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Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Solid-state multiple quantum NMR in quantum information processing: exactly solvable models

Edward B. Fel’dman, Alexey N. Pyrkov, Alexander Ivanovich Zenchuk

Published Sep 3, 2012• 19 citations
Publisher

Abstract

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.

DOI
10.1098/rsta.2011.0499

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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.
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.
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) 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.
In this work, quantum correlations in two model systems which can be invesitgated by means of solid-state NMR are considered. The first is a two-spin system in NMR spin-locking experiment, where two schemes of spin-locking are considered. Concurrence is calculated for both schemes assuming the initial thermodynamic equilibrium state. Quantum discord is calculated analytically in the high-temperature approximation. The second model system is the spin-1/2 chain with the XY interaction, where the transfer of multiple quantum (MQ) coherences from one end of the chain to another is studied. The conditions under which the density matrix of the receiver is a block-scaled version of the sender’s density matrix are found.
Quantum correlations in multiple quantum (MQ) NMR experiments are investigated in two-spin systems (dimers). In the initial moment of time one spin is in a pure quantum polarized state and the other spin is in the thermodynamic equilibrium state defined by the temperature of the sample. MQ NMR dynamics of dimers is investigated. It is shown that MQ NMR coherences of only the zeroth and second orders emerge in such a system. The intensities of those coherences are calculated. Entangled states appear in the course of the system evolution in the MQ NMR experiment. The quantum discord is obtained in the high temperature approximation.
We investigate analytically and numerically the multiple-quantum (MQ) NMR dynamics in systems of nuclear spins 1∕21∕21∕2 coupled by the dipole-dipole interactions in the case of the dipolar ordered initial state. We suggest two different methods of MQ NMR. One of them is based on the measurement of the dipolar temperature in the quasiequilibrium state which is established after the time of order ${\ensuremath{\omega}}_{\mathit{loc}}^{\ensuremath{-}1}$ (${\ensuremath{\omega}}_{\mathit{loc}}$ is the dipolar local field) after the MQ NMR experiment. The other method uses an additional resonance ${(\ensuremath{\pi}∕4)}_{y}$ pulse after the preparation period of the standard MQ NMR experiment in solids. Many-spin clusters and correlations are created faster in such experiments than in the usual MQ NMR experiments and can be used for the investigation of many-spin dynamics of nuclear spins in solids.