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Photonic materials in circuit quantum electrodynamics


Photonic synthetic materials provide an opportunity to explore the role of microscopic quantum phenomena in determining macroscopic material properties. There are, however, fundamental obstacles to overcome — in vacuum, photons not only lack mass, but also do not naturally interact with one another. Here, we review how the superconducting quantum circuit platform has been harnessed in the last decade to make some of the first materials from light. We describe the structures that are used to imbue individual microwave photons with matter-like properties such as mass, the nonlinear elements that mediate interactions between these photons, and quantum dynamic/thermodynamic approaches that can be used to assemble and stabilize strongly correlated states of many photons. We then describe state-of-the-art techniques to generate synthetic magnetic fields, engineer topological and non-topological flat bands and explore the physics of quantum materials in non-Euclidean geometries — directions that we view as some of the most exciting for this burgeoning field. Finally, we discuss upcoming prospects, and in particular opportunities to probe novel aspects of quantum thermalization and detect quasi-particles with exotic anyonic statistics, as well as potential applications in quantum information science.

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Fig. 1: A comparison of quantum matter platforms.
Fig. 2: Assembling quantum matter.
Fig. 3: The circuit QED toolbox for materials.
Fig. 4: Strongly correlated photonic matter in circuit QED.
Fig. 5: Topological lattices.
Fig. 6: Curved-space and flat-band lattices.


  1. 1.

    Walls, D. & Milburn, G. Quantum Optics (Springer, 2008).

  2. 2.

    Aspect, A., Dalibard, J. & Roger, G. Experimental test of bell’s inequalities using time-varying analyzers. Phys. Rev. Let. 49, 1804–1807 (1982).

    ADS  MathSciNet  Google Scholar 

  3. 3.

    Weihs, G., Jennewein, T., Simon, C., Weinfurter, H. & Zeilinger, A. Violation of Bell’s inequality under strict Einstein locality conditions. Phys. Rev. Lett. 81, 5039–5043 (1998).

    ADS  MathSciNet  MATH  Google Scholar 

  4. 4.

    Carusotto, I. & Ciuti, C. Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013). A review of quantum fluids of light from an interdisciplinary perspective, from exciton-polariton in microcavities to circuit-QED.

  5. 5.

    Kavokin, A., Baumberg, J., Malpuech, G. & Laussy, F. Microcavities (Oxford Univ. Press, 2017).

  6. 6.

    Deng, H., Haug, H. & Yamamoto, Y. Exciton-polariton Bose-Einstein condensation. Rev. Mod. Phys. 82, 1489–1537 (2010).

  7. 7.

    Chang, D. E., Vuletić, V. & Lukin, M. D. Quantum nonlinear optics—photon by photon. Nat. Photon. 8, 685–694 (2014).

    ADS  Google Scholar 

  8. 8.

    Schuster, D. et al. Resolving photon number states in a superconducting circuit. Nature 445, 515–518 (2007).

    ADS  Google Scholar 

  9. 9.

    Paik, H. et al. Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture. Phys. Rev. Lett. 107, 240501 (2011).

    ADS  Google Scholar 

  10. 10.

    Reagor, M. et al. Quantum memory with millisecond coherence in circuit QED. Phys. Rev. B 94, 014506 (2016).

    ADS  Google Scholar 

  11. 11.

    Houck, A. A., Türeci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nat. Phys. 8, 292–299 (2012). An authoritative earlier review on many-body physics in arrays of superconducting circuits.

  12. 12.

    Schmidt, S. & Koch, J. Circuit QED lattices: towards quantum simulation with superconducting circuits. Ann. Phys. 525, 395–412 (2013).

    Google Scholar 

  13. 13.

    Hartmann, M. J. Quantum simulation with interacting photons. J. Opt. 18, 104005 (2016).

    ADS  Google Scholar 

  14. 14.

    Noh, C. & Angelakis, D. G. Quantum simulations and many-body physics with light. Rep. Prog. Phys. 80, 016401 (2016).

    ADS  Google Scholar 

  15. 15.

    Simon, J. et al. Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature 472, 307–312 (2011).

    ADS  Google Scholar 

  16. 16.

    Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39–44 (2002).

    ADS  Google Scholar 

  17. 17.

    Grusdt, F., Letscher, F., Hafezi, M. & Fleischhauer, M. Topological growing of Laughlin states in synthetic gauge fields. Phys. Rev. Lett. 113, 155301 (2014).

    ADS  Google Scholar 

  18. 18.

