Article

Realizing the classical XY Hamiltonian in polariton simulators

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Abstract

The vast majority of real-life optimization problems with a large number of degrees of freedom are intractable by classical computers, since their complexity grows exponentially fast with the number of variables. Many of these problems can be mapped into classical spin models, such as the Ising, the XY or the Heisenberg models, so that optimization problems are reduced to finding the global minimum of spin models. Here, we propose and investigate the potential of polariton graphs as an efficient analogue simulator for finding the global minimum of the XY model. By imprinting polariton condensate lattices of bespoke geometries we show that we can engineer various coupling strengths between the lattice sites and read out the result of the global minimization through the relative phases. Besides solving optimization problems, polariton graphs can simulate a large variety of systems undergoing the U(1) symmetry-breaking transition. We realize various magnetic phases, such as ferromagnetic, anti-ferromagnetic, and frustrated spin configurations on a linear chain, the unit cells of square and triangular lattices, a disordered graph, and demonstrate the potential for size scalability on an extended square lattice of 45 coherently coupled polariton condensates. Our results provide a route to study unconventional superfluids, spin liquids, Berezinskii–Kosterlitz–Thouless phase transition, and classical magnetism, among the many systems that are described by the XY Hamiltonian.

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Acknowledgements

The authors acknowledge the support of the Skoltech NGP Program (Skoltech-MIT joint project), and the UK’s Engineering and Physical Sciences Research Council (grant EP/M025330/1 on Hybrid Polaritonics). N.G.B. is grateful to N. Prokof’ev for fruitful discussions.

Author information

Affiliations

  1. Skolkovo Institute of Science and Technology Novaya St., 100, Skolkovo 143025, Russian Federation

    • Natalia G. Berloff
    • , Kirill Kalinin
    •  & Pavlos G. Lagoudakis
  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK

    • Natalia G. Berloff
  3. Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK

    • Matteo Silva
    • , Alexis Askitopoulos
    • , Julian D. Töpfer
    • , Pasquale Cilibrizzi
    •  & Pavlos G. Lagoudakis
  4. School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, UK

    • Wolfgang Langbein

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Contributions

N.G.B. and P.G.L. designed the research and wrote the paper. M.S., A.A. and P.G.L. performed the experiments. M.S., and P.G.L. analysed the experimental data. N.G.B. and K.K. performed theoretical modelling. K.K. performed numerical simulations and analysis of numerical data. J.D.T. and P.C. contributed to the experimental apparatus and complementary measurements. W.L. and P.G.L. designed and managed the growth of the sample.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to Natalia G. Berloff or Pavlos G. Lagoudakis.

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