Abstract

Bichromatic photonic crystal structures are based on the coexistence of two different periodicities in the dielectric constant profile. They are realized starting from a photonic crystal waveguide and modifying the lattice constant only in the waveguide region. In this work, we numerically investigate the spectral and topological properties of bichromatic structures. Our calculations demonstrate that they provide a photonic analog of the integer quantum Hall state, a well-known example of a topological insulator. The nontrivial topology of the bandstructure is illustrated by the formation of strongly localized, topologically protected boundary modes when finite-sized bichromatic structures are embedded in a larger photonic crystal.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Helical edge states of topological photonic crystals with line defects

Zhen Jiang, Yongfeng Gao, Liu He, He Song, Jun Zhou, and Renjie Zhu
Appl. Opt. 58(9) 2294-2299 (2019)

Unidirectional edge states in topological honeycomb-lattice membrane photonic crystals

P. Duke Anderson and Ganapathi Subramania
Opt. Express 25(19) 23293-23301 (2017)

Topologically protected edge states in graphene plasmonic crystals

Pingping Qiu, Rui Liang, Weibin Qiu, Houbo Chen, Junbo Ren, Zhili Lin, Jia-Xian Wang, Qiang Kan, and Jiao-Qing Pan
Opt. Express 25(19) 22587-22594 (2017)