    Sørensen, A. S., Demler, E. & Lukin, M. D. Fractional quantum Hall states of atoms in optical lattices. Phys. Rev. Lett. 94, 086803 (2005).

    ADS  Google Scholar 

  19. 19.

    Hartmann, M. J., Brandao, F. G. & Plenio, M. B. Strongly interacting polaritons in coupled arrays of cavities. Nat. Phys. 2, 849–855 (2006).

    Google Scholar 

  20. 20.

    Greentree, A. D., Tahan, C., Cole, J. H. & Hollenberg, L. C. Quantum phase transitions of light. Nat. Phys. 2, 856–861 (2006).

    Google Scholar 

  21. 21.

    Angelakis, D. G., Santos, M. F. & Bose, S. Photon-blockade-induced Mott transitions and x y spin models in coupled cavity arrays. Phys. Rev. A 76, 031805 (2007).

    ADS  Google Scholar 

  22. 22.

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

    ADS  Google Scholar 

  23. 23.

    Klaers, J., Schmitt, J., Vewinger, F. & Weitz, M. Bose–Einstein condensation of photons in an optical microcavity. Nature 468, 545–548 (2010).

    ADS  Google Scholar 

  24. 24.

    Altman, E., Sieberer, L. M., Chen, L., Diehl, S. & Toner, J. Two-dimensional superfluidity of exciton polaritons requires strong anisotropy. Phys. Rev. X 5, 011017 (2015).

    Google Scholar 

  25. 25.

    Ji, K., Gladilin, V. N. & Wouters, M. Temporal coherence of one-dimensional nonequilibrium quantum fluids. Phys. Rev. B 91, 045301 (2015).

    ADS  Google Scholar 

  26. 26.

    Dagvadorj, G. et al. Nonequilibrium phase transition in a two-dimensional driven open quantum system. Phys. Rev. X 5, 041028 (2015).

    Google Scholar 

  27. 27.

    Squizzato, D., Canet, L. & Minguzzi, A. Kardar-Parisi-Zhang universality in the phase distributions of one-dimensional exciton-polaritons. Phys. Rev. B 97, 195453 (2018).

    ADS  Google Scholar 

  28. 28.

    Gerace, D., Türeci, H. E., Imamoglu, A., Giovannetti, V. & Fazio, R. The quantum-optical Josephson interferometer. Nat. Phys. 5, 281–284 (2009).

    Google Scholar 

  29. 29.

    Carusotto, I. et al. Fermionized photons in an array of driven dissipative nonlinear cavities. Phys. Rev. Lett. 103, 033601 (2009). The first proposal for a scheme to exploit driving and dissipation to generate strongly-correlated state of photons in a cavity array.

  30. 30.

    Umucalılar, R. & Carusotto, I. Fractional quantum Hall states of photons in an array of dissipative coupled cavities. Phys. Rev. Lett. 108, 206809 (2012).

    ADS  Google Scholar 

  31. 31.

    Hacohen-Gourgy, S., Ramasesh, V. V., De Grandi, C., Siddiqi, I. & Girvin, S. M. Cooling and autonomous feedback in a Bose-Hubbard chain with attractive interactions. Phys. Rev. Lett. 115, 240501 (2015).

    ADS  Google Scholar 

  32. 32.

    Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).

    ADS  Google Scholar 

  33. 33.

    Choi, S. et al. Observation of discrete time-crystalline order in a disordered dipolar many-body system. Nature 543, 221–225 (2017).

    ADS  Google Scholar 

  34. 34.

    Fausti, D. et al. Light-induced superconductivity in a stripe-ordered cuprate. Science 331, 189–191 (2011).

    ADS  Google Scholar 

  35. 35.

    Eisert, J., Friesdorf, M. & Gogolin, C. Quantum many-body systems out of equilibrium. Nat. Phys. 11, 124–130 (2015).

    Google Scholar 

  36. 36.

    Kapit, E., Hafezi, M. & Simon, S. H. Induced self-stabilization in fractional quantum hall states of light. Phys. Rev. X 4, 031039 (2014). Together with refs. 39,41–44, this work has theoretically pioneered the idea of dissipative stabilization of a non-equilibrium many-body system by means of engineered driving and losses.

  37. 37.

    Hafezi, M., Adhikari, P. & Taylor, J. Chemical potential for light by parametric coupling. Phys. Rev. B 92, 174305 (2015).

    ADS  Google Scholar 

  38. 38.

    Lebreuilly, J., Wouters, M. & Carusotto, I. Towards strongly correlated photons in arrays of dissipative nonlinear cavities under a frequency-dependent incoherent pumping. C. R. Phys. 17, 836–860 (2016).

