Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Review Article
  • Published:

Polariton condensates for classical and quantum computing

Nature Reviews Physics volume 4, pages 435–451 (2022)Cite this article

Subjects

A Publisher Correction to this article was published on 23 May 2022

This article has been updated

Abstract

Polariton lasers emit coherent monochromatic light through a spontaneous emission process. As a rare example of a system in which Bose–Einstein condensation and superfluidity are reported at room temperature, polariton lasers are interesting for fundamental research and offer potential for applications in classical and quantum information technologies. In the past 10 years, new material systems have emerged for polariton lasers, such as organic molecules, transition metal dichalcogenides, perovskites and liquid-crystal microcavities. In this Review, we discuss these emerging platforms in the context of applications in topological lasing, classical neuromorphic computing and quantum information processing.

Key points

  • Polariton lasers are coherent light emitters based on bosonic condensates of half-light, half-matter quasiparticles: exciton–polaritons.

  • Nowadays, polariton lasers with either optical or electronic injection are realized in a wide variety of organic, hybrid and inorganic systems, including two-dimensional crystals.

  • Engineering of spin–orbit coupling in polariton condensates led to the development of polariton topological insulators and lasers.

  • Phase locking in arrays of polariton condensates in planar microcavities may be used for the realization of ultrafast simulators.

  • Multistability of quasiresonantly pumped polariton condensates allows for realization of polariton neurons that pass information by means of the motion of domain walls.

  • Polariton qubits based on superposition of polariton superfluids with different orbital momenta are promising because of their high scalability and optical control.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Polariton laser structures.
Fig. 2: Topological polaritonics.
Fig. 3: Polaritonic neuromorphic computing.
Fig. 4: Polariton qubit based on a split-ring condensate.

Similar content being viewed by others

Change history

References

  1. Askitopoulos, A. et al. Robust platform for engineering pure-quantum-state transitions in polariton condensates. Phys. Rev. B 92, 035305 (2015).

    Article  ADS  Google Scholar 

  2. Ballarini, D. et al. Polaritonic neuromorphic computing outperforms linear classifiers. Nano Lett. 20, 3506–3512 (2020).

    Article  ADS  Google Scholar 

  3. Imamoğlu, A., Ram, R. J., Pau, S. & Yamamoto, Y. Nonequilibrium condensates and lasers without inversion: exciton–polariton lasers. Phys. Rev. A 53, 4250–4253 (1996).

    Article  ADS  Google Scholar 

  4. Richard, M. et al. Experimental evidence for nonequilibrium Bose condensation of exciton polaritons. Phys. Rev. B 72, 201301 (2005).

    Article  ADS  Google Scholar 

  5. Kasprzak, J. et al. Bose–Einstein condensation of exciton polaritons. Nature 443, 409–414 (2006).

    Article  ADS  Google Scholar 

  6. Dang, L. S., Heger, D., André, R., Bœuf, F. & Romestain, R. Stimulation of polariton photoluminescence in semiconductor microcavity. Phys. Rev. Lett. 81, 3920–3923 (1998).

    Article  ADS  Google Scholar 

  7. Deng, H., Weihs, G., Santori, C., Bloch, J. & Yamamoto, Y. Condensation of semiconductor microcavity exciton polaritons. Science 298, 199–202 (2002).

    Article  ADS  Google Scholar 

  8. Deng, H., Solomon, G., Hey, R., Ploog, K. & Yamamoto, Y. Spatial coherence of a polariton condensate. Phys. Rev. Lett. 99, 126403 (2007).

    Article  ADS  Google Scholar 

  9. Balili, R., Hartwell, V., Snoke, D., Pfeiffer, L. & West, K. Bose–Einstein condensation of microcavity polaritons in a trap. Science 316, 1007–1010 (2007).

    Article  ADS  Google Scholar 

  10. Lai, C. et al. Coherent zero-state and π-state in an exciton–polariton condensate array. Nature 450, 529–532 (2007).

    Article  ADS  Google Scholar 

  11. Carusotto, I. & Ciuti, C. Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013).

    Article  ADS  Google Scholar 

  12. Tassone, F., Piermarocchi, C., Savona, V., Quattropani, A. & Schwendimann, P. Bottleneck effects in the relaxation and photoluminescence of microcavity polaritons. Phys. Rev. B 56, 7554–7563 (1997).

    Article  ADS  Google Scholar 

  13. Maragkou, M., Grundy, A., Ostatnický, T. & Lagoudakis, P. Longitudinal optical phonon assisted polariton laser. Appl. Phys. Lett. 97, 111110 (2010).

    Article  ADS  Google Scholar 

  14. Porras, D., Ciuti, C., Baumberg, J. J. & Tejedor, C. Polariton dynamics and Bose–Einstein condensation in semiconductor microcavities. Phys. Rev. B 66, 085304 (2002).

    Article  ADS  Google Scholar 

  15. Liew, T. C. H., Flayac, H., Poletti, D., Savenko, I. G. & Laussy, F. P. Kinetic Monte Carlo approach to nonequilibrium bosonic systems. Phys. Rev. B 96, 125423 (2017).

    Article  ADS  Google Scholar 

  16. Christopoulos, S. et al. Room-temperature polariton lasing in semiconductor microcavities. Phys. Rev. Lett. 98, 126405 (2007).

    Article  ADS  Google Scholar 

  17. Daskalakis, K. S. et al. All-dielectric GaN microcavity: strong coupling and lasing at room temperature. Appl. Phys. Lett. 102, 101113 (2013).

    Article  ADS  Google Scholar 

  18. Bajoni, D. et al. Polariton laser using single micropillar GaAs−GaAlAs semiconductor cavities. Phys. Rev. Lett. 100, 047401 (2008).

    Article  ADS  Google Scholar 

  19. Azzini, S. et al. Ultra-low threshold polariton lasing in photonic crystal cavities. Appl. Phys. Lett. 99, 111106 (2011).

    Article  ADS  Google Scholar 

  20. Sanvitto, D. & Kéna-Cohen, S. The road towards polaritonic devices. Nat. Mater. 15, 1061–1073 (2016).

    Article  ADS  Google Scholar 

  21. Christmann, G., Butté, R., Feltin, E., Carlin, J. & Grandjean, N. Room temperature polariton lasing in a GaN/AlGaN multiple quantum well microcavity. Appl. Phys. Lett. 93, 051102 (2008).

    Article  ADS  Google Scholar 

  22. Xie, W. et al. Room-temperature polariton parametric scattering driven by a one-dimensional polariton condensate. Phys. Rev. Lett. 108, 166401 (2012).

    Article  ADS  Google Scholar 

  23. Duan, Q. et al. Polariton lasing of quasi-whispering gallery modes in a ZnO microwire. Appl. Phys. Lett. 103, 022103 (2013).

    Article  ADS  Google Scholar 

  24. Li, F. et al. From excitonic to photonic polariton condensate in a ZnO-based microcavity. Phys. Rev. Lett. 110, 196406 (2013).

    Article  ADS  Google Scholar 

  25. Kéna-Cohen, S. & Forrest, S. R. Room-temperature polariton lasing in an organic single-crystal microcavity. Nat. Photon. 4, 371–375 (2010).

