Electron readout contrast enhancement in the parallel nuclear regime of an exchange-coupled donor spin qubit system

TL;DR

This paper addresses the unexplained enhancement of electron readout contrast in exchange-coupled donor spin qubits when donor nuclei are in a parallel configuration. By modeling the two-electron energy spectrum in the $J/|\Delta| \gg 1$ parallel regime and conducting SET-based readout experiments on a pair of $^{31}$P donors, the authors show that an additional tunneling event to the SET within a single readout window accounts for the observed contrast boost. The key finding is that the entangled $\widetilde{|S\rangle}$ and $\widetilde{|T_0\rangle}$ states enable a second blip, improving spin-up detection fidelity, a result supported by detailed blip-count statistics and adiabatic ESR spectroscopy. Overall, the work provides a concrete mechanism for improved readout fidelity and enhances understanding of spin-dependent tunneling in donor-based quantum devices, with implications for scalable silicon quantum processors.

Abstract

Recent experiments on donor-based spin qubits in silicon have leveraged the exchange interaction between electrons bound to separate donor nuclei to perform two-qubit operations. A consistently observed yet unexplained phenomenon in such systems is the significant increase in electron readout contrast, measured via Elzerman-style readout to a single-electron transistor (SET) island, when the donor nuclei are initialized in a parallel spin orientation compared to an anti-parallel orientation. In this work, we present a detailed analysis of the exchange-coupled donor system in the parallel nuclear regime and propose a physical mechanism for this effect. We attribute the enhanced readout contrast to an additional electron tunneling event to the SET during a single read period, when the donor nuclei are aligned in a parallel spin configuration. These insights inform strategies for improving electron readout fidelity in these systems and contribute to a more complete understanding of spin-dependent tunnelling processes in donor-based qubit architectures.

Figures (5)

