  We consider a two-dimensional weakly interacting ultracold Bose gas whose
constituents are two-level atoms. We study the effects of a synthetic
density-dependent gauge field that arises from laser-matter coupling in the
adiabatic limit with a laser configuration such that the single-particle
zero-order vector potential corresponds to a constant synthetic magnetic field.
We find a new exotic type of current non-linearity in the Gross-Pitaevskii
equation which affects the dynamics of the order parameter of the condensate.
We investigate the rotational properties of this system, focusing in particular
on the physical conditions that make the nucleation of a quantized vortex in
the system energetically favourable with respect to the non rotating solution.
We point out that two different physical interpretations can be given to this
new non linearity: firstly it can be seen as a local modification of the mean
field coupling constant, whose value depends on the angular momentum of the
condensate. Secondly, it can be interpreted as a density modulated angular
velocity given to the cloud. Looking at the problem from both of these
viewpoints, we analyze the physical conditions that make a single vortex state
energetically favourable. In the Thomas-Fermi limit, we show that the effect of
the new nonlinearity is to induce a rotation to the condensate, where the
transition from non-rotating to rotating states depends on the density of the
cloud.