    ADS  Google Scholar 

  39. 39.

    Ma, R., Owens, C., Houck, A., Schuster, D. I. & Simon, J. Autonomous stabilizer for incompressible photon fluids and solids. Phys. Rev. A 95, 043811 (2017).

    ADS  Google Scholar 

  40. 40.

    Biella, A. et al. Phase diagram of incoherently driven strongly correlated photonic lattices. Phys. Rev. A 96, 023839 (2017).

    ADS  Google Scholar 

  41. 41.

    Lebreuilly, J. et al. Stabilizing strongly correlated photon fluids with non-Markovian reservoirs. Phys. Rev. A 96, 033828 (2017).

    ADS  Google Scholar 

  42. 42.

    Ma, R. et al. A dissipatively stabilized Mott insulator of photons. Nature 566, 51–57 (2019).This represents first experimental realization of a strongly interacting fluid of impenetrable photons.

    ADS  Google Scholar 

  43. 43.

    Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. M. & Schoelkopf, R. J. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004).

    ADS  Google Scholar 

  44. 44.

    Gu, X., Kockum, A. F., Miranowicz, A., Liu, Y.-x & Nori, F. Microwave photonics with superconducting quantum circuits. Phys. Rep. (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  45. 45.

    Koch, J. et al. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).

    ADS  Google Scholar 

  46. 46.

    Imamoḡlu, A., Schmidt, H., Woods, G. & Deutsch, M. Strongly interacting photons in a nonlinear cavity. Phys. Rev. Lett. 79, 1467–1470 (1997).

    ADS  Google Scholar 

  47. 47.

    Underwood, D. L., Shanks, W. E., Koch, J. & Houck, A. A. Low-disorder microwave cavity lattices for quantum simulation with photons. Phys. Rev. A 86, 023837 (2012).

    ADS  Google Scholar 

  48. 48.

    Kollár, A. J., Fitzpatrick, M. & Houck, A. A. Hyperbolic lattices in circuit quantum electrodynamics. Nature 571, 45–50 (2019).This work has reported the first experimental realization of an array with an intrinsically non-Euclidean geometry.

    ADS  Google Scholar 

  49. 49.

    Chen, Y. et al. Qubit architecture with high coherence and fast tunable coupling. Phys. Rev. Lett. 113, 220502 (2014).

    ADS  Google Scholar 

  50. 50.

    Roushan, P. et al. Chiral ground-state currents of interacting photons in a synthetic magnetic field.Nat. Phys. 13, 146–151 (2017).This work has reported the first experimental study of the interplay of a synthetic magnetic field and strong interactions for photons in a simplest geometry.

    Google Scholar 

  51. 51.

    Pitaevskii, L. P. & Stringari, S. Bose-Einstein Condensation and Superfluidity (Oxford Univ. Press, 2016).

  52. 52.

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

    ADS  Google Scholar 

  53. 53.

    Togan, E., Lim, H.-T., Faelt, S., Wegscheider, W. & Imamoglu, A. Enhanced interactions between dipolar polaritons. Phys. Rev. Lett. 121, 227402 (2018).

    ADS  Google Scholar 

  54. 54.

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

    Google Scholar 

  55. 55.

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

    Google Scholar 

  56. 56.

    Peyronel, T. et al. Quantum nonlinear optics with single photons enabled by strongly interacting atoms. Nature 488, 57–60 (2012).

    ADS  Google Scholar 

  57. 57.

    Jia, N. et al. A strongly interacting polaritonic quantum dot. Nat. Phys. 14, 550–554 (2018).

    Google Scholar 

  58. 58.

    Sommer, A., Büchler, H. P. & Simon, J. Quantum crystals and Laughlin droplets of cavity Rydberg polaritons. Preprint at (2015).

  59. 59.

    Clark, L. W. et al. Interacting Floquet polaritons. Nature 571, 532–536 (2019).

    Google Scholar 

  60. 60.

    Clark, L. W., Schine, N., Baum, C., Jia, N. & Simon, J. Observation of Laughlin states made of light. Preprint at (2019).

  61. 61.

    Reed, M. et al. High-fidelity readout in circuit quantum electrodynamics using the Jaynes-Cummings nonlinearity. Phys. Rev. Lett. 105, 173601 (2010).

    ADS  Google Scholar 

  62. 62.

    Walter, T. et al. Rapid high-fidelity single-shot dispersive readout of superconducting qubits. Phys. Rev. Appl. 7, 054020 (2017).

    ADS  Google Scholar 

  63. 63.

    Bakr, W. S., Gillen, J. I., Peng, A., Fölling, S. & Greiner, M. A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature 462, 74–77 (2009).

    ADS  Google Scholar 

  64. 64.

    Sherson, J. F. et al. Single-atom-resolved fluorescence imaging of an atomic Mott insulator. Nature 467, 68–72 (2010).

    ADS  Google Scholar 

  65. 65.

    Roushan, P. et al. Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science 358, 1175–1179 (2017).This work has studied the temporal dynamics of systems of few strongly interacting photons in a disordered landscape.

    ADS  MathSciNet  Google Scholar 

  66. 66.

    Cooper, K. et al. Observation of quantum oscillations between a Josephson phase qubit and a microscopic resonator using fast readout. Phys. Rev. Lett. 93, 180401 (2004).

    ADS  Google Scholar 

  67. 67.

    Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).

    ADS  Google Scholar 

  68. 68.

    Majer, J. et al. Coupling superconducting qubits via a cavity bus. Nature 449, 443–447 (2007).

    ADS  Google Scholar 

  69. 69.

    Kirchmair, G. et al. Observation of quantum state collapse and revival due to the single-photon Kerr effect. Nature 495, 205–209 (2013).

    ADS  Google Scholar 

  70. 70.

    Steffen, M. et al. Measurement of the entanglement of two superconducting qubits via state tomography. Science 313, 1423–1425 (2006).

    ADS  MathSciNet  Google Scholar 

  71. 71.

    Houck, A. A. et al. Generating single microwave photons in a circuit. Nature 449, 328–331 (2007).

    ADS  Google Scholar 

  72. 72.

    Ansmann, M. et al. Violation of Bell’s inequality in Josephson phase qubits. Nature 461, 504–506 (2009).

    ADS  Google Scholar 

  73. 73.

    Tangpanitanon, J. & Angelakis, D. G. Many-body physics and quantum simulations with strongly interacting photons. Preprint at (2019). A very recent set of lecture notes giving another perspective on strongly interacting photons.

  74. 74.

    Raftery, J., Sadri, D., Schmidt, S., Türeci, H. E. & Houck, A. A. Observation of a dissipation-induced classical to quantum transition. Phys. Rev. X 4, 031043 (2014).This work, inspired by the theoretical investigation in the next reference, provides experimental evidence of a dynamical localization transition in a dimer geometry, from an oscillatory behaviour to a self-trapped state.

    Google Scholar 

  75. 75.

    Schmidt, S., Gerace, D., Houck, A. A., Blatter, G. & Türeci, H. E. Nonequilibrium delocalization-localization transition of photons in circuit quantum electrodynamics. Phys. Rev. B 82, 100507 (2010).

    ADS  Google Scholar 

  76. 76.

    Albiez, M. et al. Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction. Phys. Rev. Lett. 95, 010402 (2005).

    ADS  Google Scholar 

  77. 77.

    Abbarchi, M. et al. Macroscopic quantum self-trapping and Josephson oscillations of exciton polaritons. Nat. Phys. 9, 275–279 (2013).

    Google Scholar 

  78. 78.

    Yan, Z. et al. Strongly correlated quantum walks with a 12-qubit superconducting processor. Science 364, 753–756 (2019).

    ADS  Google Scholar 

  79. 79.

    Ye, Y. et al. Propagation and localization of collective excitations on a 24-qubit superconducting processor. Phys. Rev. Lett. 123, 050502 (2019).

    ADS  Google Scholar 

  80. 80.

    Mazurenko, A. et al. A cold-atom Fermi–Hubbard antiferromagnet. Nature 545, 462–466 (2017).

    ADS  Google Scholar 

  81. 81.

    Umucalılar, R. & Carusotto, I. Generation and spectroscopic signatures of a fractional quantum Hall liquid of photons in an incoherently pumped optical cavity. Phys. Rev. A 96, 053808 (2017).

    ADS  Google Scholar 

  82. 82.

    Biondi, M., Blatter, G. & Schmidt, S. Emergent light crystal from frustration and pump engineering. Phys. Rev. B 98, 104204 (2018).

    ADS  Google Scholar 

  83. 83.

    Mamaev, M., Govia, L. C. G. & Clerk, A. A. Dissipative stabilization of entangled cat states using a driven Bose-Hubbard dimer. Quantum 2, 58 (2018).

    Google Scholar 

  84. 84.

