Multiple‐quantum NMR in solids

Signature of Scramblon Effective Field Theory in Random Spin Models

Information scrambling refers to the propagation of information throughout a quantum system. Its study not only contributes to our understanding of thermalization but also has wide implications in quantum information and black hole physics. Recent studies suggest that information scrambling in large-N systems with all-to-all interactions is mediated by collective modes called scramblons. However, a criterion for the validity of scramblon theory in a specific model is still missing. In this work, we address this issue by investigating the signature of the scramblon effective theory in random spin models with all-to-all interactions. We demonstrate that, in scenarios where the scramblon description holds, the late-time operator size distribution can be predicted from its early-time value, requiring no free parameters. As an illustration, we examine whether Brownian circuits exhibit a scramblon description and obtain a positive confirmation both analytically and numerically. Our findings provide a concrete experimental framework for unraveling the scramblon field theory in random spin models using quantum simulators.

Operator growth from global out-of-time-order correlators

In chaotic many-body systems, scrambling or the operator growth can be diagnosed by out-of-time-order correlators of local operators. We show that operator growth also has a sharp imprint in out-of-time-order correlators of global operators. In particular, the characteristic spacetime shape of growing local operators can be accessed using global measurements without any local control or readout. Building on an earlier conjectured phase diagram for operator growth in chaotic systems with power-law interactions, we show that existing nuclear spin data for out-of-time-order correlators of global operators are well fit by our theory. We also predict super-polynomial operator growth in dipolar systems in 3d and discuss the potential observation of this physics in future experiments with nuclear spins and ultra-cold polar molecules.

Star-topology registers: NMR and quantum information perspectives

Quantum control of large spin registers is crucial for many applications ranging from spectroscopy to quantum information. A key factor that determines the efficiency of a register for implementing a given information processing task is its network topology. One particular type, called star-topology, involves a central qubit uniformly interacting with a set of ancillary qubits. A particular advantage of the star-topology quantum registers is in the efficient preparation of large entangled states, called NOON states, and their generalized variants. Thanks to the robust generation of such correlated states, spectral simplicity, ease of polarization transfer from ancillary qubits to the central qubit, as well as the availability of large spin-clusters, the star-topology registers have been utilized for several interesting applications over the last few years. Here we review some recent progress with the star-topology registers, particularly via nuclear magnetic resonance methods.

Centerband-Only Detection of Exchange NMR with Natural-Abundance Correction Reveals an Expanded Unit Cell in Phenylalanine Crystals

The NMR pulse sequence CODEX (centerband-only detection of exchange) is a widely used method to report on the number of magnetically inequivalent spins that exchange magnetization via spin diffusion. For crystals, this rules out certain symmetries, and the rate of equilibration is sensitive to distances. Here we show that for 13 C CODEX, consideration of natural abundance spins is necessary for crystals of high complexity, demonstrated here with the amino acid phenylalanine. The NMR data rule out the C2 space group that was originally reported for phenylalanine, and are only consistent with a larger unit cell containing eight magnetically inequivalent molecules. Such an expanded cell was recently described based on single crystal data. The large unit cell dictates the use of long spin diffusion times of more than 200 seconds, in order to equilibrate over the entire unit cell volume of 1622 Å3 .

Orientational dependencies of dynamics and relaxation of multiple quantum NMR coherences in one-dimensional systems

Dynamics and relaxation, or decoherence, of multiple quantum (MQ) coherences are investigated experimentally and theoretically for different orientations of a single crystal of fluorapatite with respect to the external magnetic field. Dynamics of MQ coherences is studied as a function of the preparation period of the MQ NMR experiment. Their relaxation, or decoherence, during the evolution period is also investigated. Universal curves for dynamics and relaxation are obtained and describe the experimental data for different orientations of the investigated sample. Those curves prove the dipolar nature of the observed relaxation process. The contribution of the heteronuclear interactions to the dipolar relaxation is also investigated. The source of discrepancies between experimental data and theoretical results in the experiments with a large angle between the nuclear spin chains and the external magnetic field are discussed.

