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

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Figure 1: Schematic of the condensate density map for a five-vertex polariton graph.
Figure 2: Phase configurations of linear polariton chains.
Figure 3: Spin configurations of square polariton lattices.
Figure 4: Spin configurations of the diamond-shaped polariton lattices.
Figure 5: Spin configurations of a random polariton graph.
Figure 6: Experimental realization of extended polariton lattices.

References

  1. 1

    Pierce, N. A. & Winfree, E. Protein design is NP-hard. Protein Eng. 15, 779 (2002).

    CAS  Article  Google Scholar 

  2. 2

    Durlauf, S. N. How can statistical mechanics contribute to social science? Proc. Natl Acad. Sci. USA 96, 10582 (1999).

    CAS  Article  Google Scholar 

  3. 3

    Rojas, R. Neural Network–A Systematic Introduction (Springer, 1996).

    Google Scholar 

  4. 4

    Shehory, O. & Kraus, S. Methods for task allocation via agent coalition formation. Artif. Intell. 101, 165 (1998).

    Article  Google Scholar 

  5. 5

    Ghiringhelli, L. M. et al. Big data of materials science: critical role of the descriptor. Phys. Rev. Lett. 114, 105503 (2015).

    Article  Google Scholar 

  6. 6

    Balents, L. Spin liquids in frustrated magnets. Nature 464, 199 (2010).

    CAS  Article  Google Scholar 

  7. 7

    Cuevas, G. D. & Cubitt, T. S. Simple universal models capture all classical spin physics. Science 351, 1180 (2016).

    Article  Google Scholar 

  8. 8

    Woeginger, G. J. Exact algorithms for NP-hard problems: a survey. Combinatorial Optimization Vol. 185 (Lecture Notes in Computer Science, 2003).

    Google Scholar 

  9. 9

    Lucas, A. Ising formulations of many NP problems. Front. Phys. 2, 1 (2014).

    Article  Google Scholar 

  10. 10

    Barahona, F. On the computational complexity of Ising spin glass models. J. Phys. A 15, 3241 (1982).

    Article  Google Scholar 

  11. 11

    Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153 (2014).

    Article  Google Scholar 

  12. 12

    Zhang, S. & Huang, Y. Complex quadratic optimization and semidefinite programming. SIAM J. Optim. 16, 871 (2006).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Google Scholar 

  15. 15

    Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973).

    CAS  Article  Google Scholar 

  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).

    CAS  Article  Google Scholar 

  17. 17

    Kasprzak, J. et al. Bose–Einstein condensation of exciton polaritons. Nature 443, 409 (2006).

    CAS  Article  Google Scholar 

  18. 18

    Ohadi, H. et al. Non-trivial phase coupling in polariton multiplets. Phys. Rev. X 6, 031032 (2016).

    Google Scholar 

  19. 19

    Wouters, M. & Carusotto, I. Excitations in a nonequilibrium Bose–Einstein condensate of exciton polaritons. Phys. Rev. Lett. 99, 140402 (2007).

    Article  Google Scholar 

  20. 20

    Keeling, J. & Berloff, N. G. Spontaneous rotating vortex lattices in a pumped decaying condensate. Phys. Rev. Lett. 100, 250401 (2008).

    Article  Google Scholar 

  21. 21

    Cilibrizzi, P. et al. Polariton condensation in a strain-compensated planar microcavity with InGaAs quantum wells. Appl. Phys. Lett. 105, 191118 (2014).

    Article  Google Scholar 

  22. 22

    Tosi, G. et al. Geometrically locked vortex lattices in semiconductor quantum fluids. Nat. Commun. 3, 1243 (2012).

    CAS  Article  Google Scholar 

  23. 23

    Nussinov, Z. & van den Brink, J. Compass models: theory and physical motivations. Rev. Mod. Phys. 87, 1 (2015).

    CAS  Article  Google Scholar 

  24. 24

    Biskup, M. & Kotecky, R. True nature of long-range order in a plaquette orbital model. J. Stat. Mech. 11, 11001 (2010).

    Article  Google Scholar 

  25. 25

    Nasu, J. & Ishihara, S. Orbital compass model as an itinerant electron system. Europhys. Lett. 97, 27002 (2012).

    Article  Google Scholar 

  26. 26

    Rubinstein, M., Shrainam, B. & Nelson, D. R. Two-dimensional XY magnets with random Dzyaloshinskii–Moriya interactions. Phys. Rev. B 27, 1800 (1983).

    Article  Google Scholar 

  27. 27

    Granato, E. & Kosterlitz, J. M. Disorder in Josephson-junction arrays in a magnetic field. Phys. Rev. Lett. 62, 823 (1989).

    CAS  Article  Google Scholar 

  28. 28

    Cha, M. C. & Fertig, H. A. Orientational order and depinning of the disordered electron solid. Phys. Rev. Lett. 73, 870 (1994).

    CAS  Article  Google Scholar 

  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).

    CAS  Article  Google Scholar 

  30. 30

    Struck, J. et al. Quantum simulation of frustrated classical magnetism in triangular optical lattices. Science 333, 996 (2011).

    CAS  Article  Google Scholar 

  31. 31

    Johnson, M. W. et al. Quantum annealing with manufactured spins. Nature 473, 194 (2011).

    CAS  Article  Google Scholar 

  32. 32

    Porras, D. & Cirac, J. I. Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004).

    CAS  Article  Google Scholar 

  33. 33

    Utsunomiya, S., Takata, K. & Yamamoto, Y. Mapping of Ising models onto injection-locked laser systems. Opt. Express 19, 18091 (2011).

    Article  Google Scholar 

  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).

    CAS  Article  Google Scholar 

  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).

    Article  Google Scholar 

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

Verify currency and authenticity via CrossMark

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

Further reading

Search

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing