Sweet-spot protection of hole spins in sparse arrays via spin-dependent magneto-tunneling

TL;DR

The paper introduces a microscopic spin-dependent magneto-tunneling mechanism in a minimal hole-spin double quantum dot, revealing corrections to tunnel couplings that renormalize g-factors and create robust sweet spots even with dot-to-dot g-factor mismatches. By deriving an effective Hamiltionian and performing realistic Ge device simulations, the authors show that magneto-tunneling produces maxima and higher-order sweet spots in detuning, and modifies Rabi frequencies, thereby impacting shuttling, hopping, and flopping-mode qubits. The work provides closed-form expressions linking the magneto-tunneling terms to observable quantities and demonstrates consistency with experimental observations of unexpected sweet spots. The findings have broad implications for scaling hole-spin qubits in sparse arrays and suggest strategies to engineer robust operating points in larger quantum-dot networks.

Abstract

Recent advances in the scaling of spin qubits have led to the development of sparse architectures where spin qubits are distributed across multiple quantum dots. This distributed approach enables qubit manipulation through hopping and flopping modes, as well as protocols for spin shuttling to entangle spins beyond nearest neighbors. Therefore, understanding spin tunneling across quantum dots is fundamental for the improvement of sparse array encodings. Here, we develop a microscopic theory of a minimal sparse array formed by a hole in a double quantum dot. We show the existence of spin-dependent magnetic corrections to the tunnel couplings that help preserve existing sweet spots, even for quantum dots with different $g$-factors, and introduce new ones that are not accounted for in the simplest models. Our analytical and numerical results explain observed sweet spots in state-of-the-art shuttling and cQED experiments, are relevant to hopping and flopping modes, and apply broadly to sparse array encodings of any size.

Figures (6)

