Abstract
Improving the performance of molecular qubits is a fundamental milestone towards unleashing the power of molecular magnetism in the second quantum revolution. Taming spin relaxation and decoherence due to vibrations is crucial to reach this milestone, but this is hindered by our lack of understanding on the nature of vibrations and their coupling to spins. Here we propose a synergistic approach to study a prototypical molecular qubit. It combines inelastic Xray scattering to measure phonon dispersions along the main symmetry directions of the crystal and spin dynamics simulations based on DFT. We show that the canonical Debye picture of lattice dynamics breaks down and that intramolecular vibrations with verylow energies of 12 meV are largely responsible for spin relaxation up to ambient temperature. We identify the origin of these modes, thus providing a rationale for improving spin coherence. The power and flexibility of our approach open new avenues for the investigation of magnetic molecules with the potential of removing roadblocks toward their use in quantum devices.
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Introduction
In the last years, molecular magnetism has been giving significant contributions to the second quantum revolution providing promising systems for strategic fields like quantum computing and simulation^{1,2,3,4,5}. Indeed, magnetic molecules with two or multilevel energy structure suitable to encode qubits or qudits were synthesized and proposed for new quantum architectures and schemes^{6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21}. A pivotal step to improve the performance of molecular qubits is to reduce relaxation and decoherence due to phonons^{22,23,24,25,26}. In particular, a key stage to achieve this result is to identify the vibrational modes responsible for this irreversible dynamics and devise recipes for the synthesis of improved systems.
A full comprehension of phononinduced mechanisms in molecular nanomagnets (MNMs) requires a thorough and experimentallyassessed description of phonon modes and spinphonon couplings coefficients. Indeed, only with such a solid starting point, it is possible to disentangle the complex correlations between phonon energies, their coupling to the spins, and their role in different decoherence processes, which is crucial to obtain a sound and consistent interpretation of experimental results. A synergistic approach combining cuttingedge experimental and theoretical techniques is therefore required to investigate phonons and their role in spin relaxation, namely: i) an experimental technique able to directly access phonon energies and polarization vectors in verysmallsized and ^{1}Hrich singlecrystal samples, typical of molecular qubits; ii) stateoftheart ab initio spin dynamics simulations, which must include an accurate description of phonon modes across the entire Brillouin zone and spinphonon couplings coefficients.
Here we choose the wellcharacterized and radiationrobust [VO(TPP)] complex (VO = vanadyl, TPP = tetraphenylporphyrinate)^{27} as a benchmark to show the capabilities of a multitechnique approach with all these characteristics. [VO(TPP)] is also a very promising molecular qubit: It allows simultaneous coherent manipulation of both electronic and nuclear spins^{20,28} and it forms dimeric species where the two electronic spins are distinguishable and exchangecoupled to implement quantum gates^{29}. To fulfill point (i), we exploit Inelastic Xray Scattering (IXS). This technique has never been used before to address magnetic molecules but has several advantages with respect to more traditional spectroscopy techniques, such as inelastic neutron scattering. Crucially, IXS makes it possible to investigate very small single crystals, typical of MNMs, has energyindependent resolution and a very small background. In this work, we present a direct measurement of phonons in a molecular qubit obtained with IXS. The unique capabilities of the ID28 beamline at ESRF enable the measurement of acoustic and optical branches of [VO(TPP)] along different directions in the reciprocal space, probing both their energies and polarization vectors. Our results demonstrate that IXS has the sensitivity and the power to become the new technique of choice to investigate phonons and vibrations in molecular qubits and in MNMs in general. In particular, we find ultralowenergy optical phonon modes at about 12 meV. Even if lowenergy optical modes are a typical feature of molecular crystals of MNMs^{30,31,32,33}, [VO(TPP)], to the best of our knowledge, sets a new record, which also outdoes other ordered systems with lowenergy optical phonons, like thermoelectric materials^{34,35,36}.
Moving to point (ii), IXS results are compared with stateofthe art periodic Density Functional Theory (pDFT) calculations of phonon energies and polarization vectors. Experimental and simulated IXS crosssections are in very good agreement, validating the DFT results as the starting points for further analysis. DFT simulations also confirm the presence of ultralowenergy optical modes in [VO(TPP)] phonon dispersions, which can deeply affect coherence time. The dominant role of these modes in both low and hightemperature spin relaxation of MNMs was recently suggested by neutron scattering experiments and ab initio simulations^{25,26,31,32,33,37,38,39,40,41,42,43}. Despite these studies provide a picture of spin relaxation that substantially differ from the canonical one based on the Debye model, a direct demonstration of the role played by nonDebye lowenergy vibrations is yet to be achieved. In this work, spin dynamics simulations based on DFT are also exploited to provide a full picture of phononinduced relaxation and decoherence in the benchmark molecular qubit [VO(TPP)]. We performed a neuralnetwork based interpolation of the DFT calculations to estimate the spinphonon couplings coefficients in [VO(TPP)], revealing that these lowenergy optical phonons also possess very strong couplings to the spin. Finally, we demonstrate that ultralowenergy vibrations are responsible for magnetic relaxation up to ambient temperature. We also report calculations of phononinduced decoherence of MNMs including the important pure dephasing contribution of twophonon processes. Finally, by comparing periodic and singlemolecule DFT calculations, we also found that lowenergy optical phonons in [VO(TPP)] are mainly associated with intramolecular vibrations of specific chemical groups, pinpointing possible synthetic strategies towards the improvement of the spin coherence in this important class of molecules.
