Optimal control for electron shuttling

In this paper we apply an optimal control technique to derive control fields that transfer an electron between ends of a chain of donors or quantum dots. We formulate the transfer as an optimal steering problem, and then derive the dynamics of the optimal control. A numerical algorithm is developed to effectively generate control pulses. We apply this technique to transfer an electron between sites of a triple quantum dot and an ionized chain of phosphorus dopants in silicon. Using the optimal pulses for the spatial shuttling of phosphorus dopants, we then add hyperfine interactions to the Hamiltonian and show that a 500 G magnetic field will transfer the electron spatially as well as transferring the spin components of two of the four hyperfine states of the electron-nuclear spin pair.

Figure 1

Time dependence of site populations for electron shuttling across a triple quantum dot when acted on by time-dependent voltages optimized to achieve minimal heating (i.e., minimal pulse energy). (a) Quantum dot populations for sites 1, 2, and 3, as a function of time. Blue solid line, ${\rho }_{11}$; green dashed line, ${\rho }_{22}$; red dotted line, ${\rho }_{33}$. $J_1$ and $J_2$ were set to $-$0.07 and $-$0.14 meV, respectively. (b) Optimal control voltages. Blue solid line, ${\mu }_L$; green dashed line, ${\mu }_R$. The pulses were determined here for $N=1000$ segments. (c),(d) The Fourier transforms of ${\mu }_L$ and ${\mu }_R$, respectively, at different numbers of segments. We note that the form of the pulses converges at $N=500$, after which pulses and site populations are indistinguishable from the corresponding values obtained with $N=1000$.

Figure 2

Time dependence of site populations for an electron shuttling across a chain of three singly ionized phosphorus ions, when acted on by time-dependent voltages optimized to achieve minimal heating (i.e., minimal pulse energy). (a) Electron populations on donor sites 1, 2, and 3, as a function of time. Blue solid line, ${\rho }_{11}$; green dashed line, ${\rho }_{22}$; red dash-dotted line, ${\rho }_{33}$. The value of $\Delta$ was set to 2.7 meV. (b) Fourier transform of the optimal control pulse ${\Omega }_{12}$. The second control pulse ${\Omega }_{23}$ has the same frequency components as ${\Omega }_{12}$ with a phase difference of $-$1.719 rad. The time-domain pulses oscillate with a very high frequency, corresponding to the 2.7 meV value of $\Delta$, and so are not shown here. The optimal pulses are found converge at $N=8000$ segments.

Figure 3

(a) Time dependence of site populations for an electron shuttling across a chain of three singly ionized phosphorus atoms when the spatial shuttling Hamiltonian is supplemented by the spin Hamiltonian (32) at zero magnetic field. We show the transfer of all four hyperfine eigenstates accessible to the electron on site 1, under the pulses optimized solely for spatial shuttling in Fig. 2. The solid blue and dashed red lines show the populations on sites 1 and 3, respectively, as before. The dashed green line shows measure $D$ [Eq. (34)] of the hyperfine state transfer at site 3. (b) The same as (a), but in the presence of a finite magnetic field (500 G). The figures show that hyperfine states which align the nuclear spin with the magnetic field can be robustly transferred, independent of the magnetic field value, while transfer of the spin-flipped states is energetically forbidden. See the text for detailed explanation.