Quadrupolar Spin Research Articles

Nuclear spin quantum register in an optically active semiconductor quantum dot.

Epitaxial quantum dots (QDs) have long been identified as promising charge spin qubits offering an efficient interface to quantum light and advanced semiconductor nanofabrication technologies. However, charge spin coherence is limited by interaction with the nanoscale ensemble of atomic nuclear spins, which is particularly problematic in strained self-assembled dots. Here, we use strain-free GaAs/AlGaAs QDs, demonstrating a fully functioning two-qubit quantum register using the nanoscale ensemble of arsenic quadrupolar nuclear spins as its hardware. Tailored radio-frequency pulses allow quantum state storage for up to 20 ms, and are used for few-microsecond single-qubit and two-qubit control gates with fidelities exceeding 97%. Combining long coherence and high-fidelity control with optical initialization and readout, we implement benchmark quantum computations such as Grover's search and the Deutsch-Jozsa algorithm. Our results identify QD nuclei as a potential quantum information resource, which can complement charge spins and light particles in future QD circuits.

Orbital spin dynamics of a millisecond pulsar around a massive BH with a general mass quadrupole

ABSTRACT We investigate the spin dynamics of a millisecond pulsar (MSP) in compact orbit around a Kerr-like massive BH with an general mass quadrupole. We use the Mathisson–Papetrou–Dixon formulation to compute the orbital and spin evolution of the MSP, accounting for the non-linear interaction of the pulsar’s energy–momentum tensor on the background space–time metric. We investigate how the MSP spin and BH quadrupole moment manifest in the pulsar spin-orbital dynamics. We discuss the astrophysical observational implications of these spin and orbital dynamics on the timing of a radio pulsar in an extreme mass ratio binary, e.g. a Galactic Centre pulsar. In particular, notable timing variations in the Einstein delay and Roemer delay are observed, along with modifications to the pulsar pulse profile.

Hidden SU(2) symmetries, the symmetry hierarchy, and the emergent eightfold way in spin-1 quantum magnets

The largest allowed symmetry in a spin-1 quantum system is an $SU(3)$ symmetry rather than the $SO(3)$ spin rotation. In this work, we reveal some $SU(2)$ symmetries as subgroups of $SU(3)$ that, to the best of our knowledge, have not previously been recognized. Then, we construct $SU(2)$ symmetric Hamiltonians and explore the ground-state phase diagram in accordance with the $SU(3)\supset SU(2)\times U(1)$ symmetry hierarchy. It is natural to treat the eight generators of the $SU(3)$ symmetry on an equal footing; this approach is called the eight-fold way. We find that the spin spectral functions and spin quadrupole spectral functions share the same structure, provided that the elementary excitations are flavor waves at low energies, which serves as a clue to the eight-fold way. An emergent $S=1/2$ quantum spin liquid is proposed to coexist with gapful spin nematic order in one of the ground states. In analogy to quantum chromodynamics, we find the gap relation for hydrodynamic modes in quantum spin-orbital liquid states, which is nothing but the Gell-Mann-Okubo formula.

High-Field Magic Angle Spinning Dynamic Nuclear Polarization Using Radicals Created by γ-Irradiation.

High-field magic angle spinning dynamic nuclear polarization (MAS DNP) is often used to enhance the sensitivity of solid-state nuclear magnetic resonance experiments by transferring spin polarization from electron spins to nuclear spins. Here, we demonstrate that γ-irradiation induces the formation of stable radicals in inorganic solids, such as fused quartz and borosilicate glasses, as well as organic solids, such as glucose, cellulose, and a urea/polyethylene polymer. The radicals were then used to polarize 29Si or 1H spins in the core of some of these materials. Significant MAS DNP enhancements (ε) of more than 400 and 30 were obtained for fused quartz and glucose, respectively. For other samples, negligible values of ε were obtained, likely due to low concentrations of radicals or the presence of abundant quadrupolar spins. These results demonstrate that ionizing radiation is a promising alternative method for generating stable radicals that are suitable for high-field MAS DNP experiments.