References

  • View by:
  • |
  • |
  • |

  1. K. van Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
    [Crossref]
  2. M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010), and references therein.
    [Crossref]
  3. D. R. Hofstadter, “Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,” Phys. Rev. B 14, 2239–2249 (1976).
    [Crossref]
  4. U. Kuhl and H.-J. Stöckmann, “Microwave realization of the Hofstadter butterfly,” Phys. Rev. Lett. 80, 3232–3235 (1998).
    [Crossref]
  5. D. Jaksch and P. Zoller, “Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms,” New J. Phys. 5, 56 (2003).
    [Crossref]
  6. H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, “Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices,” Phys. Rev. Lett. 111, 185302 (2013).
    [Crossref]
  7. M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices,” Phys. Rev. Lett. 111, 185301 (2013).
    [Crossref]
  8. L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
    [Crossref]
  9. C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
    [Crossref]
  10. B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
    [Crossref]
  11. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
    [Crossref]
  12. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014), and references therein.
    [Crossref]
  13. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).
  14. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
    [Crossref]
  15. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
    [Crossref]
  16. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012).
    [Crossref]
  17. M. Minkov and V. Savona, “Haldane quantum Hall effect for light in a dynamically modulated array of resonators,” Optica 3, 200–206 (2016).
    [Crossref]
  18. M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
    [Crossref]
  19. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
    [Crossref]
  20. R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
    [Crossref]
  21. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
    [Crossref]
  22. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
    [Crossref]
  23. L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
    [Crossref]
  24. S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18, 113013 (2016).
    [Crossref]
  25. S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
    [Crossref]
  26. P. G. Harper, “Single band motion of conduction electrons in a uniform magnetic field,” Proc. Phys. Soc. London Sect. A 68, 874–878 (1955).
    [Crossref]
  27. S. Aubry and G. André, “Analicity breaking and Anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 133–164 (1980).
  28. L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
    [Crossref]
  29. Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological states and adiabatic pumping in quasicrystals,” Phys. Rev. Lett. 109, 106402 (2012).
    [Crossref]
  30. S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. 110, 180403 (2013).
    [Crossref]
  31. K. A. Madsen, E. J. Bergholtz, and P. W. Brouwer, “Topological equivalence of crystal and quasicrystal band structures,” Phys. Rev. B 88, 125118 (2013).
    [Crossref]
  32. F. Liu, S. Ghosh, and Y. D. Chong, “Localization and adiabatic pumping in a generalized Aubry-André-Harper model,” Phys. Rev. B 91, 014108 (2015).
    [Crossref]
  33. G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
    [Crossref]
  34. A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
    [Crossref]
  35. A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
    [Crossref]
  36. Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
    [Crossref]
  37. M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of topological phase transitions in photonic quasicrystals,” Phys. Rev. Lett. 110, 076403 (2013).
    [Crossref]
  38. Y. E. Kraus, Z. Ringel, and O. Zilberberg, “Four-dimensional quantum Hall effect in a two-dimensional quasicrystal,” Phys. Rev. Lett. 111, 226401 (2013).
    [Crossref]
  39. M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553, 55–58 (2018).
    [Crossref]
  40. O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553, 59–62 (2018).
    [Crossref]
  41. F. Alpeggiani, L. C. Andreani, and D. Gerace, “Effective bichromatic potential for ultra-high Q-factor photonic crystal slab cavities,” Appl. Phys. Lett. 107, 261110 (2015).
    [Crossref]
  42. A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
    [Crossref]
  43. S. Combrié, G. Lehoucq, G. Moille, A. Martin, and A. De Rossi, “Comb of high-Q resonances in a compact photonic cavity,” Laser Photon. Rev. 11, 1700099 (2017).
    [Crossref]
  44. D. Dodane, J. Bourderionnnet, S. Combrié, and A. De Rossi, “CMOS-compatible high-Q photonic crystal cavities,” in Conference on Lasers and Electro-Optics (OSA, 2018), paper STh3A.1.
  45. M. Kohmoto, “Topological invariant and the quantization of the Hall conductance,” Ann. Phys. 160, 343–354 (1985).
    [Crossref]
  46. T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances,” J. Phys. Soc. Jpn. 74, 1674–1677 (2005).
    [Crossref]
  47. J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  48. L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
    [Crossref]
  49. D. Gerace and L. C. Andreani, “Disorder-induced losses in photonic crystal waveguides with line defects,” Opt. Lett. 29, 1897–1899 (2004).
    [Crossref]
  50. J. Jágerská, H. Zhang, Z. Diao, N. L. Thomas, and R. Houdré, “Refractive index sensing with an air-slot photonic crystal nanocavity,” Opt. Lett. 35, 2523–2525 (2010).
    [Crossref]
  51. M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2014).
    [Crossref]
  52. C. Guo, M. Xiao, M. Minkov, Y. Shi, and S. Fan, “Photonic crystal slab Laplace operator for image differentiation,” Optica 5, 251–256 (2018).
    [Crossref]
  53. Y. E. Kraus and O. Zilberberg, “Topological equivalence between the Fibonacci quasicrystal and the Harper model,” Phys. Rev. Lett. 109, 116404 (2012).
    [Crossref]
  54. S. Janecek, M. Aichinger, and E. R. Hernández, “Two-dimensional Bloch electrons in perpendicular magnetic fields: an exact calculation of the Hofstadter butterfly spectrum,” Phys. Rev. B 87, 235429 (2013).
    [Crossref]

2018 (4)

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
[Crossref]

M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553, 55–58 (2018).
[Crossref]

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553, 59–62 (2018).
[Crossref]

C. Guo, M. Xiao, M. Minkov, Y. Shi, and S. Fan, “Photonic crystal slab Laplace operator for image differentiation,” Optica 5, 251–256 (2018).
[Crossref]

2017 (2)

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

S. Combrié, G. Lehoucq, G. Moille, A. Martin, and A. De Rossi, “Comb of high-Q resonances in a compact photonic cavity,” Laser Photon. Rev. 11, 1700099 (2017).
[Crossref]

2016 (2)

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18, 113013 (2016).
[Crossref]

M. Minkov and V. Savona, “Haldane quantum Hall effect for light in a dynamically modulated array of resonators,” Optica 3, 200–206 (2016).
[Crossref]

2015 (5)

M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
[Crossref]

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref]

F. Liu, S. Ghosh, and Y. D. Chong, “Localization and adiabatic pumping in a generalized Aubry-André-Harper model,” Phys. Rev. B 91, 014108 (2015).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
[Crossref]

F. Alpeggiani, L. C. Andreani, and D. Gerace, “Effective bichromatic potential for ultra-high Q-factor photonic crystal slab cavities,” Appl. Phys. Lett. 107, 261110 (2015).
[Crossref]

