Pulsed W-band electron paramagnetic resonance established in 2008 that the star-shaped single-molecule magnet [Fe4(acac)6(Br-mp)2] (an S = 5 tetrairon(III) cluster) possesses genuine quantum coherence in frozen solution, with a phase memory time T2 up to 630 ns in a proton-free carbon disulfide matrix and clear Rabi oscillations. That study, however, could not identify the microscopic spin-lattice relaxation mechanism (Raman versus Orbach), could not quantitatively assign the decoherence that remains once solvent nuclear spins are removed, and left its extrapolated microsecond-scale low-temperature coherence times untested. The open problem is a quantitative, mechanism-resolved account of spin-lattice relaxation (T1) and phase decoherence (T2) in this single-molecule magnet over an extended temperature range and across matrix and isotopic variants, sufficient to decide whether the design levers of molecular deuteration and low-temperature operation can extend coherence times toward the microsecond regime required for single-molecule-magnet qubits.

Background

A single-molecule magnet (SMM) is an exchange-coupled metal cluster with a large spin ground state and a strong uniaxial magnetic anisotropy, which together create an energy barrier against reversal of the molecule's magnetization; the archetype studied here is the star-shaped tetrairon(III) complex [FeIII4(acac)6(Br-mp)2] (acac = acetylacetonate; Br-mp3- = the deprotonated tripod 2-(bromomethyl)-2-(hydroxymethyl)-1,3-propanediolate). Three antiferromagnetically coupled peripheral Fe(III) ions surround a central Fe(III) ion, giving a total spin S = 5 ground multiplet whose eleven magnetic sublevels (spin projection MS = -5 to +5) are split even at zero applied field by the zero-field splitting parameter D = -0.34 cm^-1 (a measure of the anisotropy of the electron spin interaction energy). This makes the lowest magnetic-resonance transition fall conveniently near zero field at W-band (94 GHz) microwave frequencies. Proposed applications of SMMs in quantum information processing require that a superposition of two spin states survive for a time T2, the phase memory or spin coherence time, long enough to perform many gate operations, and that the spin can be coherently driven (Rabi oscillations, i.e. oscillations of the spin population versus the length of a resonant microwave pulse). In pulsed electron paramagnetic resonance (EPR), T2 is measured from the decay of a Hahn echo (a two-pulse refocusing sequence), while the spin-lattice relaxation time T1, the timescale on which the spin exchanges energy with the thermal vibrations (phonons) of its surroundings, is measured by inversion recovery (a sequence that inverts the spin population with a pi pulse and monitors the recovery of the magnetization, whose time constant yields T1). Spin-lattice relaxation proceeds either through the direct process (one phonon resonant with the transition), a Raman process (two phonons scattered inelastically, giving a characteristic power-law temperature dependence), or an Orbach process (absorption to and stimulated emission from a real excited spin state, giving an exponential Arrhenius-type dependence exp(-Delta/kT) that encodes the energy gap Delta of the intervening excited state). Distinguishing Raman from Orbach behavior therefore pins down the spectrum of excited spin states involved in relaxation. Decoherence of electron spins in soft matter is usually dominated by the hyperfine coupling of the electron spin to surrounding nuclear spins (superhyperfine coupling when the nuclei belong to the ligands rather than the metal ions); in the electron spin echo envelope modulation (ESEEM) technique this coupling appears as modulations of the echo decay at the nuclear Larmor frequencies, identifying which nuclei (for example solvent protons versus molecular protons) couple to the electron. Before 2008, coherence in high-spin SMMs was only estimated, at around 10 ns, although the antiferromagnetic Cr7Ni molecular ring had shown T2 of about 400 ns at 4.5 K, rising to 3.8 microseconds at 1.8 K upon full deuteration of the molecule (Ardavan et al., Phys. Rev. Lett. 98, 057201 (2007)). Schlegel, van Slageren, Manoli, Brechin and Dressel (Phys. Rev. Lett. 101, 147203 (2008)) then reported the first direct echo-detected coherence measurement and Rabi oscillations in an SMM proper: for the Fe4 cluster diluted in frozen toluene they found T2 = 307 ± 20 ns at 4.3 K (monoexponential echo decay), T1 = 1056 ± 20 ns, a strong temperature dependence of both, and ESEEM showing coupling to solvent protons; replacing toluene by deuterated toluene changed T2 very little, while the proton-free solvent CS2 raised T2 to 527 ns at zero field and to a maximum of 630 ± 30 ns at 0.1 T, accompanied by what the paper reports as an increase of T1 (T1 = 943 ns at 4.3 K and 0 T, numerically below the 1056 ± 20 ns found for toluene at the same conditions), so that no simple T1-T2 relation holds. They concluded that solvent nuclear spins, not the molecule's own 58 protons, are the primary decoherence channel in protonated matrices, but could not assign the decoherence that remains in CS2; the Raman-versus-Orbach question for T1 was left open (an Orbach fit suggested an intermediate level about 5 cm^-1 above the ground state, within the ground multiplet), the decay of the Rabi oscillations could not be separated into zero-field-splitting distributions (D-strain, i.e. molecule-to-molecule variations of D), orientational distributions, and intrinsic decoherence, and the extrapolated low-temperature coherence times (about 750 ns at 0 K for toluene, expected to be larger for CS2, potentially reaching the microsecond range suitable for qubits) were not measured. The companion study of two new Fe4 clusters (Schlegel et al., Chem. Eur. J. 16, 10178 (2010)) confirmed S = 5 ground states, exchange constants of about -8.2 to -8.5 cm^-1, and Orbach-type magnetization-relaxation barriers of about 12 cm^-1 from alternating-current susceptibility, and found that relaxation slows in frozen solution through an increased prefactor rather than a changed barrier, but did not address the pulsed-EPR relaxation mechanism or coherence times. Subsequent literature clarified decoherence mechanisms in related settings — for crystals of the SMM Fe8, phonon, nuclear-spin and intermolecular dipolar decoherence channels were quantitatively separated and verified against theory (Takahashi et al., Nature 476, 76 (2011)) — and the molecular-qubit community demonstrated long coherence in S = 1/2 mononuclear complexes: millisecond T2 in a nuclear-spin-free environment (Zadrozny et al., ACS Cent. Sci. (2015)), very long T2 and T1 in phthalocyanine complexes (Bader et al., Chem. Commun. (2016)), and a parameter-free quantitative prediction of nuclear-spin-diffusion-limited T2 for copper(II) beta-diketonates (Lenz et al., Chem. Commun. (2017)). Reviews of the tetrairon(III) SMM family (Cornia et al., Eur. J. Inorg. Chem. (2019)) and of molecular magnets for quantum technologies (Moreno-Pineda and Wernsdorfer, Nat. Rev. Phys. 3, 645 (2021)) continue to cite the 630 ns frozen-solution Fe4 result as the coherence benchmark for high-spin SMMs, while the Fe4 line itself developed toward surface grafting and devices; no published study has returned to the Fe4 frozen-solution system to resolve the relaxation mechanism or the isotope-engineering question left open in 2008.