Accurate 1H-14N distance measurements by phase-modulated RESPDOR at ultra-fast MAS.

The combination of a phase-modulated (PM) saturation pulse and symmetry-based dipolar recoupling into a rotational-echo saturation-pulse double-resonance (RESPDOR) sequence has been employed to measure 1H-14N distances. Such a measurement is challenging owing to the quadrupolar interaction of 14N nucleus and the intense 1H-1H homonuclear dipolar interactions. Thanks to the recent advances in probe technology, the homonuclear dipolar interaction can be sufficiently suppressed at a fast MAS frequency (νR ≥ 60 kHz). PM pulse is robust to large variations of parameters on quadrupolar spins, but it has not been demonstrated under very fast MAS conditions. On the other hand, the RESPDOR sequence is applicable to such condition when it employs symmetry-based pulses during the recoupling period, but a prior knowledge on the system is required. In this article, we demonstrated the PM-RESPDOR combination for providing accurate 1H-14N distances at a very fast MAS frequency of 70 kHz on two samples, namely L-tyrosine⋅HCl and N-acetyl-L-alanine. This sequence, supported by simulations and experiments, has shown its feasibility at νR = 70 kHz as well as the robustness to the 14N quadrupolar interaction. It is applicable to a wide range of 1H-14N dipolar coupling constants when a radio frequency field on the 14N channel is approximately 80 kHz or more, while the PM pulse length lasts 10 rotor periods. For the first time, multiple 1H-14N heteronuclear dipolar couplings, thus multiple quantitative distances, are simultaneously and reliably extracted by fitting the experimental fraction curves with the analytical expression. The size of the 1H-14N dipolar interaction is solely used as a fitting parameter. These determined distances are in excellent agreement with those derived from diffraction techniques.

Double-peak specific heat and spin freezing in the spin-2 triangular lattice antiferromagnet FeAl2Se4

We report the properties of a triangular lattice iron-chalcogenide antiferromagnet FeAl$_{2}$Se$_{4}$. The spin susceptibility reveals a significant antiferromagnetic interaction with a Curie-Weiss temperature {\Theta}$_{CW}$ ~ -200K and a spin-2 local moment. Despite a large spin and a large |{\Theta}$_{CW}$|, the low-temperature behaviors are incompatible with conventional classical magnets. No long-range order is detected down to 0.4K. Similar to the well-known spin-1 magnet NiGa$_{2}$S$_{4}$, the specific heat of FeAl$_{2}$Se$_{4}$ exhibits an unusual double-peak structure and a T$^{2}$ power law at low temperatures, which are attributed to the underlying quadrupolar spin correlations and the Halperin-Saslow modes, respectively. The spin freezing occurs at ~ 14K, below which the relaxation dynamics is probed by the ac susceptibility. Our results are consistent with the early theory for the spin-1 system with Heisenberg and biquadratic spin interactions. We argue that the early proposal of the quadrupolar correlation and gauge glass dynamics may be well extended to FeAl$_{2}$Se$_{4}$. Our results provide useful insights about the magnetic properties of frustrated quantum magnets with high spins.

Satellite nutation of half integer quadrupolar nuclei: Theory and practice.

For quadrupolar spin systems, interpretation of solid state NMR spectra can be hampered by the presence of resonances from both satellite and central transitions. This is particularly true for disordered systems, where many different quadrupolar sites exist, which can have strongly different quadrupolar coupling constants. If second order effects are too strong for obtaining meaningful MAS, MQMAS or STMAS spectra, an approach is needed to successfully separate central and satellite transitions. In this work, we provide a rigorous treatment of 2D quadrupolar nutation NMR for the study of central and satellite transitions in quadrupolar systems. Using this SATURN experiment (SAtellite Transition nUtation of quadRupolar Nuclei) spectral intensity can be assigned to contributions from either central or satellite transitions. We show that the experiment can be applied to any half-integer spin (3/2, 5/2, 7/2 and 9/2), and that spectra can be obtained that closely match simulations. We furthermore show that distributions in quadrupolar parameters do not hamper the assignment of central and satellite transitions from a SATURN experiment.