Results
Unveiling phonons with Inelastic Xray Scattering
The very first results on the characterizations of phonons in molecular qubits and MNMs have been obtained only very recently with the Inelastic Neutron Scattering (INS) technique^{31}. Despite the capabilities of the new generation of highflux neutron spectrometers, this technique still requires very large highquality single crystals to enable the investigation of phonon dispersions and polarization vectors along different directions in the reciprocal space. The verysmall size of molecular crystals therefore typically prevents the use of INS to access phonon dispersions. Here we demonstrate that a cuttingedge technique allowing us to overcome this hurdle is highresolution IXS^{44,45}, the possibility to use very small samples (of the order of 1 mm^{3}) being its main advantage. Further advantages of IXS are due to the orders of magnitude difference between the incident hard Xrays (E_{i} > 10 keV) and the energy scale of interest when investigating phonons and vibrations (1−10 meV, ΔE/E_{i} ~ 10^{−7}). This leads to an energyindependent resolution and to a complete decoupling between energy and momentum transfer, the latter being defined only by the scattering angle. Furthermore, IXS is essentially a backgroundfree technique, since the incoherent crosssection for Xrays involve larger energy transfers (>eV) with respect to the energy window of interest, and multiple scattering is negligible^{44}. Thus, deuterated samples are not required in IXS experiments, contrary to INS ones, where the large incoherent crosssection of hydrogen atoms, abundant in MNMs, can mask the coherent phonon signal.
Thanks to these specific features of the IXS technique, we were able to measure a [VO(TPP)] singlecystal with a size of the order of 1 × 1 × 0.5 mm^{3} and all the data were obtained by measuring just one sample with no radiation damage (same Bragg peaks before and after the experiment and no induced colour centres). The experiment was performed on ID28 at the European Syncrothron facility ESRF^{46}, one of the very few IXS beamlines providing the required resolution and lineshape profile suitable for performing phonon studies in MNMs. Its unique tradeoff between the highenergy resolution (up to δE ~ 1.5 meV) and the incident flux enables the measurement of inelastic spectra in the energy range of interest for investigating lowenergy phonons in molecular qubits. Moreover, a wide range of accessible momentum transfer allows the exploration of a wide section of the reciprocal space over specific symmetry directions in both longitudinal and transverse configurations. Measurements were performed at room temperature, with two different Xray incident energies and resolutions (see Methods for more details on ID28). By using the lowresolution configuration (δE = 3.0 meV), we performed a first exploration of [VO(TPP)] phonon modes with constantQ energy scans up to 25 meV, then we switched to the highest resolution configuration yielding δE = 1.5 meV, in order to investigate lowenergy phonons in selected portions of the reciprocal space. We explored [VO(TPP)] reciprocal space along the Γ–N, Γ–K_{x} and Γ–K_{z} symmetry directions ((h0h), (h00) and (00l), respectively in conventional cell notation, see Methods and Supplementary Note 1 for more details).
Intensities as a function of energy for some representative Q values are reported in Fig. 1 for the Γ–N direction off the (0 0 6) reciprocal lattice point (panels a, b). Longitudinal scans along the Γ–K_{x} and Γ–K_{z} directions in the same Brillouin Zone (BZ) are shown in panels (d,e) and (g,h), respectively. In Fig. 2a–c we show data along the Γ–N direction off the (6 0 0) reciprocal lattice point and the transverse scans along Γ–K_{z} and Γ–K_{x} directions. Phonon energies extracted from the data over the whole explored Q range along these directions are also reported in Fig. 1c, f, i and Fig. 2d–f, superimposed to DFT calculations (vide infra). Lowresolution data demonstrate the presence of longitudinal acoustic modes along both Γ–K_{z} and Γ–K_{x} directions, while nondispersive transverse and longitudinal optical phonon branches are clearly visible at about 810 meV and 15 meV along Γ–N and Γ–K_{z} directions, respectively. Measurements along Γ–K_{z} but with transverse Q configurations also allowed the measurements of optical phonons with ultralow energy. A nondispersive mode was in fact detected at about 2 meV (Fig. 2a, d), clearly emerging from the elastic line (panel a) and visible over the whole explored Q range/half BZ (panel d). To better resolve phonon modes down to verylow energies along other directions, we switched to the highresolution configuration with δE = 1.5 meV. The data reported in Fig. 1h for a longitudinal scan along the Γ–K_{z} direction show in fact that with this δE we are able to detect modes with energies ≤ 2 meV, here corresponding to a longitudinal acoustic and verylowlying optical modes. Phonon modes with energies of the order of 3 meV are also present along the Γ–N direction, while transverse modes with energies ≤ 2 meV are visible along the Γ–K_{x} direction, corresponding to verylowenergy optical modes. Thus, IXS data on [VO(TPP)] show the presence of ultralowenergy optical modes, clearly detectable with both low and highresolution configurations. Inelastic spectra as a function of energy for other Q values are reported in Supplementary Figs. 2–7. As discussed below, all these findings are in excellent agreement with the simulations of the IXS crosssection obtained with DFT phonon energies and polarization vectors (see colour maps of Figs. 1, 2 and the following section).
DFT simulation of IXS data
The unit cell of [VO(TPP)] is replicated three times along each crystallographic direction to obtain a 3 × 3 × 3 supercell containing 4212 atoms. As described in the Methods section, the latter is optimized with pDFT and used to compute lattice force constants and phonons across the entire BZ. Phonon dispersions of [VO(TPP)] calculated with pDFT along the same symmetry directions in the reciprocal space explored experimentally are reported in Fig. 1c, f, i and Fig. 2d–f (see also Supplementary Fig. 8). The comparison with the phonon energies extracted from the IXS data, superimposed onto the same figures, demonstrates the optimal agreement with the experimental results. Importantly, pDFT calculations confirm the presence of the optical branches lying at very low energies along the Γ–K_{z} and Γ–K_{x} directions.
The evaluation of the spinphonon couplings relies on a full description of phonon modes, comprising of both phonon energies ω_{j}(q) and polarization vectors \({{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{j}^{d}({{{{{{{\bf{q}}}}}}}})\). Therefore, the complete validation of the pDFT results requires the inspection of both these quantities before proceeding with the calculation of the spin dynamics. The phonon excitation intensities determined by the inelastic Xray crosssection directly depend on phonon polarization vectors \({{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{j}^{d}({{{{{{{\bf{q}}}}}}}})\) (see Eq. (2) in Methods), and thus, IXS experiments also probe the composition of phonon normal modes. In order to compare experimental data with DFT results, we simulated the IXS crosssections starting from DFTcalculated phonon energies ω_{j}(q) and polarization vectors \({{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{j}^{d}({{{{{{{\bf{q}}}}}}}})\). The simulated crosssections are reported as 2D colour maps in Fig. 1c, f, i and Fig. 2d–f, where the colour code represents the excitation intensity. These maps provide immediate visualization of phonon dispersions and excitation intensities along the explored symmetry directions. However, for a more direct comparison between our simulations and IXS data, we inspected the crosssection as a function of energy for some representative Q values. From the results reported in Fig. 3, the excellent agreement between experimental and calculated IXS crosssection (using the experimental linewidth) is evident for both low (panels ae,g) and highresolution data (panels f,h,i), thus validating both calculated phonon energies and polarization vectors. In particular, Fig. 3f–i highlight the contributions to the [VO(TPP)] IXS crosssection of the verylowenergy phonon modes. Only an overall rescaling of 10% has been uniformly applied to lower the phonon energies and better reproduce the IXS data. This correction is typical of pDFT calculations on molecular crystals^{31} and, especially at low energy, is mainly due to van der Waals interactions corrections, as well as to temperature effects on the simulation cell^{47}. It is also worth stressing that the calculated IXS excitation intensity of verylowenergy phonons strongly depends on the exact energy value through the phonon Bose factor n_{j}(q). This high sensitivity of the IXS crosssection at verylow energy allowed us to address small discrepancies between data and calculations, of the order of tenth of meV, along Γ–K_{z} and Γ–K_{x} directions (see Fig. 3g, i). Comparisons bewteen inelastic spectra and calculated crosssections for other Q values are reported in Supplementary Figs. 2–7.
Spinphonon coupling and decoherence simulation
The molecular qubit [VO(TPP)] contains a ^{51}V^{4+} ion, whose spin states can be described with the spin Hamiltonian
The first two terms in Eq. (1) describe the Zeeman interaction of the magnetic field with the electronic (S) and nuclear spin (I), respectively, where μ_{B} is the Bohr magneton, g is the Landé tensor, and γ_{N} is the nuclear gyromagnetic factor. The third term, instead, corresponds to the hyperfine interaction (A) between the two spins. Coupling with other magnetic nuclei (e.g., ^{1}H, ^{14}N) is here neglected. The tensors g and A are computed with DFT using the pDFToptimized structure and found in excellent agreement with experimental ones^{27}. Results are reported in Table 1.
