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...

We demonstrate experimentally the condensation of exciton-polaritons through optical trapping. The non-resonant pump profile is shaped into a ring and projected to a high quality factor microcavity where it forms a 2D repulsive optical potential originating from the interactions of polaritons with the excitonic reservoir. Increasing the population of particles in the trap eventually leads to the emergence of a confined polariton condensate that is spatially decoupled from the decoherence inducing reservoir, before any build up of coherence on the excitation region. In a reference experiment, where the trapping mechanism is switched off by changing the excitation intensity profile, polariton condensation takes place for excitation densities more than two times higher and the resulting condensate is subject to a much stronger dephasing and depletion processes.Strong coupling of cavity photons and quantum-well excitons gives rise to mixed light-matter bosonic quasiparticles called exciton-polaritons or polaritons [1]. Due to their photonic component, polaritons are several orders of magnitude lighter than atoms, which makes their condensation attainable at higher temperatures [2,3]. The manifestations of polariton condensation include polariton lasing [4], long-range spatial coherence [5,6] and stochastic vector polarisation [7]. In an ideal infinite twodimensional cavity the polariton gas is expected to undergo the Berezinsky-Kosterlitz-Thouless (BKT) phase transition [8], while in realistic structures, polaritons can condense in traps induced by random optical disorder [2] or mechanically created potentials [3,9]. Polariton condensation has also been observed in structures of lower dimensionality [10][11][12][13] where the structure itself acts as the trapping potential. Furthermore, the manipulation of polariton condensates by optically generated potentials has been previously shown [14][15][16]. In these works the condensation process was not assisted by the optical potential but used to localize an already formed polariton condensate.Here, we report on the first manifestation of polariton condensation assisted by an optically generated two dimensional potential. This scheme allows for the formation of a polariton condensate spatially separated from the excitation spot. Owing to the efficient trapping in the optical potential we observe a reduced excitation density threshold as well as higher coherence due to the decoupling of the condensate from the exciton reservoir. Polariton condensation prior to the build-up of coherence in the form of photon or polariton lasing [17,18] at the excitation area on the sample, decisively resolves the debate on the phase relation between the excitation laser and polariton condensate.We used a high quality factor GaAs/AlGaAs microcavity containing four separate triplets of 10 nm GaAs quantum wells and has a vacuum Rabi splitting of 9 meV [19], held at ∼ 6.5 K in a cold-finger cryostat and excited non-resonantly at the first reflection minimum above the cavity stop band with a s...

We report on pure-quantum-state polariton condensates in optical annular traps. The study of the underlying mechanism reveals that the polariton wavefunction always coalesces in a single purequantum-state that, counter-intuitively, is always the uppermost confined state with the highest overlap to the exciton reservoir. The tunability of such states combined with the short polariton lifetime allows for ultrafast transitions between coherent mesoscopic wavefunctions of distinctly different symmetries rendering optically confined polariton condensates a promising platform for applications such as many-body quantum circuitry and continuous-variable quantum processing. arXiv:1411.4579v2 [cond-mat.quant-gas] 10 Nov 2015

We investigate the spin dynamics of polariton condensates spatially separated from and effectively confined by the pumping exciton reservoir. We obtain a strong correlation between the ellipticity of the non-resonant optical pump and the degree of circular polarisation (DCP) of the condensate at the onset of condensation. With increasing excitation density we observe a reversal of the DCP. The spin dynamics of the trapped condensate are described within the framework of the spinor complex Ginzburg-Landau equations in the Josephson regime, where the dynamics of the system are reduced to a current-driven Josephson junction. We show that the observed spin reversal is due to the interplay between an internal Josephson coupling effect and the detuning of the two projections of the spinor condensate via transition from a synchronised to a desynchronised regime. These results suggest that spinor polariton condensates can be controlled by tuning the non-resonant excitation density offering applications in electrically pumped polariton spin switches.

Periodic incorporation of quantum wells inside a one-dimensional Bragg structure is shown to enhance coherent coupling of excitons to the electromagnetic Bloch waves. We demonstrate strong coupling of quantum well excitons to photonic crystal Bragg modes at the edge of the photonic band gap, which gives rise to mixed Bragg polariton eigenstates. The resulting Bragg polariton branches are in good agreement with the theory and allow demonstration of Bragg polariton parametric amplification.