Coupling a $^{73}$Ge nuclear spin to an electrostatically defined quantum dot

TL;DR

This work demonstrates coupling between a single isoelectronic $^{73}$Ge nuclear spin and the electron spin of a gate-defined SiMOS quantum dot, achieving tunable hyperfine interaction via electrostatic gate control. By implanting $^{73}$Ge near the Si/SiO$_2$ interface and operating a double quantum dot at millikelvin temperatures, the authors observe ten ESR transitions spaced by the hyperfine constant $A$, consistent with a spin-9/2 nucleus ($I=9/2$). The hyperfine coupling is tunable from approximately $A \in [179, 357]$ kHz by shifting the electron density toward the Ge nucleus, enabling access to the full spin-9/2 Hilbert space for future high-dimensional quantum information processing. The results establish a foundation for spin control of high-dimensional nuclear-spin qudits, with potential applications in entanglement distribution and repeated weak measurements, while benefiting from reduced decoherence during electron shuttling afforded by isoelectronic species.

Abstract

Single nuclear spins in silicon are a promising resource for quantum technologies due to their long coherence times and excellent control fidelities. Qubits and qudits have been encoded on donor nuclei, with successful demonstrations of Bell states and quantum memories on the spin-1/2 $^{31}$P and cat-qubits on the spin-7/2 $^{123}$Sb nuclei. Isoelectronic nuclear spins coupled to gate-defined quantum dots, such as the naturally occurring $^{29}$Si isotope, possess no additional charge and allow for the coupled electron to be shuttled without destroying the nuclear spin coherence. Here, we demonstrate the coupling and readout of a spin-9/2 $^{73}$Ge nuclear spin to a gate-defined quantum dot in SiMOS. The $^{73}$Ge nucleus was implanted by isotope-selective ion-implantation. We observe the hyperfine interaction (HFI) to the coupled quantum dot electron and are able to tune it from 180 kHz to 350 kHz, through the voltages applied to the lateral gate electrodes. This work lays the foundation for future spin control experiments on the spin-9/2 qudit as well as more advanced experiments such as entanglement distribution between distant nuclear spins or repeated weak measurements.

Figures (3)

• Figure 1: Device and electron-nucleus coupled system. (a) Scanning electron micrograph of a nominally identical device. The four gate deposition layers are highlighted in green, red, purple, and pink, respectively. Arrows indicate the directions of the external applied d.c. magnetic field $B_{0}$ and the a.c. magnetic field $B_{1}$ the antenna generates. The formed quantum dots under the plunger gates P1 and P2 are indicated by the blue and red circle, respectively. The green circles indicate a possible isotopically-selected $^{73}$Ge nuclei distribution in the implantation window depicted by the green dashed box. The device is operated at the dilution refrigerator base temperature of $T=50m K$. (b) Schematic of the device's cross-section in the active region, marked as the dashed white line in (a). We form the two-dimensional electron gas at the Si/SiOx interface. The single-electron transistor is confined with the SLB, SRB, and SB. The quantum dots to form qubits are laterally confined with the LCB and RCB. The single-electron dot under P1 reveals a strongly coupled $I=9/2$ nuclear spin. (c) Energy level diagram of an isoelectronic $^{73}$Ge nucleus ($I=9/2$) coupled to an electron spin ($S=1/2$) of a quantum dot. The Zeeman interaction splits the states into an electron spin-up ($m_{\mathrm{S}}=1/2$) and electron spin down ($m_{\mathrm{S}}=-1/2$) manifold, which is then further split by the hyperfine interaction based on the nuclear spin quantum number $m_{\mathrm{I}} \in \{-9/2,-7,2,...,7/2,9/2\}$ and the hyperfine interaction strength $A$. In this coupled system, the electron spin resonance (ESR) frequency $f_{\mathrm{ESR}}$ depends on the nuclear spin orientation. All $2I+1=10$ different ESR transitions are equally spaced by $A$.
• Figure 2: Nuclear spin signature in ESR. (a) Normalized even electron spin probability $P_{\mathrm{even}}$ for repeated electron spin resonance probing as a function of applied microwave signal frequency. Each frequency step is averaged over 500 shots. (b) Line-cut of the normalized electron spin-up probability $P_{\mathrm{even}}$ at repetition #1 at the dashed black line in (a). The sum of 10 Lorentzian peaks (red line) fitted with equal spacing $A$. The peak spacing is the hyperfine interaction strength $A$ between the electron and nuclear spin. The dotted lines are the individual Lorentzian peak fits. The local oscillator frequency of qubit 2 is $f_{\mathrm{LO,Q2}} = 8.401G Hz$. (c)-(h) Line-cut of the normalized electron spin-up probability $P_{\mathrm{even}}$ at repetitions #2--7 at the dashed black lines in (a). Hyperfine interaction ranges from A = 353 ± 2--357 ± 2k Hz across all repeats. Error bars represent a 95% confidence interval determined from the estimated covariance matrix.
• Figure 3: Hyperfine interaction tuning. (a) Simulated electron wavefunction of the double-dot system for three different confinement voltages $\Delta V_{\mathrm{CB}}=-100m V, 100m V, 300m V$. The white dashed lines indicate the respective wavefunction centre. (b) Hyperfine interaction strength $A$ of the quantum-dot-coupled $^{73}$Ge nuclear spin system as a function of confinement voltage $V_{\mathrm{CB}}$ (lower x-axis) and simulated mean of the electron density $\braket{y}$ (upper x-axis). The confinement voltage $V_{\mathrm{CB}}$ is the difference of the right and left confinement barrier voltage. Qualitatively, this corresponds to moving the center of the electron wavefunction closer to the $^{73}$Ge nuclear spin, leading to a larger Fermi-contact hyperfine interaction component. The red line is a linear fit with slope $m=473k Hz / V$ and vertical intercept of $A = 227k Hz$. Error bars represent a 95% confidence interval when determining the hyperfine constant from Larmor frequency time traces.