Quantum Hall states of atomic Bose gases: density profiles in single-layer and multi-layer geometries

We describe the density profiles of confined atomic Bose gases in the high-rotation limit, in single-layer and multi-layer geometries. We show that, in a local density approximation, the density in a single layer shows a landscape of quantized steps due to the formation of incompressible liquids, which are analogous to fractional quantum Hall liquids for a two-dimensional electron gas in a strong magnetic field. In a multi-layered set-up we find different phases, depending on the strength of the inter-layer tunneling t. We discuss the situation where a vortex lattice in the three-dimensional condensate (at large tunneling) undergoes quantum melting at a critical tunneling $t_{c_1}$. For tunneling well below $t_{c_1}$ one expects weakly coupled or isolated layers, each exhibiting a landscape of quantum Hall liquids. After expansion, this gives a radial density distribution with characteristic features (cusps) that provide experimental signatures of the quantum Hall liquids.