Switchable spin-photon coupling with hole spins in single-quantum dots

TL;DR

The paper develops a comprehensive 2D hole-spin model for a single quantum dot coupled to a cavity, explicitly including strain-induced inhomogeneous SOC and g-tensor variations. It identifies three spin-photon coupling channels—a vector-potential–SOC interference term, an inhomogeneous Rashba term, and strain-induced g-tensor modulation—and shows how their relative importance depends on material (Si, biaxially strained Ge, unstrained Ge) and magnetic-field orientation. For unstrained Ge, couplings reach tens of MHz, with gate-tunable switching achieved by detuning the qubit from the cavity through harmonic-length control while preserving sweet spots. Quantum-state transfer and photon-mediated two-qubit gates are analyzed, yielding >99% fidelity for QST and >90% for CZ/dispersive gates under realistic coherence times and cavity decay, positioning single-dot hole spins in unstrained Ge as a promising, compact spin-cQED platform.

Abstract

Spin qubits in semiconductor quantum dots offer a gate-tunable platform for quantum information processing. While two-qubit interactions are typically realized through exchange coupling between neighboring spins, coupling spin qubits to photons via hybrid spin-cQED devices enables long-range interactions and integration with other cQED platforms. Here, we investigate hole spin-photon coupling in compact single quantum dot setups. By incorporating ubiquitous strain inhomogeneities to our theory, we identify three main spin-photon coupling channels: a vector-potential-spin-orbit geometric mechanism--dominant for vertical magnetic fields--, an inhomogeneous Rashba term generalizing previous spin-orbit field models, and strain-induced $g$-tensor terms--most relevant for in-plane fields. Comparing Si, unstrained (relaxed) Ge, and biaxially strained Ge wells, we find that Si and unstrained Ge provide optimal coupling strengths (tens of MHz) thanks to their reduced heavy-hole, light-hole splitting. We demonstrate efficient switching of the spin-photon coupling while preserving sweet spot operation. Finally, we evaluate quantum state transfer and two-qubit gate protocols, achieving $>99\%$ fidelity for state transfer and $>90\%$ for two-qubit gates with realistic coherence times, establishing single-dot hole spins as a viable platform for compact spin-cQED architectures and highlighting unstrained Ge as a promising candidate for spin-photon interactions.

Figures (6)