    Lebreuilly, J., Aron, C. & Mora, C. Stabilizing arrays of photonic cat states via spontaneous symmetry breaking. Phys. Rev. Lett. 122, 120402 (2019).

    ADS  Google Scholar 

  85. 85.

    Bardyn, C.-E. & İmamoǧlu, A. Majorana-like modes of light in a one-dimensional array of nonlinear cavities. Phys. Rev. Lett. 109, 253606 (2012).

    ADS  Google Scholar 

  86. 86.

    Liu, Y. & Houck, A. A. Quantum electrodynamics near a photonic bandgap. Nat. Phys. 13, 48–52 (2017).

    Google Scholar 

  87. 87.

    Tomadin, A. et al. Signatures of the superfluid-insulator phase transition in laser-driven dissipative nonlinear cavity arrays. Phys. Rev. A 81, 061801 (2010).

    ADS  Google Scholar 

  88. 88.

    Le Hur, K. et al. Many-body quantum electrodynamics networks: Non-equilibrium condensed matter physics with light. C. R. Phys. 17, 808–835 (2016).

    ADS  Google Scholar 

  89. 89.

    Biondi, M., Blatter, G., Türeci, H. E. & Schmidt, S. Nonequilibrium gas-liquid transition in the driven-dissipative photonic lattice. Phys. Rev. A 96, 043809 (2017).

    ADS  Google Scholar 

  90. 90.

    Foss-Feig, M. et al. Emergent equilibrium in many-body optical bistability. Phys. Rev. A 95, 043826 (2017).

    ADS  Google Scholar 

  91. 91.

    Rota, R., Minganti, F., Ciuti, C. & Savona, V. Quantum critical regime in a quadratically driven nonlinear photonic lattice. Phys. Rev. Lett. 122, 110405 (2019).

    ADS  Google Scholar 

  92. 92.

    Vicentini, F., Minganti, F., Rota, R., Orso, G. & Ciuti, C. Critical slowing down in driven-dissipative Bose-Hubbard lattices. Phys. Rev. A 97, 013853 (2018).

    ADS  Google Scholar 

  93. 93.

    Tangpanitanon, J. et al. Hidden order in quantum many-body dynamics of driven-dissipative nonlinear photonic lattices. Phys. Rev. A 99, 043808 (2019).

    ADS  Google Scholar 

  94. 94.

    Le Boité, A., Orso, G. & Ciuti, C. Bose-Hubbard model: Relation between driven-dissipative steady states and equilibrium quantum phases. Phys. Rev. A 90, 063821 (2014).

    ADS  Google Scholar 

  95. 95.

    Wouters, M. & Carusotto, I. Absence of long-range coherence in the parametric emission of photonic wires. Phys. Rev. B 74, 245316 (2006).

    ADS  Google Scholar 

  96. 96.

    Dalla Torre, E. G., Demler, E., Giamarchi, T. & Altman, E. Quantum critical states and phase transitions in the presence of non-equilibrium noise. Nat. Phys. 6, 806–810 (2010).

    Google Scholar 

  97. 97.

    Sieberer, L. M., Buchhold, M. & Diehl, S. Keldysh field theory for driven open quantum systems. Rep. Prog. Phys. 79, 096001 (2016).

    ADS  Google Scholar 

  98. 98.

    Marino, J. & Diehl, S. Driven Markovian quantum criticality. Phys. Rev. Lett. 116, 070407 (2016).

    ADS  Google Scholar 

  99. 99.

    Lebreuilly, J., Chiocchetta, A. & Carusotto, I. Pseudothermalization in driven-dissipative non-Markovian open quantum systems. Phys. Rev. A 97, 033603 (2018).

    ADS  Google Scholar 

  100. 100.

    Jin, J., Rossini, D., Fazio, R., Leib, M. & Hartmann, M. J. Photon solid phases in driven arrays of nonlinearly coupled cavities. Phys. Rev. Lett. 110, 163605 (2013).

    ADS  Google Scholar 

  101. 101.

    Finazzi, S., Le Boité, A., Storme, F., Baksic, A. & Ciuti, C. Corner-space renormalization method for driven-dissipative two-dimensional correlated systems. Phys. Rev. Lett. 115, 080604 (2015).

    ADS  Google Scholar 

  102. 102.

    Vicentini, F., Minganti, F., Biella, A., Orso, G. & Ciuti, C. Optimal stochastic unraveling of disordered open quantum systems: Application to driven-dissipative photonic lattices. Phys. Rev. A 99, 032115 (2019).

    ADS  Google Scholar 

  103. 103.

