Electric G-Tensor Control in Quantum Systems

• Electric g-tensor control is the modulation of anisotropic Landé factors via electric fields to control quantum spin states.
• It leverages spin–orbit coupling and quantum confinement to achieve scalable, high-fidelity qubit manipulation in semiconductors and molecules.
• Device engineering, including strain tuning and symmetry breaking, enables rapid gate operations and robust resonance conditions for quantum applications.

Electric g-tensor control refers to the manipulation of the spin properties of quantum systems via modulation of the anisotropic Landé g-tensor using external electric fields. This approach enables high-fidelity, local, and scalable spin control without requiring oscillating magnetic fields, often leveraging spin-orbit coupling and quantum confinement effects. Electric g-tensor control is central to electrically-driven spin resonance (EDSR), qubit manipulation in low-dimensional semiconductors, molecular control for fundamental symmetry tests, and addressable manipulation of single atoms and artificial quantum structures.

1. Fundamental Principles of Electric G-Tensor Control

The g-tensor encodes the anisotropic coupling between the spin of a confined particle (electron, hole, or molecular spin) and an applied magnetic field. Its elements $g_{ij}$ generally depend on material composition, band structure, quantum confinement, strain, and, crucially, applied gate or electric fields. The Zeeman Hamiltonian for a Kramers doublet under electric and magnetic fields is:

$$H(\mathbf{E},\mathbf{B}) = \frac{\mu_B}{2}\,\boldsymbol{\sigma}\cdot \hat{g}(\mathbf{E})\mathbf{B}$$

With electric control, $\hat{g}(\mathbf{E})$ becomes a function of tunable gate or external fields, enabling direct electrical modulation of the effective Zeeman splitting and spin vector precession direction (Roloff et al., 2010). This sensitivity emerges through mechanisms such as:

• spin–orbit coupling, which links orbital and spin degrees of freedom,
• heavy-hole and light-hole mixing in the valence band,
• mixing of states with different orbital angular momenta or parity,
• electric-field-induced breaking of spatial or structural symmetries.

2. Mechanisms in Semiconductor Quantum Dots

2.1. Holes in Quantum Dot Molecules and Self-assembled Dots

In vertically stacked InAs/GaAs quantum dot molecules (QDMs), the $g$-tensor is electrically modulated by shifting the spatial distribution and the heavy-hole/light-hole admixture of the hole spin states. The principal components $g_{xx}(E)$, $g_{yy}(E)$, $g_{zz}(E)$ are highly anisotropic functions of electric field $E$, often described by quadratic polynomials (Pingenot et al., 2010). Purely electric universal single-qubit rotations are realized by process-tomography-optimized electric gate bias profiles, with high robustness: gate operation times of around 10 ns and fidelity loss under 1%, even in the presence of nuclear-spin and phonon-driven decoherence (Roloff et al., 2010).

For In$_{0.5}$Ga$_{0.5}$As/GaAs self-assembled dots, strong nonlinearities in the $g$-tensor components with respect to electric field yield both direct nonresonant control (via stepwise bias pulses) and subharmonic EDSR (g-tensor modulation resonance at $\omega_L/2$), with gate bias pulsation timescales down to tens of picoseconds (Pingenot et al., 2010).

2.2. g-Matrix Formalism and Symmetry Considerations

The $g$-matrix framework provides a compact, general model for electric g-tensor control, linking the Larmor and Rabi frequencies to both the $g$-tensor and its derivatives with respect to gate voltage:

$$f_R = \frac{\mu_B B V_{\mathrm{ac}}}{2 h g^*} \left|\left[\hat{g} \mathbf{b}\right]\times\left[\hat{g}' \mathbf{b}\right]\right|$$

where $g^* = |\hat{g}\mathbf{B}|/|\mathbf{B}|$ and $\hat{g}' = \partial \hat{g}/\partial V_g$ (Venitucci et al., 2018). Device symmetries, such as mirror planes, dictate the possible nonzero $g$-matrix elements and thus strongly influence the achievable Rabi rates and manipulation/decoupling points for qubit operation.

Quantum dots with engineered strain landscapes, as in Si MOS qubits, allow lateral displacement of the hole relative to nanometre-scale variations in HH-LH splitting, and gate-induced $g$-factor tuning by up to 500% is reported, with gate voltages shifting the in-plane $g$-tensor by up to $d g/dV \sim 8 \text{ V}^{-1}$ (Liles et al., 2020). Sweet spots with vanishing $d g/dV$ are observed, mitigating electrical noise-induced decoherence.

3. Experimental Realizations and Tuning Modalities

3.1. III-V Semiconductors and Quantum Nanowires

In As nanowire double quantum dots and InAs ring-like quantum dot structures, strong spin–orbit coupling and orbital effects result in giant, highly anisotropic effective $g$-tensors, tunable via gate-induced changes in the confinement potential or dot position. Experimentally, the principal values and axes of the $g$-tensor can be tuned in real time, with single- or double-dot devices demonstrating control from $g\approx 80$ down to near zero in the same charge state (Schroer et al., 2011, Potts et al., 2019). Hybridization and detuning protocols allow the orbital contribution to the $g$-tensor to be electrically quenched, enabling spin-state switching at fixed magnetic field.