2014 (3)

M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2014).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014), and references therein.
[Crossref]

2013 (12)

H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, “Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices,” Phys. Rev. Lett. 111, 185302 (2013).
[Crossref]

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices,” Phys. Rev. Lett. 111, 185301 (2013).
[Crossref]

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. 110, 180403 (2013).
[Crossref]

K. A. Madsen, E. J. Bergholtz, and P. W. Brouwer, “Topological equivalence of crystal and quasicrystal band structures,” Phys. Rev. B 88, 125118 (2013).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

S. Janecek, M. Aichinger, and E. R. Hernández, “Two-dimensional Bloch electrons in perpendicular magnetic fields: an exact calculation of the Hofstadter butterfly spectrum,” Phys. Rev. B 87, 235429 (2013).
[Crossref]

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of topological phase transitions in photonic quasicrystals,” Phys. Rev. Lett. 110, 076403 (2013).
[Crossref]

Y. E. Kraus, Z. Ringel, and O. Zilberberg, “Four-dimensional quantum Hall effect in a two-dimensional quasicrystal,” Phys. Rev. Lett. 111, 226401 (2013).
[Crossref]

2012 (4)

Y. E. Kraus and O. Zilberberg, “Topological equivalence between the Fibonacci quasicrystal and the Harper model,” Phys. Rev. Lett. 109, 116404 (2012).
[Crossref]

L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
[Crossref]

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological states and adiabatic pumping in quasicrystals,” Phys. Rev. Lett. 109, 106402 (2012).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012).
[Crossref]

2011 (2)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

2010 (2)

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010), and references therein.
[Crossref]

J. Jágerská, H. Zhang, Z. Diao, N. L. Thomas, and R. Houdré, “Refractive index sensing with an air-slot photonic crystal nanocavity,” Opt. Lett. 35, 2523–2525 (2010).
[Crossref]

2009 (1)

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref]

2008 (3)

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

2006 (1)

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[Crossref]

2005 (1)

T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances,” J. Phys. Soc. Jpn. 74, 1674–1677 (2005).
[Crossref]

2004 (1)

2003 (1)

D. Jaksch and P. Zoller, “Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms,” New J. Phys. 5, 56 (2003).
[Crossref]

1998 (1)

U. Kuhl and H.-J. Stöckmann, “Microwave realization of the Hofstadter butterfly,” Phys. Rev. Lett. 80, 3232–3235 (1998).
[Crossref]

1985 (1)

M. Kohmoto, “Topological invariant and the quantization of the Hall conductance,” Ann. Phys. 160, 343–354 (1985).
[Crossref]

1982 (1)

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[Crossref]

1980 (2)

K. van Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

S. Aubry and G. André, “Analicity breaking and Anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 133–164 (1980).

1976 (1)

D. R. Hofstadter, “Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,” Phys. Rev. B 14, 2239–2249 (1976).
[Crossref]

1955 (1)

P. G. Harper, “Single band motion of conduction electrons in a uniform magnetic field,” Proc. Phys. Soc. London Sect. A 68, 874–878 (1955).
[Crossref]

Aichinger, M.

S. Janecek, M. Aichinger, and E. R. Hernández, “Two-dimensional Bloch electrons in perpendicular magnetic fields: an exact calculation of the Hofstadter butterfly spectrum,” Phys. Rev. B 87, 235429 (2013).
[Crossref]

Aidelsburger, M.

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices,” Phys. Rev. Lett. 111, 185301 (2013).
[Crossref]

Alpeggiani, F.

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

F. Alpeggiani, L. C. Andreani, and D. Gerace, “Effective bichromatic potential for ultra-high Q-factor photonic crystal slab cavities,” Appl. Phys. Lett. 107, 261110 (2015).
[Crossref]

Amo, A.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

André, G.

S. Aubry and G. André, “Analicity breaking and Anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 133–164 (1980).

Andreani, L. C.