• Figure 1: Energy levels of a hole spin in a DQD. (a) Illustrative representation of the in-plane confinement potential and energy levels in a quasi-2D DQD strongly confined along $z$. The two dots, aligned along the $x$ axis, are detuned by the energy $\varepsilon$. Each dot is characterized by its respective $g$-matrix $\hat{g}_\mathrm{L,R}$. Tunneling between the two dots from spin $s$ to spin $s'$ ($\tau_{s,s'}$) acquires a magnetic dependence through the $g$-matrix $\hat{g}_\mathrm{T}$ (we take $\boldsymbol\mu_\mathrm{T}=\mathbf{0}$ for illustration). (b,c) Energy levels as a function of detuning for a case with large spin-charge mixing (b) and low spin-charge mixing (c). The parameters $t_c=9.6$ GHz, $t_0=1.12$ GHz, $\mathbf{t}_\mathrm{so}=(3.2,8.9,1.0)$ GHz are taken from the experiment in Ref. yu2023strong, $\hat{g}_\mathrm{L}$ and $\hat{g}_\mathrm{R}$ are explicitly given in pseudofootnote, $\mathbf{B}=(0.78,0,0)$ T in (b), and $\mathbf{B}=(0.11,0.25,0)$ T in (c). (d) Energy splitting $E_{01}$ between the lowest two spin states as a function of detuning for (b) (solid black) and (c) (solid blue). In dashed lines, we show the corrected energies after the inclusion of the magneto-tunneling mechanism with an isotropic $g$-matrix $\hat{g}_\mathrm{T}=0.2\mathds{1}$, recovering a sweet spot for the low spin-charge mixing case.
• Figure 2: Simulated $g$-factors of Ge DQD devices. (a) Top view of the simulated Germanium DQD device. $V_\mathrm{L}$ and $V_\mathrm{R}$ are the plunger gate voltages of left and right dots, while $V_{b}$ is the barrier gate voltage. The other side gates can be used to tune the asymmetry of the potential landscape. (b-d) Effective $g$-factors $g^{(\mathbf{b})}$ as a function of detuning for $\mathbf{b}=\mathbf{e}_x$ (b), $\mathbf{b}=\mathbf{e}_y$ (c), and $\mathbf{b}=\mathbf{e}_z$ (d), where $\mathbf{e}_j$ is the unit vector along the $j$ direction, in a symmetric configuration with $V_\mathrm{L}=V_\mathrm{R}=-50$ mV and $t_c=9.6$ GHz at $\varepsilon=0$. The black lines are the effective $g$-factors of the ground Kramers (GK) pair, while the blue and orange lines are those of the Wannierized left and right orbitals, respectively. Equivalently, (e-g) are the effective $g$-factors in an asymmetric configuration with $V_\mathrm{L}=-56.9$ mV, $V_\mathrm{R}=-43.1$ mV, $t_c=9.6$ GHz at zero detuning, and the top and bottom gates around the left plunger squeezing and displacing the dots ($V_{\mathrm{u,L}}=15$ mV and $V_{\mathrm{l,L}}=12$ mV).
• Figure 3: Tunability of the $g$-factors. (a-c) Effective $g$-factors $g^{(\mathbf{b})}=|\hat{g} \mathbf{b}|$ as a function of detuning for $\mathbf{b}=\mathbf{e}_x$ (a), $\mathbf{b}=\mathbf{e}_y$ (b), and $\mathbf{b}=\mathbf{e}_z$ (c), and for different values of the tunnel coupling in the same asymmetric configuration as in Fig. \ref{['fig:simexamples']}. We mark detuning sweet spots with vertical lines, where dotted lines indicate maxima and dashed lines indicate minima. (d) Effective tunneling $g$-factors $g^{(\mathbf{b})}_\mathrm{T}=|\hat{g}_\mathrm{T} \mathbf{b}|$ as a function of tunnel coupling $t_c$ for the symmetric configuration in Fig. \ref{['fig:simexamples']}. (e-f) Effective in-plane $g$-factors, $g^{(x)}$ (e) and $g^{(y)}$ (f) for symmetric configurations with different plunger gates voltages $\bar{V}=V_\mathrm{L}=V_\mathrm{R}$ with $t_c=9.6$ GHz.
• Figure 4: Higher-order sweet spots and in-plane $g$-factor behavior. (a,b) Maps of the effective $g$-factor $g^{(\mathbf{b})}=|g \mathbf{b}|$ as a function of in-plane magnetic field orientation $\phi$ and detuning energy $\varepsilon$ for the symmetric and asymmetric configurations, respectively. Sweet spot positions are marked with white lines. (c,d) Effective $g$-factor as a function of detuning for magnetic field orientations where the $g$-factors exhibit a high-level of flatness.
• Figure 5: Sweet spots and Rabi frequencies accounting for the magneto-tunneling term. (a) Sweet spot position $\varepsilon_{\mathrm{SS}}/2t_c$ with $\boldsymbol{\omega}_+=|\boldsymbol{\omega}_+|\mathbf{e}_{z}$ and $\boldsymbol{\omega}_-=\delta_-|\boldsymbol{\omega}_+|(\cos\theta_\pm\mathbf{e}_{z}+\sin\theta_\pm \mathbf{e}_{x})$, as a function of the angle $\theta_\pm$. Solid lines are the case $\tilde{g}_\mathrm{T}=0$ and were solved using Eq. (\ref{['eq:toysweet']}). Dashed lines are the case $\boldsymbol{\omega}_\mathrm{T}\approx0.05\boldsymbol{\omega}_+$ and were obtained with Eq. (\ref{['eq:sweetverygen']}). (b) Same as (a) but as a polar coordinate plot. Note that the sweet spots for $\boldsymbol{\omega}_\mathrm{T}\approx-0.05\boldsymbol{\omega}_+$ can be obtained by the symmetry transformation $\theta_{\pm}\rightarrow \theta_{\pm}+\pi$. (c) Rabi frequency normalized to the driving amplitude $(\delta\varepsilon/h)$ and to the applied magnetic field $\mathbf{B}$, as a function of detuning. The chosen parameters are $t_c=9.6$ GHz, $\boldsymbol{\omega}_+=0.2\mathbf{e}_{x}$, and $\boldsymbol{\omega}_-=0.1\mathbf{e}_{y}$. For the case $\boldsymbol{\omega}_\mathrm{T}\parallel \boldsymbol{\omega}_+$, we choose $\boldsymbol{\omega}_\mathrm{T} \approx 0.05\mathbf{e}_x$, while for the case $\boldsymbol{\omega}_\mathrm{T}\perp \boldsymbol{\omega}_+$, we choose $\boldsymbol{\omega}_\mathrm{T}=0.05\mathbf{e}_{z}$.