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arXiv:quant-ph/0510070 · Journal of Physics A Mathematical and General

Exact diagonalization of the XY-Hamiltonian of open linear chains with periodic coupling constants and its application

K. E. Feldman

Published Jan 18, 2006• 15 citations
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Abstract

A new method of diagonalization of the XY-Hamiltonian of inhomogeneous open linear chains with periodic (in space) changing Larmor frequencies and coupling constants is developed. As an example of application, multiple-quantum dynamics of an inhomogeneous chain consisting of 1001 spins is investigated. Intensities of multiple-quantum coherences are calculated for arbitrary inhomogeneous chains in the approximation of the next-nearest interactions.

DOI
10.1088/0305-4470/39/5/003

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A new method of diagonalization of the XY-Hamiltonian of inhomogeneous open linear chains with periodic (in space) changing Larmor frequencies and coupling constants is developed. As an example of application, multiple-quantum dynamics of an inhomogeneous chain consisting of 1001 spins is investigated. Intensities of multiple-quantum coherences are calculated for arbitrary inhomogeneous chains in the approximation of the next-nearest interactions.
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.
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.
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.
This article is devoted to the development of analytical and numerical approaches to the problem of the end-to-end quantum state transfer along the spin-1/2 chain using two methods: (a) a homogeneous spin chain with week end bonds and equal Larmor frequencies and (b) a homogeneous spin chain with end Larmor frequencies different from inner ones. A tridiagonal matrix representation of the XY\mathit{XY}XY Hamiltonian with nearest neighbor interactions relevant to the quantum state transfer is exactly diagonalized for a combination of the preceding two methods. In order to take into account interactions of the remote spins, we used numerical simulations of the quantum state transfer in 10-node chains. We compare the state transfer times obtained using the two preceding methods for chains governed by the both XY\mathit{XY}XY and XXZ\mathit{XXZ}XXZ Hamiltonians and using both nearest neighbor and all-node interactions.