Engineering spin Hamiltonians using multiple pulse sequences in solid state NMR spectroscopy

Multiple pulse sequences are often used to manipulate spin Hamiltonians in solid-state nuclear magnetic resonance spectroscopy. In this paper, we analyze multiple pulse sequences using the well-known average Hamiltonian theory. We first expand the resulting average Hamiltonian into a reachable set of sub-Hamiltonians and then develop a general procedure using both flip-angle and phase of the applied pulses as control variables to select any of those sub-Hamiltonians. We use this method to analyze solid-echo based sequences and to design new proton-proton homonuclear decoupling sequences in static solids. It is found that this newly designed decoupling scheme, in the presence of finite pulse length, effectively suppresses the ¹H-¹H homonuclear dipolar interactions while establishes variable scaling factors on the heteronuclear dipolar interactions and chemical shift interactions, depending on the flip-angle of the applied pulses. When the pulse flip-angle is close to 54.7°, this sequence possesses a large scaling factor with relatively low average decoupling field. When the pulse flip-angle becomes ∼120°, the scaling factor is almost zero. A static ¹⁵N-acetyl-valine crystal sample has been used as an example to confirm and validate the performance of this new decoupling scheme.

Reversibility of dynamics and multiple-quantum coherences

In spin systems, the decay of the Loschmidt echo in the time-reversal experiment (evolution-perturbation-time-reversed evolution) is linked to the generation of multiple-quantum (MQ) coherences. Unlimited growth of the MQ coherences leads to irreversibility of dynamics. In some cases, one can expect that the deviation of the Loschmidt echo and the second moment of the MQ intensities distribution are linear in time. The criteria of such behavior, called weak irreversibility, are formulated. The proposed approach can be extended beyond spin systems, in order to analyze some general aspects of reversibility of many-body quantum dynamics.

Multiple-quantum spin counting in magic-angle-spinning NMR via low-power symmetry-based dipolar recoupling

By using a symmetry-based R2(8)(1)R2(8)(-1) double-quantum (2Q) dipolar recoupling sequence, we demonstrate high-order multiple-quantum coherence (MQC) excitation at fast magic-angle spinning (MAS) frequencies up to 34 kHz. This scheme combines several attractive features, such as a relatively high dipolar scaling factor, good compensation to rf-errors, isotropic and anisotropic chemical shifts, as well as an ultra-low radio-frequency (rf) power requirement. The latter translates into nutation frequencies below 30 kHz for MAS rates up to 60 kHz, thereby permitting rf application for very long excitation periods without risk of damaging the NMR probehead or sample, while the compensation to chemical shifts improves as the MAS rate increases. (31)P MQC spin counting is demonstrated on powders of calcium hydroxyapatite (Ca5(PO4)3OH) and anhydrous sodium diphosphate (Na4P2O7), from which all even coherence orders up to 30 and 14 were detected, respectively, over the respective MAS ranges of 15-24 kHz and 20-34 kHz. The amplitude distributions among the (31)P MQC orders depend on the precise nutation frequency during recoupling, despite that the highest detected order was relatively insensitive to this parameter. An observed gradual transition from a Gaussian to exponential functionality of the MQC amplitude-profile is discussed in relation to the prevailing approach to derive spin-cluster sizes by fitting the MQC amplitude-distribution to a Gaussian decay, where minor systematic deviations between the model and experimental data are frequently reported.

Effects of experimental imperfections on a spin counting experiment

Spin counting NMR is an experimental technique that allows a determination of the size and time evolution of networks of dipolar coupled nuclear spins. This work reports on an average Hamiltonian treatment of two spin counting sequences and compares the efficiency of the two cycles in the presence of flip errors, RF inhomogeneity, phase transients, phase errors, and offset interactions commonly present in NMR experiments. Simulations on small quantum systems performed using the two cycles reveal the effects of pulse imperfections on the resulting multiple quantum spectra, in qualitative agreement with the average Hamiltonian calculations. Experimental results on adamantane are presented, demonstrating differences in the two sequences in the presence of pulse errors.