The simulation of spinphonon relaxation with electronic structure methods requires the calculation of the effect of phonons on the Hamiltonian of Eq. (1). Here, we consider the modulation of the leading terms of Eq. (1), i.e. g and A. This corresponds to computing first and secondorder derivatives of these tensors with respect to phonon displacements, q_{αq}, i.e. V_{αq} = (∂A/∂q_{αq}), \({{{{{{{{\bf{V}}}}}}}}}_{\alpha {{{{{{{\bf{q}}}}}}}}\beta {{{{{{{{\bf{q}}}}}}}}}^{{\prime} }}=({\partial }^{2}{{{{{{{\bf{A}}}}}}}}/\partial {q}_{\alpha {{{{{{{\bf{q}}}}}}}}}\partial {q}_{\beta {{{{{{{{\bf{q}}}}}}}}}^{{\prime} }})\) and similarly for g^{26,38}. The indexes α and q point to the phonon’s band index and reciprocal space vector, respectively. The total number of secondorder derivatives scales quadratically with the number of molecular degrees of freedom and for [VO(TPP)], it would require a minimum number of 10^{5} DFT calculations. This volume of simulations is not sustainable with modernday computational hardware and software. We solve this technical challenge by employing neural networks to efficiently interpolate DFT results^{48}. The relation between the structure of [VO(TPP)] and the value of A and g is sampled 2000 times by applying random perturbations within the range ± 0.05 Å to [VO(TPP)]’s optimized structure. Several neural networks with up to four hidden layers are trained on 1600 samples to predict the two tensors as a function of [VO(TPP)] atomic distortions. The prediction of the bestperforming models is tested against DFT calculations for 200 samples not used at the training stage, revealing their excellent accuracy with a root mean squared error of 0.45 MHz and 8.2 × 10^{−5} on the prediction of A and g, respectively (see Supplementary Figs. 9–11). The neural networks are then used to compute a 36points numerical differentiation of the spin Hamiltonian coefficients with respect to all pairs of molecular degrees of freedom, for a total of 10^{6} evaluations of A and g (see Supplementary Fig. 12).
Figure 4 shows the comparison between the simulated phonon density of states and the norm of linear spinphonon coupling coefficients, ∣V(ω_{α})∣ (defined in the Methods section), for the A tensor as a function of the vibrations energy ℏω_{αq}. Several important conclusions can be drawn from this result. Most importantly, this analysis shows a sizeable spinphonon coupling for ultralowenergy vibrations. In addition, the computed spinphonon coupling norm and the phonons density of states show remarkably different profiles. This observation is in agreement with previous reports in other molecular complexes and stems from the fact that vibrational modes are varied in nature and depending on their symmetry^{39} and localization on the first coordination shell^{22,26,42}, they will be more or less effective in coupling to the spin. Finally, the vibrational DOS does not follow the canonical Debye profile with ~ω^{2}, but instead follows a linear dependency overlapped to a complex structure due to the many optical transitions falling at very low energy^{25}. This demonstrates a breakdown of the simple Debye picture. The inset of Fig. 4 shows the comparison between the lowenergy DOS of [VO(TPP)] and [VO(acac)_{2}]. Differently from the former, the latter shows a more typical Debyelike profile at low energy in virtue of the higher energy of optical phonons, with the first one computed at ~6 meV at the Γpoint^{25,31}.
According to recent literature^{26,38}, spinphonon relaxation at temperature above ~10−20 K is due to a twophonon Raman mechanism, where a spin transition is triggered by the absorption of one phonon and the simultaneous emission of a second phonon with similar energy. We simulate spin relaxation and decoherence due to this mechanism by solving the secondorder secular Redfield equation including quadratic spinphonon coupling terms \({{{{{{{{\bf{V}}}}}}}}}_{\alpha {{{{{{{\bf{q}}}}}}}}\beta {{{{{{{{\bf{q}}}}}}}}}^{{\prime} }}\)^{26,38} as implemented in the software MolForge^{26} (see Supplementary Note 2 for more details on the workflow). As observed previously for vanadyl compounds^{26,38}, the simulations show the contribution of the modulation of the A tensor to be the leading relaxation mechanism in moderate magnetic fields such as 0.3 T employed in Xband EPR experiments. We note that the contributions of A and g are relatively close to one another in [VO(TPP)] for the values of the field considered and their contribution to spinphonon relaxation follows qualitatively similar trends (see Supplementary Figs. 14–19).