    Yoshioka, N. & Hamazaki, R. Constructing neural stationary states for open quantum many-body systems. Phys. Rev. B 99, 214306 (2019).

    ADS  Google Scholar 

  104. 104.

    Hartmann, M. J. & Carleo, G. Neural-network approach to dissipative quantum many-body dynamics. Phys. Rev. Lett. 122, 250502 (2019).

    ADS  Google Scholar 

  105. 105.

    Strathearn, A., Kirton, P., Kilda, D., Keeling, J. & Lovett, B. W. Efficient non-Markovian quantum dynamics using time-evolving matrix product operators. Nat. Commun. 9, 3322 (2018).

    ADS  Google Scholar 

  106. 106.

    Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: Many-body localization, thermalization, and entanglement. Rev. Mod. Phys. 91, 021001 (2019).

    ADS  MathSciNet  Google Scholar 

  107. 107.

    Xu, K. et al. Emulating many-body localization with a superconducting quantum processor. Phys. Rev. Lett. 120, 050507 (2018).

    ADS  Google Scholar 

  108. 108.

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

    ADS  Google Scholar 

  109. 109.

    Cooper, N., Dalibard, J. & Spielman, I. Topological bands for ultracold atoms. Rev. Mod. Phys. 91, 015005 (2019).

    ADS  MathSciNet  Google Scholar 

  110. 110.

    Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).This article reviews the field of topological photonics from a cross-platform perspective, highlighting the links with other areas of topological condensed-matter physics.

    ADS  MathSciNet  Google Scholar 

  111. 111.

    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).Together with the experimental implementation in ref. 112, this work has highlighted that the quantum Hall effect is not restricted to fermionic electrons, thus opening the field of topological photonics.

    ADS  Google Scholar 

  112. 112.

    Wang, Z., Chong, Y., Joannopoulos, J. & Soljačić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009).

    ADS  Google Scholar 

  113. 113.

    Koch, J., Houck, A. A., Le Hur, K. & Girvin, S. Time-reversal-symmetry breaking in circuit-QED-based photon lattices. Phys. Rev. A 82, 043811 (2010).

    ADS  Google Scholar 

  114. 114.

    Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon. 6, 782–787 (2012).

    ADS  Google Scholar 

  115. 115.

    Ningyuan, J., Owens, C., Sommer, A., Schuster, D. & Simon, J. Time-and site-resolved dynamics in a topological circuit. Phys. Rev. X 5, 021031 (2015).

    Google Scholar 

  116. 116.

    Albert, V. V., Glazman, L. I. & Jiang, L. Topological properties of linear circuit lattices. Phys. Rev. Lett. 114, 173902 (2015).

    ADS  MathSciNet  Google Scholar 

  117. 117.

    Lu, Y. et al. Probing the Berry curvature and fermi arcs of a Weyl circuit. Phys. Rev. B 99, 020302 (2019).

    ADS  Google Scholar 

  118. 118.

    Imhof, S. et al. Topolectrical-circuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).

    Google Scholar 

  119. 119.

    Anderson, B. M., Ma, R., Owens, C., Schuster, D. I. & Simon, J. Engineering topological many-body materials in microwave cavity arrays. Phys. Rev. X 6, 041043 (2016).

    Google Scholar 

  120. 120.

    Owens, C. et al. Quarter-flux Hofstadter lattice in a qubit-compatible microwave cavity array. Phys. Rev. A 97, 013818 (2018).This work has reported the experimental realization of an ɑ = 1/4 Harper-Hofstadter model for photons on a qubit compatible platform.

    ADS  Google Scholar 

  121. 121.

    Tai, M. E. et al. Microscopy of the interacting Harper–Hofstadter model in the two-body limit. Nature 546, 519–523 (2017).

    ADS  Google Scholar 

  122. 122.

    Cai, W. et al. Observation of topological magnon insulator states in a superconducting circuit. Phys. Rev. Lett. 123, 080501 (2019).

    ADS  Google Scholar 

  123. 123.

    Cho, J., Angelakis, D. G. & Bose, S. Fractional quantum Hall state in coupled cavities. Phys. Rev. Lett. 101, 246809 (2008).

    ADS  Google Scholar 

  124. 124.

    de Léséleuc, S. et al. Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms. Science 365, 775–780 (2019).An experimental study of topological states in synthetic quantum matter using an alternative platform consisting of a gas of spin excitations in an array of Rydberg atoms trapped by optical tweezers.

    ADS  MathSciNet  Google Scholar 

  125. 125.

