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Dynamics of N-qubit chains

Mr Courtney Mewton

Published May 1, 2008• 0 citations
Publisher

Abstract

We investigate various properties of a linear chain of qubits coupled throughthe dipole-dipole interaction. Most of the analysis is performed in the nearest-neighbor approximation. The Hamiltonian is ¯rst diagonalized, and we ex-plore some of the mathematical properties of the eigenstates. The patternof radiation emission from the single-excitation states is given, along withthe demonstration that most of these single-excitation states are subradiant,making them useful in applications such as quantum information storage andprocessing. The entanglement dynamics of the chain is also investigated,mainly showing how very long chains lose initial concurrence, as opposed toshort chains which do not.We conclude with an investigation into trying to move beyond the nearest-neighbor approximation, and derive the characteristic polynomial of theHamiltonian eigenvalue equation in analytic form, and use it to derive apower series approximation for the energy eigenvalues.

DOI
10.14264/159582

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We investigate various properties of a linear chain of qubits coupled throughthe dipole-dipole interaction. Most of the analysis is performed in the nearest-neighbor approximation. The Hamiltonian is ¯rst diagonalized, and we ex-plore some of the mathematical properties of the eigenstates. The patternof radiation emission from the single-excitation states is given, along withthe demonstration that most of these single-excitation states are subradiant,making them useful in applications such as quantum information storage andprocessing. The entanglement dynamics of the chain is also investigated,mainly showing how very long chains lose initial concurrence, as opposed toshort chains which do not.We conclude with an investigation into trying to move beyond the nearest-neighbor approximation, and derive the characteristic polynomial of theHamiltonian eigenvalue equation in analytic form, and use it to derive apower series approximation for the energy eigenvalues.
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. 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.
We calculate the radiative properties for a linear dipole-coupled chain of qubits. Using the explicit energy eigenstates of the system, we find the radiation patterns for spontaneous transitions from the one-photon eigenstates to the ground state of the system. We show that depending on the excitation of a specific atom, the radiation tends to be focused either along or perpendicular to the chain. We conclude with a derivation of the total decay rate of the one-photon eigenstates, and find the interesting result that for systems where the photon wavenumber is not much larger than the interatomic spacing, up to 94% of the eigenstates are subradiant, that is, they decay significantly slower than a single atom in isolation.
We present a technique to identify exact analytic expressions for the multiquantum eigenstates of a linear chain of coupled qubits. A choice of Hilbert subspaces is described that allows an exact solution of the stationary Schr\"odinger equation without imposing periodic boundary conditions and without neglecting end effects, fully including the dipole-dipole nearest-neighbor interaction between the atoms. The treatment is valid for an arbitrary coherent excitation in the atomic system, any number of atoms, any size of the chain relative to the resonant wavelength and arbitrary initial conditions of the atomic system. The procedure we develop is general enough to be adopted for the study of excitation in an arbitrary array of atoms including spin chains and one-dimensional Bose-Einstein condensates.
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