3.2. Carbon Nanotube QDs and Gate-Based Axes Control

The $g$-tensor in bent carbon nanotube single and double quantum dots is electrically controlled by translating the dot along the tube via side- and back-gate voltages, rotating the principal axes and modulating anisotropy. Kondo peak splitting and cotunneling spectroscopy provide sensitive readout of the evolving $g$-tensor, which tracks the spatial geometry of the dot along the nanotube bend (Lai et al., 2012).

3.3. Quantum Point Contacts

In GaAs (311)A QPCs, the $g$-tensor possesses off-diagonal components $g_{xz}$ originating from HH-LH mixing and spin-orbit interactions unique to the interface orientation. Experimentally, both the magnitude and the sign of the $g$-factor are reversed by purely electric means via gate tuning, with quantitative linkages to the in-plane momentum and subband index. “Spin off” states with $g^*=0$ are accessible for qubit initialization (Srinivasan et al., 2017).

4. Advanced Theoretical Considerations and Formal Limitations

The standard g-tensor formalism ($g$-TF) is valid for resonant, monochromatic electric driving and for bichromatic modulation on a single gate. Expressing the Rabi frequency and Bloch–Siegert shifts in terms of $g$, $g'$, and $g''$ accurately predicts spin dynamics in these cases. However, for bichromatic driving with two distinct gates, $g$-TF breaks down: the Rabi frequency depends on additional all-electric parameters beyond $g$, $g'$, $g''$ arising from non-adiabatic effects in the Schrieffer–Wolff transformation, requiring a more generalized treatment, particularly for scalable crossbar architectures (György et al., 8 Apr 2025).

A compact summary table (columns: Regime, Validity of $g$-TF, Required Parameters):

Driving Regime	$g$-TF Suffices?	Additional Required Parameters
Monochromatic, 1 or 2 gates	Yes	$g$, $g'$
Bichromatic, single gate	Yes	$g$, $g'$, $g''$
Bichromatic, two distinct gates	No	$g$, $g'$, $g''$, plus $\vec{\Upsilon}$

5. G-Tensor Control in Molecules and Single Atoms

Electric g-tensor control extends beyond semiconductor systems:

• In YbOH, a candidate for electron EDM searches, electric fields mix $l$-doublet parity states, minimizing the $g$-factor difference $\Delta g$ between levels. Stark mixing leads to tunability of $\Delta g$ to $<4 \times 10^{-5}$ relative to $g$, suppressing magnetic field systematics in high-precision measurements (Petrov, 5 Aug 2024).
• For single adatoms (e.g., Ti–H on MgO), an RF STM tip drives electrically induced piezoelectric displacement, which modulates the crystal field and thus $g$-tensor anisotropy. This mechanism yields Rabi frequencies of order 0.5–1 MHz (for Ti–H), with possible enhancement for heavier adatoms with larger spin-orbit coupling (Ferrón et al., 2019).

6. Material and Device Engineering for Optimized G-Tensor Modulation

Optimal electric g-tensor control requires:

• Strong spin–orbit coupling (holes in III–V and group-IV materials),
• Deliberately engineered strain or symmetry breaking for maximal $g$-tensor response,
• Device geometries that maximize the coupling of electric fields to the relevant orbital or spin degrees of freedom (e.g., lateral confinement direction near [110] in Si for enhanced $\partial g/\partial E$ (Qvist et al., 2021)),
• Gate architectures that facilitate both large-tunability points (“manipulation” bias) and sweet spots for noise robustness (“decoupling” bias) (Liles et al., 2020, Venitucci et al., 2018).

Material-specific considerations include the role of Luttinger parameter anisotropy, relative strengths of heavy-hole and light-hole contributions, and the presence of parasitic Dresselhaus SOI and nuclear spins.

7. Applications and Outlook

Electric g-tensor control provides a scalable and rapid interface to qubit manipulation and readout in a broad set of solid-state, molecular, and atomic platforms. Its key roles include:

• Universal single-qubit gate operations with fidelities exceeding 99% and operation times below 10 ns in QDMs (Roloff et al., 2010).
• Qubit initialization, state transfer, and noise-robust “decoupling” via tunable sweet spots in silicon devices (Liles et al., 2020).
• Suppression of systematic errors in eEDM searches by minimization of Stark-tunable $g$-factor differences (Petrov, 5 Aug 2024).
• Local addressability of spins in crossbar-type quantum computing architectures and STM-driven manipulation of individual adatoms (György et al., 8 Apr 2025, Ferrón et al., 2019).

Ongoing research is elucidating the limits of $g$-tensor-based models for increasingly complex drive schemes and is guiding the development of devices with engineered $g$-tensor landscapes for both quantum information processing and precision measurement applications.