  One of the recently established paradigms in condensed matter physics is
examining a system's behaviour in artificially constructed potentials, giving
insight into physical phenomena of quantum fluids in hard-to-reach settings. A
prominent example is the matter-wave scatterer lattice, also known as the
barrier lattice or repulsive Dirac comb. There, high energy matter waves
undergo transmission and reflection through narrow width barriers leading to
stringent phase matching conditions with subsequent lattice band formation. It
is one of the most well taught system in quantum mechanics but its realisation
for macroscopic matter-wave fluids has remained elusive, in contrast to
evanescently coupled lattice sites or waveguides. Here, we implement and study
a system of exciton-polariton condensates in a non-Hermitian Lieb lattice of
scatterer potentials by optically injecting incoherent exciton clouds which
both emit, and interact with traveling polariton waves. By fine tuning the
lattice parameters, we reveal a nonequilibrium phase transition between two
distinct regimes of polariton condensation: a scatterer lattice of gain guided
polaritons condensing on the lattice potential maxima, and trapped polaritons
condensing in the lattice potential minima. The transition is characterised by
multimodal condensation due to gain competition between the two regimes. Energy
tomography on the polariton emission enables us to measure the intricate band
structure of the optically induced lattices. Our results pave the way towards
unexplored physics of non-Hermitian fluids in non-stationary mixtures of
confined and freely expanding waves.
