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Journal of Experimental and Theoretical Physics

Reversible and Irreversible Propagation of Quantum Information and Its Manifestation in Multiple-Quantum NMR Spectra in Solids

V. E. Zobov, A. A. Lundin

Published Aug 1, 2020• 2 citations
Publisher

Abstract

Abstract On the basis of the earlier developed statistical theory of the growth of the effective size of correlated clusters (the number of correlated spins), an expression for the shape of the multiple-quantum (MQ) NMR spectrum is obtained that takes into account the loss of coherence in a spin system due, for example, to the controlled intervention of the experimenter. It is shown that the scrambling and decoherence processes in the MQ spectrum of the multiparticle system of a solid are not separated, unlike the corresponding spectra of some large isolated molecules [27] in a solution. The relations obtained allow one to extract the necessary information about the above processes (scrambling and decoherence) from the dependence of the MQ spectra on experimental parameters.

DOI
10.1134/s1063776120060096

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Abstract On the basis of the earlier developed statistical theory of the growth of the effective size of correlated clusters (the number of correlated spins), an expression for the shape of the multiple-quantum (MQ) NMR spectrum is obtained that takes into account the loss of coherence in a spin system due, for example, to the controlled intervention of the experimenter. It is shown that the scrambling and decoherence processes in the MQ spectrum of the multiparticle system of a solid are not separated, unlike the corresponding spectra of some large isolated molecules [27] in a solution. The relations obtained allow one to extract the necessary information about the above processes (scrambling and decoherence) from the dependence of the MQ spectra on experimental parameters.
Multiple-quantum (MQ) solid-state NMR spectroscopy allows the growth of multiple-spin correlations and, thus, the spreading of quantum information in the object under study to be observed. Recently, in [11] it was proposed to control this process through a controlled perturbation added to the effective Hamiltonian that causes degradation of correlated spin clusters with a rate determined by the number of spins K in a cluster. However, this perturbation can also lead to degradation whose rate is determined by the coherence order M. In this paper, to investigate the influence of a small added perturbation, we used an expansion into orthogonal operators that allowed the cluster size distribution to be taken into account. In our calculations we realized a simple model with known amplitudes of the expansion into a complete set of orthogonal operators in the absence of a perturbation. We performed numerical calculations of the “preparation time” dependences of the MQ spectra, their second moments, and the coherence orders at which the MQ spectra decrease by a factor of e as well as the average correlated spin cluster sizes Kˉ\bar {K}Kˉ . The coherence-order-dependent contribution to the degradation is shown to change the shape of the MQ spectrum. In particular, as the preparation time increases, the MQ spectrum can be stabilized, while the growth of Kˉ\bar {K}Kˉ is retained. Due to the change in the shape of the MQ spectrum, the relations of its characteristics to the number Kˉ\bar {K}Kˉ change compared to those for the Gaussian function (traditionally used to process the experiments). These changes should be taken into account when studying the spreading of quantum information through MQ spectroscopy.
Multiple quantum (MQ) NMR is an effective tool for the generation of a large cluster of correlated particles, which, in turn, represent a basis for quantum information processing devices. Studying the available exactly solvable models clarifies many aspects of the quantum information. In this study, we consider two exactly solvable models in the MQ NMR experiment: (i) the isolated system of two spin-1/2 particles (dimers) and (ii) the large system of equivalent spin-1/2 particles in a nanopore. The former model is used to describe the quantum correlations and their relations with the MQ NMR coherences, whereas the latter helps one to model the creation and decay of large clusters of correlated particles.
As opposed to traditional methods of multiple-quantum NMR spectroscopy authors of the article [G.A. Alvarez, D. Suter, Phys. Rev. A 84, 012320 (2011)] generated the effective double-quantum Hamiltonian, with the slight adding usual secular dipole-dipole Hamiltonian to the first one at the stage of the correlations preparing (so the perturbation appeared). At this framework the width of MQ spectra reveals tending to any constant amount with the growing up of the time. Also there were shown that the spectral width decreases with increasing perturbation strength. The growing up of the cluster size as it was supposed by G.A. Alvarez and D. Suter was restricted by mentioned perturbation. We are assuming an alternative explanation. The growing up of cluster size still goes on but the width of MQ spectrum becomes stable because of different decay rate of MQ-spectral components in dependence on theirs location in spectrum. Here we are calculating the widths of MQ spectra in dependence on the time of "preparation" for different values of the perturbation strength and by this way we get the dependence of stabilized amount of MQ-spectrum width on mentioned strength. So, the excellent compliance obtains with the analogues experimental relation observed in Ref. [Phys. Rev. A 84, 012320 (2011)].
Multiple quantum (MQ) NMR methods1 are applied to the analysis of various problems of quantum information processing. It is shown that the two-spin/two-quantum Hamiltonian1 describing MQ NMR dynamics is related to the flip-flop Hamiltonian of a one-dimensional spin system in the approximation of the nearest neighbor interactions. As a result, it is possible to organize quantum state transfer along a linear chain. MQ NMR experiments are performed on quasi-one-dimensional chains of 19F nuclei in calcium fluorapatite Ca5(PO4)3F. Relaxation of the MQ NMR coherences is considered as the simplest model of decoherence processes. A theory of the dipolar relaxation of the MQ NMR coherences in one-dimensional systems is developed. A good agreement of the theoretical predictions and the experimental data is obtained.
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.
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.