Attractive ultracold bosons in a necklace optical lattice

We study the ground-state properties of the Bose-Hubbard model with attractive interactions on an M-siteone-dimensional periodic--necklacelike--lattice, whose experimental realization in terms of ultracold atoms ispromised by a recently proposed optical trapping scheme, as well as by the control over the atomic interactionsand tunneling amplitudes granted by well-established optical techniques. We compare the properties of thequantum model to a semiclassical picture based on a number-conserving suM coherent state, which results ina set of modified discrete nonlinear Schrödinger equations. We show that, owing to the presence of a correctionfactor ensuing from number conservation, the ground-state solution to these equations provides a remarkablysatisfactory description of its quantum counterpart not only--as expected--in the weak-interaction, superfluidregime, but even in the deeply quantum regime of large interactions and possibly small populations. Inparticular, we show that in this regime, the delocalized, Schrödinger-cat-like quantum ground state can be seenas a coherent quantum superposition of the localized, symmetry-breaking ground state of the variationalapproach. We also show that, depending on the hopping to interaction ratio, three regimes can be recognizedboth in the semiclassical and quantum picture of the system.