Coherent manipulation of three-qubit states in a molecular single-ion magnet

Abstract

We study the quantum spin dynamics of nearly isotropic ${\mathrm{Gd}}^{3+}$ ions entrapped in polyoxometalate molecules and diluted in crystals of a diamagnetic ${\mathrm{Y}}^{3+}$ derivative. The full energy-level spectrum and the orientations of the magnetic anisotropy axes have been determined by means of continuous-wave electron paramagnetic resonance experiments, using X-band (9--10 GHz) cavities and on-chip superconducting waveguides and 1.5-GHz resonators. The results show that seven allowed transitions between the $2S+1$ spin states can be separately addressed. Spin coherence ${\mathrm{T}}_{2}$ and spin-lattice relaxation ${\mathrm{T}}_{1}$ rates have been measured for each of these transitions in properly oriented single crystals. The results suggest that quantum spin coherence is limited by residual dipolar interactions with neighbor electronic spins. Coherent Rabi oscillations have been observed for all transitions. The Rabi frequencies increase with microwave power and agree quantitatively with predictions based on the spin Hamiltonian of the molecular spin. We argue that the spin states of each ${\mathrm{Gd}}^{3+}$ ion can be mapped onto the states of three addressable qubits (or, alternatively, of a $d=8$-level ``qudit''), for which the seven allowed transitions form a universal set of operations. Within this scheme, one of the coherent oscillations observed experimentally provides an implementation of a controlled-controlled-NOT (or Toffoli) three-qubit gate.