  We suggest a scheme for the preparation of highly correlated Laughlin (LN)
states in the presence of synthetic gauge fields, realizing an analogue of the
fractional quantum Hall effect in photonic or atomic systems of interacting
bosons. It is based on the idea of growing such states by adding weakly
interacting composite fermions (CF) along with magnetic flux quanta one-by-one.
The topologically protected Thouless pump ("Laughlin's argument") is used to
create two localized flux quanta and the resulting hole excitation is
subsequently filled by a single boson, which, together with one of the flux
quanta forms a CF. Using our protocol, filling 1/2 LN states can be grown with
particle number N increasing linearly in time and strongly suppressed number
fluctuations. To demonstrate the feasibility of our scheme, we consider
two-dimensional (2D) lattices subject to effective magnetic fields and strong
on-site interactions. We present numerical simulations of small lattice systems
and discuss also the influence of losses.