    Article  ADS  Google Scholar 

  26. Plumhof, J., Stöferle, T., Mai, L., Scherf, U. & Mahrt, R. Room-temperature Bose–Einstein condensation of cavity exciton–polaritons in a polymer. Nat. Mater. 13, 247–252 (2014).

    Article  ADS  Google Scholar 

  27. Su, R. et al. Room-temperature polariton lasing in all-inorganic perovskite nanoplatelets. Nano Lett. 17, 3982–3988 (2017).

    Article  ADS  Google Scholar 

  28. Su, R. et al. Direct measurement of a non-Hermitian topological invariant in a hybrid light–matter system. Sci. Adv. 7, eabj8905 (2021).

    Article  Google Scholar 

  29. Shan, H. et al. Spatial coherence of room-temperature monolayer WSe2 exciton–polaritons in a trap. Nat. Commun. 12, 6406 (2021).

    Article  ADS  Google Scholar 

  30. Zhao, J. et al. Ultralow threshold polariton condensate in a monolayer semiconductor microcavity at room temperature. Nano Lett. 21, 3331–3339 (2021).

    Article  ADS  Google Scholar 

  31. Daskalakis, K., Maier, S., Murray, R. & Kéna-Cohen, S. Nonlinear interactions in an organic polariton condensate. Nat. Mater. 13, 271–278 (2014).

    Article  ADS  Google Scholar 

  32. Christmann, G. et al. Impact of disorder on high quality factor III–V nitride microcavities. Appl. Phys. Lett. 89, 261101 (2006).

    Article  ADS  Google Scholar 

  33. Tischler, J., Bradley, M., Bulović, V., Song, J. & Nurmikko, A. Strong coupling in a microcavity LED. Phys. Rev. Lett. 95, 036401 (2005).

    Article  ADS  Google Scholar 

  34. Khalifa, A. A., Love, A. P. D., Krizhanovskii, D., Skolnick, M. S. & Roberts, J. S. Electroluminescence emission from polariton states in GaAs-based semiconductor microcavities. Appl. Phys. Lett. 92, 061107 (2008).

    Article  ADS  Google Scholar 

  35. Tsintzos, S. I., Pelekanos, N. T., Konstantinidis, G., Hatzopoulos, Z. & Savvidis, P. G. A GaAs polariton light-emitting diode operating near room temperature. Nature 453, 372–375 (2008).

    Article  ADS  Google Scholar 

  36. Bajoni, D. et al. Polariton light-emitting diode in a GaAs-based microcavity. Phys. Rev. B 77, 113303 (2008).

    Article  ADS  Google Scholar 

  37. Schneider, C. et al. An electrically pumped polariton laser. Nature 497, 348–352 (2013).

    Article  ADS  Google Scholar 

  38. Bhattacharya, P., Xiao, B., Das, A., Bhowmick, S. & Heo, J. Solid state electrically injected exciton–polariton laser. Phys. Rev. Lett. 110, 206403 (2013).

    Article  ADS  Google Scholar 

  39. Suchomel, H. et al. Spatio-temporal coherence in vertically emitting GaAs-based electrically driven polariton lasers. Appl. Phys. Lett. 116, 171103 (2020).

    Article  ADS  Google Scholar 

  40. Bhattacharya, P. et al. Room temperature electrically injected polariton laser. Phys. Rev. Lett. 112, 236802 (2014).

    Article  ADS  Google Scholar 

  41. Schneider, C. et al. Exciton–polariton trapping and potential landscape engineering. Rep. Prog. Phys. 80, 016503 (2016).

    Article  ADS  Google Scholar 

  42. Amthor, M. et al. Electro-optical switching between polariton and cavity lasing in an InGaAs quantum well microcavity. Opt. Express 22, 31146–31153 (2014).

    Article  ADS  Google Scholar 

  43. Tsotsis, P. et al. Tuning the energy of a polariton condensate via bias-controlled rabi splitting. Phys. Rev. Appl. 2, 014002 (2014).

    Article  ADS  Google Scholar 

  44. Brodbeck, S. et al. Impact of lateral carrier confinement on electro-optical tuning properties of polariton condensates. Appl. Phys. Lett. 107, 041108 (2015).

    Article  ADS  Google Scholar 

  45. Mak, K. F., Lee, C., Hone, J., Shan, J. & Heinz, T. F. Atomically thin MoS2: a new direct-gap semiconductor. Phys. Rev. Lett. 105, 136805 (2010).

    Article  ADS  Google Scholar 

  46. Mak, K. F. & Shan, J. Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides. Nat. Photon. 10, 216–226 (2016).

    Article  ADS  Google Scholar 

  47. Chernikov, A. et al. Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2. Phys. Rev. Lett. 113, 076802 (2014).

    Article  ADS  Google Scholar 

  48. Wang, G. et al. Colloquium: excitons in atomically thin transition metal dichalcogenides. Rev. Mod. Phys. 90, 021001 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  49. Liu, X. et al. Strong light–matter coupling in two-dimensional atomic crystals. Nat. Photon. 9, 30–34 (2015).

    Article  ADS  Google Scholar 

  50. Dufferwiel, S. et al. Exciton–polaritons in van der Waals heterostructures embedded in tunable microcavities. Nat. Commun. 6, 8579 (2015).

    Article  ADS  Google Scholar 

  51. Lundt, N. et al. Room-temperature Tamm-plasmon exciton–polaritons with a WSe2 monolayer. Nat. Commun. 7, 13328 (2016).

    Article  ADS  Google Scholar 

  52. Zhang, L., Gogna, R., Burg, W., Tutuc, E. & Deng, H. Photonic-crystal exciton–polaritons in monolayer semiconductors. Nat. Commun. 9, 713 (2018).

    Article  ADS  Google Scholar 

  53. Liu, W. et al. Strong exciton–plasmon coupling in MoS2 coupled with plasmonic lattice. Nano Lett. 16, 1262–1269 (2016).

    Article  ADS  Google Scholar 

  54. Schneider, C., Glazov, M. M., Korn, T., Höfling, S. & Urbaszek, B. Two-dimensional semiconductors in the regime of strong light–matter coupling. Nat. Commun. 9, 2695 (2018).

    Article  ADS  Google Scholar 

  55. Basov, D. N., Asenjo-Garcia, A., Schuck, P. J., Zhu, X. & Rubio, A. Polariton panorama. Nanophotonics 10, 549–577 (2021).

    Article  Google Scholar 

  56. Gu, J., Chakraborty, B., Khatoniar, M. & Menon, V. M. A room-temperature polariton light-emitting diode based on monolayer WS2. Nat. Nanotechnol. 14, 1024–1028 (2019).

    Article  ADS  Google Scholar 

  57. Mak, K. F., Xiao, D. & Shan, J. Light–valley interactions in 2D semiconductors. Nat. Photon. 12, 451–460 (2018).

    Article  ADS  Google Scholar 

  58. Lundt, N. et al. Valley polarized relaxation and upconversion luminescence from Tamm-plasmon trion–polaritons with a MoSe2 monolayer. 2D Mater. 4, 025096 (2017).

