Pulse error compensating symmetric magic-echo trains

Nuclear magnetic resonance diffraction with subangstrom precision

Prior to the development of MRI, Mansfield and Grannell proposed NMR diffraction (NMRd) as a method to investigate the structure of crystalline materials. When realized on the atomic scale, NMRd would be a powerful tool to study the structure of materials, utilizing the spectroscopic capabilities of NMR. The main challenge to achieving this goal lies in the ability to encode large relative phase differences between neighboring nuclear spins on the atomic scale. Utilizing key advances in nanoMRI technology, we demonstrate the ability to encode and detect angstrom-scale modulation of approximately 2 million ³¹P spins in an indium-phosphide (InP) nanowire with subangstrom precision. The work represents a significant step toward the realization of atomic-scale NMRd.

We have combined ultrasensitive force-based spin detection with high-fidelity spin control to achieve NMR diffraction (NMRd) measurement of ~2 million 31P spins in a (50 nm)3 volume of an indium-phosphide (InP) nanowire. NMRd is a technique originally proposed for studying the structure of periodic arrangements of spins, with complete access to the spectroscopic capabilities of NMR. We describe two experiments that realize NMRd detection with subangstrom precision. In the first experiment, we encode a nanometer-scale spatial modulation of the z-axis magnetization of 31P spins and detect the period and position of the modulation with a precision of <0.8 Å. In the second experiment, we demonstrate an interferometric technique, utilizing NMRd, to detect an angstrom-scale displacement of the InP sample with a precision of 0.07 Å. The diffraction-based techniques developed in this work extend the Fourier-encoding capabilities of NMR to the angstrom scale and demonstrate the potential of NMRd as a tool for probing the structure and dynamics of nanocrystalline materials.

Static Solid Relaxation Ordered Spectroscopy: SS-ROSY

A two-dimensional pulse sequence is introduced for correlating nuclear magnetic resonance anisotropic chemical shifts to a relaxation time (e.g., T₁) in solids under static conditions. The sequence begins with a preparatory stage for measuring relaxation times, and is followed by a multiple pulse sequence for homonuclear dipolar decoupling. Data analysis involves the use of Fourier transform, followed by a one-dimensional inverse Laplace transform for each frequency index. Experimental results acquired on solid samples demonstrate the general approach, and additional variations involving heteronuclear decoupling and magic angle spinning are discussed.

Microscale 3D imaging by magnetic resonance force microscopy using full-volume Fourier- and Hadamard-encoding

Three-dimensional spatially resolved full-volume imaging by magnetic resonance force microscopy at room temperature is described. Spatial resolution in z-dimension is achieved by using the magnetic-field gradient of a ferromagnetic particle that is also used for the force detection of the magnetic resonance. The gradient of the radiofrequency pulses generated by two separate wire-bonded microcoils is used for spatial resolution in x- and y-dimension. To enhance the sensitivity of our measurement Hadamard- and Fourier-encoding schemes are applied due to their multiplex effect. Measurements were taken on a patterned (NH₄)₂SO₄ crystal sample. From the calculated magnetic field distributions, a 3D image was reconstructed with a voxel volume of about 5 μm³ (1.2 μm × 3.0 μm × 1.4 μm in x-, y- and z-dimension).

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.

Comparing the efficacy of solid and magic-echo refocusing sequences: Applications to 1H NMR echo spectroscopy of shale rock

Quantitative evaluation of the solid and viscous components of unconventional shale rock, namely kerogen and bitumen, is important for understanding reservoir quality. Short transverse coherence times, due to strong ¹H-¹H dipolar interactions, motivates the application of solid state refocusing pulse sequences that allow for investigating components of the free-induction decay that are otherwise obscured by instrumental effects such as probe ringdown. This work reports on static, wide-line ¹H spectroscopy of shale rock and their extracted components, which include kerogen and bitumen, by the application of solid echo and magic echo pulse sequences. We characterize the efficiency of these cycles as a function of the radio frequency power and inter-pulse spacing. Magic echos are shown to provide superior refocusing in comparison to solid echo based experiments, as can be understood from the truncation of the Magnus expansion and ability to also refocus any I_z Hamiltonians (e.g. static field inhomogeneity). We characterize the optimal echo spacing and RF power for two shale samples of different maturity, motivating routine core and cuttings analysis and applications.

Experimental quantification of decoherence via the Loschmidt echo in a many spin system with scaled dipolar Hamiltonians

We performed Loschmidt echo nuclear magnetic resonance experiments to study decoherence under a scaled dipolar Hamiltonian by means of a symmetrical time-reversal pulse sequence denominated Proportionally Refocused Loschmidt (PRL) echo. The many-spin system represented by the protons in polycrystalline adamantane evolves through two steps of evolution characterized by the secular part of the dipolar Hamiltonian, scaled down with a factor |k| and opposite signs. The scaling factor can be varied continuously from 0 to 1/2, giving access to a range of complexity in the dynamics. The experimental results for the Loschmidt echoes showed a spreading of the decay rates that correlate directly to the scaling factors |k|, giving evidence that the decoherence is partially governed by the coherent dynamics. The average Hamiltonian theory was applied to give an insight into the spin dynamics during the pulse sequence. The calculations were performed for every single radio frequency block in contrast to the most widely used form. The first order of the average Hamiltonian numerically computed for an 8-spin system showed decay rates that progressively decrease as the secular dipolar Hamiltonian becomes weaker. Notably, the first order Hamiltonian term neglected by conventional calculations yielded an explanation for the ordering of the experimental decoherence rates. However, there is a strong overall decoherence observed in the experiments which is not reflected by the theoretical results. The fact that the non-inverted terms do not account for this effect is a challenging topic. A number of experiments to further explore the relation of the complete Hamiltonian with this dominant decoherence rate are proposed.

