Geometry of the dephasing sweet spots of spin-orbit qubits

Abstract

The dephasing time of spin-orbit qubits is limited by the coupling with electrical and charge noise. However, there may exist ``dephasing sweet spots'' where the qubit decouples (to first order) from the noise so that the dephasing time reaches a maximum. Here we discuss the nature of the dephasing sweet spots of a spin-orbit qubit electrically coupled to some fluctuator. We characterize the Zeeman energy ${E}_{\mathrm{Z}}$ of this qubit by the tensor $G$ such that ${E}_{\mathrm{Z}}={\ensuremath{\mu}}_{B}\sqrt{{\mathbf{B}}^{\mathrm{T}}G\mathbf{B}}$ (with ${\ensuremath{\mu}}_{B}$ the Bohr magneton and $\mathbf{B}$ the magnetic field), and its response to the fluctuator by the derivative ${G}^{\ensuremath{'}}$ of $G$ with respect to the fluctuating field. The geometrical nature of the sweet spots on the unit sphere describing the magnetic field orientation depends on the sign of the eigenvalues of ${G}^{\ensuremath{'}}$. We show that sweet spots usually draw lines on this sphere. We then discuss how to characterize the electrical susceptibility of a spin-orbit qubit with test modulations on the gates. We apply these considerations to a Ge/GeSi spin-qubit heterostructure, and discuss the prospects for the engineering of sweet spots.