    Article  Google Scholar 

  59. Dufferwiel, S. et al. Valley-addressable polaritons in atomically thin semiconductors. Nat. Photon. 11, 497–501 (2017).

    Article  Google Scholar 

  60. Sun, Z. et al. Optical control of room-temperature valley polaritons. Nat. Photon. 11, 491–496 (2017).

    Article  Google Scholar 

  61. Chen, Y.-J., Cain, J. D., Stanev, T. K., Dravid, V. P. & Stern, N. P. Valley-polarized exciton–polaritons in a monolayer semiconductor. Nat. Photon. 11, 431–435 (2017).

    Article  ADS  Google Scholar 

  62. Lundt, N. et al. Optical valley Hall effect for highly valley-coherent exciton–polaritons in an atomically thin semiconductor. Nat. Nanotechnol. 14, 770–775 (2019).

    Article  ADS  Google Scholar 

  63. Kavokin, A., Malpuech, G. & Glazov, M. Optical spin Hall effect. Phys. Rev. Lett. 95, 136601 (2005).

    Article  ADS  Google Scholar 

  64. Onga, M., Zhang, Y., Ideue, T. & Iwasa, Y. Exciton Hall effect in monolayer MoS2. Nat. Mater. 16, 1193–1197 (2017).

    Article  ADS  Google Scholar 

  65. Wurdack, M. et al. Motional narrowing, ballistic transport, and trapping of room-temperature exciton polaritons in an atomically-thin semiconductor. Nat. Commun. 12, 5366 (2021).

    Article  ADS  Google Scholar 

  66. Shahnazaryan, V., Iorsh, I., Shelykh, I. A. & Kyriienko, O. Exciton-exciton interaction in transition-metal dichalcogenide monolayers. Phys. Rev. B 96, 115409 (2017).

    Article  ADS  Google Scholar 

  67. Stepanov, P. et al. Exciton-exciton interaction beyond the hydrogenic picture in a MoSe2 monolayer in the strong light-matter coupling regime. Phys. Rev. Lett. 126, 167401 (2021).

    Article  ADS  Google Scholar 

  68. Gu, J. et al. Enhanced nonlinear interaction of polaritons via excitonic Rydberg states in monolayer WSe2. Nat. Commun. 12, 2269 (2021).

    Article  ADS  Google Scholar 

  69. Emmanuele, R. P. A. et al. Highly nonlinear trion-polaritons in a monolayer semiconductor. Nat. Commun. 11, 3589 (2020).

    Article  ADS  Google Scholar 

  70. Tan, L. B. et al. Interacting polaron-polaritons. Phys. Rev. X 10, 021011 (2020).

    Google Scholar 

  71. Zhang, L. et al. Van der Waals heterostructure polaritons with moiré-induced nonlinearity. Nature 591, 61–65 (2021).

    Article  ADS  Google Scholar 

  72. Cadiz, F. et al. Excitonic linewidth approaching the homogeneous limit in MoS2-based van der Waals heterostructures. Phys. Rev. X 7, 021026 (2017).

    Google Scholar 

  73. Fang, H. H. et al. Control of the exciton radiative lifetime in van der Waals heterostructures. Phys. Rev. Lett. 123, 067401 (2019).

    Article  ADS  Google Scholar 

  74. Waldherr, M. et al. Observation of bosonic condensation in a hybrid monolayer MoSe2-GaAs microcavity. Nat. Commun. 9, 3286 (2018).

    Article  ADS  Google Scholar 

  75. Anton-Solanas, C. et al. Bosonic condensation of exciton–polaritons in an atomically thin crystal. Nat. Mater. 20, 1233–1239 (2021).

    Article  ADS  Google Scholar 

  76. Lidzey, D. G. et al. Strong exciton–photon coupling in an organic semiconductor microcavity. Nature 395, 53–55 (1998).

    Article  ADS  Google Scholar 

  77. Wenus, J. et al. Optical strong coupling in microcavities containing J-aggregates absorbing in near-infrared spectral range. Org. Electron. 8, 120–126 (2007).

    Article  Google Scholar 

  78. Gambino, S. et al. Exploring light–matter interaction phenomena under ultrastrong coupling regime. ACS Photonics 1, 1042–1048 (2014).

    Article  Google Scholar 

  79. Dietrich, C. P. et al. An exciton–polariton laser based on biologically produced fluorescent protein. Sci. Adv. 2, e1600666 (2016).

    Article  ADS  Google Scholar 

  80. Betzold, S. et al. Coherence and interaction in confined room-temperature polariton condensates with Frenkel excitons. ACS Photonics 7, 384–392 (2019).

    Article  ADS  Google Scholar 

  81. Dusel, M. et al. Room temperature organic exciton–polariton condensate in a lattice. Nat. Commun. 11, 2863 (2020).

    Article  ADS  Google Scholar 

  82. Dusel, M. et al. Room-temperature topological polariton laser in an organic lattice. Nano Lett. 21, 6398–6405 (2021).

    Article  ADS  Google Scholar 

  83. Zasedatelev, A. V. et al. A room-temperature organic polariton transistor. Nat. Photon. 13, 378–383 (2019).

    Article  ADS  Google Scholar 

  84. Zasedatelev, A. V. et al. Single-photon nonlinearity at room temperature. Nature 597, 493–497 (2021).

    Article  ADS  Google Scholar 

  85. Brehier, A., Parashkov, R., Lauret, J.-S. & Deleporte, E. Strong exciton-photon coupling in a microcavity containing layered perovskite semiconductors. Appl. Phys. Lett. 89, 171110 (2006).

    Article  ADS  Google Scholar 

  86. Lanty, G. et al. Hybrid cavity polaritons in a ZnO-perovskite microcavity. Phys. Rev. B 84, 195449 (2011).

    Article  ADS  Google Scholar 

  87. Evans, T. J. et al. Continuous-wave lasing in cesium lead bromide perovskite nanowires. Adv. Opt. Mater. 6, 1700982 (2018).

    Article  Google Scholar 

  88. Du, W. et al. Strong exciton–photon coupling and lasing behavior in all-inorganic CsPbBr3 micro/nanowire Fabry-Pérot cavity. ACS Photonics 5, 2051–2059 (2018).

    Article  Google Scholar 

  89. Wang, J. et al. Lasing from lead halide perovskite semiconductor microcavity system. Nanoscale 10, 10371–10376 (2018).

    Article  Google Scholar 

  90. Park, K. et al. Light–matter interactions in cesium lead halide perovskite nanowire lasers. J. Phys. Chem. Lett. 7, 3703–3710 (2016).

    Article  Google Scholar 

  91. Su, R. et al. Room temperature long-range coherent exciton polariton condensate flow in lead halide perovskites. Sci. Adv. 4, eaau0244 (2018).

    Article  ADS  Google Scholar 

  92. Shang, Q. et al. Surface plasmon enhanced strong exciton–photon coupling in hybrid inorganic–organic perovskite nanowires. Nano Lett. 18, 3335–3343 (2018).

