Low temperature MQ NMR dynamics in dipolar ordered state

We investigate analytically and numerically the Multiple Quantum (MQ) NMR dynamics in dipolar ordered spin systems of nuclear spins 1/2 at low temperatures. We consider two different methods of MQ NMR. One of them is based on the measurement of the dipolar energy. The other method uses an additional resonance (π/4)_y-pulse after the preparation period of the standard MQ NMR experiment in solids and allows one to measure the Zeeman energy. Both considered methods are sensitive to the contribution of remote spins in the interaction and to the spin system structure. The QS method is sensitive to the spin number in the molecule while the PS method gives very similar time dependencies of the intensities of MQ coherences for different spin numbers. It is shown that the use of the dipolar ordered initial state has the advantage of exciting the highest order MQ coherences in clusters of 4m identical spins, where m=1,2,3,…, that is impossible to do with the standard MQ method. MQ NMR methods based on the dipolar ordered initial states at low temperatures complement the standard NMR spectroscopy for better studying structures and dynamic processes in solids.     

Dipolar temperature and multiple-quantum NMR dynamics in dipolar ordered-spin systems

We investigate analytically and numerically the multiple-quantum (MQ) NMR dynamics in systems of nuclear spins 1/2 coupled by dipole-dipole interactions in the case of the dipolar-ordered initial state. We suggest a new method of MQ NMR based on the measurement of the dipolar temperature in the quasi-equilibrium state, which establishes after the time of order ω _loc ⁻¹ (ω_loc is the dipolar local field) after the MQ NMR experiment. Manyspin clusters and correlations are created faster in such an experiment than in usual MQ NMR experiments and can be used for the investigation of the many-spin dynamics of nuclear spins in solids. The text was submitted by the authors in English.     

Evolution of spin entanglement and an entanglement witness in multiple-quantum NMR experiments

We investigate the evolution of entanglement in multiple-quantum (MQ) NMR experiments in crystals with pairs of close nuclear 1/2-spins. The initial thermodynamic equilibrium state of the system in a strong external magnetic field evolves under the nonsecular part of the dipolar Hamiltonian. As a result, MQ coherences of the zeroth and plus/minus second orders appear. A simple condition for the emergence of entanglement is obtained. We show that the measure of the spin-pair entanglement—concurrence—coincides qualitatively with the intensity of MQ coherences of the plus/minus second order and, hence, the entanglement can be studied with MQ NMR methods. We introduce an entanglement witness using MQ NMR coherences of the plus/minus second order. The text was submitted by the authors in English.     

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.     

Multiple Quantum Coherences (MQ) NMR and Entanglement Dynamics in the Mixed-Three-Spin XXX Heisenberg Model with Single-Ion Anisotropy

We analytically investigate Multiple Quantum (MQ) NMR dynamics in a mixed-three-spin (1/2,1,1/2) system with XXX Heisenberg model at the front of an external homogeneous magnetic field B. A single-ion anisotropy property ζ is considered for the spin-1. The intensities dependence of MQ NMR coherences on their orders (zeroth and second orders) for two pairs of spins (1,1/2) and (1/2,1/2) of the favorite tripartite system are obtained. It is also investigated dynamics of the pairwise quantum entanglement for the bipartite (sub)systems (1,1/2) and (1/2,1/2) permanently coupled by, respectively, coupling constants J₁ and J₂, by means of concurrence and fidelity. Then, some straightforward comparisons are done between these quantities and the intensities of MQ NMR coherences and ultimately some interesting results are reported. We also show that the time evolution of MQ coherences based on the reduced density matrix of the pair spins (1,1/2) is closely connected with the dynamics of the pairwise entanglement. Finally, we prove that one can introduce MQ coherence of the zeroth order corresponds to the pair spins (1,1/2) as an entanglement witness at some special time intervals.     

Line shapes of multiple quantum NMR coherences in one-dimensional quantum spin chains in solids

General formulae for intensities of multiple quantum (MQ) NMR coherences in systems of nuclear spins coupled by the dipole-dipole interactions are derived. The second moments of the MQ coherences of zero- and second orders are calculated for infinite linear chains in the approximation of the nearest neighbor interactions. Supercomputer simulations of intensities of MQ coherences of linear chains are performed at different times of preparation and evolution periods of MQ NMR experiments. The second moments obtained from the developed theory are compared with the results of the supercomputer analysis of MQ NMR dynamics. The linewidth information in MQ NMR experiments is discussed.     