• Figure 1: Hybrid spin-photon system. (a) Anisotropic 2D harmonic quantum dot with lengths $\ell_{x,y}=\sqrt{\hbar/(m_\parallel\omega_{x,y})}$. The cavity field displaces the hole wave function along $x$ by $eF_\text{zpf}(a+a^\dagger)$, where $a^\dagger$ ($a$) creates (annihilates) a cavity photon. A magnetic field $\mathbf{B}=B(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ sets the Zeeman splitting of the ground-state spin doublet. (b) Shear-strain component $\varepsilon_{xz}(x,y)$ at $z=0$ for the strained Ge/GeSi device of Ref. abadillo2023hole with a well thickness of 16 nm, a circular Al gate of 50 nm radius (grey circle) separated from the quantum well by 5 nm of insulating Al$_2$O$_3$ and 50 nm thick upper GeSi barrier. (c) Inverse inhomogeneous spin-orbit length $1/\ell_\text{so}^{(x)}(x,y)$ at $z=0$ for motion along $x$ in the same device.
• Figure 2: Larmor frequency and spin-photon coupling for Ge with and without biaxial strain. The dot is circular with $\ell_x=\ell_y=30$ nm and $F_\text{zpf}=30$ V/m. We take $\Delta_\mathrm{LH}=70$ meV (strained Ge) and $\Delta_\mathrm{LH}=3.5$ meV (unstrained Ge) costa2025buried, and $\beta_{yx}^\text{(uGe)}\approx -4.1\times 10^{-6}$ nm$^{-3}$; $\beta_{yx}^\text{(Ge)}\approx -0.31\times 10^{-6}$ nm$^{-3}$; $p_{xz}^\text{(uGe)}\approx 3.9\times 10^{-6}$ nm$^{-1}$; $p_{xz}^\text{(Ge)}\approx 4.9\times 10^{-6}$ nm$^{-1}$; see Table \ref{['tab:compact_fits']}. For each case, the plots are symmetric under $\phi\!\to\!-\phi$. (a) $|\boldsymbol{\omega}_L|/2\pi$ versus magnetic field orientation $\mathbf{B}=B(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ at $B=0.2$ T. (b) Magnetic field required to reach $|\boldsymbol{\omega}_L|/2\pi=2.5$ GHz as a function of field orientation. (c) Transverse coupling $|\boldsymbol{\lambda}_\perp|/2\pi$ versus field orientation at $B=1$ T. (d) Longitudinal coupling $\lambda_\parallel\cos\varphi_\parallel/2\pi$ versus field orientation at $B=1$ T. Note the two different scales in the colorbar referring to the strained (left) and unstrained (right) cases.
• Figure 3: Transverse spin-photon coupling and required magnetic field versus harmonic confinement. Parameters: $F_\text{zpf}=30$ V/m for Ge and uGe devices; $F_\text{zpf}=90$ V/m since Si lever arms can be three times larger than in Ge yu2023strong$; \Delta_\mathrm{LH}=70$ meV (strained Ge) and $\Delta_\mathrm{LH}=3.5$ meV (unstrained Ge and Si); $\beta_{yx}^\text{(Si)}\approx -3.8\times 10^{-6}$ nm$^{-3}$; see Table \ref{['tab:compact_fits']}; and we assume the resonance condition $\omega_R=|\boldsymbol{\omega}_L|$. (a) $|\boldsymbol{\lambda}_\perp|/2\pi$ versus harmonic length for circular dots ($\ell_x=\ell_y=\ell$) at $B_z=0.1$ T (solid) and $0.2$ T (dashed), from the analytical formulas in Eq. \ref{['eq:specificspinphoton']}. The data points show tight-binding simulations based on Eq. \ref{['eq:hybrid2D']} (Appendix \ref{['app:TB']}). (b) As in (a) but for in-plane fields $B_x=0.5$ T (solid) and $1$ T (dashed). (c) $|\boldsymbol{\lambda}_\perp|/2\pi$ versus anisotropy ratio $\ell_y/\ell_x$ at fixed $\ell_x=15$ nm and $B_x=1$ T. (d) Magnetic field along $x$ required to reach $|\boldsymbol{\omega}_L|/2\pi=2.5$ GHz versus $\ell_x$ for Si (blue), Ge (orange), and unstrained Ge (green), for several anisotropy ratios $\ell_y/\ell_x$.
• Figure 4: Gate control of $|\boldsymbol{\omega}_L|$ and spin-photon couplings in planar devices for circular dots and with parameters relevant for Ge. In all panels: $\ell_x=\ell_y=30$ nm and $F_\text{zpf}=30$ V/m; $B_x=1$ T except in (e) and (f), where $B_y=1$ T. (a) Change in Larmor frequency $\Delta|\boldsymbol{\omega}_L|/2\pi$ versus in-plane electric field $F_x$. (b) $|\boldsymbol{\lambda}_\perp|/2\pi$ (solid, left axis) and $\lambda_\parallel/2\pi$ (dashed, right axis) versus $F_x$. (c,d) Same as (a,b) but versus $F_y$. (e,f) Same as (a,b) for diagonal bias $F_x=F_y$ with $B_y=1$ T. (g,h) Same as (a,b) but versus harmonic length.
• Figure 5: Quantum state transfer and two-qubit gate protocols. (a) Pulse sequences: QST between qubit and photon (top), sideband-based CZ (middle), and dispersive $\sqrt{i\text{SWAP}}$ (bottom). Time and frequency are schematic. (b) QST average infidelity versus cavity decay rate $\kappa'$ for different $T_2^*$. (c) CZ average infidelity versus $\kappa'$ for the same $T_2^*$. (d) $\sqrt{i\text{SWAP}}$ average infidelity versus $\kappa'$ for the same $T_2^*$.