    Article  ADS  Google Scholar 

  93. Wang, J. et al. Spontaneously coherent orbital coupling of counterrotating exciton polaritons in annular perovskite microcavities. Light Sci. Appl. 10, 45 (2021).

    Article  ADS  Google Scholar 

  94. Su, R. et al. Observation of exciton polariton condensation in a perovskite lattice at room temperature. Nat. Phys. 16, 301–306 (2020).

    Article  Google Scholar 

  95. Bouteyre, P. et al. Room-temperature cavity polaritons with 3D hybrid perovskite: toward large-surface polaritonic devices. ACS Photonics 6, 1804–1811 (2019).

    Article  Google Scholar 

  96. Fieramosca, A. et al. Two-dimensional hybrid perovskites sustaining strong polariton interactions at room temperature. Sci. Adv. 5, eaav9967 (2019).

    Article  ADS  Google Scholar 

  97. Wu, J. et al. Nonlinear parametric scattering of exciton polaritons in perovskite microcavities. Nano Lett. 21, 3120–3126 (2021).

    Article  ADS  Google Scholar 

  98. Wu, J. et al. Perovskite polariton parametric oscillator. Adv. Photon. 3, 055003 (2021).

    Article  ADS  Google Scholar 

  99. Kim, J. Y., Lee, J.-W., Jung, H. S., Shin, H. & Park, N.-G. High-efficiency perovskite solar cells. Chem. Rev. 120, 7867–7918 (2020).

    Article  Google Scholar 

  100. Lekenta, K. et al. Tunable optical spin Hall effect in a liquid crystal microcavity. Light Sci. Appl. 7, 74 (2018).

    Article  ADS  Google Scholar 

  101. Rechcińska, K. et al. Engineering spin-orbit synthetic Hamiltonians in liquid-crystal optical cavities. Science 366, 727–730 (2019).

    Article  ADS  Google Scholar 

  102. Król, M. et al. Observation of second-order meron polarization textures in optical microcavities. Optica 8, 255–261 (2021).

    Article  ADS  Google Scholar 

  103. Król, M. et al. Realizing optical persistent spin helix and Stern-Gerlach deflection in an anisotropic liquid crystal microcavity. Phys. Rev. Lett. 127, 190401 (2021).

    Article  ADS  Google Scholar 

  104. Kokhanchik, P., Sigurdsson, H., Piętka, B., Szczytko, J. & Lagoudakis, P. G. Photonic Berry curvature in double liquid crystal microcavities with broken inversion symmetry. Phys. Rev. B 103, L081406 (2021).

    Article  ADS  Google Scholar 

  105. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  ADS  Google Scholar 

  106. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).

    Article  ADS  Google Scholar 

  107. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    Article  ADS  Google Scholar 

  108. König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).

    Article  ADS  Google Scholar 

  109. Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).

    Article  ADS  Google Scholar 

  110. Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011).

    Article  Google Scholar 

  111. Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  112. Ota, Y. et al. Active topological photonics. Nanophotonics 9, 547–567 (2020).

    Article  Google Scholar 

  113. Karzig, T., Bardyn, C.-E., Lindner, N. H. & Refael, G. Topological polaritons. Phys. Rev. X 5, 031001 (2015).

    Google Scholar 

  114. Bardyn, C.-E., Karzig, T., Refael, G. & Liew, T. C. H. Topological polaritons and excitons in garden-variety systems. Phys. Rev. B 91, 161413 (2015).

    Article  ADS  Google Scholar 

  115. Nalitov, A. V., Solnyshkov, D. D. & Malpuech, G. Polariton \({\mathbb{Z}}\) topological insulator. Phys. Rev. Lett. 114, 116401 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  116. Jacqmin, T. et al. Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons. Phys. Rev. Lett. 112, 116402 (2014).

    Article  ADS  Google Scholar 

  117. Amo, A. & Bloch, J. Exciton–polaritons in lattices: a non-linear photonic simulator. C. R. Phys. 17, 934–945 (2016).

    Article  ADS  Google Scholar 

  118. Suchomel, H. et al. Platform for electrically pumped polariton simulators and topological lasers. Phys. Rev. Lett. 121, 257402 (2018).

    Article  ADS  Google Scholar 

  119. Solnyshkov, D. D., Nalitov, A. V. & Malpuech, G. Kibble–Zurek mechanism in topologically nontrivial zigzag chains of polariton micropillars. Phys. Rev. Lett. 116, 046402 (2016).

    Article  ADS  Google Scholar 

  120. Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989).

    Article  ADS  Google Scholar 

  121. Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979).

    Article  ADS  Google Scholar 

  122. St-Jean, P. et al. Lasing in topological edge states of a one-dimensional lattice. Nat. Photon. 11, 651–656 (2017).

    Article  ADS  Google Scholar 

  123. Harder, T. H. et al. Coherent topological polariton laser. ACS Photonics 8, 1377–1384 (2021).

    Article  Google Scholar 

  124. Yi, K. & Karzig, T. Topological polaritons from photonic Dirac cones coupled to excitons in a magnetic field. Phys. Rev. B 93, 104303 (2016).

    Article  ADS  Google Scholar 

  125. Zhang, Y., Kartashov, Y. V., Zhang, Y., Torner, L. & Skryabin, D. V. Inhibition of tunneling and edge state control in polariton topological insulators. APL Photonics 3, 120801 (2018).

    Article  ADS  Google Scholar 

  126. Li, C. et al. Lieb polariton topological insulators. Phys. Rev. B 97, 081103 (2018).

    Article  ADS  Google Scholar 

  127. Sun, M., Ko, D., Leykam, D., Kovalev, V. M. & Savenko, I. G. Exciton–polariton topological insulator with an array of magnetic dots. Phys. Rev. Appl. 12, 064028 (2019).

    Article  ADS  Google Scholar 

  128. Hofmann, D. & Sentef, M. A. Resonant laser excitation and time-domain imaging of chiral topological polariton edge states. Phys. Rev. Res. 2, 033386 (2020).

    Article  Google Scholar 

  129. Klembt, S. et al. Exciton–polariton topological insulator. Nature 562, 552–556 (2018).

    Article  ADS  Google Scholar 

  130. Bahari, B. et al. Nonreciprocal lasing in topological cavities of arbitrary geometries. Science 358, 636–640 (2017).

    Article  ADS  Google Scholar 

  131. Harari, G. et al. Topological insulator laser: theory. Science 359, eaar4003 (2018).

    Article  Google Scholar 

  132. Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).

    Article  Google Scholar 

  133. Dikopoltsev, A. et al. Topological insulator vertical-cavity laser array. Science 373, 1514–1517 (2021).

    Article  ADS  Google Scholar 

  134. Amelio, I. & Carusotto, I. Theory of the coherence of topological lasers. Phys. Rev. X 10, 041060 (2020).

    Google Scholar 

  135. Kartashov, Y. V. & Skryabin, D. V. Two-dimensional topological polariton laser. Phys. Rev. Lett. 122, 083902 (2019).

    Article  ADS  Google Scholar 

  136. Kartashov, Y. V. & Skryabin, D. V. Modulational instability and solitary waves in polariton topological insulators. Optica 3, 1228–1236 (2016).

