Electric-field control and noise protection of the flopping-mode spin qubit

We propose and analyze a “flopping-mode” mechanism for electric dipole spin resonance based on the delocalization of a single electron across a double quantum dot confinement potential. Delocalization of the charge maximizes the electronic dipole moment compared to the conventional single-dot spin resonance configuration. We present a theoretical investigation of the flopping-mode spin qubit properties through the crossover from the double- to the single-dot configuration by calculating effective spin Rabi frequencies and single-qubit gate fidelities. The flopping-mode regime optimizes the artificial spin-orbit effect generated by an external micromagnet and draws on the existence of an externally controllable sweet spot, where the coupling of the qubit to charge noise is highly suppressed. We further analyze the sweet spot behavior in the presence of a longitudinal magnetic field gradient, which gives rise to a second-order sweet spot with reduced sensitivity to charge fluctuations.

Figure 1

(a) Schematic illustration of the flopping-mode EDSR mechanism, where the spin of an electron (shown as green circles) delocalized between two QDs is driven via an electric field (purple line) in a magnetic field gradient (represented with red arrows). (b) Energy levels $E_{0,\cdots ,3}$ of the Hamiltonian (17) as a function of the interdot detuning $\varepsilon$, calculated with $t_c=20\mu \text{eV}, E_z=24\mu \text{eV}, g{\mu }_B{b}_x=15\mu \text{eV}$, and $g{\mu }_B{b}_z=4\mu \text{eV}$. The asymmetry with respect to $\varepsilon$ is due to the longitudinal magnetic field gradient. Around zero detuning, $\left|\varepsilon \right|\ll 2t_c$, the electron delocalizes across the DQD, yielding a larger electric dipole moment $p$ compared to the single-dot regime. The arrows represent the electrically addressable spin (solid line), charge (dashed line), and spin-charge (dotted line) transitions.

Figure 2

(a) Ratio between the spin Rabi frequency ${\mathrm{\Omega }}_s$ and the charge Rabi frequency ${\mathrm{\Omega }}_c$ as a function of detuning $\varepsilon$ for $t_c=15\mu \text{eV}$. The spin Rabi frequency is maximized for $\varepsilon =0$. (b) Single-qubit average gate infidelity as a function of $\varepsilon$ and $t_c$. As expected, $\overline{F}$ is symmetric about $\varepsilon =0$, with the highest values achieved at $\varepsilon =0$ and slightly away from the line with maximal spin-charge mixing, $\mathrm{\Omega }=E_z$ (black dashed line). The other parameters are chosen to be $E_z=24\mu \text{eV}, g{\mu }_B{b}_x=2\mu \text{eV}, {\mathrm{\Omega }}_c/2\pi =500\text{MHz}, {\gamma }_{1,c}/2\pi =18\text{MHz}, {\gamma }_{\varphi }/2\pi =600\text{MHz}$, and ${\gamma }_M/2\pi =2\text{MHz}$.

Figure 3

Spin qubit energy splitting $E_s$ as a function of the detuning $\varepsilon$, for various values of the longitudinal gradient field ($g{\mu }_B{b}_z$), as indicated, increasing from top to bottom. The interdot tunnel splitting amounts to (a) $2t_c=18\mu \text{eV}$ and (b) $2t_c=30\mu \text{eV}$, while the homogeneous field Zeeman energy is $E_z=24\mu \text{eV}$ and the transverse inhomogeneous component is $g{\mu }_B{b}_x=2\mu \text{eV}$. The thin shaded areas indicate first-order sweet spots for the corresponding color line and the wide blue shaded area in (b) indicates the region around the second-order sweet spot for $g{\mu }_B{b}_z=0.3\mu \text{eV}$. The discontinuity in (a) occurs at $\mathrm{\Omega }=E_z$ due to a level crossing of the upper qubit state.

Figure 4

Second derivative ${\partial }_{\varepsilon }^2{E}_s$ of the spin qubit energy splitting with respect to the detuning $\varepsilon$ as a function of $t_c$ and $\varepsilon$ for (a) $g{\mu }_B{b}_z=0.16\mu \text{eV}$ and (b) $g{\mu }_B{b}_z=0.5\mu \text{eV}$. The black dashed lines indicate the first-order sweet spot positions and the circle in panel (b) indicates the position of the second-order sweet spot. The homogeneous field Zeeman energy is $E_z=24\mu \text{eV}$ and the transverse inhomogeneous component is $g{\mu }_B{b}_x=2\mu \text{eV}$.

Figure 5

Single-qubit average gate infidelity $1-\overline{F}$ as a function of detuning $\varepsilon$ and interdot tunnel coupling $t_c$ for (a) $g{\mu }_B{b}_z=0.16\mu \text{eV}$ and (b) $g{\mu }_B{b}_z=0.5\mu \text{eV}$. The homogeneous field Zeeman energy is $E_z=24\mu \text{eV}$ and the transverse inhomogeneous component is $g{\mu }_B{b}_x=2\mu \text{eV}$. The other parameters are chosen to be ${\mathrm{\Omega }}_c/2\pi =500\text{MHz}, {\gamma }_{1,c}/2\pi =18\text{MHz}, {\gamma }_{\varphi }/2\pi =600\text{MHz}$, and ${\gamma }_M/2\pi =2\text{MHz}$. The black dashed lines indicate the first-order sweet spot positions. In panel (b) the squares mark the position of the first-order sweet spots for $t_c=13\mu \text{eV}$ and the circle indicates the position of the second-order sweet spot.