Interplay between exchange interaction and magnetic field gradient in a double quantum dot with two individual electron spin qubits

We studied experimentally the coherent exchange interaction between two antiparallel electron spins in a double quantum dot with an increasing magnetic field gradient between the two dots. We observed an increase of the oscillation frequency and a reduction of the oscillation contrast. Both observations are consistent with a simple model of two spins interacting via exchange interaction. The demonstrated control of the two-electron spin states is interesting to produce gates for individual spin qubits.

Figure 1

(a) Scanning electron microscopy (SEM) image of the device. The red and blue gates are used to tune the chemical potential of both dots and are connected to high frequency lines. (b) Charge stability diagram close to the (1,1)-(0,2) crossing. Each charge configuration corresponds to a measured conductance. Along the detuning line $\epsilon$, the relative detuning between the chemical potential of the two dots can be controlled with $V_{g1}$ and $V_{g2}$ and the system can then be tuned from the (1,1) to the (0,2) charge configuration through the (1,1)-(0,2) crossing. The conversion from voltage to energy is calibrated via standard transport measurements.

Figure 2

Energy diagram of the two-electrons spin states in a double quantum dot for weak (a) and strong (b) gradient $\Delta B_z$. A positive pulse in $\epsilon$ results in a fast branching of the exchange interaction. For weak $\Delta B_z$, a change of the antiparallel eigenstates results in coherent oscillations between $|\uparrow ,\downarrow \rangle$ and $|\downarrow ,\uparrow \rangle$. For strong $\Delta B_z$, the eigenstates remain the same during the detuning pulse and therefore no oscillation is expected. (c) Coherent exchange oscillations on the probability to detect the system in the (0,2) charge configuration as a fuction of the pulse duration $\tau$ for different exchange pulse amplitudes ${\epsilon }_0$. On top of (c), a scheme of the multistep procedure is presented. The dashed line represents the detuning at which the analysis with respect to $\Delta B_z$ is performed (data presented in Fig. 3). The inset presents the result of a fit procedure that allows the determination of the dependence between exchange energy $J$ and ${\epsilon }_0$.

Figure 3

Schematics of the experimental detuning pulses used in the experiment to analyze the evolution of the oscillation contrast as a function of the magnetic field gradient (a) $S\text{−}{T}_0$ coherent oscillations and (b) coherent exchange oscillations. The energy diagram of the two electron spins in a tunnel coupled double dot is represented on top of each figure.

Figure 4

(a) $S\text{−}{T}_0$ coherent oscillations as a function of the pulse duration ${\tau }_R$ for different DNP sequence duration $T_{\mathrm{pump}}$. The system is initialized in the singlet state in the (1,1) charge configuration. The longer the DNP sequence, the faster the coherent oscillations. (b) Coherent exchange oscillations as a function of the exchange pulse duration $\tau$ for different DNP sequence durations [the pulse amplitude is set at ${\epsilon }_0=-0.6$ meV as shown by the dashed line in Fig. 2]. The system is initialized in the lowest energy antiparallel spin state via adiabatic transfer from the singlet ground state of the (0,2) region to the (1,1) charge configuration. A slight decrease of the oscillation frequency is observed as $T_{\mathrm{pump}}$ is increased. Solid lines are fits obtained by solving the Schrödinger equation numerically (see text).

Figure 5

(a) Evolution of the $\Delta B_z$ extracted from the fitting procedure of the $S\text{−}{T}_0$ coherent oscillations [Fig. 4] as a function of the DNP sequence duration. (b) Evolution of the amplitude of $S\text{−}{T}_0$ [Fig. 4] and exchange coherent oscillations [Fig. 4] extracted from the fitting procedure as a function of $\Delta B_z$. The $S\text{−}{T}_0$ amplitude with $\Delta B_z$ decays almost linearly (linear fit in solid green line), whereas the amplitude of the coherent exchange oscillations decays nonlinearly. The blue solid line corresponds to the expected amplitude dependence in $\Delta B_z$ extracted from the numerical computation of the coherent exchange oscillations (see Fig. 4). It is worth noting that all the parameters have been fixed during the calibration procedure and the only remaining parameter is an overall amplitude factor.