    Article  ADS  Google Scholar 

  137. Gulevich, D. R., Yudin, D., Skryabin, D. V., Iorsh, I. V. & Shelykh, I. A. Exploring nonlinear topological states of matter with exciton–polaritons: edge solitons in kagome lattice. Sci. Rep. 7, 1780 (2017).

    Article  ADS  Google Scholar 

  138. Bardyn, C.-E., Karzig, T., Refael, G. & Liew, T. C. H. Chiral Bogoliubov excitations in nonlinear bosonic systems. Phys. Rev. B 93, 020502 (2016).

    Article  ADS  Google Scholar 

  139. Mandal, S., Ge, R. & Liew, T. C. H. Antichiral edge states in an exciton polariton strip. Phys. Rev. B 99, 115423 (2019).

    Article  ADS  Google Scholar 

  140. Banerjee, R., Mandal, S. & Liew, T. C. H. Optically induced topological spin-valley Hall effect for exciton polaritons. Phys. Rev. B 103, L201406 (2021).

    Article  ADS  Google Scholar 

  141. Sigurdsson, H., Li, G. & Liew, T. C. H. Spontaneous and superfluid chiral edge states in exciton–polariton condensates. Phys. Rev. B 96, 115453 (2017).

    Article  ADS  Google Scholar 

  142. Banerjee, R., Mandal, S. & Liew, T. C. H. Coupling between exciton–polariton corner modes through edge states. Phys. Rev. Lett. 124, 063901 (2020).

    Article  ADS  Google Scholar 

  143. Liu, W. et al. Generation of helical topological exciton–polaritons. Science 370, 600–604 (2020).

    Article  MathSciNet  Google Scholar 

  144. Wu, L.-H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015).

    Article  ADS  Google Scholar 

  145. Li, M. et al. Experimental observation of topological Z2 exciton–polaritons in transition metal dichalcogenide monolayers. Nat. Commun. 12, 4425 (2021).

    Article  ADS  Google Scholar 

  146. Lackner, L. et al. Tunable exciton–polaritons emerging from WS2 monolayer excitons in a photonic lattice at room temperature. Nat. Commun. 12, 4933 (2021).

    Article  ADS  Google Scholar 

  147. Pickup, L., Sigurdsson, H., Ruostekoski, J. & Lagoudakis, P. G. Synthetic band-structure engineering in polariton crystals with non-Hermitian topological phases. Nat. Commun. 11, 4431 (2020).

    Article  ADS  Google Scholar 

  148. Pieczarka, M. et al. Topological phase transition in an all-optical exciton–polariton lattice. Optica 8, 1084–1091 (2021).

    Article  ADS  Google Scholar 

  149. El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).

    Article  Google Scholar 

  150. Gao, T. et al. Observation of non-Hermitian degeneracies in a chaotic exciton–polariton billiard. Nature 526, 554–558 (2015).

    Article  ADS  Google Scholar 

  151. Gao, W., Li, X., Bamba, M. & Kono, J. Continuous transition between weak and ultrastrong coupling through exceptional points in carbon nanotube microcavity exciton–polaritons. Nat. Photon. 12, 362–367 (2018).

    Article  ADS  Google Scholar 

  152. Khurgin, J. B. Exceptional points in polaritonic cavities and subthreshold Fabry–Perot lasers. Optica 7, 1015–1023 (2020).

    Article  ADS  Google Scholar 

  153. Comaron, P., Shahnazaryan, V., Brzezicki, W., Hyart, T. & Matuszewski, M. Non-Hermitian topological end-mode lasing in polariton systems. Phys. Rev. Res. 2, 022051 (2020).

    Article  Google Scholar 

  154. Mandal, S., Banerjee, R., Ostrovskaya, E. A. & Liew, T. C. H. Nonreciprocal transport of exciton polaritons in a non-Hermitian chain. Phys. Rev. Lett. 125, 123902 (2020).

    Article  ADS  Google Scholar 

  155. Hatano, N. & Nelson, D. R. Localization transitions in non-Hermitian quantum mechanics. Phys. Rev. Lett. 77, 570–573 (1996).

    Article  ADS  Google Scholar 

  156. Xu, X. et al. Interaction-induced double-sided skin effect in an exciton–polariton system. Phys. Rev. B 103, 235306 (2021).

    Article  ADS  Google Scholar 

  157. Sich, M. et al. Observation of bright polariton solitons in a semiconductor microcavity. Nat. Photon. 6, 50–55 (2012).

    Article  ADS  Google Scholar 

  158. Tanese, D. et al. Polariton condensation in solitonic gap states in a one-dimensional periodic potential. Nat. Commun. 4, 1749 (2013).

    Article  ADS  Google Scholar 

  159. Cristofolini, P. et al. Optical superfluid phase transitions and trapping of polariton condensates. Phys. Rev. Lett. 110, 186403 (2013).

    Article  ADS  Google Scholar 

  160. Sich, M. et al. Effects of spin-dependent interactions on polarization of bright polariton solitons. Phys. Rev. Lett. 112, 046403 (2014).

    Article  ADS  Google Scholar 

  161. Pinsker, F. & Flayac, H. On-demand dark soliton train manipulation in a spinor polariton condensate. Phys. Rev. Lett. 112, 140405 (2014).

    Article  ADS  Google Scholar 

  162. De Giorgi, M. et al. Control and ultrafast dynamics of a two-fluid polariton switch. Phys. Rev. Lett. 109, 266407 (2012).

    Article  ADS  Google Scholar 

  163. Ballarini, D. et al. All-optical polariton transistor. Nat. Commun. 4, 1778 (2013).

    Article  ADS  Google Scholar 

  164. Cerna, R. et al. Ultrafast tristable spin memory of a coherent polariton gas. Nat. Commun. 4, 2008 (2013).

    Article  ADS  Google Scholar 

  165. Mirek, R. et al. Neuromorphic binarized polariton networks. Nano Lett. 21, 3715–3720 (2021).

    Article  ADS  Google Scholar 

  166. Dreismann, A. et al. A sub-femtojoule electrical spin-switch based on optically trapped polariton condensates. Nat. Mater. 15, 1074–1078 (2016).

    Article  ADS  Google Scholar 

  167. Zasedatelev, A. V. et al. A room-temperature organic polariton transistor. Nat. Photon. 13, 378–383 (2019).

    Article  ADS  Google Scholar 

  168. Wertz, E. et al. Propagation and amplification dynamics of 1D polariton condensates. Phys. Rev. Lett. 109, 216404 (2012).

    Article  ADS  Google Scholar 

  169. Liao, L. et al. Propagation of a polariton condensate in a one-dimensional microwire at room temperature. Appl. Phys. Express 12, 052009 (2019).

    Article  ADS  Google Scholar 

  170. Lerario, G. et al. High-speed flow of interacting organic polaritons. Light Sci. Appl. 6, e16212 (2017).

    Article  Google Scholar 

  171. Marković, D., Mizrahi, A., Querlioz, D. & Grollier, J. Physics for neuromorphic computing. Nat. Rev. Phys. 2, 499–510 (2020).