Figure 5 reports the predictions of T_{1} and T_{2} in [VO(TPP)] against the experimental results obtained by inversion recovery and Hahn echo at Xband frequencies^{27} (see plotted data points in Supplementary Tables 5, 6). We observe a good agreement between experimental and simulated T_{1} timescale above ~20 K and especially in the hightemperature regime, where simulations reveal a T^{−2} profile, typical of twophonon relaxation^{26,38,49}. At lower temperatures, experiments are affected by crossrelaxation mediated by spinspin dipolar interactions and active due to the relatively high concentration (2%) of the V^{4+} magnetic ions inside the isostructural diamagnetic Ti^{4+} host^{27}.
The simulation of a hightemperature Raman profile (T^{−2}) from 10 K supports an interpretation of spin relaxation due to ultralowenergy vibrations. We further test this hypothesis by manually removing the contribution to T_{1} from all the vibrations with ℏω_{α} < 6 meV. These results are also reported in Fig. 5 and show a largely suppressed spinphonon relaxation in these artificial conditions, thus confirming the relevance of lowenergy phonons in limiting spin lifetime. This result is in agreement with the relevance of lowenergy phonons in limiting spin lifetime, especially if they are strongly coupled to the spin, as shown for the investigated compound by our calculations. Since these phonons are always more populated than highenergy ones, they maintain a dominant role in the spin dynamics in any temperature condition.
Here we also report the DFT simulation of T_{2} due to phonons and show that it follows the exact same temperature profile of T_{1} and a trend T_{2} ≤ T_{1}. In particular, we simulate T_{2} ~ T_{1} for Raman relaxation due to the modulation of the hyperfine coupling (see inset of Fig. 5) and T_{2} ~ 0.8T_{1} in the presence of the modulation of the gtensor (see Supplementary Fig. 19). This finding is in stark contrast with the canonical relation T_{2} = 2T_{1} for phononlimited spin coherence. We attribute this deviation to the presence of the pure dephasing contribution (\({T}_{2}^{*}\)) to T_{2}, usually neglected for spinphonon processes. As we show in Supplementary Note 4, \({T}_{2}^{*}\) vanishes for onephonon processes, but becomes finite for twophonon processes, like the ones considered here. The phononinduced puredephasing decoherence mechanism has a simple physical interpretation. \({T}_{2}^{*}\) results from energyconserving phonon processes, where a pair of degenerate phonons exchange energy among them, i.e. one phonon is absorbed by the spin and another one is simultaneously emitted. If the two phonons are not equally coupled to the spin, this process generates an effective magnetic noise at the spin site that leads to dephasing. Interestingly, there is an elegant parallel between this process and the one experienced by the spin coupled to a spin bath via dipole interactions. In the latter scenario, the cause for decoherence is energyconserving flipflop spin processes, where spins with an equal gyromagnetic factor exchange energy among them and cause dephasing of the central spin^{50,51}.
Importantly, our simulations of T_{2} are in agreement with literature results that usually show T_{2} < T_{1}^{27,52,53,54,55} and reproduce the temperature dependence of the experimental decoherence time of [VO(TPP)] up to small rescaling factor. The discrepancy we observe in Fig. 5 is likely due to the hereneglected coupling of the electronic spin with other nuclei^{56} (e.g., first coordination sphere ^{14}N), contributing to the decoherence rate with additional pure dephasing terms with a similar temperature dependence. However, the ratio between T_{2} and T_{1} observed in the literature appears to be systemdependent. Intriguingly, a study on NitrogenVacancy diamond defects has shown that T_{2} ~ 0.5T_{1} after a dynamical decoupling pulse sequence was applied to entirely remove the contribution of spinspin interactions and reveal the true phononlimited T_{2}^{57}. These results point to the urgent need to further investigate the relation between spin relaxation and decoherence to verify if there is a universal trend among T_{2} and T_{1}.
In order to provide some chemical insight into the nature of the relevant phonons for spin relaxation, we study the molecular distortions associated with the first mode at the Γpoint, for [VO(TPP)] at ~1.5 meV. As it can be appreciated from Fig. 6, this vibration corresponds to a large distortion of the phenyl rings overlapped to a bending of the porphyrin ring. Supplementary Movie 1 also shows that this molecular distortion is further combined with intermolecular translations and rotations, where different molecules in the crystals change their reciprocal orientation and position. The visualization of these lowenergy phonons thus shows that they are affected by both intra and intermolecular contributions.
