  We study the ground states of 2D lattice bosons in an artificial gauge field.
Using state of the art DMRG simulations we obtain the zero temperature phase
diagram for hardcore bosons at densities $n_b$ with flux $n_\phi$ per unit
cell, which determines a filling $\nu=n_b/n_\phi$. We find several robust
quantum Hall phases, including (i) a bosonic integer quantum Hall phase (BIQH)
at $\nu=2$, that realizes an interacting symmetry protected topological phase
in 2D (ii) bosonic fractional quantum Hall phases including robust states at
$\nu=2/3$ and a Laughlin state at $\nu=1/2$. The observed states correspond to
the bosonic Jain sequence ($\nu=p/(p+1)$) pointing towards an underlying
composite fermion picture. In addition to identifying Hamiltonians whose ground
states realize these phases, we discuss their preparation beginning from
independent chains, and ramping up interchain couplings. Using time dependent
DMRG simulations, these are shown to reliably produce states close to the
ground state for experimentally relevant system sizes. Besides the
wave-function overlap, we utilize a simple physical signature of these phases,
the non-monotonic behavior of a two-point correlation, a direct consequence of
edge states in a finite system, to numerically assess the effectiveness of the
preparation scheme. Our proposal only utilizes existing experimental
capabilities.