Multiple Quantum NMR in One-Dimensional and Nano-Scale Systems: Theory and Computer Simulations

We investigate several exactly solvable models of multiple quantum (MQ) NMR dynamics. We study MQ NMR dynamics of one-dimensional systems (chains and rings) and nano-voids filled with spin-carrying atoms or molecules in a strong external magnetic field. Our focus is the problem of the profile of MQ NMR coherences (i.e., the dependence of the intensities of MQ NMR coherences on their orders). We find that integrable one-dimensional systems exhibit peculiar behavior that does not shed much light on a more generic situation. On the other hand, important lessons can be extracted from our solution for the nano-scale systems. We find that the profile of the intensities of the MQ NMR coherences is rather exponential than Gaussian as was proposed earlier. We also address the cluster growth in the course of the evolution of MQ NMR coherences.     

Exact results on spin dynamics and multiple quantum NMR dynamics in alternating spin-1/2 chains with <Emphasis Type="Italic">XY</Emphasis> Hamiltonian at high temperatures

We extend the picture of a transfer of nuclear spin-1/2 polarization along a homogeneous one-dimensional chain with the XY Hamiltonian to the inhomogeneous chain with alternating nearest neighbor couplings and alternating Larmor frequencies. To this end, we exactly calculate the spectrum of the spin-1/2 XY Hamiltonian of the alternating chain with an odd number of sites. The exact spectrum of the XY Hamiltonian is also applied to study the multiple quantum (MQ) NMR dynamics of the alternating spin-1/2 chain. MQ NMR spectra are shown to have the MQ coherences of zero and ± second orders just as in the case of a homogeneous chain. The intensities of the MQ coherences 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.     

Multiple-quantum dynamics of one-dimensional nuclear spin systems in solids

Multiple-quantum spin dynamics is studied using analytic and numerical methods for one-dimensional finite linear chains and rings of nuclear spins 1/2 coupled by dipole-dipole interactions. An approximation of dipole-dipole interaction between nearest neighbors having the same constants is used to obtain exact expressions for the intensities of the multiple-quantum coherences in the spin systems studied, which are initially in thermal equilibrium and whose evolution is described by a two-spin two-quantum Hamiltonian. An approximation of nearest neighbors with arbitrary dipole-dipole interaction constants is used to establish a simple relationship between the multiple-quantum dynamics and the dynamics of spin systems with an XY Hamiltonian. Numerical methods are developed to calculate the intensities of multiple-quantum coherences in multiple-quantum NMR spectroscopy. The integral of motion is obtained to expand the matrix of the two-spin two-quantum Hamiltonian into two independent blocks. Using the nearest-neighbor approximation the Hamiltonian is factorized according to different values of the projection operator of the total spin momentum on the direction of the external magnetic field. Results of calculations of the multiple-quantum dynamics in linear chains of seven and eight nuclear spins and a six-spin ring are presented. It is shown that the evolution of the intensities of the lowest-order multiple-quantum coherences in linear chains is accurately described allowing for dipole-dipole interaction of nearest and next-nearest neighbors only. Numerical calculations are used to compare the contributions of nearest and remote spins to the intensities of the multiple-quantum coherences.     

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.     

Dipolar Relaxation of Multiple Quantum NMR Coherences as a Model of Decoherence of Many-Qubit Coherent Clusters

Dipolar relaxation of multiple quantum (MQ) nuclear magnetic resonance (NMR) coherence is investigated on the evolution period of the MQ NMR experiment in chains of ¹⁹F nuclei in a single crystal of calcium fluorapatite. The dependence of the relaxation time of the MQ coherence of the second order on the size of the coherent spin cluster formed on the preparation period is obtained. The dipolar relaxation of MQ NMR coherences is considered as a model for the investigation of decoherence of quantum states of many-qubit spin clusters.     

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.     

Decay of multispin multiple-quantum coherent states in the NMR of a solid and the stabilization of their intensity profile with time

Variations, experimentally observed in [14], in the intensity profiles of multiple-quantum (MQ) coherences in the presence of two special types of perturbations are explained on the basis of the theory, earlier developed by the authors, of the growth of the effective size of correlated clusters (the number of correlated spins) and the relaxation of MQ coherent states [23]. The intensity and the character of perturbation were controlled by the experimenters. It is shown that the observed stabilization of profiles with time is not associated with the stabilization of the cluster size. Quite the contrary, a cluster of correlated spins monotonically grows, while the observed variations in the intensity profile and its stabilization with time are attributed to the dependence of the decay rate of an MQ coherence on its order (its position in the MQ spectrum). The results of the theory are in good agreement with the experimental data.     

Quantum coherence dynamics of spin pairs in many-spin cluster

We consider the dynamics of a quantum coherence of two chosen spins in systems of dipolar coupled nuclear spins s=1/2 in solid. With the purpose to study this coherence we suggest two different methods. One of them uses the partial trace technique and reduced density matrix. The second method is based on the calculation the intensity of multiple quantum coherences using two-spin operator and the density matrix of the whole spin system. Results of calculations of the multiple-quantum dynamics in spin clusters of various dimensionalities are presented. It is shown that the whole density matrix method is more informative than the method based on the reduced density matrix.     