Discussion
Although previous experiments and simulations have shown that lowenergy optical modes are a typical feature of MNMs^{25,30,31,32,33,37,38}, a full characterization of the role of lowenergy optical phonons in the spin decoherence of a molecular qubit has never been provided before. In this work, we demonstrated the capabilities of our synergetic approach that combines IXS measurements and DFT spin dynamics simulations for a quantitative investigation of phonons in molecular qubits and their role in spin decoherence. To benchmark this approach we have chosen one of the most promising molecular qubits, the wellcharacterized [VO(TPP)] complex, being also one of the most radiationrobust of the vanadyl family. In particular, we have demonstrated the presence of very soft optical modes in [VO(TPP)], never seen before at such low energies, making this compound the ideal testbed for investigating their contribution to spin decoherence. We have then provided a complete account of the contributions to spin decoherence, including the pure dephasing induced by twophonon Raman processes, a mechanism never discussed previously. Our analysis made it possible to uncover a wealth of additional insights on the spin dynamics of this molecular qubit that can hardly be extracted from the sole relaxation data. Indeed, the latter approach, based on a simple fitting of the BronsVanVleck formula, failed to recognize the importance of lowenergy vibrations and instead identified highenergy phonons as responsible for spin relaxation. Although it is not possible to exclude additional relaxation mechanisms taking place^{26,58}, the ones simulated here clearly represent an important contribution to T_{1} and T_{2} and they must therefore be addressed for improving coherence times.
Beyond evidencing the presence of the lowenergy vibration modes that fully break down the Debye model in [VO(TPP)], we also identified their origin in the soft torsional degrees of freedom involving the rotation of the phenyl groups. These results point to the removal of the four phenyls rings from [VO(TPP)] (as in vanadyl porphyrin^{52}) as a potential chemical strategy for tailoring the intramolecular motions and thus slowing down relaxation. This conclusion is also supported by the fact that [VO(TPP)] is characterized by shorter spinphonon relaxation times T_{1} with respect to other VObased systems^{53,54} (see Fig. 7), where these phenyl groups are absent (for comparison with other VObased systems in frozen solution^{14} and decoherence times T_{2} see Supplementary Figs. 22–23, respectively).
Since the majority of experimental studies on the relaxation dynamics of MNMs are performed in crystals or polycristalline samples, the study of phonons in molecular crystals is the natural first step to understand spin relaxation and benchmark theoretical models. However, the longterm goal for applications is the embedding of single molecules in quantum devices. To check the persistence of the lowenergy vibrations in single molecules and test the potential of the proposed chemical strategy, we therefore performed also gasphase vibration calculations of [VO(TPP)] and of the vanadyl porphyrin (see Supplementary Movie 2 provided as Supplementary Material). Conversely, these simulations show that the gasphase vibrations of [VO(TPP)] also exhibit lowenergy modes with comparable frequency as the lattice’s Γpoint. On the other hand, once the phenyl rings are removed, the lowest energy vibrational mode in the gas phase is shifted to 6.5 meV, a fourfold increase in frequency. This result points to a potential increase in T_{1} as the one reported in Fig. 5 after the removal of lowenergy phonons. However, it should be stressed that this value represents an upper limit to what can possibly be achieved, as the presence of any condensedmatter environment will eventually reintroduce some lowenergy vibrations due to the admixing of inter and intramolecular displacements.
In conclusion, our work revealed the vibrational contributions to spin decoherence in a prototypical molecular qubit, highlighting the importance of lowenergy phonons for both spin relaxation and pure spin dephasing. Moreover, we have shown that ultralow optical phonons in molecular crystals are likely originated by the presence of lowenergy modes at the molecular level further shifted down in energy once combined with lattice vibrations, and that chemical engineering of molecular structures have the potential to considerably reduce the lowenergy vibrational contributions and improve spinrelaxation time. The unprecedented insight into the nature of vibrational states, their coupling to spin, and their role in spin decoherence demonstrates that the IXS+DFT spin dynamics approach is the new technique of choice for investigating spinphonon dynamics in molecular compounds and for the design of new systems.
Methods
Sample preparation
The [VO(TPP)] molecular qubit was synthesized according to the experimental procedure reported in ref. ^{27}, and purified by thermal sublimation at 533 K (260^{∘}C) and 10^{−6} mbar for 48 h. After purification, single crystals suitable for IXS experiments (about 1 mm^{3}) were obtained by slow evaporation of a CH_{2}Cl_{2}/nheptane (95:5) solution over two weeks.
IXS experiment
The ID28 beamline at the European Syncrotron Facility is an inelastic Xrays spectrometer with energy and momentum transfer ranges particularly suited for studying phonons dispersions. The very small beam size (of the order of a few tens of μm) allows the investigation of systems available only in small quantities. The optical layout is based on the tripleaxis principle, composed of the very highenergy resolution monochromator (first axis), the sample goniometer (second axis) and the crystal analyser (third axis). Thanks to its backscattering geometry and its length, ID28 is able to acquire a sufficient beam offset between the incident photon beam from the Xray source and the very highenergy resolution beam focused at the sample position.
