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Communications in Theoretical Physics

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

Hamid Arian Zad

Published Dec 1, 2016• 2 citations

Abstract

Abstract 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 J 1 and J 2 , 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.

DOI: 10.1088/0253-6102/66/6/629

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The dynamics of entanglement is investigated on the basis of exactly solvable models of multiple quantum (MQ) NMR spin dynamics. It is shown that the time evolution of MQ coherences of systems of coupled nuclear spins in solids is directly connected with dynamics of the quantum entanglement. We studied analytically the dynamics of entangled states for two- and three-spin systems coupled by the dipole-dipole interaction. In this case the dynamics of the quantum entanglement is uniquely determined by the time evolution of MQ coherences of the second order. The real part of the density matrix describing MQ dynamics in solids is responsible for MQ coherences of the zeroth order while its imaginary part is responsible for the second order. Thus, one can conclude that the dynamics of the entanglement is connected with transitions from the real part of the density matrix to the imaginary one, and vice versa. A pure state which generalizes the Greenberger-Horne-Zeilinger (GHZ) and W states is found. Different measures of the entanglement of this state are analyzed for tripartite systems.

We investigated dynamics of entangled states on the basis of multiple-quantum (MQ) NMR methods. A time evolution of MQ coherences of nuclear spins coupled by the dipole-dipole interactions (DDI) in solids is simply connected with dynamics of quantum entangled states. We studied analytically dynamics of the entangled states for two- and three-spin systems. In this case dynamics of the quantum entanglement is uniquely determined by the time evolution of MQ coherences of the second order. The real part of the density matrix describing MQ dynamics in solids is responsible for MQ coherences of the zeroth order while its imaginary part is responsible for the second order. Thus, one can conclude that dynamics of the entanglement is connected with transitions from the real part of the density matrix to the imaginary one and vice versa.

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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.

With the purpose to reveal consistency between multiple quantum (MQ) coherences and entanglement, we investigate numerically the dynamics of these phenomena in one-dimensional linear chains and ring of nuclear spins 1/2 coupled by dipole dipole interactions. As opposed to the calculation of the MQ coherence intensity based on the density matrix describing the spin system as a whole, we consider the "differentiated" intensity related only to the chosen spin pair based on the reduced density matrix. It is shown that the entanglement and the MQ coherence have similar dynamics only for nearest neighbors while we did not obtained any consistency for remote spins.

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