Radiofrequency to Microwave Coherent Manipulation of an Organometallic Electronic Spin Qubit Coupled to a Nuclear Qudit

We report here a comprehensive characterization of a 3d organometallic complex, [V(Cp)2Cl2] (Cp = cyclopentadienyl), which can be considered as a prototypical multilevel nuclear qudit (nuclear spin I = 7/2) hyperfine coupled to an electronic qubit (electronic spin S = 1/2). By combining complementary magnetic resonant techniques, such as pulsed electron paramagnetic resonance (EPR) and broadband nuclear magnetic resonance (NMR), we extensively characterize its Spin Hamiltonian parameters and its electronic and nuclear spin dynamics. Moreover, we demonstrate the possibility to manipulate the qubit–qudit multilevel structure by resonant microwave and radiofrequency pulses, driving coherent Rabi oscillations between targeted electronuclear states. The obtained results demonstrate that this simple complex is a promising candidate for quantum computing applications.

. Overlay of the molecular structure of [V(Cp) 2 Cl 2 ] (orange, monoclinic P21/c space group) and of [Ti(Cp) 2 Cl 2 ] (blue, triclinic P1 space group). Figure S2. View of the crystal structure of 1 along the b crystallographic axis with the crystal packing of the molecule along the (10-1) plane highlighted. Figure S3. Comparison between experimental (see legend) and simulated (black line, triclinic space group crystal structure of 2) PXRD patterns (540°, 2) for 3a-3c. Figure S4. Optical image (left) and schematic representation (right) of a single crystal of 3c used for EPR and NMR measurements with crystal face indexing. Figure S5. Experimental EDFS (blue line) and CW (black lines) X-band EPR spectra for 3a (a) and 3b (b) recorded at T = 20 K. To ease the comparison with CW data, EDFS data were translated to account for frequency difference. The spectral simulations corresponding to the spin Hamiltonian parameters reported in the text are shown in red. In the experimental spectrum of 3a a broad band is clearly observed probably due to a clustering of the paramagnetic components that has not been included in the simulation (red line) and does not contribute to the EDFS spectrum (blue line).

Table S1
Nominal crystallographic directions and refined orientations of the applied magnetic field with respect to the xyz magnetic tensor frames in term of the azimuthal and polar angles and , for each configuration.

Table S2
Director cosines of the principal directions of g and A tensors in the ab'c* cartesian reference system.

Figure S11
Pulsed EPR inversion recovery traces recorded for 3c recorded at different temperatures and best fit curves based on Eq.
(2) of main text. Table S3. Best-fit parameters of the models used to simulate the temperature dependence of e T 1 extracted by pulsed EPR experiments.

Effect of electron nuclear mixing on n T 2 and Rabi frequency
Nuclear Spin decoherence and Rabi frequencies are both influenced by the electro-nuclear mixing. Indeed, in the examined magnetic field range, the mixing between electronic and nuclear spin wave-function provides the dominant contribution for both processes. On the one hand, the Rabi frequency is proportional to matrix elements of transverse spin operators between pairs of eigenstates, while n T 2 is proportional to the difference of + expectation values of between the pair of examined states. In both cases, since + , the small electronic mixing of the wave-function ( ) is responsible of ≫ ∝ 1/ almost the entire effect. Hence, the nuclear n T 2 acquires some of the electron spin decoherence.