Problem Statement

Construct a quantitative, mechanism-resolved account of spin relaxation and decoherence of the S = 5 single-molecule magnet [Fe4(acac)6(Br-mp)2] diluted in frozen solution, as left open by the 2008 W-band pulsed-EPR study. The account must jointly deliver the following four mandatory deliverables, as one linked scientific objective: (1) identification of the microscopic spin-lattice relaxation mechanism governing T1 over an extended temperature range - discrimination between a Raman power law and an Orbach process, including the energy and coupling strengths of the intervening excited spin state(s) within or near the S = 5 ground multiplet; (2) a quantitative decomposition of the decoherence that remains when solvent nuclear spins are eliminated (the CS2 matrix), determining the contribution of the molecule's own 58 protons (and other superhyperfine-coupled nuclei) relative to other intrinsic channels, and in particular a quantitative determination of whether a fully or partially deuterated Fe4 analogue would exhibit a longer T2 than the protonated molecule under otherwise identical conditions; (3) the relation between the T1 energy-relaxation channels and the phase decoherence T2 in these glassy matrices, which the source data show is not simple; and (4) a quantitative low-temperature determination of the predicted increase of T2, including whether the extrapolated microsecond-scale coherence times suitable for qubit operation are actually achieved, together with the separation of inhomogeneous broadening (D-strain and orientational distributions) from intrinsic decoherence that this requires. The account is required for the parent [Fe4(acac)6(Br-mp)2] complex in each of the frozen matrices examined in 2008 (toluene, deuterated toluene and CS2); the molecule's chemically substituted and isotopically labelled analogues enter only through deliverable (2). Where the source literature measures no value of a required quantity - in particular the T2 of a deuterated analogue and the low-temperature trend of T2 - the deliverable is a quantitative determination or prediction from the available data and established theory, with explicitly stated assumptions. The problem is the one stated by the source work and its immediate companion literature; it is not the general engineering of molecular qubits, which later literature pursued on mononuclear S = 1/2 platforms instead of resolving it for this high-spin cluster.

