The characteristic polynomial of the next-nearest-neighbour qubit chain for single excitations

Mewton, C. J. and Ficek, Z. (2008) The characteristic polynomial of the next-nearest-neighbour qubit chain for single excitations. Journal of Physics A: Mathematical and Theoretical, 41 44 Article Number: 445201: 445201-1-445201-18. doi:10.1088/1751-8113/41/44/445201


Author Mewton, C. J.
Ficek, Z.
Title The characteristic polynomial of the next-nearest-neighbour qubit chain for single excitations
Journal name Journal of Physics A: Mathematical and Theoretical   Check publisher's open access policy
ISSN 1751-8113
Publication date 2008-11-07
Sub-type Article (original research)
DOI 10.1088/1751-8113/41/44/445201
Volume 41
Issue 44 Article Number: 445201
Start page 445201-1
End page 445201-18
Total pages 18
Editor M.T. Batchelor
Place of publication Bristol, England
Publisher Institute of Physics Publishing
Collection year 2009
Language eng
Subject C1
970102 Expanding Knowledge in the Physical Sciences
029999 Physical Sciences not elsewhere classified
Abstract The characteristic polynomial for a chain of dipole–dipole coupled two-level atoms with nearest-neighbour and next-nearest-neighbour interactions is developed for the study of eigenvalues and eigenvectors for single-photon excitations. Such a system is mathematically equivalent to an XX spin chain in an external magnetic field. We find the exact form of the polynomial in terms of the Chebyshev polynomials of the second kind that is valid for an arbitrary number of atoms and coupling strengths. We then propose a technique for expressing the roots of the polynomial as a power series in the coupling constants. The general properties of the solutions are also explored, to shed some light on the general properties that the exact, analytic form of the energy eigenvalues should have. A method for deriving the eigenvectors of the Hamiltonian is also outlined.
Keyword N-Atom system
Heisenberg-antiferromagnet
Quantum dots
Model
Q-Index Code C1
Q-Index Status Confirmed Code
Additional Notes Published 7 October 2008. -- Article Number: 445201

 
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Created: Mon, 23 Mar 2009, 13:09:27 EST by Jo Hughes on behalf of School of Mathematics & Physics