    Article  Google Scholar 

  172. Zhu, J., Zhang, T., Yang, Y. & Huang, R. A comprehensive review on emerging artificial neuromorphic devices. Appl. Phys. Rev. 7, 011312 (2020).

    Article  ADS  Google Scholar 

  173. Song, S. et al. Recent progress of optoelectronic and all-optical neuromorphic devices: a comprehensive review of device structures, materials, and applications. Adv. Intell. Syst. 3, 2000119 (2021).

    Article  Google Scholar 

  174. Schuman, C. D. et al. A survey of neuromorphic computing and neural networks in hardware. Preprint at https://arxiv.org/abs/1705.06963 (2017).

  175. Liew, T. C. H., Kavokin, A. V. & Shelykh, I. A. Optical circuits based on polariton neurons in semiconductor microcavities. Phys. Rev. Lett. 101, 016402 (2008).

    Article  ADS  Google Scholar 

  176. Baas, A., Karr, J. P., Eleuch, H. & Giacobino, E. Optical bistability in semiconductor microcavities. Phys. Rev. A 69, 023809 (2004).

    Article  ADS  Google Scholar 

  177. Whittaker, D. M. Effects of polariton-energy renormalization in the microcavity optical parametric oscillator. Phys. Rev. B 71, 115301 (2005).

    Article  ADS  Google Scholar 

  178. Espinosa-Ortega, T. & Liew, T. C. H. Complete architecture of integrated photonic circuits based on and and not logic gates of exciton polaritons in semiconductor microcavities. Phys. Rev. B 87, 195305 (2013).

    Article  ADS  Google Scholar 

  179. Koniakhin, S. V. et al. Stationary quantum vortex street in a driven-dissipative quantum fluid of light. Phys. Rev. Lett. 123, 215301 (2019).

    Article  ADS  Google Scholar 

  180. Lerario, G. et al. Parallel dark-soliton pair in a bistable two-dimensional exciton–polariton superfluid. Phys. Rev. Res. 2, 042041 (2020).

    Article  Google Scholar 

  181. Espinosa-Ortega, T., Liew, T. C. H. & Shelykh, I. A. Optical diode based on exciton–polaritons. Appl. Phys. Lett. 103, 191110 (2013).

    Article  ADS  Google Scholar 

  182. Banerjee, R. & Liew, T. C. H. Artificial life in an exciton–polariton lattice. New J. Phys. 22, 103062 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  183. Byrnes, T., Koyama, S., Yan, K. & Yamamoto, Y. Neural networks using two-component Bose-Einstein condensates. Sci. Rep. 3, 2531 (2013).

    Article  ADS  Google Scholar 

  184. Espinosa-Ortega, T. & Liew, T. C. H. Perceptrons with Hebbian learning based on wave ensembles in spatially patterned potentials. Phys. Rev. Lett. 114, 118101 (2015).

    Article  ADS  Google Scholar 

  185. Montavon, G., Orr, G. & Müller, K.-R. Neural Networks: Tricks of the Trade (Springer, 2012).

  186. Kudithipudi, D., Saleh, Q., Merkel, C., Thesing, J. & Wysocki, B. Design and analysis of a neuromemristive reservoir computing architecture for biosignal processing. Front. Neurosci. 9, 502 (2016).

    Article  Google Scholar 

  187. Du, C. et al. Reservoir computing using dynamic memristors for temporal information processing. Nat. Commun. 8, 2204 (2017).

    Article  ADS  Google Scholar 

  188. Vandoorne, K. et al. Experimental demonstration of reservoir computing on a silicon photonics chip. Nat. Commun. 5, 3541 (2014).

    Article  ADS  Google Scholar 

  189. Duport, F., Schneider, B., Smerieri, A., Haelterman, M. & Massar, S. All-optical reservoir computing. Opt. Express 20, 22783–22795 (2012).

    Article  ADS  Google Scholar 

  190. Paquot, Y. et al. Optoelectronic reservoir computing. Sci. Rep. 2, 287 (2012).

    Article  Google Scholar 

  191. Larger, L. et al. Photonic information processing beyond turing: an optoelectronic implementation of reservoir computing. Opt. Express 20, 3241–3249 (2012).

    Article  ADS  Google Scholar 

  192. Brunner, D., Soriano, M. C., Mirasso, C. R. & Fischer, I. Parallel photonic information processing at gigabyte per second data rates using transient states. Nat. Commun. 4, 1364 (2013).

    Article  ADS  Google Scholar 

  193. Larger, L. et al. High-speed photonic reservoir computing using a time-delay-based architecture: million words per second classification. Phys. Rev. X 7, 011015 (2017).

    Google Scholar 

  194. Opala, A., Ghosh, S., Liew, T. C. & Matuszewski, M. Neuromorphic computing in Ginzburg–Landau polariton-lattice systems. Phys. Rev. Appl. 11, 064029 (2019).

    Article  ADS  Google Scholar 

  195. Matuszewski, M. et al. Energy-efficient neural network inference with microcavity exciton polaritons. Phys. Rev. Appl. 16, 024045 (2021).

    Article  ADS  Google Scholar 

  196. Fujii, K. & Nakajima, K. Harnessing disordered-ensemble quantum dynamics for machine learning. Phys. Rev. Appl. 8, 024030 (2017).

    Article  ADS  Google Scholar 

  197. Negoro, M., Mitarai, K., Fujii, K., Nakajima, K. & Kitagawa, M. Machine learning with controllable quantum dynamics of a nuclear spin ensemble in a solid. Preprint at https://arxiv.org/abs/1806.10910 (2018).

  198. Delteil, A. et al. Towards polariton blockade of confined exciton–polaritons. Nat. Mater. 18, 219–222 (2019).

    Article  Google Scholar 

  199. Muñoz-Matutano, G. et al. Emergence of quantum correlations from interacting fibre-cavity polaritons. Nat. Mater. 18, 213–218 (2019).

    Article  Google Scholar 

  200. Ghosh, S., Nakajima, K., Krisnanda, T., Fujii, K. & Liew, T. C. H. Quantum neuromorphic computing with reservoir computing networks. Adv. Quantum Technol. 4, 2100053 (2021).

    Article  Google Scholar 

  201. Bloch, J. et al. Strong-coupling regime in pillar semiconductor microcavities. Superlattices Microstruct. 22, 371–374 (1997).

    Article  ADS  Google Scholar 

  202. Kaitouni, R. I. et al. Engineering the spatial confinement of exciton polaritons in semiconductors. Phys. Rev. B 74, 155311 (2006).

    Article  ADS  Google Scholar 

  203. Cerda-Méndez, E. A. et al. Polariton condensation in dynamic acoustic lattices. Phys. Rev. Lett. 105, 116402 (2010).

    Article  ADS  Google Scholar 

  204. Masumoto, N. et al. Exciton–polariton condensates with flat bands in a two-dimensional kagome lattice. New J. Phys. 14, 065002 (2012).

    Article  ADS  Google Scholar 

  205. Kim, N. Y. et al. Exciton–polariton condensates near the Dirac point in a triangular lattice. New J. Phys. 15, 035032 (2013).

