Injection and decay of particles in an inhomogeneous quantum condensate can significantly change its behaviour. We model trapped, pumped, decaying condensates by a complex Gross-Pitaevskii equation and analyse the density and currents in the steady state. With homogeneous pumping, rotationally symmetric solutions are unstable. Stability may be restored by a finite pumping spot. However if the pumping spot is larger than the Thomas-Fermi cloud radius, then rotationally symmetric solutions are replaced by solutions with spontaneous arrays of vortices. These vortex arrays arise without any rotation of the trap, spontaneously breaking rotational symmetry.PACS numbers: 03.75. Kk,47.37.+q,71.36.+c,71.35.Lk While much of the possible physics of quantum condensates has been examined in experiments on atomic gases, superfluid Helium and superconductors, there has recently been much interest in examples of condensates of quasiparticle excitations, such as excitons [1,2] (bound electron-hole pairs), exciton-polaritons [3,4,5] (superpositions of quantum well excitons and microcavity photons), and magnons (spin-wave excitations) both in magnetic insulating crystals [6,7] [33] and in superfluid 3 He [8,9,10]. One particular difference shown by these systems is that the quasiparticles have finite lifetimes, and as a result, they can be made to form condensates out of equilibrium, which are best understood as a steady state balance between pumping and decay, rather than true thermal equilibrium. The effects of pumping and decay in these condensates have been the subject of several recent works [5,11,12,13,14,15,16,17,18,19,20] which have shown that even when collisions can rapidly thermalise the energy distribution of a system, there may yet be noticeable effects associated with the energy scale introduced by the pumping and decay.The Gross-Pitaevskii equation (GPE) has been applied to successfully describe many features of equilibrium condensates when far in the condensed regime, including density profiles, the dynamics of vortices, hydrodynamic modes -see e.g. [21] and Refs. therein. Using a meanfield description of the condensate, e.g. [18,19,20], one can recover a complex Gross-Pitaevskii equation (cGPE), including terms representing gain, loss and an external trapping potential. This letter studies the interplay between pumping and decay and the external trapping potential in the context of the cGPE in order to illustrate some of the differences between equilibrium and nonequilibrium condensates. In the absence of trapping, this is the celebrated complex Ginzburg-Landau equation that describes a vast variety of phenomena [22] from nonlinear waves to second-order phase transitions, from superconductivity to liquid crystals and cosmic strings and binary fluids [23]. What is of interest in this letter is how pumping and decay, described in the cGPE modify behaviour compared to the regular GPE as is widely applied to spatially inhomogeneous equilibrium quantum condensates [21]. Spatial inhomogeneity, due to either engineere...

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...

Large scale numerical simulations of the Gross-Pitaevskii equation are used to elucidate the self-evolution of a Bose gas from a strongly nonequilibrium initial state. The stages of the process confirm and refine the theoretical scenario of Bose-Einstein condensation developed by Svistunov and co-workers ͓J. Mosc. Phys. Soc. 1, 373 ͑1991͒; Sov. Phys. JETP 75, 387 ͑1992͒; 78, 187 ͑1994͔͒: the system evolves from the regime of weak turbulence to superfluid turbulence via states of strong turbulence in the long-wavelength region of energy space.

Seeing macroscopic quantum states directly remains an elusive goal. Particles with boson symmetry can condense into such quantum fluids producing rich physical phenomena as well as proven potential for interferometric devices [1-10]. However direct imaging of such quantum states is only fleetingly possible in high-vacuum ultracold atomic condensates, and not in superconductors. Recent condensation of solid state polariton quasiparticles, built from mixing semiconductor excitons with microcavity photons, offers monolithic devices capable of supporting room temperature quantum states [11-14] that exhibit superfluid behaviour [15,16]. Here we use microcavities on a semiconductor chip supporting two-dimensional polariton condensates to directly visualise the formation of a spontaneously oscillating quantum fluid. This system is created on the fly by injecting polaritons at two or more spatially-separated pump spots. Although oscillating at tuneable THz-scale frequencies, a simple optical microscope can be used to directly image their stable archetypal quantum oscillator wavefunctions in real space. The self-repulsion of polaritons provides a solid state quasiparticle that is so nonlinear as to modify its own potential. Interference in time and space reveals the condensate wavepackets arise from non-equilibrium solitons. Control of such polariton condensate wavepackets demonstrates great potential for integrated semiconductor-based condensate devices.Comment: accepted in Nature Physic