Tuning Methods for Semiconductor Spin Qubits

We present efficient methods to reliably characterize and tune gate-defined semiconductor spin qubits. Our methods are developed for double quantum dots in $\mathrm{Ga}\mathrm{As}$ heterostructures, but they can easily be adapted to other quantum-dot-based qubit systems. These tuning procedures include the characterization of the interdot tunnel coupling, the tunnel coupling to the surrounding leads, and the identification of various fast initialization points for the operation of the qubit. Since semiconductor-based spin qubits are compatible with standard semiconductor process technology and hence promise good prospects of scalability, the challenge of efficiently tuning the dot’s parameters will only grow in the near future, once the multiqubit stage is reached. With the anticipation of being used as the basis for future automated tuning protocols, all measurements presented here are fast-to-execute and easy-to-analyze characterization methods. They result in quantitative measures of the relevant qubit parameters within a couple of seconds and require almost no human interference.

Figure 1

False color SEM image of a device similar to the one used in this work, including the contacting scheme. Applying static voltages to the gray gates confines two electrons in a double dot potential in the 2DEG of a $\mathrm{Ga}\mathrm{As}/{\mathrm{Al}}_{0.31}{\mathrm{Ga}}_{0.69}\mathrm{As}$ heterostructure. The blue gates, RFX and RFY, are used exclusively for fast manipulation. The dot on the left is used for charge sensing of the double dot and is embedded in an impedance-matching circuit as the resistive element. The crossed boxes represent Ohmic contacts to the leads.

Figure 2

(a) Charge stability diagram of the double dot measured using rf reflectometry. Different values of $V_{\mathrm{rf}}$ correspond to different charge ground states. (b) Fit of the stability diagram using the model described in Sec. 3a. Circles and lines mark the automatically detected triple points and the positions of the charge transition. The gray arrow represents the direction of the detuning $ϵ$. In both figures, we subtracted a background value due to the direct influence of the charge sensor extracted from the fit.

Figure 3

(a) To determine the influence of the various dc gates on the position of a lead transition, we apply two different voltages to each one of the dc gates in turn. For each set of voltages, we measure the occupation of the double dot as we sweep across two regions in the charge stability diagram. (b),(c) To measure the tunneling times to the leads, we apply megahertz-frequency square voltage pulses across the respective lead transition, forcing an electron to be exchanged with the respective reservoir. The tunneling time can be extracted from the rise time of the response signal. (d) The interdot tunnel coupling is extracted by sweeping along the detuning $ϵ$, recording the average charge occupancy, and measuring the broadening of the transition.

Figure 4

(a) High-resolution charge-stability diagram around the (2,0)-(1,1) transition used to define the two-electron spin qubit. Important points used to initialize the qubit in different states and for measurement are marked by dots and further explained in the main text. Transitions are labeled in white. (b) Diagram showing the energy relaxation cascade used to initialize the (1,1) ground state $|\uparrow \uparrow ⟩$ at point $T_{+}$ (see Secs. 3f). $S_L\otimes \downarrow (\uparrow )$ denotes the state of a singlet state in the left quantum dot and a down (up) state in the right dot. Arrows indicate relaxation via electron exchange with the (2,1) charge configuration. (c) Charge stability diagram measured using the pulse sequence $M\text{−}{R}_1\text{−}{R}_2\text{−}M$ (see Secs. 3f). The measurement trapezoid appears as an area where the readout signal $V_{\mathrm{rf}}$ is between the values corresponding to the $S$(2,0) and to the (1,1) configurations (yellowish turquoise area). The blurred boundaries of the readout trapezoid reflect a failure of the random load pulse sequence rather than instabilities. (d) Adding a waiting time at point $S$ after the pulse sequence from (c) maps out the “mouse bite” (yellow area within the measurement trapezoid), i.e., the region of singlet reload within the measurement triangle (see Sec. 3f).

Figure 5

(a) To determine the optimal position of the singlet reload point, we shift the position of point $S$ in the pulse sequence $M\text{−}{R}_1\text{−}{R}_2\text{−}M\text{−}S\text{−}M$ along the direction ${\delta }_S$ [see text and Fig. 4]. The optimal position for $S$ is in the middle of the plateau of low $V_{\mathrm{rf}}$ values. (b) The singlet reload time is extracted by applying the pulse sequence $M\text{−}{R}_1\text{−}{R}_2\text{−}M\text{−}S\text{−}M$ and measuring the triplet return probability as a function of the waiting time $t$ at point $S$. (c),(d) The positions of the $T_{+}$ reload point and of the $S{T}_{+}$ transition can be determined using the pulse sequences described in Secs. 3f and 3g, respectively, and result in peaks in the measured $V_{\mathrm{rf}}$ signals as a function of the displacement ${\delta }_{\mathrm{T}}$ and of the detuning $ϵ$.

Figure 6

Characteristic charge-stability diagram of the sensing dot, measured with rf reflectometry. Depending monotonically on the conductance through the dot, $V_{\mathrm{rf}}$ shows Coulomb oscillations once the source and drain barriers are sufficiently opaque. Tuning the sensor to a sensitive position (see circle) allows for charge sensing of the nearby double quantum dot.

Figure 7

(a) Honeycomb pattern of the double dot resolved using the sensing dot. Background oscillations are caused by an imperfect compensation of the response of the sensor to gates B1 and B2. (b) Direct transport measurement through the double dot. Coulomb peaks are visible only for not too negative voltages. (c) Same as in (a), but this time using the side gates S1 and S2 instead of B1 and B2. This typically reduces the background oscillation in the transconductance of the sensing dot. The different intensities of the lines delineating the honeycomb pattern reflect the transparency of the tunnel barriers to the external leads.