Modelling electric field control in a 4f molecular qudit with hyperfine coupling

Understanding the fundamental principles of spin-electric coupling in molecules with hyperfine-coupled electronic and nuclear spins offers a route to electric field-based molecular quantum information. We recently addressed the electronic degrees of freedom in $\left[\mathrm{Tm}{\left\{N{\left({\mathrm{Si}}^i{\mathrm{Pr}}_3\right)}_2\right\}}_2\right]$ . Here, we treat both electronic and I = 1/2 nuclear spins explicitly to investigate the possibility of electric field control of the nuclear degrees of freedom. Furthermore, since the hyperfine coupling breaks Kramers degeneracy and therefore spin-electric coupling arises at zeroth-order, we investigate if this the inclusion of the nuclear spin strongly influences the overall coupling. Transitions are classified as EPR-, NMR-, or mixed/forbidden character, revealing that EPR-like transitions couple more strongly to electric fields than NMR-like ones, as crystal-field modulation dominates over hyperfine modulation. The anisotropy of the electric field effect agrees with previous results, but magnetic-field orientation dependence is suppressed by zeroth-order spin-electric coupling. Dissipative spin-dynamics simulations show that experimentally feasible electric field strengths and relaxation times permit coherent manipulation of both the electronic and nuclear spins, demonstrating an experimentally viable pathway for electric field control in $\left[\mathrm{Tm}{\left\{N{\left({\mathrm{Si}}^i{\mathrm{Pr}}_3\right)}_2\right\}}_2\right]$ .