Magnetization transfer from protons to quadrupolar nuclei in solid-state NMR using PRESTO or dipolar-mediated refocused INEPT methods.

In solid-state NMR spectroscopy, the through-space transfer of magnetization from protons to quadrupolar nuclei is employed to probe proximities between those isotopes. Furthermore, such transfer, in conjunction with Dynamic Nuclear Polarization (DNP), can enhance the NMR sensitivity of quadrupolar nuclei, as it allows the transfer of DNP-enhanced 1H polarization to surrounding nuclei. We compare here the performances of two approaches to achieve such transfer: PRESTO (Phase-shifted Recoupling Effects a Smooth Transfer of Order), which is currently the method of choice to achieve the magnetization transfer from protons to quadrupolar nuclei and which has been shown to supersede Cross-Polarization under Magic-Angle Spinning (MAS) for quadrupolar nuclei and D-RINEPT (Dipolar-mediated Refocused Insensitive Nuclei Enhanced by Polarization Transfer) using symmetry-based SR412 recoupling, which has already been employed to transfer the magnetization in the reverse way from half-integer quadrupolar spin to protons. We also test the PRESTO sequence with R1676 recoupling using 270090180 composite π-pulses as inversion elements. This recoupling scheme, which has previously been proposed to reintroduce 1H Chemical Shift Anisotropy (CSA) at high MAS frequencies with high robustness to rf-field inhomogeneity, has not so far been employed to reintroduce dipolar couplings with protons. These various techniques to transfer magnetization from protons to quadrupolar nuclei are analyzed using (i) an average Hamiltonian theory, (ii) numerical simulations of spin dynamics, and (iii) experimental 1H → 27Al and 1H → 17O transfers in as-synthesized AlPO4-14 and 17O-labelled fumed silica, respectively. The experiments and simulations are done at two magnetic fields (9.4 and 18.8 T) and several spinning speeds (15, 18-24 and 60 kHz). This analysis indicates that owing to its γ-encoded character, PRESTO yields the highest transfer efficiency at low magnetic fields and MAS frequencies, whereas owing to its higher robustness to rf-field inhomogeneity and chemical shifts, D-RINEPT is more sensitive at high fields and MAS frequencies, notably for protons exhibiting large offset or CSA, such as those involved in hydrogen bonds.

63,65Cu NQR Spectra and Spin–Lattice Relaxation in Thermoelectric CuAlO2

The 63,65Cu nuclear quadrupole resonance spectra and spin–lattice relaxation rate (1/T1) have been measured in the semiconductor compound CuAlO2. The value of the nuclei quadrupole interaction constant QCC = 56.24(6) MHz (T = 298 K) has been obtained. The broad maximum has been found in the temperature dependence of 1/T1 in the low-temperature region (below 276 K). This maximum can be associated with the presence of energy levels in the forbidden band. The activation energy has been estimated in CuAlO2 [EA = 45(2) meV], assuming the activation character of the mobility of holes.

Quadrupolar 14N NMR Relaxation from Force-Field and Ab Initio Molecular Dynamics in Different Solvents.

Quadrupolar NMR spin relaxation rates and corresponding line widths were computed for the quadrupolar nucleus 14N for neat acetonitrile as well as for 1-methyl-1,3-imidazole and 1-methyl-1,3,4-triazole in different solvents. Molecular dynamics (MD) was performed with forces from the Kohn-Sham (KS) theory (ab initio MD) and force-field molecular mechanics (classical MD), followed by KS electric field gradient (EFG) calculations. For acetonitrile the agreement of the 14N line width with experiment is very good. Relative line widths for the azole nitrogens are improved over simpler approximations used previously in conjunction with single-point calculations at the multiconfigurational self-consistent field level. Overall, the NMR line widths are computed within a factor of 2 of the experimental values, giving access to reasonable estimates both of the dynamic EFG variance in the solvated systems as well as the associated correlation times that determine the relaxation rates.

Quadrupolar Exchange-Only Spin Qubit.

