# Realizing the classical XY Hamiltonian in polariton simulators

## 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.

## Access options

from$8.99 All prices are NET prices. ## References 1. 1 Pierce, N. A. & Winfree, E. Protein design is NP-hard. Protein Eng. 15, 779 (2002). 2. 2 Durlauf, S. N. How can statistical mechanics contribute to social science? Proc. Natl Acad. Sci. USA 96, 10582 (1999). 3. 3 Rojas, R. Neural Network–A Systematic Introduction (Springer, 1996). 4. 4 Shehory, O. & Kraus, S. Methods for task allocation via agent coalition formation. Artif. Intell. 101, 165 (1998). 5. 5 Ghiringhelli, L. M. et al. Big data of materials science: critical role of the descriptor. Phys. Rev. Lett. 114, 105503 (2015). 6. 6 Balents, L. Spin liquids in frustrated magnets. Nature 464, 199 (2010). 7. 7 Cuevas, G. D. & Cubitt, T. S. Simple universal models capture all classical spin physics. Science 351, 1180 (2016). 8. 8 Woeginger, G. J. Exact algorithms for NP-hard problems: a survey. Combinatorial Optimization Vol. 185 (Lecture Notes in Computer Science, 2003). 9. 9 Lucas, A. Ising formulations of many NP problems. Front. Phys. 2, 1 (2014). 10. 10 Barahona, F. On the computational complexity of Ising spin glass models. J. Phys. A 15, 3241 (1982). 11. 11 Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153 (2014). 12. 12 Zhang, S. & Huang, Y. Complex quadratic optimization and semidefinite programming. SIAM J. Optim. 16, 871 (2006). 13. 13 Man-Cho So, A., Zhang, J. & Ye, Y. On approximating complex quadratic optimization problems via semidefinite programming relaxations. Math. Program. B 110, 93–110 (2007). 14. 14 Berezinskii, V. L. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems. Sov. Phys. JETP 32, 493 (1971). 15. 15 Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973). 16. 16 Weisbuch, C., Nishioka, M., Ishikawa, A. & Arakawa, Y. Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity. Phys. Rev. Lett. 69, 3314 (1992). 17. 17 Kasprzak, J. et al. Bose–Einstein condensation of exciton polaritons. Nature 443, 409 (2006). 18. 18 Ohadi, H. et al. Non-trivial phase coupling in polariton multiplets. Phys. Rev. X 6, 031032 (2016). 19. 19 Wouters, M. & Carusotto, I. Excitations in a nonequilibrium Bose–Einstein condensate of exciton polaritons. Phys. Rev. Lett. 99, 140402 (2007). 20. 20 Keeling, J. & Berloff, N. G. Spontaneous rotating vortex lattices in a pumped decaying condensate. Phys. Rev. Lett. 100, 250401 (2008). 21. 21 Cilibrizzi, P. et al. Polariton condensation in a strain-compensated planar microcavity with InGaAs quantum wells. Appl. Phys. Lett. 105, 191118 (2014). 22. 22 Tosi, G. et al. Geometrically locked vortex lattices in semiconductor quantum fluids. Nat. Commun. 3, 1243 (2012). 23. 23 Nussinov, Z. & van den Brink, J. Compass models: theory and physical motivations. Rev. Mod. Phys. 87, 1 (2015). 24. 24 Biskup, M. & Kotecky, R. True nature of long-range order in a plaquette orbital model. J. Stat. Mech. 11, 11001 (2010). 25. 25 Nasu, J. & Ishihara, S. Orbital compass model as an itinerant electron system. Europhys. Lett. 97, 27002 (2012). 26. 26 Rubinstein, M., Shrainam, B. & Nelson, D. R. Two-dimensional XY magnets with random Dzyaloshinskii–Moriya interactions. Phys. Rev. B 27, 1800 (1983). 27. 27 Granato, E. & Kosterlitz, J. M. Disorder in Josephson-junction arrays in a magnetic field. Phys. Rev. Lett. 62, 823 (1989). 28. 28 Cha, M. C. & Fertig, H. A. Orientational order and depinning of the disordered electron solid. Phys. Rev. Lett. 73, 870 (1994). 29. 29 Gingras, M. J. P. & Huse, D. A. Topological defects in the random-field XY model and the pinned vortex lattice to vortex glass transition in type-II superconductors. Phys. Rev. B 53, 15193 (1996). 30. 30 Struck, J. et al. Quantum simulation of frustrated classical magnetism in triangular optical lattices. Science 333, 996 (2011). 31. 31 Johnson, M. W. et al. Quantum annealing with manufactured spins. Nature 473, 194 (2011). 32. 32 Porras, D. & Cirac, J. I. Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004). 33. 33 Utsunomiya, S., Takata, K. & Yamamoto, Y. Mapping of Ising models onto injection-locked laser systems. Opt. Express 19, 18091 (2011). 34. 34 Marandi, A., Wang, Z., Takata, K., Byer, R. L. & Yamamoto, Y. Network of time-multiplexed optical parametric oscillators as a coherent Ising machine. Nat. Photon. 8, 937–942 (2014). 35. 35 Nixon, M., Ronen, E., Friesem, A. A. & Davidson, N. Observing geometric frustration with thousands of coupled lasers. Phys. Rev. Lett. 110, 184102 (2013). Download references ## 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 Authors ### 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. ### Corresponding authors Correspondence to Natalia G. Berloff or Pavlos G. Lagoudakis. ## Ethics declarations ### Competing interests The authors declare no competing financial interests. ## Supplementary information ### Supplementary Information Supplementary Information (PDF 277 kb) ## Rights and permissions Reprints and Permissions ## About this article ### Cite this article Berloff, N., Silva, M., Kalinin, K. et al. Realizing the classical XY Hamiltonian in polariton simulators. Nature Mater 16, 1120–1126 (2017). https://doi.org/10.1038/nmat4971 Download citation • Received: • Accepted: • Published: • Issue Date: ## Further reading • ### Toward Arbitrary Control of Lattice Interactions in Nonequilibrium Condensates • Kirill P. Kalinin • & Natalia G. Berloff Advanced Quantum Technologies (2020) • ### Nanolaser-based emulators of spin Hamiltonians • Midya Parto • , William E. Hayenga • , Alireza Marandi • , Demetrios N. Christodoulides • & Mercedeh Khajavikhan Nanophotonics (2020) • ### Improved Phase Locking of Laser Arrays with Nonlinear Coupling • Simon Mahler • , Matthew L. Goh • , Chene Tradonsky • , Asher A. Friesem • & Nir Davidson Physical Review Letters (2020) • ### Synthetic band-structure engineering in polariton crystals with non-Hermitian topological phases • L. Pickup • , H. Sigurdsson • , J. Ruostekoski • & P. G. Lagoudakis Nature Communications (2020) • ### Coherent population oscillations and an effective spin-exchange interaction in a$\mathcal{PT}\$ symmetric polariton mixture

• P. A. Kalozoumis
• , G. M. Nikolopoulos
•  & D. Petrosyan

EPL (Europhysics Letters) (2020)