Electrically driven spin resonance with bichromatic driving

Electrically driven spin resonance (EDSR) is an established tool for controlling semiconductor spin qubits. Here, we theoretically study a frequency-mixing variant of EDSR, where two driving tones with different drive frequencies are applied, and the resonance condition connects the spin Larmor frequency with the sum of the two drive frequencies. Focusing on flopping-mode operation of a single electron in a double quantum dot with spin-orbit interaction, we calculate the parameter dependence of the Rabi frequency and the Bloch-Siegert shift. A shared-control spin qubit architecture could benefit from this bichromatic EDSR scheme, as it enables simultaneous single-qubit gates.

Figure 1

Bichromatic driving used for selective qubit addressing in 2D qubit array. (a) Vertical arrangement, inspired by Ref. [31]. Qubits (grey ellipses) are defined by the charge or spin of single electrons confined in double quantum dots (black dots). Selective qubit addressing is realized by bichromatic driving, where ac voltages with frequency $\omega$ and $\overline{\omega }$ are applied on two orthogonal plunger gates (blue and red lines). If $\omega +\overline{\omega }$ matches the qubit splitting, then only the qubit at the intersection of the two driven gates (i.e., the central qubit in the figure) performs Rabi oscillations. Electron confinement might require further gates that are not drawn here. (b) Lateral arrangement, inspired by Ref. [32]. When performing single-qubit gates, the grey barrier gates allow hybridization between quantum dots, while the black ones do not. Qubits are defined by the charge or spin of electrons confined in a double quantum dot. The qubit addressed by bichromatic driving is represented by the two grey disks. On-site energy is controlled by the diagonal plunger gates. Selective addressing of the grey qubit is realized by bichromatic driving, where ac voltages with frequency $\omega$ and $\overline{\omega }$ are switched on two orthogonal plunger gates (red and blue lines). If $\omega +\overline{\omega }$ matches the qubit splitting, then only the selected (grey) qubit is controlled.

Figure 2

Energy levels of the flopping-mode Hamiltonian as a function of the detuning. The flopping-mode spin qubit with splitting ${\omega }_{\text{split}}$ is formed by the two lowest energy states. On the right-hand side we show the dominant charge ($g$round or $e$xcited) and spin (up or down) configurations of the states. Parameters: $t_0=21\mu \mathrm{eV}$, $\hslash {\omega }_z=24\mu \mathrm{eV}$, $\hslash {\mathrm{\Omega }}_{\text{SO}}=2\mu \mathrm{eV}$, $\hslash \delta {\omega }_z=0\mu \mathrm{eV}$, and $\hslash {\mathrm{\Omega }}_z=0\mu \mathrm{eV}$.

Figure 3

Bichromatic Rabi frequency as function of detuning for the flopping-mode spin qubit. Blue-solid line: Analytical result (17). Red dots: Result from the numerical simulation, where $\overline{\omega }$ was optimised numerically for each $\epsilon$ value to obtain complete Rabi oscillations. Parameters: $t_0=21\mu \mathrm{eV}$, $\hslash {\omega }_z=24\mu \mathrm{eV}$, $\hslash {\mathrm{\Omega }}_y=2\mu \mathrm{eV}$, $\hslash {\mathrm{\Omega }}_x=\hslash {\mathrm{\Omega }}_z=0\mu \mathrm{eV}$, $\hslash \delta {\omega }_z=0\mu \mathrm{eV}$, $\omega =0.7{\omega }_{\text{split}}$, and $E_{\text{ac}}={\overline{E}}_{\text{ac}}=2\mu \mathrm{eV}$.

Figure 4

Bichromatic Rabi frequency as function of the frequency of one of the driving fields, $\omega$. Blue-solid line: Analytical result (17). Red dots: Result from the numerical simulation, where $\overline{\omega }$ was optimised numerically for each $\omega$ value to obtain complete Rabi oscillations. Parameters: $\epsilon =21\mu \mathrm{eV}$, $t_0=21\mu \mathrm{eV}$, $\hslash {\omega }_z=24\mu \mathrm{eV}$, $\hslash {\mathrm{\Omega }}_{\text{SO}}=2\mu \mathrm{eV}$, ${\mathrm{\Omega }}_z=0\mu \mathrm{eV}$, $\hslash \delta {\omega }_z=0\mu \mathrm{eV}$, and $E_{\text{ac}}={\overline{E}}_{\text{ac}}=2\mu \mathrm{eV}$.

Figure 5

Bichromatic Bloch-Siegert shift as function of the strength of one of the two electric fields $E_{\mathrm{ac}}$. Blue solid line: Analytical result (18). The other electric field strength is fixed, ${\overline{E}}_{\mathrm{ac}}=2\mu \mathrm{eV}$. The remaining parameters: $\hslash {\omega }_z=24\mu \mathrm{eV}$, $\hslash {\mathrm{\Omega }}_y=2\mu \mathrm{eV}$, $\hslash {\mathrm{\Omega }}_x=\hslash {\mathrm{\Omega }}_z=0\mu \mathrm{eV}$, $\epsilon =30\mu \mathrm{eV}$, $t_0=21\mu \mathrm{eV}$, $\hslash \delta {\omega }_z=3\mu \mathrm{eV}$, and $\omega =0.7{\omega }_{\mathrm{split}}$. ${\omega }_{\mathrm{split}}$ was determined numerically, ${\omega }_{\mathrm{split}}/\left(2\pi \right)=5.36026$ GHz. $\overline{\omega }$ was optimised numerically for each $E_{\text{ac}}$ value to obtain complete Rabi oscillations.

Figure 6

Damping of the Rabi oscillation due to charge noise and $g$-factor asymmetry. Frequency $\overline{\omega }$ was optimised numerically to maximize the fidelity, yielding $\overline{\omega }/\left(2\pi \right)=1.73442$ GHz for $\delta {\omega }_z=0$ (red curve), and $\overline{\omega }/\left(2\pi \right)=1.71347$ GHz for $\hslash \delta {\omega }_z=0.5\mu \mathrm{eV}$ (black curve). The remaining parameters: $t_0=21\mu \mathrm{eV}$, ${\epsilon }_0=30\mu \mathrm{eV}$, $\hslash {\omega }_z=24\mu \mathrm{eV}$, $\hslash {\mathrm{\Omega }}_y=2\mu \mathrm{eV}$, $\hslash {\mathrm{\Omega }}_x=\hslash {\mathrm{\Omega }}_z=0\mu \mathrm{eV}$, $E_{\text{ac}}={\overline{E}}_{\text{ac}}=2\mu \mathrm{eV}$, and $\omega =0.7{\omega }_{\text{split}}$. Splitting frequency ${\omega }_{\text{split}}$ was calculated from Eq. (14). Marks on lines are only guides to the eye.

Figure 7

Parallelization of single-qubit gates in a 2D-qubit array with shared control. Blue (red) qubits in the first (second) row are driven bichromatically by the blue (red) ac fields. By an appropriate choice of the durations (${\tau }_{1,1},\cdots ,{\tau }_{2,2}$) and the initial phases of pulses through the vertical control lines, any combination of single-qubit gate is achievable. The voltage on the vertical-control lines is the sum of ac voltage pulses with different frequencies.