Computing Interior Eigenvalues of a Schroedinger Equation

William Mitchell

NIST, Gaithersburg, MD 20899-8910


Abstract

We have applied multigrid methods to solve a two-dimensional Schroedinger equation in order to study the feasibility of a quantum computer based on extremely-cold neutral alkali-metal atoms. Qubits are implemented as motional states of an atom trapped in a single well of an optical lattice of counter-propagating laser beams. Quantum gates are constructed by bringing two atoms together in a single well leaving the interaction between the atoms to cause entanglement. For special geometries of the optical lattices and thus shape of the wells, quantifying the entanglement reduces to solving for selected eigenfunctions of a Schroedinger equation that contains a two-dimensional Laplacian, a trapping potential that describes the optical well, and a short-ranged interaction potential. The desired eigenfunctions correspond to eigenvalues that are deep in the interior of the spectrum where the trapping potential becomes significant. The finite element discretization of the Schroedinger equation produces a generalized eigenproblem Ax = lambda*Mx where A is the stiffness matrix and M is the mass matrix. A spectral transformation transforms this problem to an eigenproblem in which the matrix is composed from M, the inverse of A, and a constant that is close to the desired eigenvalues. We use an iterative eigensolver, Arnoldi's method, to solve this eigenproblem for a few of the smallest eigenvalues and corresponding eigenfunctions, which correspond to the desired eigenfunctions. Arnoldi's method does not require the actual matrix of the eigenproblem, only the action of multiplying by that matrix. Multiplication by M is straight forward. Multiplication by the inverse of A is achieved by a multigrid method.