Geometry of the dephasing sweet spots of spin-orbit qubits

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 $E_\mathrm{Z}=\mu_B\sqrt{\vec{B}^\mathrm{T}G\vec{B}}$ (with $\mu_B$ the Bohr magneton and $\vec{B}$ the magnetic field), and its response to the fluctuator by the derivative $G^\prime$ 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 $G^\prime$. 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.