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Phonon-induced relaxation of a molecular qubit unravelled by inelastic X-ray scattering and spin dynamics simulations


Inelastic X-ray scattering (IXS) measurements and periodic Density Functional Theory (pDFT) calculations reveal the relaxation dynamics of a molecular qubit by directly accessing its phonon dispersions and polarisation vectors. The critical role played by ultra-low-energy modes, detected using the high-resolution IXS beamline ID28, was demonstrated by an international team of researchers.

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Molecular nanomagnets (MNMs) are very promising for quantum information processing. Indeed, the molecular spin of these systems can encode a qubit (the basic unit of a quantum computer), which can be coherently manipulated with electromagnetic fields. By taking advantage of the high degree of chemical control of their molecular structure, the energy levels of MNMs can be tailored to match the envisaged technological applications. For instance, the presence of a sizeable number of low-energy levels enables one to define “self-correcting” qubits [1]. However, the use of MNMs as qubits is undermined by molecular vibrations and spin-phonon interactions, playing an important role in magnetic relaxation and determining the coherence times of superposition of states. The key to construct a reliable model of phonon-induced relaxation in MNMs is to have experimental access to phonon dispersions and eigenvectors, which are necessary for a quantitative evaluation of spin-phonon coupling coefficients.

The very first results on the characterisations of phonons in molecular qubits and MNMs were obtained recently by inelastic neutron scattering [2]. This technique requires very large, high-quality single crystals, preferably deuterated. On the contrary, molecular qubits are typically rich in 1H and are grown in small-sized molecular crystals, thus preventing the use of neutrons. Moreover, incoherent scattering significantly limits the energy and momentum space that can be explored.

In this work, IXS was used at beamline ID28 to measure phonon dispersions in a crystal of molecular qubits. The main advantage of IXS compared to neutron scattering is the possibility of using very small samples, of the order of 1 mm3 volume. This, combined with an energy-independent resolution of about 1.5 meV, a complete decoupling between energy and momentum transfer, and the background-free signal, makes IXS a very powerful tool to study phonons in MNMs.



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Fig. 1: Representative ID28 data on [VO(TPP)] (black scatters), obtained at T = 300 K and measured with longitudinal scans along the symmetry direction Γ–Kz (a) at the (0 0 6.6) point with a resolution of δE = 3 meV, and (b) at the (0 0 6.2) point with δE = 1.5 meV. Solid lines are results of the fit obtained with FIT28, a customised data analysis software developed on ID28 ( c) IXS cross-section (colour map) simulated with DFT phonon energies and polarisation vectors along the Γ–Kz directions. Cyan/white dots/squares are experimental IXS excitation energies extracted from the complete set of data over the whole explored Q range. Adapted with permission from E. Garlatti et al., Nat. Commun. 14, 1653 (2023), under a Creative Commons Attribution 4.0 International License.

The material studied in this work was a molecular crystal of [VO(TPP)] (VO = vanadyl, TPP = tetraphenylporphyrinate). This radiation-robust molecule is a very promising molecular qubit, embedding both electronic and nuclear spins suitable for implementing quantum gates [1]. On ID28, acoustic and optical branches of [VO(TPP)] were measured along different directions in the reciprocal space, probing both their energies and polarisation vectors (Figure 1). In particular, the high resolution of the beamline made it possible to detect ultra-low-lying optical modes at about 1-2 meV (Figure 1b,c), fully breaking down the Debye picture (a model for estimating the phonon contribution to the specific heat capacity in a solid) and deeply affecting magnetic relaxation.



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Fig. 2: a) Spin-phonon coupling coefficients calculated as a function of the frequency of the vibration (red line) compared to vibrational density of states (black line) of [VO(TPP)]. b) Computed spin-phonon relaxation time T1 (red line-full squares) and coherence time T2 (inset, continuous green line-empty squares). The blue line-filled triangles represent the simulated T1 after removal of all low-energy phonons. Black full (empty) circles are used to report the inversion recovery (Hahn echo) experimental data on [VO(TPP)]. c) Molecular distortions associated with the first optical mode at the Γ-point (red, orange) compared with the equilibrium molecular structure (yellow). Reprinted with permission from E. Garlatti et al., Nat. Commun. 14, 1653 (2023), under a Creative Commons Attribution 4.0 International License.