A statistical approach for analyzing the development of 1H multiple-quantum coherence in solids

A novel statistical approach for analyzing (1)H multiple-quantum (MQ) spin dynamics in so-called spin-counting solid-state NMR experiments is presented. The statistical approach is based on the percolation theory with Monte Carlo methods and is examined by applying it to the experimental results of three solid samples having unique hydrogen arrangement for 1-3 dimensions: the n-alkane/d-urea inclusion complex as a one-dimensional (1D) system, whose (1)H nuclei align approximately in 1D, and magnesium hydroxide and adamantane as a two-dimensional (2D) and a three-dimensional (3D) system, respectively. Four lattice models, linear, honeycomb, square and cubic, are used to represent the (1)H arrangement of the three samples. It is shown that the MQ dynamics in adamantane is consistent with that calculated using the cubic lattice and that in Mg(OH)2 with that calculated using the honeycomb and the square lattices. For n-C20H42/d-urea, these 4 lattice models fail to express its result. It is shown that a more realistic model representing the (1)H arrangement of n-C20H42/d-urea can describe the result. The present approach can thus be used to determine (1)H arrangement in solids.

Hydrogen Microstructure in Amorphous Semiconductors

The effects of deposition parameters on the H microstructure of plasma deposited amorphous silicon (a-Si:H) and amorphous silicon carbide (a-SiC:H) are measured via multiple quantum nuclear magnetic resonance (MQ NMR). These studies indicate clusters of 5 to 7H atoms exist in a-Si:H films prepared at temperatures ranging from 113 to 324°C. In the range from 270 to 324°C, only these small clusters exist, but lower temperature films also contain larger clusters. Annealing studies indicate H rearranges in a-Si:H prior to evolution. Deposition temperature and annealing temperature have similar effects on H concentration in a-Si:H, but deposition temperature control the density and microstructure of the film. The addition of dopant atoms also affects the H microstructure, with phosphorous causing larger H clusters to form, and boron reducing clustering in a-Si:H films. This perturbation of the film's microstructure suggests that the effects of dopant addition are more complex in amorphous than in crystalline semiconductors. The concentration of carbon atoms also effects H microstructure of a-SiC:H in a complex way. We surmise that H microstructure, rather than H content, determines amorphous semiconductors properties.

Oligomeric structure, dynamics, and orientation of membrane proteins from solid-state NMR

Solid-state NMR is a versatile and powerful tool for determining the dynamic structure of membrane proteins at atomic resolution. I review the recent progress in determining the orientation, the internal and global protein dynamics, the oligomeric structure, and the ligand-bound structure of membrane proteins with both alpha-helical and beta sheet conformations. Examples are given that illustrate the insights into protein function that can be gained from the NMR structural information.

Detection of multiple-quantum coherences with projective nuclear magnetic resonance measurement

It is shown that in nuclear magnetic resonance, multiple-quantum (MQ) coherences can be detected "instantly" by exploiting the principle of quantum-mechanical projective measurement. Therefore, the mixing period, which involves collective multispin dynamics and converts MQ coherences into observable single-quantum coherence (magnetization), is not necessary. The experimental examples are given for two finite clusters: benzene in liquid crystal and liquid crystal 4'-n-pentyl-4-cyanobiphenyl, and for solid adamantane with an infinite network of dipolar couplings.

Reduced decoherence in large quantum registers

Among the biggest obstacles for building larger (and thus more powerful) quantum-information processors is decoherence, the decay of quantum-information by the coupling between the quantum register and its environment. Procedures for reducing decoherence processes will be essential for successful operation of larger quantum processors. We study model quantum registers consisting of up to 4900 qubits and measure their decay as a function of the register size. We demonstrate that appropriate sequences of qubit rotations reduce the coupling between system and environment for all sizes of the quantum register, thus preserving the quantum-information 50 times longer than without decoupling.

