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Abstract
We develop computational tools necessary to extend the application of Krylov complexity beyond the simple Hamiltonian systems considered thus far in the literature. As a first step toward this broader goal, we show how the Lanczos algorithm that iteratively generates the Krylov basis can be augmented to treat coherent states associated with the Jacobi group, the semi-direct product of the 3-dimensional real Heisenberg-Weyl group H1, and the symplectic group, Sp(2, ℝ) ≃ SU(1, 1). Such coherent states are physically realized as squeezed states in, for example, quantum optics [1]. With the Krylov basis for both the SU(1, 1) and Heisenberg-Weyl groups being well understood, their semi-direct product is also partially analytically tractable. We exploit this to benchmark a scheme to numerically compute the Lanczos coefficients which, in principle, generalizes to the more general Jacobi group Hn ⋊ Sp(2n, ℝ).
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Details
1 University of Cape Town, The Laboratory for Quantum Gravity & Strings, Department of Mathematics & Applied Mathematics, Cape Town, South Africa (GRID:grid.7836.a) (ISNI:0000 0004 1937 1151); The National Institute for Theoretical and Computational Sciences, Matieland, South Africa (GRID:grid.7836.a)
2 University of Cape Town, The Laboratory for Quantum Gravity & Strings, Department of Mathematics & Applied Mathematics, Cape Town, South Africa (GRID:grid.7836.a) (ISNI:0000 0004 1937 1151)