The role of these low-energy modes in the relaxation dynamics of [VO(TPP)] was then investigated by combining the experimental results obtained by IXS with spin dynamics simulations based on state-of-the-art pDFT. These calculations provided a complete picture of phonon-induced relaxation and, for the first time, decoherence. By performing a neural network-based interpolation of the DFT calculations to estimate the spin-phonon couplings coefficients [3], it was shown that low-energy optical phonons in [VO(TPP)] possess very strong couplings (Figure 2a) and are crucial in magnetic relaxation up to ambient temperature (Figure 2b). By inspecting the nature of the relevant phonons for spin relaxation (Figure 2c), it was also possible to provide a chemical strategy for tailoring the intramolecular motions in [VO(TPP)] and thus slowing down relaxation.

These results demonstrate that the synergistic combination of the IXS technique with pDFT spin dynamics simulations is able to yield an unprecedented insight into the nature of phonons and vibrations, as well as their role in magnetic relaxation and decoherence of molecular qubits.

This research was funded by the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement No 862893 (FET-OPEN project FATMOLS) and by the European Research Council (ERC) (grant agreement No. [948493]). It was also supported by the Italian MIUR with the Progetto Dipartimenti di Eccellenza 2018-2022 (ref. B96C1700020008), by Fondazione Cariparma and the National Recovery and Resilience Plan, National Center for HPC, Big Data and Quantum Computing.


Principal publication and authors
The critical role of ultra-low-energy vibrations in the relaxation dynamics of molecular qubits, E. Garlatti (a,b), A. Albino (c), S. Chicco (a), V.H.A. Nguyen (d), F. Santanni (c), L. Paolasini (e), C. Mazzoli (f), R. Caciuffo (g), F. Totti (c), P. Santini (a,b), R. Sessoli (c), A. Lunghi (d), S. Carretta (a,b), Nat. Commun. 14, 1653 (2023);
(a) Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma and UdR Parma, INSTM, Parma (Italy)
(b) INFN, Sezione di Milano- Bicocca, gruppo collegato di Parma, Parma (Italy)
(c) Dipartimento di Chimica ‘Ugo Schiff’, Università Degli Studi di Firenze and UdR Firenze, INSTM, Sesto Fiorentino (Italy)
(d) School of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2 (Ireland)
(e) ESRF
(f) National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY (USA)
(g) INFN, Sezione di Genova, Genova (Italy)

[1] E. Garlatti et al., Nat. Commun. 11, 1751 (2020).
[2] S. Chicco et al., Chem. Sci. 12, 12046 (2021).
[3] A. Lunghi, Appl. Magn. Reson. 51, 1343 (2020).


About the beamline: ID28

Beamline ID28 is dedicated to the study of phonon dispersion in condensed matter at momentum transfers, Q, and energy transfers, E, characteristic of collective atom motions. Inelastic X-ray scattering (IXS) is therefore closely related to inelastic neutron scattering (INS) techniques, and consequently largely shares the scientific questions addressed in fields ranging from life science to materials research. IXS is particularly suited for the study of disordered systems such as liquids and glasses in a Q,E-range inaccessible to INS and of samples only available in very small quantities (<< 1 mm3) and/or submitted to very high pressures (up to 100 GPa).

Determination of the phonon dispersion, or more generally, of the high-frequency (THz) collective dynamics, allows to access various material properties such as sound velocities, elastic constants, interatomic force constants, phonon-phonon interactions, phonon-electron coupling, dynamical instabilities, relaxation phenomena etc.

Applications of IXS from phonons on ID28 can be divided into the following categories:

  • Phonon dispersion in crystalline materials, only available in very small quantity, or otherwise incompatible with inelastic neutron scattering techniques (high-temperature superconductors, large bandgap semiconductors , actinides)
  • Determination of the high-frequency collective dynamics in disordered systems (hydrogen-bonded liquids, liquid metals, molten salts, glass formers, quantum liquids, biological materials)
  •  Phonon dispersion under extreme conditions of very high pressure (up to 100 GPa) (geophysically relevant materials, metals, liquids)
  • Lattice dynamics in thin films and interfaces