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A bosonic analogue of the fractional quantum Hall eff ect occurs in rapidlyrotating trapped Bose gases: There is a transition from uncorrelated Hartreestates to strongly correlated states such as the Laughlin wave function. Thisphysics may be described by eff ective Hamiltonians with delta interactionsacting on a bosonic N-body Bargmann space of analytic functions. In a previouspaper [N. Rougerie, S. Serfaty, J. Yngvason, Phys. Rev. A 87, 023618 (2013)] westudied the case of a quadratic plus quartic trapping potential and derivedconditions on the parameters of the model for its ground state to beasymptotically strongly correlated. This relied essentially on energy upperbounds using quantum Hall trial states, incorporating the correlations of theBose-Laughlin state in addition to a multiply quantized vortex pinned at theorigin. In this paper we investigate in more details the density of these trialstates, thereby substantiating further the physical picture described in [N.Rougerie, S. Serfaty, J. Yngvason, Phys. Rev. A 87, 023618 (2013)], improvingour energy estimates and allowing to consider more general trapping potentials.Our analysis is based on the interpretation of the densities of quantum Halltrial states as Gibbs measures of classical 2D Coulomb gases (plasma analogy).New estimates on the mean- field limit of such systems are presented.

Quantum Hall phases and plasma analogy in rotating trapped Bose gases