Nuclear spin state narrowing via gate-controlled Rabi oscillations in a double quantum dot

We study spin dynamics for two electrons confined to a double quantum dot under the influence of an oscillating exchange interaction. This leads to driven Rabi oscillations between the $\mid \uparrow \downarrow ⟩$ state and the $\mid \downarrow \uparrow ⟩$ state of the two-electron system. The width of the Rabi resonance is proportional to the amplitude of the oscillating exchange. A measurement of the Rabi resonance allows one to narrow the distribution of nuclear spin states and thereby to prolong the spin decoherence time. Further, we study decoherence of the two-electron states due to the hyperfine interaction and give requirements on the parameters of the system in order to initialize in the $\mid \uparrow \downarrow ⟩$ state and to perform a $\sqrt{\mathrm{SWAP}}$ operation with unit fidelity.

Figure 1

(a) This figure illustrates the projection obtained through an ideal complete measurement of the Rabi-resonance line shape. All the different Lorentzian resonances corresponding to different nuclear spin eigenstates add up to a Gaussian line shape. (b) Through a perfect complete measurement of the line shape of the Rabi resonance, which involves many single measurements of ${\tau }^z$, the superposition collapses and we are left with one single Lorentzian centered around $2x_0^{\prime }={\Omega }_n$, which in general is different from $2x_0$.

Figure 2

In this figure, we show a typical[39] sequence of the rescaled probability density of eigenvalues $\pi \left(x\right)={\rho }_I^{\left(\right\{M_i\},\{{\alpha }_i^{-}\},\{{\omega }_i\left\}\right)}\left(x\right)∕\max \left[{\rho }_I^{\left(\right\{M_i\},\{{\alpha }_i^{-}\},\{{\omega }_i\left\}\right)}\right(x\left)\right]$ for the adaptive conditional scheme. Here, ${\rho }_I^{\left(\right\{M_i\},\{{\alpha }_i^{-}\},\{{\omega }_i\left\}\right)}\left(x\right)$ is given in Eq. (33). We have $x=\delta h_n^z+\delta b^z$, $j∕{\sigma }_0=1∕10$ and in (a)–(c) the initial Gaussian distribution (with FWHM $2{\sigma }_0\sqrt{2\ln 2}\approx 2{\sigma }_0$) is plotted for reference. (a) Up to $M=50$ measurements the outcome is never $\mid -⟩$ and thus each measurement “burns a hole” into the distribution where it previously had its maximum. (b) In the 51st measurement the outcome is $\mid -⟩$, which leads to a narrowed distribution of nuclear spin eigenvalues (peak centered at $\approx 0.5$) with a FWHM that is reduced by a factor $\approx j∕4{\sigma }_0$. (c) Adapting the driving frequency $\omega$ to this peak, i.e., setting $\omega ∕2=x_{\max }$ in subsequent measurements, leads to further narrowing every time $\mid -⟩$ is measured. In this example, the final FWHM is $\approx {\sigma }_0∕100$, i.e., the distribution has been narrowed by a factor $\approx j∕10{\sigma }_0$. (d) The probability $P^{-}$ to measure $\mid -⟩$ jumps up after the 51st measurement, and after $\mid -⟩$ is measured several more times, this probability saturates close to $1∕2$.