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arXiv:1211.4121

Lower bound for ground state energy of BEC in a rotating optical lattice

Yousef Azizi

Published Nov 17, 2012• 0 citations
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Abstract

We use the frustrated XY model approximation of BEC in a rotating optical lattice and formulate the problem of the ground state in terms of eigenvectors and eigenvalues of frustrated adjacency matrix (coupling matrix). By using this formulation, we show that there is a lower bound for ground state energy in terms of maximum eigenvalue of this matrix.

DOI
10.48550/arxiv.1211.4121

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We use the frustrated XY model approximation of BEC in a rotating optical lattice and formulate the problem of the ground state in terms of eigenvectors and eigenvalues of frustrated adjacency matrix (coupling matrix). By using this formulation, we show that there is a lower bound for ground state energy in terms of maximum eigenvalue of this matrix.
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We discuss the ground state of the two-dimensional Bose-Hubbard (BH) Hamiltonian, relevant for rotating gaseous Bose-Einstein condensates, by employing U(1) quantum rotor approach and the topologically constrained path integral that includes a summation over U(1) topological charge. We derive an effective quantum action for the BH model, which enables a nonperturbative treatment of the zero-temperature phase transition. We calculate the ground-state phase diagram, analytically deriving maximum repulsive energy for several rational values of the frustration rotation parameter f=0f=0f=0, 1/2, 1/3, 1/4, and 1/6 for the square and triangular lattices, which improves upon previous theoretical treatments. The ground state of the rotating Bose-Einstein condensates on a triangular lattice appears to be most stable against the effects of rotation. Performed calculations revealed strong dependence of the critical ratio of the kinetic energy to the repulsive on-site energy, which separates the global coherent from the insulating state, on topology of the lattice.
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Following a novel experimental arrangement which can rotate a two-dimensional optical lattice at frequencies up to several kilohertz we discuss the ground state of the two-dimensional Bose-Hubbard Hamiltonian, relevant for rotating gaseous Bose-Einstein condensates, by employing U(1) quantum rotor approach and the topologically constrained path integral.Ultracold atoms in such a rotating lattice can be used for the direct quantum simulation of strongly correlated systems under large effective magnetic fields.We derive an effective quantum action for the Bose-Hubbard model, which enables a non-perturbative treatment of the zero-temperature phase transition in the rotating frame.We calculate the ground-state phase diagram, analytically deriving maximum repulsive energy for several rational values of the frustration rotation parameter f = 0, 1/2, 1/3, 1/4, and 1/6 for the square and triangular lattice.Performed calculations revealed strong non-monotonical dependence of the critical ratio of the kinetic energy to the repulsive on-site energy, that separates the global coherent from the insulating state, on topology of the lattice.
We study ground, symmetric and central vortex states, as well as their energy and chemical potential diagrams, in rotating Bose-Einstein condensates (BEC) analytically and numerically.We start from the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term, scale it to obtain a four-parameter model, reduce it to a 2D GPE in the limiting regime of strong anisotropic confinement and present its semiclassical scaling and geometrical optics.We discuss the existence/nonexistence problem for ground states (depending on the angular velocity) and find that symmetric and central vortex states are independent of the angular rotational momentum.We perform numerical experiments computing these states using a continuous normalized gradient flow (CNGF) method with a backward Euler finite difference (BEFD) discretization.Ground, symmetric and central vortex states, as well as their energy configurations, are reported in 2D and 3D for a rotating BEC.Through our numerical study, we find various configurations with several vortices in both 2D and 3D structures, energy asymptotics in some limiting regimes and ratios between energies of different states in a strong replusive interaction regime.Finally we report the critical angular velocity at which the ground state loses symmetry, numerical verification of dimension reduction from 3D to 2D, errors for the Thomas-Fermi approximation, and spourous numerical ground states when the rotation speed is larger than the minimal trapping frequency in the xy plane.
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