We propose a quadrupolar exchange-only spin qubit that is highly robust against charge noise and nuclear spin dephasing, the dominant decoherence mechanisms in quantum dots. The qubit consists of four electrons trapped in three quantum dots, and operates in a decoherence-free subspace to mitigate dephasing due to nuclear spins. To reduce sensitivity to charge noise, the qubit can be completely operated at an extended charge noise sweet spot that is first-order insensitive to electrical fluctuations. Because of on-site exchange mediated by the Coulomb interaction, the qubit energy splitting is electrically controllable and can amount to several GHz even in the "off" configuration, making it compatible with conventional microwave cavities.

Pushing the limit of NMR-based distance measurements – retrieving dipolar couplings to spins with extensively large quadrupolar frequencies

Dipolar recoupling under magic-angle spinning allows to measure accurate inter-nuclear distances provided that the two interacting spins can be efficiently and uniformly excited. Alexander (Lex) Vega has shown that adiabatic transfers of populations in quadrupolar spins during the application of constant-wave (cw) radio-frequency pulses lead to efficient and quantifiable dipolar recoupling curves. Accurate distance determination within and beyond the adiabatic regime using cw pulses is limited by the size of the quadrupolar coupling constant. Here we show that using the approach of long-pulse phase modulation, dipolar recoupling and accurate distances can be obtained for nuclei having extensively large quadrupolar frequencies of 5–10 MHz. We demonstrate such results by obtaining a 31P-79/81Br distance in a compound for which bromine-79 (spin-3/2) has a quadrupolar coupling constant of 11.3 MHz, and a 13C-209Bi distance where the bismuth (spin-9/2) has a quadrupolar coupling constant of 256 MHz, equaling a quadrupolar frequency of 10.7 MHz. For Bromine, we demonstrate that an analytical curve based on the assumption of complete spin saturation fits the data. In the case of bismuth acetate, a C-Bi3 spin system must be used in order to match the correct saturation recoupling curve, and results are in agreement with the crystallographic structure.

Effective Floquet Hamiltonian theory of multiple-quantum NMR in anisotropic solids involving quadrupolar spins: Challenges and Perspectives.

The response of a quadrupolar nucleus (nuclear spin with I >12) to an oscillating radio-frequency pulse/field is delicately dependent on the ratio of the quadrupolar coupling constant to the amplitude of the pulse in addition to its duration and oscillating frequency. Consequently, analytic description of the excitation process in the density operator formalism has remained less transparent within existing theoretical frameworks. As an alternative, the utility of the "concept of effective Floquet Hamiltonians" is explored in the present study to explicate the nuances of the excitation process in multilevel systems. Employing spin I =32 as a case study, a unified theoretical framework for describing the excitation of multiple-quantum transitions in static isotropic and anisotropic solids is proposed within the framework of perturbation theory. The challenges resulting from the anisotropic nature of the quadrupolar interactions are addressed within the effective Hamiltonian framework. The possible role of the various interaction frames on the convergence of the perturbation corrections is discussed along with a proposal for a "hybrid method" for describing the excitation process in anisotropic solids. Employing suitable model systems, the validity of the proposed hybrid method is substantiated through a rigorous comparison between simulations emerging from exact numerical and analytic methods.

Saturation capability of short phase modulated pulses facilitates the measurement of longitudinal relaxation times of quadrupolar nuclei