Absence of exponential sensitivity to small perturbations in nonintegrable systems of spins 1/2

We show that macroscopic nonintegrable lattices of spins 1/2, which are often considered to be chaotic, do not exhibit the basic property of classical chaotic systems, namely, exponential sensitivity to small perturbations. We compare chaotic lattices of classical spins and nonintegrable lattices of spins 1/2 in terms of their magnetization responses to an imperfect reversal of spin dynamics known as Loschmidt echo. In the classical case, magnetization is exponentially sensitive to small perturbations with a characteristic exponent equal to twice the value of the largest Lyapunov exponent of the system. In the case of spins 1/2, magnetization is only power-law sensitive to small perturbations.

Applications of Floquet-Magnus expansion, average Hamiltonian theory and Fer expansion to study interactions in solid state NMR when irradiated with the magic-echo sequence

This work presents the possibility of applying the Floquet-Magnus expansion and the Fer expansion approaches to the most useful interactions known in solid-state nuclear magnetic resonance using the magic-echo scheme. The results of the effective Hamiltonians of these theories and average Hamiltonian theory are presented.

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.

Quantum simulation via filtered Hamiltonian engineering: application to perfect quantum transport in spin networks

We propose a method for Hamiltonian engineering that requires no local control but only relies on collective qubit rotations and field gradients. The technique achieves a spatial modulation of the coupling strengths via a dynamical construction of a weighting function combined with a Bragg grating. As an example, we demonstrate how to generate the ideal Hamiltonian for perfect quantum information transport between two separated nodes of a large spin network. We engineer a spin chain with optimal couplings starting from a large spin network, such as one naturally occurring in crystals, while decoupling all unwanted interactions. For realistic experimental parameters, our method can be used to drive almost perfect quantum information transport at room temperature. The Hamiltonian engineering method can be made more robust under decoherence and coupling disorder by a novel apodization scheme. Thus, the method is quite general and can be used to engineer the Hamiltonian of many complex spin lattices with different topologies and interactions.

Investigation of the Effect of Finite Pulse Errors on BABA Pulse Sequence Using Floquet-Magnus Expansion Approach

This paper presents the study of finite pulse widths for the BABA pulse sequence using the Floquet-Magnus expansion (FME) approach. In the FME scheme, the first order F1 is identical to its counterparts in average Hamiltonian theory (AHT) and Floquet theory (FT). However, the timing part in the FME approach is introduced via the Λ1 (t) function not present in other schemes. This function provides an easy way for evaluating the spin evolution during "the time in between" through the Magnus expansion of the operator connected to the timing part of the evolution. The evaluation of Λ1 (t) is useful especially for the analysis of the non-stroboscopic evolution. Here, the importance of the boundary conditions, which provides a natural choice of Λ1 (0) is ignored. This work uses the Λ1 (t) function to compare the efficiency of the BABA pulse sequence with δ - pulses and the BABA pulse sequence with finite pulses. Calculations of Λ1 (t) and F1 are presented.

Multispin correlations and pseudo-thermalization of the transient density matrix in solid-state NMR: free induction decay and magic echo

Quantum unitary evolution typically leads to thermalization of generic interacting many-body systems. There are very few known general methods for reversing this process, and we focus on the magic echo, a radio-frequency pulse sequence known to approximately "rewind" the time evolution of dipolar coupled homonuclear spin systems in a large magnetic field. By combining analytic, numerical, and experimental results we systematically investigate factors leading to the degradation of magic echoes, as observed in reduced revival of mean transverse magnetization. Going beyond the conventional analysis based on mean magnetization we use a phase encoding technique to measure the growth of spin correlations in the density matrix at different points in time following magic echoes of varied durations and compare the results to those obtained during a free induction decay (FID). While considerable differences are documented at short times, the long-time behavior of the density matrix appears to be remarkably universal among the types of initial states considered - simple low order multispin correlations are observed to decay exponentially at the same rate, seeding the onset of increasingly complex high order correlations. This manifestly athermal process is constrained by conservation of the second moment of the spectrum of the density matrix and proceeds indefinitely, assuming unitary dynamics.

