Ultracoherent operation of spin qubits with superexchange coupling

With the use of nuclear-spin-free materials such as silicon and germanium, spin-based quantum bits (qubits) have evolved to become among the most coherent systems for quantum information processing. The new frontier for spin qubits has therefore shifted to the ubiquitous charge noise and spin-orbit interaction, which are limiting the coherence times and gate fidelities of solid-state qubits. In this paper we investigate superexchange, as a means of indirect exchange interaction between two single electron spin qubits, each embedded in a single semiconductor quantum dot (QD), mediated by an intermediate, empty QD. Our results suggest the existence of “supersweet spots”, in which the qubit operations implemented by superexchange interaction are simultaneously first-order-insensitive to charge noise and to errors due to spin-orbit interaction. The proposed spin-qubit architecture is scalable and within the manufacturing capabilities of semiconductor industry.

Figure 1

(a) The geometry of the system, where $B$ denotes the direction of the external magnetic field, $t_{LC}$ are spin-conserving hoppings between the left $\left(L\right)$ and center $\left(C\right)$ dot (marked with dotted blue lines), $t_{CR}$ are spin-conserving hoppings between the $C$ and the right $\left(R\right)$ dot (marked with dashed red lines), and $\left[110\right]$ and $\left[\overline{1}10\right]$ are the crystallographic axes. (b) The scheme of all possible superexchange paths in absence of spin-orbit interactions. All superexchange paths involve four tunneling events, two between the $L$ and the $C$ QDs $t_{LC}$ and two between the $C$ and the $R$ QDs $t_{CR}$. $\uparrow$ stands for a spin-up state, $\downarrow$ for a spin-down state and fields in the parentheses denote charge occupancies of the $(L,C,R)$ QDs.

Figure 2

Level diagram, where $E$ denotes the energy, $x$ the position, $\epsilon$ is the energy difference between the outer dots $(L$ and $R)$, and $\delta$ the energy between the average energy of the outer dots $(L$ and $R)$ and the center $\left(C\right)$ QD.

Figure 3

The superexchange in the absence of spin-orbit interaction $J_{\mathrm{SE}}$ as a function of the detuning parameters $\delta$ and $\epsilon$. The points represent the superexchange sweet spots $J_1({\delta }_1,{\epsilon }_1)=0.08,J_2({\delta }_2,{\epsilon }_2)=64.65$, and $J_3({\delta }_3,{\epsilon }_3)=J_4({\delta }_4,{\epsilon }_4)=13.62$ in units of $t^4/U^3$, where $t$ is the tunneling and $U$ is Coulomb repulsion. The black line marks $J_{\mathrm{SE}}=0$, black dashed line $J_{\mathrm{SE}}=-10$ and white dashed line $J_{\mathrm{SE}}=10$. The white regions represent areas in which the energy difference $\mathrm{\Delta }E$ between the $(2,0,0)$ $(0,2,0)$ $(0,0,2)$ $(1,1,0)$ $(0,1,1)$ charge states and superexchange states $\left(101\right)$ becomes comparable to $t$, and therefore no superexchange takes place. Here, we chose $t=17.8\mu \mathrm{eV}$.

Figure 4

(a) The strength of the spin-orbit contribution $D$ around the “sweet spot” $J_3$ as a function of the detuning parameter $\epsilon$ and $\delta$ for $E_{\mathrm{z}}=0.38U$. The dashed black line marks a path along which $D=0$. (b) Superexchange as a function of $\delta$ for $\epsilon =-1.34\text{U}$ in the case of vanishing spin orbit. (Inset) Magnitude of different exchange paths in the context of Table 1 in the case of vanishing spin orbit. The horizontal black dashed line represents the point ${\delta }_0$ in which $J_{\mathrm{SE}}=0$. (c) Coherent superexchange oscillations as a function of the detuning $\delta$ and time $T$ in the case of vanishing spin-orbit interaction. The probability to occupy the $(\downarrow ,0,\uparrow )$ state is not displayed because $P_{\uparrow 0\downarrow }=1-P_{\downarrow 0\uparrow }$. Parameters of the plots are tunneling $t=17.8\mu \mathrm{eV}$, detuning $\epsilon =-1.34\text{U}$, the Coulomb repulsion $U=1\text{meV}$.