    Boada, O., Celi, A., Rodríguez-Laguna, J., Latorre, J. I. & Lewenstein, M. Quantum simulation of non-trivial topology. N. J. Phys. 17, 045007 (2015).

    Google Scholar 

  126. 126.

    Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Sarma, S. D. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  127. 127.

    Leykam, D., Andreanov, A. & Flach, S. Artificial flat band systems: from lattice models to experiments. Adv. Phys. X 3, 1473052 (2018).

    Google Scholar 

  128. 128.

    Casteels, W., Rota, R., Storme, F. & Ciuti, C. Probing photon correlations in the dark sites of geometrically frustrated cavity lattices. Phys. Rev. A 93, 043833 (2016).

    ADS  Google Scholar 

  129. 129.

    Kollár, A. J., Fitzpatrick, M., Sarnak, P. & Houck, A. A. Line-graph lattices: Euclidean and non-Euclidean flat bands, and implementations in circuit quantum electrodynamics. Commun. Math. Phys. (2019).

  130. 130.

    Biggs, N. Algebraic Graph Theory 2nd edn (Cambridge Univ. Press, 1993).

  131. 131.

    Shirai, T. The spectrum of infinite regular line graphs. Trans. Am. Math. Soc. 352, 115–132 (1999).

    MathSciNet  MATH  Google Scholar 

  132. 132.

    Irvine, W. T., Vitelli, V. & Chaikin, P. M. Pleats in crystals on curved surfaces. Nature 468, 947–951 (2010).

    ADS  Google Scholar 

  133. 133.

    Kinsey, L. C. Topology of Surfaces (Springer, 1997).

  134. 134.

    Can, T., Laskin, M. & Wiegmann, P. Fractional quantum hall effect in a curved space: Gravitational anomaly and electromagnetic response. Phys. Rev. Lett. 113, 046803 (2014).

    ADS  Google Scholar 

  135. 135.

    Schine, N., Chalupnik, M., Can, T., Gromov, A. & Simon, J. Electromagnetic and gravitational responses of photonic landau levels. Nature 565, 173–179 (2019).

    ADS  Google Scholar 

  136. 136.

    Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).This work reviews the perspectives of using the synthetic dimension concept to investigate new states of topological quantum matter using either atoms or photons.

    Google Scholar 

  137. 137.

    Irvine, W. T. & Vitelli, V. Geometric background charge: dislocations on capillary bridges. Soft Matter 8, 10123–10129 (2012).

    ADS  Google Scholar 

  138. 138.

    Breuckmann, N. P. & Terhal, B. M. Constructions and noise threshold of hyperbolic surface codes. IEEE Trans. Inf. Theory 62, 3731–3744 (2016).

    MathSciNet  MATH  Google Scholar 

  139. 139.

    Abuwasib, M., Krantz, P. & Delsing, P. Fabrication of large dimension aluminum air-bridges for superconducting quantum circuits. J. Vac. Sci. Technol. B 31, 031601 (2013).

    Google Scholar 

  140. 140.

    Foxen, B. et al. Qubit compatible superconducting interconnects. Quantum Sci. Technol. 3, 014005 (2018).

    ADS  Google Scholar 

  141. 141.

    Berkley, A. J., Johnson, M. W. & Bunyk, P. I. Systems and methods for superconducting integrated circuits. US Patent 9,355,365 (2016).

  142. 142.

    Holland, E. T. et al. Single-photon-resolved cross-Kerr interaction for autonomous stabilization of photon-number states. Phys. Rev. Lett. 115, 180501 (2015).

    ADS  Google Scholar 

  143. 143.

    Collodo, M. C. et al. Observation of the crossover from photon ordering to delocalization in tunably coupled resonators. Phys. Rev. Lett. 122, 183601 (2019).

    ADS  Google Scholar 

  144. 144.

    Burnell, F., Parish, M. M., Cooper, N. & Sondhi, S. L. Devil’s staircases and supersolids in a one-dimensional dipolar bose gas. Phys. Rev. B 80, 174519 (2009).

    ADS  Google Scholar 

  145. 145.

    Sameti, M., Poto c čnik, A., Browne, D. E., Wallraff, A. & Hartmann, M. J. Superconducting quantum simulator for topological order and the toric code. Phys. Rev. A 95, 042330 (2017).

    ADS  Google Scholar 

  146. 146.

    Marcos, D., Rabl, P., Rico, E. & Zoller, P. Superconducting circuits for quantum simulation of dynamical gauge fields. Phys. Rev. Lett. 111, 110504 (2013).

    ADS  Google Scholar 

  147. 147.

