Inverted singlet-triplet qubit coded on a two-electron double quantum dot

The $s_z=0$ spin configuration of two electrons confined at a double quantum dot (DQD) encodes the singlet-triplet qubit (STQ). We introduce the inverted STQ (ISTQ) that emerges from the setup of two quantum dots (QDs) differing significantly in size and out-of-plane magnetic fields. The strongly confined QD has a two-electron singlet ground state, but the weakly confined QD has a two-electron triplet ground state in the $s_z=0$ subspace. Spin-orbit interactions act nontrivially on the $s_z=0$ subspace and provide universal control of the ISTQ together with electrostatic manipulations of the charge configuration. GaAs and InAs DQDs can be operated as ISTQs under realistic noise conditions.

Figure 1

STQ coded on an asymmetric DQD. Each QD contains one electron. Modifications of the confining potentials allow an electron transfer to reach a doubly occupied QD. ${\text{QD}}_L$ has a two-electron singlet ground state; for ${\text{QD}}_R$ the spinless triplet state is lower in energy. The out-of-plane component $B_z$ of the magnetic field favors triplets, but the confining energy favors singlets. The magnetic field $\mathit{B}$ is tilted by the angle $\varphi$ from the dot-connection axis ${\mathit{e}}_x$ and by the out-of-plane angle $\theta$ from ${\mathit{e}}_z$. The $[1,0,0]$ direction of the lattice is rotated by the angle $\xi$ from ${\mathit{e}}_x$. We introduce additionally the rotation angle $\rho$ between $\mathit{B}$ and the $[1,0,0]$ direction.

Figure 2

Energy diagram of a STQ as a function of the electrostatic bias $\epsilon$ according to Eqs. (7) and (9). The blue and red lines (gray lines in the print version) describe the energies of the lowest singlet $E_S$ and spinless triplet $E_T$; black lines show excited states. The left dotted line labels the charge transition point at $\epsilon =-U_L$, where $E_{S_{1,1}}$ and $E_{S_{2,0}}$ have the same energies (similarly $E_{T_{1,1}}$ and $E_{T_{0,2}}$ have equal energies at $\epsilon =U_R)$. We obtain a $(2,0)$ singlet ground state at $\epsilon <0$, while $\epsilon >0$ favors the $(0,2)$ triplet. $E_S$ and $E_T$ cross at ${\epsilon }^{*}$. SOIs couple $E_S$ and $E_T$. The inset shows the region around ${\epsilon }^{*}$. The dashed curves are energy levels in the absence of SOIs.

Figure 3

Sketch of the energy levels $|S\rangle$ and $|T\rangle$ that encode the ISTQ. The energy levels are shifted compared to Fig. 2, while the energy differences between $E_S$ and $E_T$ remain unchanged at each $\epsilon$. $|S\rangle$ and $|T\rangle$ have equal orbital energies at ${\epsilon }^{*}$. SOIs lift the degeneracy and cause an anticrossing ${\Delta }_{\text{so}}$. $|S\rangle$ is the ground state for $\epsilon <{\epsilon }^{*}$, but $|S\rangle$ is the excited state for $\epsilon >{\epsilon }^{*}$. We label one point deep in $(2,0)$ by ${\epsilon }_{(2,0)}$ with the energy splitting ${\Omega }_{(2,0)}$ [similarly, $|S\rangle$ and $|T\rangle$ have the energy splitting ${\Omega }_{(0,2)}$ at ${\epsilon }_{(0,2)}$ in $(0,2)\text{]}$.

Figure 4

Two DQDs [labeled by $\left(L\right)$ and $\left(R\right)\text{]}$ encode two ISTQs, which are coupled using Coulomb interactions. ${\text{QD}}_R^{\left(L\right)}$ and ${\text{QD}}_L^{\left(R\right)}$ are closest to each other and the electron configurations at these QDs $(n_R^{\left(L\right)}$ and $n_L^{\left(R\right)})$ dominate the interaction between the qubits [cf. Eq. (14)].