Spin shuttling in a silicon double quantum dot

The transport of quantum information between different nodes of a quantum device is among the challenging functionalities of a quantum processor. In the context of spin qubits, this requirement can be met by coherent electron spin shuttling between semiconductor quantum dots. Here we theoretically study a minimal version of spin shuttling between two quantum dots. To this end, we analyze the dynamics of an electron during a detuning sweep in a silicon double quantum dot (DQD) occupied by one electron. Possibilities and limitations of spin transport are investigated. Spin-orbit interaction and the Zeeman effect in an inhomogeneous magnetic field play an important role for spin shuttling and are included in our model. Interactions that couple the position, spin, and valley degrees of freedom open a number of avoided crossings in the spectrum allowing for diabatic transitions and interfering paths. The outcomes of single and repeated spin shuttling protocols are explored by means of numerical simulations and an approximate analytical model based on the solution of the Landau-Zener problem. We find that a spin infidelity as low as $1-F_s\lesssim 0.002$ with a relatively fast level velocity of $\alpha =600\mu {\mathrm{eV}\mathrm{ns}}^{-1}$ is feasible for optimal choices of parameters or by making use of constructive interference.

Figure 1

Energy level diagram of the system under consideration: two quantum dots (QDs) filled with one electron with spin $\sigma$. The vector $\mathit{d}$ points from the center of the left dot to the center of the right dot and thus determines the orientation of the DQD in the crystal. An electron is shuttled from the left QD to the empty right QD in the presence of a global magnetic field $\mathit{B}$. A micromagnet can cause a magnetic field difference $2\mathit{b}=2(b_x,b_y,b_z)$ between the left and right QD resulting in different total field ${\mathit{B}}_{L,R}=\mathit{B}\pm \mathit{b}$ and Zeeman splittings $B_{L\left(R\right)}$. The time dependence of the level detuning $\varepsilon \left(t\right)$ is chosen such that it conveys the electron from left to right. In addition to spin-conserving hopping $t_{\mathrm{sc}}$ a spin-flip tunneling term $t_{\mathrm{sf}}$ occurs due to the SOI and noncollinear magnetic fields in the two dots.

Figure 2

Spectrum of the Hamiltonian $H^{\prime }$, Eq. (6), as a function of the detuning $\varepsilon$, with diabatic (dashed) and adiabatic (solid) states. Spin-conserving tunneling $t_c$ opens avoided crossings at $\varepsilon \approx \pm b_z$ and the spin-flipping interactions $a$ and $\mathit{b}$ open avoided crossings at $\varepsilon \approx \pm B$. Plot parameters are $t_c=21\mu \mathrm{eV}, B=30\mu \mathrm{eV}, \mathit{b}=(30\mu \mathrm{eV},0,0)$, and $a=\sqrt{2}(i-1)\mu \mathrm{eV}$. The spin ground state is labeled with $\downarrow$ and the excited spin state with $\uparrow$, respectively; by $(\sigma ,0)$ [$(0,\sigma )$] we denote that an electron with spin $\sigma$ is localized in the left [right] dot.

Figure 3

Spin shuttling infidelity without valley degeneracy. (a) Logarithm of the spin shuttling infidelity $1-F_s$ for initialization in the excited spin state, $|\mathrm{in}\rangle =|\uparrow ,0\rangle$, as a function of the transverse magnetic field difference $b_x$ and the Zeeman splitting $B$. Free parameters are chosen as $t_c=21\mu \mathrm{eV},{\varepsilon }_0=8\mathrm{meV}$, and $b_z=0$. With increasing $b_x$ the spin-flip tunneling $t_{\mathrm{sf}}$ leads to diabatic transitions, and $1-F_s$ shows local maxima (minima) due to destructive (constructive) LZ interference. The cyan line indicates the minimal $|t_{\mathrm{sf}}|$ for each $B$. (b) Spin infidelity $1-F_s$ along a cut through panel (a) indicated by the dashed line ($B=2t_c=42\mu \mathrm{eV}$) for the excited spin state $|\mathrm{in}\rangle =|\uparrow ,0\rangle$ (blue, solid) and the ground state, $|\mathrm{in}\rangle =|\downarrow ,0\rangle$ (red, solid). The dashed lines correspond to the case without SOI, $a=0$. The ground state does not show interference in a single passage.

