Several platforms are currently being explored for simulating physical systems whose complexity increases faster than polynomially with the number of particles or degrees of freedom in the system. Defects and vacancies in semiconductors or dielectric materials [1,2], magnetic impurities embedded in solid helium [3], atoms in optical lattices [4,5], photons [6], trapped ions [7,8] and superconducting q-bits [9] are among the candidates for predicting the behaviour of spin glasses, spin-liquids, and classical magnetism among other phenomena with practical technological applications. Here we investigate the potential of polariton graphs as an efficient simulator for finding the global minimum of the XY Hamiltonian. By imprinting polariton condensate lattices of bespoke geometries we show that we can simulate a large variety of systems undergoing the U(1) symmetry breaking transitions. We realise various magnetic phases, such as ferromagnetic, anti-ferromagnetic, and frustrated spin configurations on unit cells of various lattices: square, triangular, linear and a disordered graph. Our results provide a route to study unconventional superfluids, spin-liquids, Berezinskii-KosterlitzThouless phase transition, classical magnetism among the many systems that are described by the XY Hamiltonian.Many properties of strongly correlated spin systems, such as spin liquids and unconventional superfluids are difficult to study as strong interactions between n particles become intractable for n as low as 30 [10]. Feynman envisioned that a quantum simulator -a special-purpose analogue processor -could be used to solve such problems [11]. It is expected that quantum simulators would lead to accurate modelling of the dynamics of chemical reactions, motion of electrons in materials, new chemical compounds and new materials that could not be obtained with classical computers using advanced numerical algorithms [12]. More generally, quantum simulators can be used to solve hard optimization problems that are at the heart of almost any multicomponent system: new materials for energy, pharmaceuticals, and photosynthesis, among others [13]. Many hard optimisation problems do not necessitate a quantum simulator as only recently realised through a network of optical parametric oscillators (OPOs) that simulated the Ising Hamiltonian of thousands of spins [14,15]. The Ising model corresponds to the n = 1 case of the n-vector model of classical unit vector spins s i with the Hamiltonian H I = − ij J ij s i · s j , where J ij is the coupling between the sites labelled i and j. For n = 2 the n-vector Hamiltonian becomes the XY Hamiltonian H XY = − ij J ij cos(θ i − θ j ), where we have parameterized unit planar vectors using the polar coordinates s i = (cos θ i , sin θ i ). Since H XY is invariant under rotation of all spins by the same angle θ i → θ i + φ the XY model is the simplest model that undergoes the U (1) symmetry-breaking transition. As such, it is used * correspondence address: pavlos.lagoudakis@soton.ac.uk to emulate Berezinskii-Kos...

Non-linearity and finite signal propagation speeds are omnipresent in nature, technologies, and real-world problems, where efficient ways of describing and predicting the effects of these elements are in high demand. Advances in engineering condensed matter systems, such as lattices of trapped condensates, have enabled studies on non-linear effects in manybody systems where exchange of particles between lattice nodes is effectively instantaneous. Here, we demonstrate a regime of macroscopic matter-wave systems, in which ballistically expanding condensates of microcavity exciton-polaritons act as picosecond, microscale nonlinear oscillators subject to time-delayed interaction. The ease of optical control and readout of polariton condensates enables us to explore the phase space of two interacting condensates up to macroscopic distances highlighting its potential in extended configurations. We demonstrate deterministic tuning of the coupled-condensate system between fixed point and limit cycle regimes, which is fully reproduced by time-delayed coupled equations of motion similar to the Lang-Kobayashi equation.

We demonstrate deterministic control of the nearest and next-nearest neighbor coupling in the unit cell of a square lattice of microcavity exciton-polariton condensates. We tune the coupling in a continuous and reversible manner by optically imprinting potential barriers of variable height, in the form of spatially localized incoherent exciton reservoirs that modify the particle flow between condensates. By controlling the couplings in a 2 × 2 polariton cluster, we realize ferromagnetic, anti-ferromagnetic and paired ferromagnetic phases, and demonstrate the potential scalability of the system.