We study single-and multi-quantum transitions of the nuclear spins of ionized arsenic donors in silicon and find quadrupolar effects on the coherence times, which we link to fluctuating electrical field gradients present after the application of light and bias voltage pulses. To determine the coherence times of superpositions of all orders in the 4-dimensional Hilbert space, we use a phasecycling technique and find that, when electrical effects were allowed to decay, these times scale as expected for a field-like decoherence mechanism such as the interaction with surrounding 29 Si nuclear spins.Aiming for the realization of a scalable quantum technology, the electron and nuclear spins of phosphorus donors in silicon have been studied extensively [1][2][3][4][5][6][7][8]. Due to different interaction strengths with their surroundings, they form a powerful combination of a fast, but more volatile electron spin and a slower, but very coherent nuclear spin qubit [9][10][11]. This nuclear spin is I = 1/2 for phosphorus, but systems with a higher nuclear spin can be realized by simply replacing phosphorus by the other hydrogenic donors As (I = 3/2) [12,13], Sb (5/2 and 7/2) [14,15], and Bi (9/2) [16][17][18]. Several advantages of the d-dimensional Hilbert spaces of such systems, sometimes called qudits, have been proposed, such as the realization of simpler and more efficient gates [19,20] or more secure quantum cryptography [21]. In addition, one higher order system can replace several qubits, simplifying their physical implementation [22]. These concepts usually require coherent superpositions of higher orders, which show specific interactions with their surroundings that can be reflected in the observed coherence times. In particular, the additional quadrupole interaction with electric field gradients arising from strain [13,23,24] or defect states [25], has to be considered for heavier dopants. In this work, we study first-and higher-order coherences of ionized As donors in silicon. By including light and voltage pulses in the experiment, we are able to link the additional quadrupolar decoherence effect to the electrical environment of the nucleus and show that it vanishes, when the sample is allowed to relax electrically. In addition, we use a phase cycling technique to study superpositions of all orders and find that the coherence times scale inversely proportional to the coherence order, as expected for a field-like decoherence mechanism such as the interaction with surrounding 29 Si nuclear spins. The Hamiltonian H characterizing ionized arsenic donors with nuclear spin I = {I x , I y , I z } in a magnetic field B z can be written as where ν 0 = γ n B z with the nuclear gyromagnetic ratio γ n and h is Planck's constant. The second term on the right hand side of (1) describes the nuclear quadrupole interaction with an effective electric field gradient V 33 , here approximated to first order [26], withwhere Q is the nuclear quadrupole moment, e is the elementary charge, and ϑ describes the angle between B z and...