Exact solutions in the dynamics of alternating open chains of spins s = 1/2 with the XY Hamiltonian and their application to problems of multiple-quantum dynamics and quantum information theory

A method for exactly diagonalizing the XY Hamiltonian of an alternating open chain of spins s = 1/2 has been proposed on the basis of the Jordan-Wigner transformation and analysis of the dynamics of spinless fermions. The multiple-quantum spin dynamics of alternating open chains at high temperatures has been analyzed and the intensities of multiple-quantum coherences have been calculated. The problem of the transfer of a quantum state from one end of the alternating chain to the other is studied. It has been shown that the ideal transfer of qubits is possible in alternating chains with a larger number of spins than that in homogeneous chains.     

A study of quantum error correction by geometric algebra and liquid-state NMR spectroscopy

Quantum error correcting codes enable the information contained in a quantum state to be protected from decoherence due to external perturbations. Applied to NMR, this procedure does not alter normal relaxation, but rather converts the state of a 'data' spin into multiple quantum coherences involving additional ancilla spins. These multiple quantum coherences relax at differing rates, thus permitting the original state of the data to be approximately reconstructed by mixing them together in an appropriate fashion. This paper describes the operation of a simple, three-bit quantum code in the product operator formalism, and uses geometric algebra methods to obtain the error-corrected decay curve in the presence of arbitrary correlations in the external random fields. These predictions are confirmed in both the totally correlated and uncorrelated cases by liquid-state NMR experiments on ¹³C-labelled alanine, using gradient-diffusion methods to implement these idealized decoherence models. Quantum error correction in weakly polarized systems requires that the ancilla spins be prepared in a pseudo-pure state relative to the data spin, which entails a loss of signal that exceeds any potential gain through error correction. Nevertheless, this study shows that quantum coding can be used to validate theoretical decoherence mechanisms, and to provide detailed information on correlations in the underlying NMR relaxation dynamics.     

Distance measurements in spin-1/2 systems by 13C and 31P solid-state NMR in dense dipolar networks

In this article solid-state NMR methods for the determination of internuclear dipole-dipole couplings between homonuclear spin-1/2 nuclei are presented. They are suitable for relatively dense dipolar networks which are still dominated by 2-spin interactions. C-/R-symmetry theory is applied to create a double-quantum average Hamiltonian using phase-modulated radio-frequency irradiation and magic-angle sample-rotation. Symmetry derived pulse sequences with improved compensation against chemical shift anisotropies were found assuming a small isotropic chemical shift difference and using numerical calculations of the spin dynamics. Moreover it is shown that a constant time procedure can be used to acquire reliable double-quantum build-up curves even in systems in which damping obscures oscillations in their symmetric build-up curve. This technique is demonstrated on four crystalline model compounds with 31P and 13C spin systems typical for inorganic and biological applications. Comparison to crystal structure data indicates that the distances derived this way from 31P and 13C double-quantum NMR carry only small systematic errors caused for example by anisotropic J-coupling, dipolar contributions from adjacent spins and relaxation.     

Quantum Treatment of Intermolecular Multiple-Quantum Coherences with IntramolecularJCoupling in Solution NMR

A recently introduced density matrix picture for dipolar effects in solution NMR (1996,J. Chem. Phys.105,874) gave complete solutions for intermolecular multiple-quantum coherences for single-component samples without scalar couplings. This paper, for the first time, shows that this quantum picture can lead to explicit signal expressions for multicomponent samples of molecules with internal scalar couplings (here assumed to generate a first-order spectrum) and long-range dipolar couplings. Experimental observation of a triplet in the indirectly detected dimension for a heteronuclear CRAZED sequence (¹³CHCl₃sample, ZQ or 2Q coherences) gives clear evidence that the coupling is due to the intermolecular dipolar coupling. We also make comparisons with classical pictures which introduce the dipolar demagnetization field in multicomponent spin systems.     

Fidelity enhancement by logical qubit encoding

We demonstrate coherent control of two logical qubits encoded in a decoherence free subspace (DFS) of four dipolar-coupled protons in an NMR quantum information processor. A pseudopure fiducial state is created in the DFS, and a unitary logical qubit entangling operator evolves the system to a logical Bell state. The four-spin molecule is partially aligned by a liquid crystal solvent, which introduces strong dipolar couplings among the spins. Although the system Hamiltonian is never fully specified, we demonstrate high fidelity control over the logical degrees of freedom. In fact, the DFS encoding leads to higher fidelity control than is available in the full four-spin Hilbert space.