The ability of various pulse types, which are commonly applied for distance measurements, to saturate or invert quadrupolar spin polarization has been compared by observing their effect on magnetization recovery curves under magic-angle spinning. A selective central transition inversion pulse yields a bi-exponential recovery for a diamagnetic sample with a spin-3/2, consistent with the existence of two processes: the fluctuations of the electric field gradients with identical single (W1) and double (W2) quantum quadrupolar-driven relaxation rates, and spin exchange between the central transition of one spin and satellite transitions of a dipolar-coupled similar spin. Using a phase modulated pulse, developed for distance measurements in quadrupolar spins (Nimerovsky et al., JMR 244, 2014, 107–113) and suggested for achieving the complete saturation of all quadrupolar spin energy levels, a mono-exponential relaxation model fits the data, compatible with elimination of the spin exchange processes. Other pulses such as an adiabatic pulse lasting one-third of a rotor period, and a two-rotor-period long continuous-wave pulse, both used for distance measurements under special experimental conditions, yield good fits to bi-exponential functions with varying coefficients and time constants due to variations in initial conditions. Those values are a measure of the extent of saturation obtained from these pulses. An empirical fit of the recovery curves to a stretched exponential function can provide general recovery times. A stretching parameter very close to unity, as obtained for a phase modulated pulse but not for other cases, suggests that in this case recovery times and longitudinal relaxation times are similar. The results are experimentally demonstrated for compounds containing 11B (spin-3/2) and 51V (spin-7/2). We propose that accurate spin lattice relaxation rates can be measured by a short phase modulated pulse (<1–2ms), similarly to the "true T1" measured by saturation with an asynchronous pulse train (Yesinowski, JMR 252, 2015, 135–144).

Analysis of large-anisotropy-spin recoupling pulses for distance measurement under magic-angle spinning NMR shows the superiority and robustness of a phase modulated saturation pulse.

The distance between a spin one-half and an attached spin possessing a large anisotropy can be obtained using different dipolar recoupling sequences that are based on the rotational-echo double resonance technique under magic-angle spinning solid-state NMR. The general difference between these sequences with respect to the coupled spin is the set of pulses applied in order to drive this spin out of equilibrium, thereby recoupling the dipolar interaction. Since complete inversion is practically not possible due to the coupled-spin anisotropy, using one or another pulse depends on the experimental and spin conditions: the spinning speed, the strength of the radio frequency field, the size of the anisotropic interaction (quadrupolar or chemical shiftanisotropy couplings), the offset, and the accuracy of setting the magic angle. Here we present a detailed description of the behavior of the anisotropic spin magnetization, including the macroscopic level transition probabilities, the degree of inversion, and the microscopic and macroscopic magnetizations during the applications of these pulses under different experimental conditions. As simulations show, a complete randomization of spin populations under a wide range of experimental conditions occurs under a specific phase modulation of the recoupling pulse while for all other cases dependence on experimental conditions is large and the achievable bandwidth is limited. A result of this detailed analysis is that the extension of the phase modulated pulse extends even further its robustness. The saturation capability is demonstrated experimentally for the quadrupolar spin of boron-11 in 4-methoxyphenylboronic acid.

Communication: Dissolution DNP reveals a long-lived deuterium spin state imbalance in methyl groups.

We report the generation and observation of long-lived spin states in deuterated methyl groups by dissolution DNP. These states are based on population imbalances between manifolds of spin states corresponding to irreducible representations of the C3v point group and feature strongly dampened quadrupolar relaxation. Their lifetime depends on the activation energies of methyl group rotation. With dissolution DNP, we can reduce the deuterium relaxation rate by a factor up to 20, thereby extending the experimentally available time window. The intrinsic limitation of NMR spectroscopy of quadrupolar spins by short relaxation times can thus be alleviated.

Low-energy physics of three-orbital impurity model with Kanamori interaction

We discuss the low-energy physics of the three-orbital Anderson impurity model with the Coulomb interaction term of the Kanamori form which has orbital SO(3) and spin SU(2) symmetry and describes systems with partially occupied ${t}_{2g}$ shells. We focus on the case with two electrons in the impurity that is relevant to Hund's metals. Using the Schrieffer-Wolff transformation we derive an effective Kondo model with couplings between the bulk and impurity electrons expressed in terms of spin, orbital, and orbital quadrupole operators. The bare spin-spin Kondo interaction is much smaller than the orbit-orbit and spin-orbital couplings or is even ferromagnetic. Furthermore, the perturbative scaling equations indicate faster renormalization of the couplings related to orbital degrees of freedom compared to spin degrees of freedom. Both mechanisms lead to a slow screening of the local spin moment. The model thus behaves similarly to the related quantum impurity problem with a larger SU(3) orbital symmetry (Dworin-Narath interaction) where this was first observed. We find that the two problems actually describe the same low-energy physics since the SU(3) symmetry is dynamically established through the renormalization of the splittings between the orbital and quadrupole coupling constants to zero. The perturbative renormalization group results are corroborated with the numerical-renormalization group (NRG) calculations. The dependence of spin Kondo temperatures and orbital Kondo temperatures as a function of interaction parameters, the hybridization, and the impurity occupancy is calculated and discussed.

