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Utilizing dispersive gate sensing (DGS), we investigate the spin-orbit field($\textbf{B}_{SO}$) orientation in a many-electron double quantum dot (DQD)defined in an InSb nanowire. While characterizing the inter-dot tunnelcouplings, the measured dispersive signal depends on the electron chargeoccupancy, as well as on the amplitude and orientation of the external magneticfield. The dispersive signal is mostly insensitive to the external fieldorientation when a DQD is occupied by a total odd number of electrons. For aDQD occupied by a total even number of electrons, the dispersive signal isreduced when the finite external magnetic field aligns with the effective$\textbf{B}_{SO}$ orientation. This fact enables the identification of$\textbf{B}_{SO}$ orientations for different DQD electron occupancies. The$\textbf{B}_{SO}$ orientation varies drastically between charge transitions,and is generally neither perpendicular to the nanowire nor in the chip plane.Moreover, $\textbf{B}_{SO}$ is similar for pairs of transitions involving thesame valence orbital, and varies between such pairs. Our work demonstrates thepracticality of DGS in characterizing spin-orbit interactions in quantum dotsystems, without requiring any current flow through the device.

Variable and orbital-dependent spin-orbit field orientations in a InSb double quantum dot characterized via dispersive gate sensing