Electric quadrupole shifts of the precession frequencies of $^{131}$Xe atoms in rectangular cells

We study an atomic comagnetometer design based on the spin precessions of $^{129}$Xe and $^{131}$Xe atoms in glass cells. The quadrupole splittings in the precession spectrum of $^{131}$Xe are fully resolved, allowing a precise determination of the magnetic-dipole precession frequency. The transverse asymmetry of quadrupole interactions, due to both the geometry and surface properties of the cell, characterized by a non-zero asymmetry parameter $\eta$, modifies the dependence of the quadrupole splittings on the relative orientation between the cell axes and the bias magnetic field, and lead to additional corrections in the precession frequencies of $^{131}$Xe atoms. We examine these effects both theoretically and experimentally, and develop methods to quantify and control such shifts.