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arXiv:1110.0991

The effect of spin-lattice relaxation on multiple-quantum NMR coherences in solids

Edward B. Fel’dman, Alexey N. Pyrkov

Published Oct 5, 2011• 2 citations
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Abstract

A phenomenological theory of spin-lattice relaxation of multiple-quantum coherences in systems of two dipolar coupled spins at low temperatures is developed. Intensities of multiple-quantum NMR coherences depending on the spin-lattice relaxation time are obtained. It is shown that the theory is also applicable to finite spin chains when the approximation of nearest neighbour interaction is used. An application of this theory to an estimation of the influence of decoherence processes on quantum entanglement and its fluctuations is briefly discussed.

DOI
10.48550/arxiv.1110.0991

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A phenomenological theory of spin-lattice relaxation of multiple-quantum coherences in systems of two dipolar coupled spins at low temperatures is developed. Intensities of multiple-quantum NMR coherences depending on the spin-lattice relaxation time are obtained. It is shown that the theory is also applicable to finite spin chains when the approximation of nearest neighbour interaction is used. An application of this theory to an estimation of the influence of decoherence processes on quantum entanglement and its fluctuations is briefly discussed.
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We investigate numerically the time dependence of the multiple quantum coherences and entanglement in linear chains up to nine nuclear spins of 1∕21∕21∕2 coupled by the dipole-dipole interactions. Two models are considered: (1) a spin chain with nearest-neighbor dipole-dipole interactions and (2) a more realistic model with interactions between all spins. It is shown that the entangled states appear between remote particles which do not interact directly (model 1), while the interaction between all spins (model 2) not always results in entanglement between remote spins.
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.
Multiple quantum experiments on nuclear magnetic resonance on spins of 19 F in calcium fluorapatite have been performed. It has been shown that more than 97% of the observed signal refers to the multiple quantum coherences of the zeroth and plus/minus second orders. The dependences of the intensities of these coherences on the time of the preparation period of the multiple quantum NMR experiment have been obtained. The dipole relaxation of multiple quantum coherences on the evolution period of the multiple quantum nuclear magnetic resonance experiment has been studied. A probe providing short radio-frequency pulses with the duration up to 0.3 μs has been developed for experiments. The experimental results show that the studied system can be used for the transmission of quantum information and the study of the decoherence process in strongly correlated multispin clusters.
Multiple quantum (MQ) nuclear magnetic resonance (NMR) spin dynamics are investigated analytically in infinite one-dimensional (1D) chains of spins 1/2. The representation of spin 1/2 operators with fermion field operators allows to calculate exactly the spin density operator, and hence NMR observables, under a variety of different conditions for 1D spin systems. The exact expressions are valid for all times and for a macroscopic number of coupled spins. The calculations for a 1D spin system initially at thermal equilibrium, and evolving under a 2-quantum/2-spin average dipolar Hamiltonian, in the presence of nearest-neighbor dipolar interactions yield MQ NMR spectra with 0- and 2-quantum coherences only. For a nonequilibrium initial condition with transverse magnetization, the analogous spin dynamics calculations produce MQ NMR spectra with all possible coherences of odd orders. Calculations at the level of perturbation theory, which include next-nearest-neighbor dipolar interactions, generate MQ spectra with higher even order coherences for equilibrium initial condition and evolution under a 2-quantum/2-spin propagator. Consideration of multiple spin correlations, 0-quantum coherences, and rf pulse imperfections are also presented. The relevance and implications of these theoretical results for comparison with the recent MQ NMR experiments of Yesinowski et al. on materials with quasi-one-dimensional distributions of spins, and for MQ NMR of solids in general are discussed.
We study the decay of multiple quantum (MQ) NMR coherences in systems with the large number of equivalent spins. As being created on the preparation period of the MQ NMR experiment, they decay due to the dipole-dipole interactions (DDI) on the evolution period of this experiment. It is shown that the relaxation time decreases with the increase in MQ coherence order (according to the known results) and in the number of spins. We also consider the modified preparation period of the MQ NMR experiment [G. A. Alvarez and D. Suter, Phys. Rev. Lett. 104, 230403 (2010)] concatenating the short evolution periods under the secular DDI Hamiltonian (the perturbation) with the evolution period under the nonsecular averaged two-spin/two-quantum Hamiltonian. The influence of the perturbation on the decoherence rate is investigated for the systems consisting of 200-600 equivalent spins.