Constants of motion in NMR spectroscopy

We present a general method for constructing a subset of the constants of motion in terms of products of spin operators. These operators are then used to give insight into the multi-spin orders comprising the quasi-equilibrium state formed under a Jeener-Broekaert sequence in small, dipolar-coupled, spin systems. We further show that constants of motion that represent single-quantum coherences are present due to the symmetry of the dipolar Hamiltonian under 180 degrees spin rotations, and that such coherences contribute a DC component to the FID which vanishes in the absence of the flip-flop terms and is only present for spin clusters with an odd number of spins.

A DOQSY approach for the elucidation of torsion angle distributions in biopolymers: application to silk

Silk from the wild silkworm Samia cynthia ricini with a molecular mass of about 300kDa consists of alternating repeats of nano-crystalline poly-(Ala) and non-crystalline glycine-rich domains. The backbone torsion angles between pairs of these two amino acids is determined by DOQSY solid-state NMR spectroscopy: the alanine-rich domains are predominantly in a beta-sheet conformation, whereas the glycine-rich domains are found to be partially in an extended beta-sheet conformation and partially in an approximately 3(1)-helical conformation. In the cast film from liquid silk significantly different secondary structures were found: the alanine-rich domains are alpha-helical conformation, whereas the results for glycine-labelled sample are explained by a random-coil state. A detailed error analysis of the technique is presented.

Multiple-quantum NMR spin dynamics of inhomogeneous one-dimensional systems in solids

Multiple-quantum NMR spin dynamics of inhomogeneous one-dimensional systems in solids is investigated by analytical and numerical methods. A fermion approach for MQ spin dynamics of one-dimensional inhomogeneous systems is developed in the approximation of the dipole-dipole interactions (DDI) of nearest neighbors. It is shown that only MQ coherences of the zeroth and plus/minus second orders appear in the approximation of the DDI of the nearest neighbors even in inhomogeneous one-dimensional systems. We also investigate MQ dynamics of inhomogeneous chains numerically. Intensities of MQ NMR coherences for a linear chain consisting of 3000 spins are calculated.

Triple-quantum dynamics in multiple-spin systems undergoing magic-angle spinning: application to 13C homonuclear correlation spectroscopy

We analyze the multiple-quantum dynamics governed by a new homonuclear recoupling strategy effecting an average dipolar Hamiltonian comprising three-spin triple-quantum operators (e.g., S(p)+S(q)+S(r)+) under magic-angle spinning conditions. Analytical expressions are presented for polarization transfer processes in systems of three and four coupled spins-1/2 subject to triple-quantum filtration (3QF), and high-order multiple-quantum excitation is investigated numerically in moderately large clusters, comprising up to seven spins. This recoupling approach gives highly efficient excitation of triple-quantum coherences: ideally, up to 67% of the initial polarization may be recovered by 3QF in three-spin systems in polycrystalline powders. Two homonuclear 2D correlation strategies are demonstrated experimentally on powders of uniformly 13C-labeled alanine and tyrosine: the first correlates the single-quantum spectrum in the first dimension with the corresponding 3QF spectrum along the other. The second protocol correlates triple-quantum coherences with their corresponding single-quantum coherences within triplets of coupled spins.

Entanglement assisted metrology

We propose a new approach to the measurement of a single spin state, based on nuclear magnetic resonance (NMR) techniques and inspired by the coherent control over many-body systems envisaged by quantum information processing. A single target spin is coupled via the magnetic dipolar interaction to a large ensemble of spins. Applying radio frequency pulses, we can control the evolution so that the spin ensemble reaches one of two orthogonal states whose collective properties differ depending on the state of the target spin and are easily measured. We first describe this measurement process using quantum gates; then we show how equivalent schemes can be defined in terms of the Hamiltonian and thus implemented under conditions of real control, using well established NMR techniques. We demonstrate this method with a proof of principle experiment in ensemble liquid state NMR and simulations for small spin systems.

