  All-to-all interacting, disordered quantum many-body models have a wide range
of applications across disciplines, from spin glasses in condensed-matter
physics, over holographic duality in high-energy physics, to annealing
algorithms in quantum computing. Typically, these models are abstractions that
do not find unambiguous physical realisations in nature. Here, we realise an
all-to-all interacting, disordered spin system by subjecting an atomic cloud in
a cavity to a controllable light shift. Adjusting the detuning between atom
resonance and cavity mode, we can tune between disordered versions of a
central-mode model and a Lipkin-Meshkov-Glick model. By spectroscopically
probing the low-energy excitations of the system, we explore the competition of
interactions with disorder across a broad parameter range. We show how disorder
in the central-mode model breaks the strong collective coupling, making the
dark state manifold cross over to a random distribution of weakly-mixed
light-matter, "grey", states. In the Lipkin-Meshkov-Glick model the
ferromagnetic finite-size ground state evolves towards a paramagnet as disorder
is increased. In that regime, semi-localised eigenstates emerge, as we observe
by extracting bounds on the participation ratio. These results present
significant steps towards freely programmable cavity-mediated interactions for
the design of arbitrary spin Hamiltonians.
