3D tomography of exchange phase in a Si/SiGe quantum dot device

Abstract

The exchange interaction is a foundational building block for the operation of spin-based quantum processors. Extracting the exchange interaction coefficient $J(\mathbf{V})$, as a function of gate electrode voltages, is important for understanding disorder, faithfully simulating device performance, and operating spin qubits with high fidelity. Typical coherent measurements of exchange in spin qubit devices yield a modulated cosine of an accumulated phase, which in turn is the time integral of exchange. As such, extracting $J(\mathbf{V})$ from experimental data is difficult due to the ambiguity of inverting a cosine, the sensitivity to noise when unwrapping phase, as well as the problem of inverting the integral. As a step toward obtaining $J(\mathbf{V})$, we tackle the first two challenges to reveal the accumulated phase, $φ(\mathbf{V})$. We incorporate techniques from a wide range of fields to robustly extract and model a 3D phase volume for spin qubit devices from a sequence of 2D measurements. In particular, we present a measurement technique to obtain the wrapped phase, as done in phase-shifting digital holography, and utilize the max-flow/min-cut phase unwrapping method (PUMA) to unwrap the phase in 3D voltage space. We show this method is robust to the minimal observed drift in the device, which we confirm by increasing scan resolution. Upon building a model of the extracted phase, we optimize over the model to locate a minimal-gradient $π$ exchange pulse point in voltage space. Our measurement protocol may provide detailed information useful for understanding the origins of device variability governing device yield, enable calibrating device models to specific devices during operation for more sophisticated error attribution, and enable a systematic optimization of qubit control. We anticipate that the methods presented here may be applicable to other qubit platforms.

Figures (6)

• Figure 1: Device operation. (a) Schematic illustration of a 12-QD Intel Si/SiGe Tunnel Falls device with magnified view of device region used for this experiment. QDs are formed under plunger gates (P1, P2, P3) on the top half of the device, as indicated by green (addressed during state preparation and measurement) and blue (gauge spin) circles, and are separated by barrier gates (B0, B1, B2). (b) Schematic electrostatic potential diagram for the experimental section of the device under idle conditions (grey) and during an exchange pulse (black). (c) Qubit bloch sphere, where $J_n$ is the exchange interaction between QD2 and QD3, and $J_z$ is the exchange interaction between QD1 and QD2. (d) Example pulse control sequence during a phase-shifted fingerprint measurement.
• Figure 2: Measurements. In figure (a) we have an example of a phase raster scan at fixed $V_{B2}$ greater than $V_{B2_{\textrm{idle}}}$. In figure (b) we have an example of a fingerprint scan taken at $0\degree$ rotation in detuning space, c.f. (c). In figure (c) we show the trajectories through detuning space used to collect fingerprints in (d). Each black line indicates the $V_{P2}$ and $V_{P3}$ coordinates along the detuning ($V_{P2}$-$V_{P3}$) axis of a fingerprint corresponding to that detuning angle. Trajectories are plotted over a phase raster measured with $V_{B2} = V_{B2_{\textrm{idle}}}$ and prolonged wait times to show the stable detuning limits for this charge state, for reference. The red circle indicates a cylindrical core of data removed when model fitting. Figure (d) shows the fingerprint scans as cross-sectional scans in 3D voltage space, for $0\pi$ phase shift.
• Figure 3: Data processing pipeline for $0\degree$ detuning. The pipeline steps leading up to a mask applied to the wrapped phase. We use the masks on the unwrapped phases to fit our model to the full 3D data.
• Figure 4: Model. In (a), we show a rendering of the 3D phase spline model. The mesh outline indicates the convex hull of the confident phase region. From the bottom contour to top, the contours represent $0.1$, $\pi / 4$, $\pi / 2$, and $\pi$ radians. We have placed a small marble at the location of the symmetric operating point for a $\pi$ exchange pulse where phase gradients with respect to P2 and P3 vanish. In (b), we show a volumetric slice comparison of the experimental data and the spline model.
• Figure 5: Time-dependence of phase shift operations on the prepared singlet state. Phase shift outcomes are calculated based on the singlet return probability for 11 random idle coordinates, defined by $V_{B2}=$0 V and $\lvert V_{P2} - V_{P3}\rvert <$ 14 mV, in each fingerprint measurement. Error bars represent standard deviation. To partially correct for drift in the SET output signal, SET reference measurements are recorded during qubit reset, and the difference in SET output during reset and measurement is pre-calibrated and assumed to be constant. (a) Unreferenced and (b) Referenced measurements of each phase shift outcome, as a function of time since the first measurement in a series for a given phase shift condition. (c) Unreferenced and (d) Referenced measurements of each phase shift outcome as a function of detuning angle condition.