Multiple quantum spin counting techniques with quadrupolar nuclei

Phase incremented and continuous irradiation multiple spin correlation methods are applied to spin [Formula: see text] nuclei with small quadrupole couplings such as (7)Li in LiCl and are shown to successfully produce a coherently coupled dipolar spin network. Application to the analogous Na salt shows successful spin correlation evolving at a slower rate due to the weaker homonuclear dipolar coupling strength between Na nuclei. The results are analysed using a statistical approach. Spin counting is non-trivial as not only multiple quantum coherences between spins are generated but also within the quadrupolar spin levels. Na(2)C(2)O(4) is investigated as a material with non-negligible quadrupole coupling and it is in this limit that the spin correlation techniques are found to break down.

Structure and ionic interactions of organic-inorganic composite polymer electrolytes studied by solid-state NMR and Raman spectroscopy

Solid-state NMR studies of composite polymer electrolytes are reported. The materials consist of polyethylene oxide and an organic inorganic composite, together with a lithium salt, and are candidates for electrolytes in solid-state lithium ion batteries. Silicon and aluminum MAS and multiple quantum MAS are used to characterize the network character of the organic-inorganic composite, and spin diffusion measurements are used to determine the nanostructure of the polymer/composite blending. Multiple quantum spin counting is used to measure the ion aggregation. The NMR results are supported by Raman spectra, calorimetry, and impedance spectroscopy. From these experiments it is concluded that the composite suppresses polymer crystallization without suppressing its local mobility, and also suppresses the tendency for the ions to aggregate. This polymer composite thus appears very promising for application in lithium ion batteries.

Sensitivity enhancement in multiple-quantum NMR experiments with CPMG detection

We present a modified multiple-quantum (MQ) experiment, which implements the Carr-Purcell-Meiboom-Gill (CPMG) detection scheme in the static MQ NMR experiment proposed by W. S. Warren et al. (1980, J. Chem. Phys.73, 2084-2099) and exploited further by O. N. Antzutkin and R. Tycko (1999, J. Chem. Phys.110, 2749-2752). It is demonstrated that a significant enhancement in the sensitivity can be achieved by acquiring echo trains in the MQ experiments for static powder samples. The modified scheme employing the CPMG detection was superior to the original MQ experiment, in particular for the carbonyl carbon with a very large chemical shift anisotropy.

Biomolecular solid state NMR: advances in structural methodology and applications to peptide and protein fibrils

Solid state nuclear magnetic resonance (NMR) methods can provide atomic-level structural constraints on peptides and proteins in forms that are not amenable to characterization by other high-resolution structural techniques, owing to insolubility, high molecular weight, noncrystallinity, or other characteristics. Important examples include peptide and protein fibrils and membrane-bound peptides and proteins. Recent advances in solid state NMR methodology aimed at structural problems in biological systems are reviewed. The power of these methods is illustrated by experimental results on amyloid fibrils and other protein fibrils.

Multiple quantum solid-state NMR indicates a parallel, not antiparallel, organization of beta-sheets in Alzheimer's beta-amyloid fibrils

Senile plaques associated with Alzheimer's disease contain deposits of fibrils formed by 39- to 43-residue beta-amyloid peptides with possible neurotoxic effects. X-ray diffraction measurements on oriented fibril bundles have indicated an extended beta-sheet structure for Alzheimer's beta-amyloid fibrils and other amyloid fibrils, but the supramolecular organization of the beta-sheets and other structural details are not well established because of the intrinsically noncrystalline, insoluble nature of amyloid fibrils. Here we report solid-state NMR measurements, using a multiple quantum (MQ) (13)C NMR technique, that probe the beta-sheet organization in fibrils formed by the full-length, 40-residue beta-amyloid peptide (Abeta(1-40)). Although an antiparallel beta-sheet organization often is assumed and is invoked in recent structural models for full-length beta-amyloid fibrils, the MQNMR data indicate an in-register, parallel organization. This work provides site-specific, atomic-level structural constraints on full-length beta-amyloid fibrils and applies MQNMR to a significant problem in structural biology.

