Improved measurement of the Land\'{e} g factor of the 5D$_{5/2}$ state of Ba II with a single trapped ion

We report an improved measurement of the Land\'e g factor of the 5D$_{5/2}$ level of singly ionized barium. The measurement was performed by interrogation of Doppler-cooled $^{138}$Ba$^{+}$ ions in a linear Paul trap. The 5D$_{5/2}$ g factor is extracted from radiofrequency spectroscopy of both the ground level and the metastable 5D$_{5/2}$ level. We measure the ground and 5D$_{5/2}$ Zeeman splittings at multiple trap rf voltages and magnetic field strengths in order to ensure that ac Zeeman shifts from trap currents do not contaminate the measurement. Other systematics including effects from the power line phase, magnetic field gradients, and magnetic field drifts are addressed. The g factor is found to be 1.20040(4), an increase in precision of an order of magnitude over our previous measurement. This increased precision will enable a new measurement of the octupole moment of the $^{137}$Ba nucleus, reducing the dependence on atomic theory calculations to below experimental error.