Spin and quadrupolar orders in the spin-1 bilinear-biquadratic model for iron-based superconductors

Motivated by recent experimental and theoretical progresses of the magnetic properties of iron-based superconductors, we provide a comprehensive analysis of the spin-1 bilinear-biquadratic (BBQ) model on the square lattice. Using variational approach in the mean-field level, we identify the existence of various magnetic phases, including conventional spin dipolar orderings (ferro- or antiferromagnet), novel quadrupolar (spin nematic) orderings and mixed dipolar-quadrupolar orderings. In contrast to the usual Heisenberg model, the elementary excitations of the spin-1 BBQ model are described by the flavor-wave theory within the SU(3) representation. By fitting the experimental spin-wave dispersion, we determine the refined exchange couplings corresponding to the collinear antiferromagnetic iron pnictides. We also present the dynamic structure factors of both spin dipolar and quadrupolar components with connections to the future experiments.

Site-resolved multiple-quantum filtered correlations and distance measurements by magic-angle spinning NMR: Theory and applications to spins with weak to vanishing quadrupolar couplings

We discuss and analyze four magic-angle spinning solid-state NMR methods that can be used to measure internuclear distances and to obtain correlation spectra between a spin I = 1/2 and a half-integer spin S > 1/2 having a small quadrupolar coupling constant. Three of the methods are based on the heteronuclear multiple-quantum and single-quantum correlation experiments, that is, high rank tensors that involve the half spin and the quadrupolar spin are generated. Here, both zero and single-quantum coherence of the half spins are allowed and various coherence orders of the quadrupolar spin are generated, and filtered, via active recoupling of the dipolar interaction. As a result of generating coherence orders larger than one, the spectral resolution for the quadrupolar nucleus increases linearly with the coherence order. Since the formation of high rank tensors is independent of the existence of a finite quadrupolar interaction, these experiments are also suitable to materials in which there is high symmetry around the quadrupolar spin. A fourth experiment is based on the initial quadrupolar-driven excitation of symmetric high order coherences (up to p = 2S, where S is the spin number) and subsequently generating by the heteronuclear dipolar interaction higher rank (l + 1 or higher) tensors that involve also the half spins. Due to the nature of this technique, it also provides information on the relative orientations of the quadrupolar and dipolar interaction tensors. For the ideal case in which the pulses are sufficiently strong with respect to other interactions, we derive analytical expressions for all experiments as well as for the transferred echo double resonance experiment involving a quadrupolar spin. We show by comparison of the fitting of simulations and the analytical expressions to experimental data that the analytical expressions are sufficiently accurate to provide experimental (7)Li-(13)C distances in a complex of lithium, glycine, and water. Discussion of the regime for which such an approach is valid is given.

Accessing long lived 1H states via 2H couplings

In this paper we demonstrate long-lived states involving a pair of chemically equivalent protons, with lifetimes ∼30 times T1 up to a total lifetime of ∼117s at high field (8.45T). This is demonstrated on trans-ethylene-d2 in solution, where magnetic inequivalence gives access to the long-lived states. It is shown that the remaining J-coupling between the two quadrupolar deuterium spins, JQQ, splits the conditions for optimally generating proton singlet states. Detailed simulations of the spin evolution are performed, shedding light on the coherent evolution during singlet–triplet conversion as well as on the incoherent evolution that causes relaxation. Subsequently, the simulations are compared with experimental results validating the theoretical insights. Possible applications include storage of hyperpolarization in the proton long-lived state. Of particular interest may be utilization of parahydrogen induced polarization to directly induce the examined long-lived states.