A single crystal of [VO(TPP)] with size of the order of 1 × 1 × 0.5 mm^{3} was mounted on ID28 side station in order to explore the reciprocal space with diffuse scattering and identify a suitable scattering plane. The sample was oriented in order to have (h0l) as the scattering plane (in conventional cell notation, see Supplementary Tables 1–3 and Supplementary Fig. 1) and explore the [VO(TPP)] reciprocal space along the Γ–N, Γ–K_{z} and Γ–K_{x} directions ((h0h), (00l) and (h00) directions, respectively), in both longitudinal and transverse configurations. In particular, we focused on the 600 and 006 Bragg reflections. The sample was glued on a standard sample holder and placed on the ID28 sample stage, at the temperature of 300 K. We worked in transmission geometry and we exploited two different configurations of the ID28 silicon monochromator: Si(9 9 9), selecting an incoming energy of E = 17.794 keV with an energy resolution δE = 3.0 meV and Si(12 12 12) with an incoming energy of E = 23.725 keV and an energy resolution δE = 1.5 meV. The spectrometer layout, with 9 different analysers within the same scattering plane (~0.75 deg spacing), enables the detection of 9 different momentum transfer simultaneously. This characteristic of the ID28 instrument increases significantly the BZ sampling, obtained when the additional analysers aligns along a symmetry direction. Data on [VO(TPP)] were collected performing constantQ energy scans for several Q values along the selected highsymmetry directions. Given the long tails of the Lorentzianlike lineshape of the elastic signal (described with a dampedharmonicoscillator function), we can only measure ∣q∣ ≥ 0.1 Å^{−1}, i.e., not to close to the selected Bragg peaks/Γ points. Each dataset has also been normalized by the incident Xray flux measured by the beam monitor before the sample.
DFT calculations
The unitcell Xray structure of [VO(TPP)]^{27} was used as starting points for a periodic DFT optimization of a 3 × 3 × 3 supercell with the software CP2K^{59} (see Supplementary Table 4). Density functional theory (DFT) with the PBE functional^{60}, including Grimme’s D3 van der Waals corrections^{61}, was used together with a doublezeta polarized (DZVP) MOLOPT basis set. A planewave cutoff of 1875 Ry was used. Lattice force constants and phonons were computed with the software MolForge^{26}. Lattice force constants were computed with a twopoint numerical differentiation of atomic forces with step of 0.01 Å. The tensors A and g were computed with the software ORCA^{62} using DFT with the hybrid functional PBE0^{63} and DKHdef2SVP basis for C and H atoms and DKHdef2TZVPP for all other atoms. All basis sets were decontracted and RIJCOSX was used as approximation for Coulomb and HartreeFock Exchange.
Data analysis and simulations
IXS spectra as a function of energy were fitted by means of the custom beamline software FIT28 (see ref. ^{46}). The spectral components (elastic line and phonon excitations) were modelled with singledampedharmonicoscillator functions with the experimental resolution of each specific instrumental configuration. The Stokes and antiStokes intensities are corrected for the Bose–Einstein thermal population factor. It is worth stressing that the presence of more than one phonon mode within the experimental resolution with a nonzero crosssection results in an effective broadening of excitation lines, emerging from the summation of excitations close in energy.
The onephonon IXS crosssection is defined as^{45}
where ω_{j}(q) is the energy of the jth phonon branch. The momentum conservation law of the inelastic scattering events involve the exchanged scattering vector Q, the phonon quasimomentum q and the reciprocal lattice vector G, while the energy conservation depends on the energy transfer ω. \({n}_{j}({{{{{{{\bf{q}}}}}}}})={(\exp (\beta {\omega }_{j}({{{{{{{\bf{q}}}}}}}}))1)}^{1}\) is the phonon Bose factor with \(\beta={({K}_{B}T)}^{1}\). The function F_{j,q}(Q) is the onephonon structure factor, taking into account interference effects between the different atoms d in the unit cell:
where f_{d}(Q) is the Xray atomic form factor, m_{d} the mass and R_{d} the position vector in the real space of each atom, while \({{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{j}^{d}({{{{{{{\bf{q}}}}}}}})\) are the phonon polarization vectors.
Data simulations were performed by calculating the scattering crosssection in Eq. (2) with pDFTcalculated phonon energies ω_{j}(q) and normal modes polarization vectors \({{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{j}^{d}({{{{{{{\bf{q}}}}}}}})\). A rescaling of about 10% was uniformly applied to all the calculated phonon energies along all the symmetry directions. We then assumed a singledampedharmonicoscillator lineshape with the FWHM of the corresponding dataset (for the colour maps we used a FWHM = 0.1 meV, in order to distinguish the contribution of single phonon branches with a nonzero crosssection within the experimental resolution).