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

F. Alpeggiani, L. C. Andreani, and D. Gerace, “Effective bichromatic potential for ultra-high Q-factor photonic crystal slab cavities,” Appl. Phys. Lett. 107, 261110 (2015).
[Crossref]

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[Crossref]

D. Gerace and L. C. Andreani, “Disorder-induced losses in photonic crystal waveguides with line defects,” Opt. Lett. 29, 1897–1899 (2004).
[Crossref]

Ashoori, R. C.

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

Atala, M.

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices,” Phys. Rev. Lett. 111, 185301 (2013).
[Crossref]

Aubry, S.

S. Aubry and G. André, “Analicity breaking and Anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 133–164 (1980).

Barik, S.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18, 113013 (2016).
[Crossref]

Barreiro, J. T.

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices,” Phys. Rev. Lett. 111, 185301 (2013).
[Crossref]

Bergholtz, E. J.

K. A. Madsen, E. J. Bergholtz, and P. W. Brouwer, “Topological equivalence of crystal and quasicrystal band structures,” Phys. Rev. B 88, 125118 (2013).
[Crossref]

Bloch, I.

M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553, 55–58 (2018).
[Crossref]

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices,” Phys. Rev. Lett. 111, 185301 (2013).
[Crossref]

Bourderionnnet, J.

D. Dodane, J. Bourderionnnet, S. Combrié, and A. De Rossi, “CMOS-compatible high-Q photonic crystal cavities,” in Conference on Lasers and Electro-Optics (OSA, 2018), paper STh3A.1.

Brouwer, P. W.

K. A. Madsen, E. J. Bergholtz, and P. W. Brouwer, “Topological equivalence of crystal and quasicrystal band structures,” Phys. Rev. B 88, 125118 (2013).
[Crossref]

Burton, W. C.

H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, “Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices,” Phys. Rev. Lett. 111, 185302 (2013).
[Crossref]

Cai, T.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
[Crossref]

Cai, X.

L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
[Crossref]

Carusotto, I.

R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

Chen, K. P.

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553, 59–62 (2018).
[Crossref]

Chen, S.

L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
[Crossref]

Chong, Y. D.

F. Liu, S. Ghosh, and Y. D. Chong, “Localization and adiabatic pumping in a generalized Aubry-André-Harper model,” Phys. Rev. B 91, 014108 (2015).
[Crossref]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

Combrié, S.

S. Combrié, G. Lehoucq, G. Moille, A. Martin, and A. De Rossi, “Comb of high-Q resonances in a compact photonic cavity,” Laser Photon. Rev. 11, 1700099 (2017).
[Crossref]

D. Dodane, J. Bourderionnnet, S. Combrié, and A. De Rossi, “CMOS-compatible high-Q photonic crystal cavities,” in Conference on Lasers and Electro-Optics (OSA, 2018), paper STh3A.1.

D’Errico, C.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

Davidson, N.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref]

De Rossi, A.

S. Combrié, G. Lehoucq, G. Moille, A. Martin, and A. De Rossi, “Comb of high-Q resonances in a compact photonic cavity,” Laser Photon. Rev. 11, 1700099 (2017).
[Crossref]

D. Dodane, J. Bourderionnnet, S. Combrié, and A. De Rossi, “CMOS-compatible high-Q photonic crystal cavities,” in Conference on Lasers and Electro-Optics (OSA, 2018), paper STh3A.1.

Dean, C. R.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

DeGottardi, W.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18, 113013 (2016).
[Crossref]

Demler, E. A.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

den Nijs, M.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[Crossref]

Diao, Z.

Dodane, D.

D. Dodane, J. Bourderionnnet, S. Combrié, and A. De Rossi, “CMOS-compatible high-Q photonic crystal cavities,” in Conference on Lasers and Electro-Optics (OSA, 2018), paper STh3A.1.

Dorda, G.

K. van Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

Elias, D. C.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Fal’ko, V. I.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Fallani, L.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
[Crossref]

Fan, S.