    Article  ADS  Google Scholar 

  206. Kusudo, K. et al. Stochastic formation of polariton condensates in two degenerate orbital states. Phys. Rev. B 87, 214503 (2013).

    Article  ADS  Google Scholar 

  207. Cerda-Méndez, E. A. et al. Exciton–polariton gap solitons in two-dimensional lattices. Phys. Rev. Lett. 111, 146401 (2013).

    Article  ADS  Google Scholar 

  208. Milićević, M. et al. Edge states in polariton honeycomb lattices. 2D Mater. 2, 034012 (2015).

    Article  Google Scholar 

  209. Tanese, D. et al. Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential. Phys. Rev. Lett. 112, 146404 (2014).

    Article  ADS  Google Scholar 

  210. Baboux, F. et al. Measuring topological invariants from generalized edge states in polaritonic quasicrystals. Phys. Rev. B 95, 161114 (2017).

    Article  ADS  Google Scholar 

  211. Baboux, F. et al. Bosonic condensation and disorder-induced localization in a flat band. Phys. Rev. Lett. 116, 066402 (2016).

    Article  ADS  Google Scholar 

  212. Winkler, K. et al. Collective state transitions of exciton–polaritons loaded into a periodic potential. Phys. Rev. B 93, 121303 (2016).

    Article  ADS  Google Scholar 

  213. Sala, V. G. et al. Spin-orbit coupling for photons and polaritons in microstructures. Phys. Rev. X 5, 011034 (2015).

    Google Scholar 

  214. Goblot, V. et al. Nonlinear polariton fluids in a flatband reveal discrete gap solitons. Phys. Rev. Lett. 123, 113901 (2019).

    Article  ADS  Google Scholar 

  215. Zhang, L. et al. Weak lasing in one-dimensional polariton superlattices. Proc. Natl Acad. Sci. USA 112, E1516–E1519 (2015).

    Google Scholar 

  216. Rodriguez, S. R. K. et al. Interaction-induced hopping phase in driven-dissipative coupled photonic microcavities. Nat. Commun. 7, 11887 (2016).

    Article  ADS  Google Scholar 

  217. Saito, H., Aioi, T. & Kadokura, T. Order-disorder oscillations in exciton–polariton superfluids. Phys. Rev. Lett. 110, 026401 (2013).

    Article  ADS  Google Scholar 

  218. Kovalev, V. M., Savenko, I. G. & Iorsh, I. V. Ultrafast exciton–polariton scattering towards the Dirac points. J. Phys. Cond. Matt. 28, 105301 (2016).

    Article  ADS  Google Scholar 

  219. Ozawa, T., Amo, A., Bloch, J. & Carusotto, I. Klein tunneling in driven-dissipative photonic graphene. Phys. Rev. A 96, 013813 (2017).

    Article  ADS  Google Scholar 

  220. Nalitov, A. V., Liew, T. C. H., Kavokin, A. V., Altshuler, B. L. & Rubo, Y. G. Spontaneous polariton currents in periodic lateral chains. Phys. Rev. Lett. 119, 067406 (2017).

    Article  ADS  Google Scholar 

  221. Askitopoulos, A. et al. Polariton condensation in an optically induced two-dimensional potential. Phys. Rev. B 88, 041308 (2013).

    Article  ADS  Google Scholar 

  222. Ohadi, H. et al. Spontaneous spin bifurcations and ferromagnetic phase transitions in a spinor exciton–polariton condensate. Phys. Rev. X 5, 031002 (2015).

    Google Scholar 

  223. Ohadi, H. et al. Nontrivial phase coupling in polariton multiplets. Phys. Rev. X 6, 031032 (2016).

    Google Scholar 

  224. Ohadi, H. et al. Spin order and phase transitions in chains of polariton condensates. Phys. Rev. Lett. 119, 067401 (2017).

    Article  ADS  Google Scholar 

  225. Sigurdsson, H. et al. Driven-dissipative spin chain model based on exciton–polariton condensates. Phys. Rev. B 96, 155403 (2017).

    Article  ADS  Google Scholar 

  226. Ohadi, H. et al. Synchronization crossover of polariton condensates in weakly disordered lattices. Phys. Rev. B 97, 195109 (2018).

    Article  ADS  Google Scholar 

  227. Berloff, N. G. et al. Realizing the classical xy Hamiltonian in polariton simulators. Nat. Mater. 16, 1120–1126 (2017).

    Article  ADS  Google Scholar 

  228. De las Cuevas, G. & Cubitt, T. S. Simple universal models capture all classical spin physics. Science 351, 1180–1183 (2016).

    Article  ADS  Google Scholar 

  229. Lagoudakis, P. G. & Berloff, N. G. A polariton graph simulator. New J. Phys. 19, 125008 (2017).

    Article  ADS  Google Scholar 

  230. 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).

    Article  ADS  Google Scholar 

  231. Kalinin, K. P. & Berloff, N. G. Networks of non-equilibrium condensates for global optimization. New J. Phys. 20, 113023 (2018).

    Article  ADS  Google Scholar 

  232. Kalinin, K. P. & Berloff, N. G. Global optimization of spin Hamiltonians with gain-dissipative systems. Sci. Rep. 8, 17791 (2018).

    Article  ADS  Google Scholar 

  233. Kalinin, K. P. & Berloff, N. G. Polaritonic network as a paradigm for dynamics of coupled oscillators. Phys. Rev. B 100, 245306 (2019).

    Article  ADS  Google Scholar 

  234. Kalinin, K. P. & Berloff, N. G. Simulating Ising and n-state planar Potts models and external fields with nonequilibrium condensates. Phys. Rev. Lett. 121, 235302 (2018).

    Article  ADS  Google Scholar 

  235. Kalinin, K. P., Amo, A., Bloch, J. & Berloff, N. G. Polaritonic xy-Ising machine. Nanophotonics 9, 4127–4138 (2020).

    Article  Google Scholar 

  236. Kyriienko, O., Sigurdsson, H. & Liew, T. C. H. Probabilistic solving of NP-hard problems with bistable nonlinear optical networks. Phys. Rev. B 99, 195301 (2019).

    Article  ADS  Google Scholar 

  237. Xue, Y. et al. Split-ring polariton condensates as macroscopic two-level quantum systems. Phys. Rev. Res. 3, 013099 (2021).

    Article  Google Scholar 

  238. Sedov, E. S., Lukoshkin, V. A., Kalevich, V. K., Savvidis, P. G. & Kavokin, A. V. Circular polariton currents with integer and fractional orbital angular momenta. Phys. Rev. Res. 3, 013072 (2021).

    Article  Google Scholar 

  239. Ma, X. et al. Realization of all-optical vortex switching in exciton–polariton condensates. Nat. Commun. 11, 897 (2020).

    Article  ADS  Google Scholar 

  240. Berger, B., Kahlert, M., Schmidt, D. & Assmann, M. Spectroscopy of fractional orbital angular momentum states. Opt. Express 26, 32248–32258 (2018).

    Article  ADS  Google Scholar 

  241. Leblanc, C., Malpuech, G. & Solnyshkov, D. D. High-frequency exciton–polariton clock generator. Phys. Rev. B 101, 115418 (2020).