    Sterdyniak, A., Regnault, N. & Möller, G. Particle entanglement spectra for quantum Hall states on lattices. Phys. Rev. B 86, 165314 (2012).

    ADS  Google Scholar 

  148. 148.

    Gerster, M., Rizzi, M., Silvi, P., Dalmonte, M. & Montangero, S. Fractional quantum Hall effect in the interacting Hofstadter model via tensor networks. Phys. Rev. B 96, 195123 (2017).

    ADS  Google Scholar 

  149. 149.

    Rosson, P., Lubasch, M., Kiffner, M. & Jaksch, D. Bosonic fractional quantum Hall states on a finite cylinder. Phys. Rev. A 99, 033603 (2019).

    ADS  Google Scholar 

  150. 150.

    Macaluso, E. et al. Charge and statistics of lattice quasiholes from density measurements: a tree tensor network study. Phys. Rev. Res. 2, 013145 (2020).This work reports a numerical study of a fractional quantum Hall state in a lattice of realistic size, highlighting schemes to detect the anyonic statistics of quasi-holes.

    Google Scholar 

  151. 151.

    Grusdt, F., Yao, N. Y., Abanin, D., Fleischhauer, M. & Demler, E. Interferometric measurements of many-body topological invariants using mobile impurities. Nat. Commun. 7, 11994 (2016).

    ADS  Google Scholar 

  152. 152.

    Umucalılar, R. & Carusotto, I. Many-body braiding phases in a rotating strongly correlated photon gas. Phys. Lett. A 377, 2074–2078 (2013).

    ADS  MathSciNet  MATH  Google Scholar 

  153. 153.

    Umucalılar, R., Macaluso, E., Comparin, T. & Carusotto, I. Time-of-flight measurements as a possible method to observe anyonic statistics. Phys. Rev. Lett. 120, 230403 (2018).

    ADS  Google Scholar 

  154. 154.

    Macaluso, E., Comparin, T., Mazza, L. & Carusotto, I. Fusion channels of non-Abelian anyons from angular-momentum and density-profile measurements. Phys. Rev. Lett. 123, 266801 (2019).

    ADS  Google Scholar 

  155. 155.

    Stern, A. Anyons and the quantum Hall effect—a pedagogical review. Ann. Phys. 323, 204–249 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  156. 156.

    Song, C. et al. Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit. Phys. Rev. Lett. 121, 030502 (2018).

    ADS  Google Scholar 

  157. 157.

    Alicki, R. & Kosloff, R. Thermodynamics in the Quantum Regime (eds Binder F. et al) Ch. 1 (Springer, 2018).

  158. 158.

    Leviatan, E., Pollmann, F., Bardarson, J. H., Huse, D. A. & Altman, E. Quantum thermalization dynamics with matrix-product states. Preprint at (2017).

  159. 159.

    Zurek, W. H. in Quantum Decoherence (eds Duplantier B., Raimond J. M. & Rivasseau V.) Ch. 1 (Birkhäuser, 2006).

  160. 160.

    Gardner, G. C., Fallahi, S., Watson, J. D. & Manfra, M. J. Modified MBE hardware and techniques and role of gallium purity for attainment of two dimensional electron gas mobility >35×106 cm2/V s in AlGaAs/GaAs quantum wells grown by MBE. J. Cryst. Growth 441, 71–77 (2016).

  161. 161.

    Dean, C. et al. Intrinsic gap of the ν = 5/2 fractional quantum Hall state. Phys. Rev. Lett. 100, 146803 (2008).

    ADS  Google Scholar 

  162. 162.

    Dial, O. et al. Bulk and surface loss in superconducting transmon qubits. Supercond. Sci. Tech. 29, 044001 (2016).

    ADS  Google Scholar 

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A.K. and A.H. acknowledge financial support from the National Science Foundation via the Princeton Center for Complex Materials DMR-1420541 and by the ARO MURI W911NF-15-1-0397. I.C. acknowledges financial support from the Provincia Autonoma di Trento and from the FET-Open Grant MIR-BOSE (737017) and Quantum Flagship Grant PhoQuS (820392) of the European Union. The work of J.S. and D.I.S. was partially supported by the University of Chicago Materials Research Science and Engineering Center, which is funded by the National Science Foundation under award number DMR-1420709. J.S. and D.I.S. also acknowledge support from ARO MURI grant W911NF-15-1-0397.

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Carusotto, I., Houck, A.A., Kollár, A.J. et al. Photonic materials in circuit quantum electrodynamics. Nat. Phys. 16, 268–279 (2020).

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