References

  1. C. Schlegel, J. van Slageren, M. Manoli, E. K. Brechin, M. Dressel, 'Direct observation of quantum coherence in single-molecule magnets', Physical Review Letters 101, 147203 (2008). DOI: 10.1103/PhysRevLett.101.147203; arXiv:0807.0303; https://doi.org/10.1103/PhysRevLett.101.147203
  2. A. Ardavan, O. Rival, J. J. L. Morton, S. J. Blundell, A. M. Tyryshkin, G. A. Timco, R. E. P. Winpenny, 'Will spin-relaxation times in molecular magnets permit quantum information processing?', Physical Review Letters 98, 057201 (2007). DOI: 10.1103/PhysRevLett.98.057201; https://doi.org/10.1103/PhysRevLett.98.057201
  3. C. Schlegel, E. Burzurí, F. Luis, F. Moro, M. Manoli, E. K. Brechin, M. Murrie, J. van Slageren, 'Magnetic properties of two new Fe4 single-molecule magnets in the solid state and in frozen solution', Chemistry - A European Journal 16, 10178-10185 (2010). DOI: 10.1002/chem.200903505; https://doi.org/10.1002/chem.200903505
  4. F. El Hallak, P. Neugebauer, A.-L. Barra, J. van Slageren, M. Dressel, A. Cornia, 'Torque-detected ESR of a tetrairon(III) single molecule magnet', Journal of Magnetic Resonance 223, 55-60 (2012). DOI: 10.1016/j.jmr.2012.08.008; https://doi.org/10.1016/j.jmr.2012.08.008
  5. S. Takahashi, I. S. Tupitsyn, J. van Tol, C. C. Beedle, D. N. Hendrickson, P. C. E. Stamp, 'Decoherence in crystals of quantum molecular magnets', Nature 476, 76-79 (2011). DOI: 10.1038/nature10314; https://doi.org/10.1038/nature10314
  6. J. M. Zadrozny, J. Niklas, O. G. Poluektov, D. E. Freedman, 'Millisecond coherence time in a tunable molecular electronic spin qubit', ACS Central Science 1, 488-492 (2015). DOI: 10.1021/acscentsci.5b00338; https://doi.org/10.1021/acscentsci.5b00338
  7. K. Bader, M. Winkler, J. van Slageren, 'Tuning of molecular qubits: very long coherence and spin-lattice relaxation times', Chemical Communications 52, 3623-3626 (2016). DOI: 10.1039/c6cc00300a; https://doi.org/10.1039/c6cc00300a
  8. S. Lenz, K. Bader, H. Bamberger, J. van Slageren, 'Quantitative prediction of nuclear-spin-diffusion-limited coherence times of molecular quantum bits based on copper(II)', Chemical Communications 53, 4477-4480 (2017). DOI: 10.1039/c6cc07813c; https://doi.org/10.1039/c6cc07813c
  9. A. Cornia, M. Mannini, R. Sessoli, D. Gatteschi, 'Propeller-shaped Fe4 and Fe3M molecular nanomagnets: a journey from crystals to addressable single molecules', European Journal of Inorganic Chemistry 2019, issue 5, 552-568 (2019). DOI: 10.1002/ejic.201801266; https://doi.org/10.1002/ejic.201801266
  10. E. Moreno-Pineda, W. Wernsdorfer, 'Measuring molecular magnets for quantum technologies', Nature Reviews Physics 3, 645-659 (2021). DOI: 10.1038/s42254-021-00340-3; https://doi.org/10.1038/s42254-021-00340-3