Neural Networks
The neural networks used to interpolate spin Hamiltonian coefficients were built using the Keras API and Tensorflow library. The input layer contains 234 nodes, corresponding to 3N Cartesian coordinates with N being the number of atoms in the [VO(TPP)] molecule, and the output layer contains 9 nodes, corresponding to the tensor components of A or g. The number of hidden layers and the number of nodes in each hidden layers are varied to obtain the model bestsuited for A and g. The sigmoid function is used as activation function for the hidden layers. The selected model for A has 2 hidden layers with 128 and 64 nodes in each hidden layer, while the model selected for g has 3 hidden layers with 128, 64, and 32 nodes in each hidden layer. The model obtained is trained with 1600 configurations of randomly distorted [VO(TPP)] molecule. The regularization hyperparameter is optimized for each model using a validation set of 158 configurations of [VO(TPP)] and the performance of the models is evaluated with a test set of 200 configurations that the models have never seen before (see Supplementary Note 2 and Supplementary Figs. 9–11 for further details).
Spinrelaxation simulations
The trained machine learning models are used to calculate the first and secondorder derivatives, V_{αq} = (∂A/∂q_{αq}) and \({{{{{{{{\bf{V}}}}}}}}}_{\alpha {{{{{{{\bf{q}}}}}}}}\beta {{{{{{{{\bf{q}}}}}}}}}^{{\prime} }}= ({\partial }^{2}{{{{{{{\bf{A}}}}}}}}/\partial {q}_{\alpha {{{{{{{\bf{q}}}}}}}}}\partial {q}_{\beta {{{{{{{{\bf{q}}}}}}}}}^{{\prime} }})\), numerically (Supplementary Fig. 12). The norm of the spinphonon coupling coefficients for the modulation of A is defined as
and similarly for the gtensor. The coefficients are then used to model the spin relaxation with MolForge^{26} (see Supplementary Note 3 for more details). The relaxation time T_{1} is fitted from the M_{z} dependence on time with a double exponential function (Supplementary Fig. 13a, c). The coherence time T_{2} is fitted from the M_{⊥} dependence on time with a double exponential function (Supplementary Fig. 13b, d). The external static magnetic field is set at 0.33 T along the zdirection. T_{1} values were converged with respect to the number of q points used to sample the Brillouin zone and the size of the Gaussian smearing used to represent Dirac’s delta function appearing in the Redfield equations^{38,49} (Supplementary Figs. 14–20). A final mesh of 4 × 4 × 4 q points and smear of 10 cm^{−1} was used to obtain the temperature profile of T_{1} and T_{2}.
Data availability
Raw data from the IXS experiment were generated at the ESRF (proposal number HC4312) and are available from the corresponding authors upon reasonable request. They will also be available on the ESRF Data Portal from 08/10/2023.
Code availability
Matlab codes for data simulations are available from the corresponding authors upon reasonable request. The code MolForge was used to simulate phonons and spin dynamics, and it is available at “github.com/LunghiGroup/MolForge" and at https://doi.org/10.5281/zenodo.7596042.
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Acknowledgements
Dr. Yong Cai is gratefully acknowledged for useful discussions on IXS experiments. This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement No 862893 (FETOPEN project FATMOLS) (S.C.) and from the European Research Council (ERC) (grant agreement No. [948493]) (A.L.). It was also supported by the Italian MIUR with the Progetto Dipartimenti di Eccellenza 20182022 (ref. B96C1700020008) (F.T., R.S.), by Fondazione Cariparma (S.C.) and The National Recovery and Resilience Plan, Mission 4 Component 2  Investment 1.4  NATIONAL CENTER FOR HPC, BIG DATA AND QUANTUM COMPUTING—funded by the European Union—NextGeneration EU  CUP B83C22002830001 (F.T.). We also acknowledge the European Synchrotron Radiation Source for instrument time on the ID28 beamline (proposal number HC4312) (E.G.) and the Trinity College Research IT and the Irish Centre for HighEnd Computing (ICHEC) for computational resources.
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E.G., P.S., R.S. and S.C. proposed the use of inelastic Xray scattering to measure phonons in molecular qubits. E.G., S.Ch. and L.P. performed the experiment after discussion with C.M. and R.C. on a single crystal sample synthesized by F.S. Data treatment was made by S.Ch. and L.P., while A.A., F.T. and A.L. performed DFT calculations. Data analysis and simulations were made by E.G., S.Ch. and S.C. A.L. developed the neuralnetwork approach and performed spin dynamics simulations with V.H.A.N. E.G., A.L. and S.C. wrote the manuscript with inputs from all coauthors.
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Garlatti, E., Albino, A., Chicco, S. et al. The critical role of ultralowenergy vibrations in the relaxation dynamics of molecular qubits. Nat Commun 14, 1653 (2023). https://doi.org/10.1038/s4146702336852y
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DOI: https://doi.org/10.1038/s4146702336852y
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