    Article  ADS  Google Scholar 

  242. Byrnes, T., Wen, K. & Yamamoto, Y. Macroscopic quantum computation using Bose–Einstein condensates. Phys. Rev. A 85, 040306 (2012).

    Article  ADS  Google Scholar 

  243. Verger, A., Ciuti, C. & Carusotto, I. Polariton quantum blockade in a photonic dot. Phys. Rev. B 73, 193306 (2006).

    Article  ADS  Google Scholar 

  244. Ghosh, S. & Liew, T. C. H. Quantum computing with exciton–polariton condensates. npj Quantum Inf. 6, 16 (2020).

    Article  ADS  Google Scholar 

  245. Kyriienko, O. & Liew, T. C. H. Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems. Phys. Rev. A 90, 063805 (2014).

    Article  ADS  Google Scholar 

  246. Kyriienko, O. & Liew, T. C. H. Exciton–polariton quantum gates based on continuous variables. Phys. Rev. B 93, 035301 (2016).

    Article  ADS  Google Scholar 

  247. Liew, T. C. H. & Rubo, Y. G. Quantum exciton–polariton networks through inverse four-wave mixing. Phys. Rev. B 97, 041302 (2018).

    Article  ADS  Google Scholar 

  248. Einstein, A. Strahlungs-Emission und Absorption nach der Quantentheorie. Verh. Deutsch. Phys. Gesell. 18, 318–323 (1916).

    ADS  Google Scholar 

  249. Klaas, M. et al. Evolution of temporal coherence in confined exciton–polariton condensates. Phys. Rev. Lett. 120, 017401 (2018).

    Article  ADS  Google Scholar 

  250. Kim, S. et al. Coherent polariton laser. Phys. Rev. X 6, 011026 (2016).

    Google Scholar 

  251. Coldren, L. & Corzine, S. Diode Lasers and Photonic Integrated Circuits (Wiley, 2012).

  252. Bloch, I., Hänsch, T. W. & Esslinger, T. Atom laser with a cw output coupler. Phys. Rev. Lett. 82, 3008–3011 (1999).

    Article  ADS  Google Scholar 

  253. Kavokin, A., Liew, T. C. H., Schneider, C. & Höfling, S. Bosonic lasers: The state of the art (Review Article). Low Temp. Phys. 42, 323 (2016).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

A.K. acknowledges the Rosatom Road Map for Quantum Computing programme. T.C.H.L. was supported by the Ministry of Education (Singapore) Tier 2 project MOE2019-T2-1-004. C.S. acknowledges funding provided by the European Research Council (ERC project 679288, unlimit-2D) as well as the German Research Foundation (DFG) (Project SCHN1376 14.1). S.K. and S.H. acknowledge financial support from the German Research Foundation (DFG) through the Würzburg–Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter “ct.qmat” (EXC 2147, project ID 390858490). S.K. acknowledges funding provided by the German Research Foundation (DFG) (Project KL3124/3.1).

Author information

Authors and Affiliations

  1. Key Laboratory for Quantum Materials of Zhejiang Province, School of Science, Westlake University, Hangzhou, China

    Alexey Kavokin

  2. Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Timothy C. H. Liew

  3. Institute of Physics, University of Oldenburg, Oldenburg, Germany

    Christian Schneider

  4. Skolkovo Institute of Science and Technology, Moscow, Russia

    Pavlos G. Lagoudakis

  5. Department of Physics and Astronomy, University of Southampton, Southampton, UK

    Pavlos G. Lagoudakis

  6. Technische Physik and Wilhelm-Conrad-Röntgen-Research Center for Complex Material Systems, University of Würzburg, Würzburg, Germany

    Sebastian Klembt & Sven Hoefling

  7. Würzburg–Dresden Cluster of Excellence ct.qmat, University of Würzburg, Würzburg, Germany

    Sebastian Klembt & Sven Hoefling

Authors
  1. Alexey Kavokin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Timothy C. H. Liew
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pavlos G. Lagoudakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Sebastian Klembt
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Sven Hoefling
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed to all aspects of the article.

Corresponding author

Correspondence to Alexey Kavokin.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Reviews Physics thanks Edgar Cerda-Mendez and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Glossary

Bragg mirror

A periodic dielectric structure that reflects light owing to the optical interference effect. Dielectric Bragg mirrors may provide higher reflectivities than metallic mirrors, which is why they are widely used in laser structures and microcavities.

Microcavity

A thin layer of semiconductor or dielectric material sandwiched between two mirrors. In contrast to conventional optical cavities, microcavities confine a limited number of optical modes, frequently only one mode, which is why they are characterized by a very high finesse.

Tamm-plasmon

An optical mode confined between a dielectric Bragg mirror and a metallic layer.

Frenkel exciton

An elementary excitation in a molecular crystal. Frenkel excitons are characterized by relatively small Bohr radii (of the order of a lattice constant) and large binding energies (typically of the order of 1 eV).

Wannier–Mott exciton

An elementary excitation in an inorganic semiconductor crystal. Wannier–Mott excitons are characterized by large Bohr radii (several tens of lattice constants) and relatively small binding energies (typically of the order of 10 meV).

TE–TM splitting

The energy splitting between optical modes having their electric field (TE) or magnetic field (TM) vectors in the plane of the cavity, respectively. The splitting is defined by Maxwell boundary conditions at the boundaries of the cavity. It strongly affects the polarization dynamics of exciton–polaritons, creating a kind of effective magnetic field acting upon polariton pseudospin (Stokes vector).

Su–Schrieffer–Heeger (SSH) model

This model predicts formation of spatially localized electronic states at the ends of molecular chains. It has generalizations for a large variety of one-dimensional systems.

Zak-phase

A topological number that refers to the Berry’s phase picked up by a particle moving across the Brillouin zone in a one-dimensional crystal.

Soliton

A solitary wave formed in a nonlinear system that preserves its shape when propagating. One can distinguish between bright and dark optical solitons characterized by a peak and a trough of intensity of the electromagnetic field. Bright solitons have been observed in polariton flows.

Polariton blockade

A formation of a polariton state with a fixed number of particles due to the nonlinear absorption of pumping laser light. It is expected to occur in the case of quasiresonant optical excitation where the efficiency of absorption of the laser light becomes strongly dependent on the occupation number of a polariton mode.

NP-hard problems

An important class of mathematical problems that are more complex than the most complex of NP problems, with NP standing for nondeterministic polynomial time. NP problems represent the set of decision problems solvable in polynomial time by a non-deterministic Turing machine.

Polariton superfluid

A polariton condensate demonstrating features of a superfluid such as quantized vortices, persistent currents, solitons, lack of scattering and viscosity, and Bogolyubov-like linear dispersion of excitations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kavokin, A., Liew, T.C.H., Schneider, C. et al. Polariton condensates for classical and quantum computing. Nat Rev Phys 4, 435–451 (2022). https://doi.org/10.1038/s42254-022-00447-1

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42254-022-00447-1

This article is cited by

Search

Advanced search

Quick links

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