• Figure 1: Eigenstates of the exchange-coupled donor system. (a) Schematic of the exchange-coupled donor system, consisting of two donor nuclei: n$_{1}$ and n$_{2}$, each possessing a bound electron, e$_{1}$ and e$_{2}$, respectively. These electrons are coupled together with an exchange-interaction, $J$. (b) Projection of one of the electron eigenstates of the exchange-coupled system onto the pure state $\ket{\uparrow \downarrow}$ and onto the $\ket{T_0}$ state, as a function of the ratio between the electron coupling strength, $J$, and the electron detuning, $|\Delta|$. The black dashed lines show the typical $J/|\Delta|$ values for the case of the nuclei in either a parallel (10$^{-1}$) or anti-parallel (10$^{2}$) spin orientation. (c) Projection of the other electron eigenstates of the exchange-coupled system onto the pure state $\ket{\downarrow \uparrow }$ and onto the $\ket{S}$ state as a function of the ratio between the electron coupling strength, $J$, and the electron detuning, $|\Delta|$. (d) Energy eigenstates of the electrons in the parallel nuclei regime, where $J/|\Delta| \approx 10^{-1}$ . $\ket{T_{-}} = \ket{\downarrow \downarrow}$ and $\ket{T_{+}} = \ket{\uparrow \uparrow}$ represent two of the triplet states, while $\widetilde{\ket{S}}$ and $\widetilde{\ket{T_{0}}}$ represent a hybridised 'singlet-like' or 'triplet-like' state. (e) Schematic of the expected ESR transitions present in the parallel nuclear regime.
• Figure 2: Tunneling to the SET in the parallel vs anti-parallel nuclear regime. (a) Two-electron energy levels with respect to the electrochemical potential of the SET, $\mu_{\text{SET}}$. Labels (i - iv) depict the numbered steps associated with the electron readout process for the case of the nuclei in a parallel spin orientation. The subscripts for each of the two-electron energy levels depict the initial (green) and final (yellow) states following a tunneling event. (b) Tunneling scheme for the case of the anti-parallel nuclei. In this case the eigenstates of the two electrons are separable and hence the tunneling scheme of electron 1 is the same as for a single-donor electron. (c) Histogram of average number of blips per readout for the case of the nuclei being in a parallel (purple) or anti-parallel (red) state. The dashed lines represent the mean value of each histogram. The error bar for this value, and throughout this work, is given by one standard deviation, $1\sigma$. Histogram of the number of blips per readout for the case of the (d) anti-parallel or (e) parallel nuclei. (f) Histogram of the average number of blips per readout trace when initializing in either the $\ket{T_-}$, $\ket{T_{X}}$ or $\ket{T_+}$ state, where $\ket{T_{X}}$ is a mixture of $\widetilde{\ket{{T_0}}}$, $\ket{T_-}$ and $\ket{T_+}$ (see Appendix C). (g),(h),(i) Histogram of the number of blips per readout for the case of parallel nuclei, when the electrons are initialized in a (g) $\ket{T_-}$, (h) $\widetilde{\ket{{T_0}}}$ and (i) $\ket{T_+}$ state. (j) Adiabatic frequency spectrum of one of the electrons in the exchange-coupled donor pair, for the case of the nuclei being initialized in either an anti-parallel spin orientation (red line) or a parallel spin orientation (purple line). The black dashed line shows the expected up proportion for the case of the parallel nuclear regime, given in equation \ref{['up_proportion_formula']}, where $P_{(\Downarrow \Downarrow / \Uparrow \Uparrow)}$ and $P_{(\Downarrow \Uparrow/\Uparrow \Downarrow)}$ represent the electron spin up proportions for the parallel and anti-parallel nuclear states, respectively.
• Figure 3: Elzerman-style readout of the donor electron spin. (a) False-colored scanning electron microscope image of the gate structure of a device nominally the same as the device used in this work. The donor atom schematics shows the location of the donor implantation window in the device. The single-electron transistor (SET) region is highlighted by the dashed black box. (b) Schematics of the Fermi-Dirac distributions that describe the distribution of states in the SET island for the case of a temperature of T = 0 K (left) and temperature T > 0 K (right). $\mu_{\text{SET}}$ represents the electrochemical potential of the SET island. (c) SET current trace during readout, showing a 'blip' of current, indicating the tunneling of an $\ket{\uparrow}$ electron to the SET island and re-initialization into the $\ket{\downarrow}$ state. The dashed green line indicates an example of a threshold current, which is used to determine whether or not a blip occurred during a readout trace.
• Figure 4: Tunneling time calculation. (a) 100 raw SET current traces. (b) Histogram of the tunneling time of the electron from the SET island to the donor taken from 100 measured SET current traces. The red line shows the exponential fit to this histogram, from which we extracted the tunnel time. The blue box shows the bandwidth of the instruments, which was approximately 50 kHz in these experiments. Durations below the bandwidth are not accurately determined and are denoted with a dashed box.
• Figure 5: Electron spin time evolution simulation in the parallel nuclear regime. (a) Simulation of the driving of the $\alpha$ transition, at a frequency centered between the $\ket{T_-}$$\leftrightarrow$$\ket{T_0}$ and $\ket{T_0}$$\leftrightarrow$$\ket{T_+}$ transition, which differ due to a different hyperfine between the two donors. The simulation was carried out with $J$ = 10 MHz and $\Delta A$ = 90 kHz and a Rabi frequency of $f_R =$ 1 MHz. The electrons were initialized in the $T_- = \ket{\downarrow_1 \downarrow_2}$ state before applying the driving pulse. This plot shows the projection of the two electrons along the Z-axis during the time evolution. (b) The same time evolution shown in (a) but with the projection on each of the two-electron eigenstates plotted over time. Vertical black dashed line denotes the $\frac{\pi}{2}$ duration at which the state $\ket{T_{X}}$ is prepared. (c) Husimi functions of the $\ket{T_-}$,$\ket{T_{X}}$ and $\ket{T_{+}}$ states.