Narrowing of protein NMR spectral lines broadened by chemical exchange

Broadening of spectral lines is a signature of chemical exchange phenomena on microsecond to millisecond time scales but has deleterious effects on spectral resolution and sensitivity. A multipulse method based on chemical shift scaling that reduces chemical exchange broadening during frequency-encoding periods of liquid-state multidimensional NMR experiments is described. The proposed scheme utilizes low-power radiofrequency pulses, which offer the advantages of short cycle times and minimal sample heating. The method is suitable for biological macromolecules, as relaxation not resulting from chemical exchange is reduced by placing the magnetization along the z axis for part of the evolution trajectory. The resolution and sensitivity enhancement for resonances broadened by chemical exchange is demonstrated on the protein ribonuclease A. The work demonstrates the feasibility of applying coherent averaging techniques, which were originally developed in solid-state NMR spectroscopy, to biological NMR spectroscopy in the liquid state for resolution enhancement and facilitates the detection of resonances that are severely broadened by chemical exchange processes.

Total compensation of pulse transients inside a resonator

The profile of rf pulses that nuclear spins experience inside a resonator deviates from that of rf voltage signals generated by a NMR spectrometer according to users' pulse programming, when change of the profile in time is comparable to or shorter than the time constant of the resonator. In our previous work [Takeda et al., J. Magn. Reson. 197 (2009) 242-244], we proposed active compensation of rf pulse transients, in which the amplitude transient of the rf pulse can be suppressed without sacrificing the Q factor of the probe. Here we extend the idea of active compensation toward total compensation of the amplitude as well as phase transients. By measuring the transient response of the probe to a given excitation using a pickup coil, the response function determining the transient behavior of the probe is numerically obtained. Then, by numerically solving the convolution equation with the help of Laplace transformation, one can obtain the amplitude and phase profiles of the pulse that should be programmed in the spectrometer in order to apply the rf pulses to the nuclear spins as intended. Accurate rf pulsing based on this idea is experimentally demonstrated, and prospect and requirements for coping with the receiver dead-time problem are discussed.

Self-refocused slice selection by magic echo DANTE with 270 degrees flipping Gaussian RF modulation

The method of slice selection proposed for solid-state MRI by combining DANTE selective excitation with magic echo (ME) line narrowing requires a rephasing period ca. 0.6 times the DANTE excitation period. The added rephasing period results in a significant loss of sensitivity due to transverse relaxation. To solve the sensitivity problem, we make use of the self-refocusing effect of the 270 degrees Gaussian-shaped soft pulse by introducing a 270 degrees flipping Gaussian modulation to the ME DANTE method. This eliminates the rephasing period. The utility of the improved method is demonstrated by experiments performed on test samples of adamantane and polycarbonate.

Simulations of information transport in spin chains

Transport of quantum information in linear spin chains has been the subject of much theoretical work. Experimental studies by NMR in solid state spin systems (a natural implementation of such models) is complicated since the dipolar Hamiltonian is not solely comprised of nearest-neighbor XY-Heisenberg couplings. We present here a similarity transformation between the XY Hamiltonian and the double-quantum Hamiltonian, an interaction which is achievable with the collective control provided by radio-frequency pulses. Not only can this second Hamiltonian simulate the information transport in a spin chain, but it also creates coherent states, whose intensities give an experimental signature of the transport. This scheme makes it possible to study experimentally the transport of polarization beyond exactly solvable models and explore the appearance of quantum coherence and interference effects.

Tailored slice selection in solid-state MRI by DANTE under magic-echo line narrowing

We propose a method of slice selection in solid-state MRI by combining DANTE selective excitation with magic-echo (ME) line narrowing. The DANTE RF pulses applied at the ME peaks practically do not interfere with the ME line narrowing in the combined ME DANTE sequence. This allows straightforward tailoring of the slice profile simply by introducing an appropriate modulation, such as a sinc modulation, into the flip angles of the applied DANTE RF pulses. The utility of the method has been demonstrated by preliminary experiments performed on a test sample of adamantane.

On the application of magic echo cycles for quadrupolar echo spectroscopy of spin-1 nuclei

Magic echo cycles are introduced for performing quadrupolar echo spectroscopy of spin-1 nuclei. An analysis is performed via average Hamiltonian theory showing that the evolution under chemical shift or static field inhomogeneity can be refocused simultaneously with the quadrupolar interaction using these cycles. Due to the higher convergence in the Magnus expansion, with sufficient RF power, magic echo based quadrupolar echo spectroscopy outperforms the conventional two pulse quadrupolar echo in signal to noise. Experiments highlighting a signal to noise enhancement over the entire bandwidth of the quadrupolar pattern of a powdered sample of deuterated polyethelene are shown.

Spin diffusion of correlated two-spin states in a dielectric crystal

Reciprocal space measurements of spin diffusion in a single crystal of calcium fluoride (CaF2) have been extended to dipolar ordered states. The experimental results for the component of the spin diffusion rate parallel to the external field are D(parallel)(D)=29+/-3x10(-12) cm(2)/s for the [001] direction and D(parallel)(D)=33+/-4x10(-12) cm(2)/s for the [111] direction. The measured diffusion rates for dipolar order are faster than those for Zeeman order and are considerably faster than predicted by simple theoretical models. It is suggested that constructive interference in the transport of the two-spin states is responsible for this enhancement. As expected, the anisotropy in the diffusion rates is observed to be significantly less for dipolar order compared to the Zeeman case.