phonon interaction as a function of the cycle time $\Delta t\left(\Delta t=\pi \hslash ∕\epsilon \right)$. The distance between the dot centers was $L=50\text{nm}$ for all the cases considered. For all the gate-engineered quantum dots, the effective radius was $a=25\text{nm}$. The parameters for the GaAs dots were $\Xi =7\text{eV}$, $s=5.14\times {10}^3\mathrm{m}∕\mathrm{s}$, $\rho =5.31\mathrm{g}∕{\text{cm}}^3$, $M=e{e}_{14}∕\left({ϵ}_0\kappa \right)$, where $e_{14}=0.16\mathrm{C}∕{\mathrm{m}}^2$, $\kappa =12.8$ [20]. As in Ref. 13, for silicon the following parameters were used: $a=3\text{nm}$ for phosphorus impurity states, effective deformation potential $\Xi =3.3\text{eV}$, $s=9.0\times {10}^3\mathrm{m}∕\mathrm{s}$, $\rho =2.33\mathrm{g}∕{\text{cm}}^3$. The results are shown as follows: (1) the double-donor structure in silicon; (2) decoherence in GaAs due to piezointeraction; (3) the contribution to decoherence due to deformation interaction in GaAs; (4) the gate-engineered dots in Si. The dashed lines denote $D_A$, the dotted lines denote $D_P$, and the solid lines are $D$. The relaxation rate for the double-donor structure in silicon is not seen because it is very small in the given range of times.