Figure 4

(a) Spin infidelity $1-F_s$ for $|\mathrm{in}\rangle =|\uparrow ,0\rangle$ (blue) and $|\mathrm{in}\rangle =|\downarrow ,0\rangle$ (red) as a function of the number of round trips $N$ in the DQD, where $B=25\mu \mathrm{eV}, b_x=-3.4\mu \mathrm{eV}, b_z=1.56\mu \mathrm{eV}, t_c=21\mu \mathrm{eV}$, and ${\varepsilon }_0=0.8\mathrm{meV}$. The interference pattern in the excited state is more complex since the multiple anticrossings passed per transition give rise to several oscillating terms. (b) Repeated shuttling of $|\mathrm{in}\rangle =|\uparrow ,0\rangle$. The dark blue curve is the same as in (a), the light blue curve additionally takes into account the coupling to reservoirs described by Eq. (17) with $t_r=t_c/10$ and Coulomb repulsion between the neighboring dots $U_c=5\mathrm{meV}$ and temperature $T=0.1\mathrm{K}$. The effects of spin-flip cotunneling (red) and decoherence (orange) are indicated with arrows. The $(1,0)\leftrightarrow (0,1)$ charge transition is crossed in the middle between the triple points involving the (0,0) and (1,1) regimes.

Figure 5

Spectrum of $H_v$ as a function of the detuning $\varepsilon$ (solid) and diabatic basis states (dashed) for $B=54\mu \mathrm{eV}, t_c=21\mu \mathrm{eV}, a=b_x=b_z=0$ and $E_{v,L}=76\mu \mathrm{eV}, E_{v,R}=58\mu \mathrm{eV}$, (a) $\vartheta =0$, (b) $\vartheta =\pi /6$, (c) $\vartheta =\pi /2$. In the limit $\vartheta \to 0$ ($\vartheta \to \pi /2$) the valley-flip (valley-conserving) tunneling matrix elements $\propto sin\vartheta$ ($\propto cos\vartheta$) vanish. Pink and green ellipses in (b) indicate where spin-conserving valley transitions allow for two interfering paths for both spin states. Arrows indicate upper and lower paths for the spin ground state in the excited valley which splits at $A1$ and can interfere at $A2$.

Figure 6

Spin infidelity with valley degree of freedom. (a) Plotted is the logarithm of the spin infidelity $1-F_s$ as a function of the magnetic field $B$ and the transverse magnetic field gradient $b_x$ for $|\mathrm{in}\rangle =|\uparrow 2,0\rangle$ with $a=b_z=0, t_c=21\mu \mathrm{eV}, {\varepsilon }_0=0.8\mathrm{meV}$ and $E_{v,L}=76\mu \mathrm{eV}, E_{v,R}=58\mu \mathrm{eV}, \vartheta =\pi /4$. Interference extrema of first and second order due to both, spin and valley transitions are present. The fine ripples are finite-time LZ effects due to the relatively short ramp. (Inset) Spin-valley infidelity $1-F_{s,v}$ (blue) and charge infidelity $1-F_c$ (red) as function of the valley mixing angle $\vartheta$ for $|\mathrm{in}\rangle =|\uparrow 2,0\rangle$ and parameters as in the main plot, $B=54\mathrm{meV}, b_x=0$. At $\vartheta =\pi /2$ the valley-conserving tunneling matrix elements vanish. The transport of spin and valley is limited by the charge shuttling infidelity $1-F_c$ whose maxima are due to destructive LZ interference suppressing the charge transfer. Green dots indicate spots with vanishing infidelity for which closed analytic expressions were found in Sec. 4c. (b) Spin (charge) infidelity plotted as solid (dashed) lines for $|\mathrm{in}\rangle =|\uparrow 1,0\rangle$ (blue) and $|\mathrm{in}\rangle =|\downarrow 1,0\rangle$ (red) as a function of the transverse magnetic gradient $b_x$ with $B=42\mu \mathrm{eV}$, remaining parameters as in (a). In the case $|\mathrm{in}\rangle =|\uparrow 1,0\rangle$ a large spin shuttling error may come from a transition to $|\mathrm{out}\rangle =|0,\downarrow 2\rangle$ while the charge is transported faithfully.

Figure 7

Charge infidelity $1-F_c$ for $|\mathrm{in}\rangle =|\sigma 2,0\rangle$ as a function of $(E_{v,L},E_{v,R})$ for $B=54\mu \mathrm{eV}, a=b_z=b_x=0, t_c=21\mu \mathrm{eV},\vartheta =\pi /4$, and ${\varepsilon }_0=8\mathrm{meV}$. The density plot is a numerical result, blue (red) solid lines indicate maxima (minima) predicted by the analytical model after a constant shift $E_{v,j}\mapsto E_{v,j}+k$. For small valley splitting the agreement is good, however, the second order minimum of $1-F_s$ has a large deviation already, as seen in the upper right corner.