Solid-state NMR as a probe of amyloid fibril structure

Amyloid fibrils are intrinsically noncrystalline, insoluble, high-molecular-weight aggregates of peptides and proteins, with considerable biomedical and biophysical significance. Solid-state NMR techniques are uniquely capable of providing high-resolution, site-specific structural constraints for amyloid fibrils, at the level of specific interatomic distances and torsion angles. So far, a relatively small number of solid-state NMR studies of amyloid fibrils have been reported. These have addressed issues about the supramolecular organization of beta-sheets in the fibrils and the peptide conformation in the fibrils, and have concentrated on the beta-amyloid peptide of Alzheimer's disease. Many additional applications of solid-state NMR to amyloid fibrils from a variety of sources are anticipated in the near future, as these systems are ideally suited for the technique and are of widespread current interest.

Solid-state NMR determination of the secondary structure of Samia cynthia ricini silk

Silks are fibrous proteins that form heterogeneous, semi-crystalline solids. Silk proteins have a variety of physical properties reflecting their range of functions. Spider dragline silk, for example, has high tensile strength and elasticity, whereas other silks are better suited to making housing, egg sacs or the capture spiral of spiders' webs. The differing physical properties arise from variation in the protein's primary and secondary structure, and their packing in the solid phase. The high mechanical performance of spider dragline silk, for example, is probably due to a beta-sheet conformation of poly-alanine domains, embedded as small crystallites within the fibre. Only limited structural information can be obtained from diffraction of silks, so further characterization requires spectroscopic studies such as NMR. However, the classical approach to NMR structure determination fails because the high molecular weight, repetitive primary structure and structural heterogeneity of solid silk means that signals from individual amino-acid residues cannot be resolved. Here we adapt a recently developed solid-state NMR technique to determine torsion angle pairs (phi, psi) in the protein backbone, and we study the distribution of conformations in silk from the Eri silkworm, Samia cynthia ricini. Although the most probable conformation in native fibres is an anti-parallel beta-sheet, film produced from liquid directly extracted from the silk glands appears to be primarily alpha-helical.

Article

The behavior of multiple spin dynamics under time-reversal excitation is investigated in strongly dipolar coupled spin networks. It is demonstrated that the typical physical intuition that NMR spectroscopists have of time-reversal in such systems is erroneous. By studying the equation of motion of the spin system in an infinite one-dimensional quantum spin chain with nearest-neighbor dipolar interactions and evolving under time-reversal excitation, a more appropriate physical picture emerges. Implications of these new findings are briefly discussed in the context of error propagation and teleportation in NMR quantum information processing.Key words: time-reversal, NMR, nuclear spin dynamic, quantum information processing.

Selection rules for multiple quantum NMR excitation in solids: derivation from time-reversal symmetry and comparison with simulations and (13)C NMR experiments

New derivations of selection rules for excitation and detection of multiple quantum coherences in coupled spin-1/2 systems are presented. The selection rules apply to experiments in which the effective coupling Hamiltonian used for multiple quantum excitation is both time-reversal invariant and time-reversible by a phase shift of the radiofrequency pulse sequence that generates the effective couplings. The selection rules are shown to be consequences of time-reversal invariance and time-reversibility and otherwise independent of the specific form of the effective coupling Hamiltonian. Numerical simulations of multiple quantum NMR signal amplitudes and experimental multiple quantum excitation spectra are presented for the case of a multiply (13)C-labeled helical polypeptide. The simulations and experiments confirm the selection rules and demonstrate their impact on multiple quantum (13)C NMR spectra in this biochemically relevant case.