C. Guo, M. Xiao, M. Minkov, Y. Shi, and S. Fan, “Photonic crystal slab Laplace operator for image differentiation,” Optica 5, 251–256 (2018).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012).
[Crossref]

Fang, K.

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012).
[Crossref]

Fattori, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

Flower, C.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
[Crossref]

Forsythe, C.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Fort, C.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

Fromherz, T.

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

Fukui, T.

T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances,” J. Phys. Soc. Jpn. 74, 1674–1677 (2005).
[Crossref]

Galli, M.

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

Ganeshan, S.

S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. 110, 180403 (2013).
[Crossref]

Gao, Y.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Geim, A. K.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Gerace, D.

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

F. Alpeggiani, L. C. Andreani, and D. Gerace, “Effective bichromatic potential for ultra-high Q-factor photonic crystal slab cavities,” Appl. Phys. Lett. 107, 261110 (2015).
[Crossref]

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[Crossref]

D. Gerace and L. C. Andreani, “Disorder-induced losses in photonic crystal waveguides with line defects,” Opt. Lett. 29, 1897–1899 (2004).
[Crossref]

Ghahari, F.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Ghosh, S.

F. Liu, S. Ghosh, and Y. D. Chong, “Localization and adiabatic pumping in a generalized Aubry-André-Harper model,” Phys. Rev. B 91, 014108 (2015).
[Crossref]

Goldman, N.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

Gorbachev, R. V.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Grigorieva, I. V.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Guglielmon, J.

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553, 59–62 (2018).
[Crossref]

Guinea, F.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Guo, C.

Hafezi, M.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18, 113013 (2016).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
[Crossref]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

Harper, P. G.

P. G. Harper, “Single band motion of conduction electrons in a uniform magnetic field,” Proc. Phys. Soc. London Sect. A 68, 874–878 (1955).
[Crossref]

Hasan, M. Z.

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010), and references therein.
[Crossref]

Hatsugai, Y.

T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances,” J. Phys. Soc. Jpn. 74, 1674–1677 (2005).
[Crossref]

Hernández, E. R.

S. Janecek, M. Aichinger, and E. R. Hernández, “Two-dimensional Bloch electrons in perpendicular magnetic fields: an exact calculation of the Hofstadter butterfly spectrum,” Phys. Rev. B 87, 235429 (2013).
[Crossref]

Hofstadter, D. R.

D. R. Hofstadter, “Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,” Phys. Rev. B 14, 2239–2249 (1976).
[Crossref]

Hone, J.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Houdré, R.

Hu, X.

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref]

Huang, S.

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553, 59–62 (2018).
[Crossref]

Hunt, B.

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

Inguscio, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

Ishigami, M.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Ivchenko, E. L.

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref]

Jágerská, J.

Jaksch, D.

D. Jaksch and P. Zoller, “Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms,” New J. Phys. 5, 56 (2003).
[Crossref]

Jalil, R.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Janecek, S.

S. Janecek, M. Aichinger, and E. R. Hernández, “Two-dimensional Bloch electrons in perpendicular magnetic fields: an exact calculation of the Hofstadter butterfly spectrum,” Phys. Rev. B 87, 235429 (2013).
[Crossref]

Jarillo-Herrero, P.

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

Joannopoulos, J.

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014), and references therein.
[Crossref]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

Johnson, S.

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Kane, C. L.

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010), and references therein.
[Crossref]

Karasahin, A.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
[Crossref]

Katoch, J.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Kennedy, C. J.

H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, “Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices,” Phys. Rev. Lett. 111, 185302 (2013).
[Crossref]

Kessler, S.

Ketterle, W.

H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, “Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices,” Phys. Rev. Lett. 111, 185302 (2013).
[Crossref]

Kim, P.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Kohmoto, M.

M. Kohmoto, “Topological invariant and the quantization of the Hall conductance,” Ann. Phys. 160, 343–354 (1985).
[Crossref]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[Crossref]

Koshino, M.

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Kraus, Y. E.

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553, 59–62 (2018).
[Crossref]

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of topological phase transitions in photonic quasicrystals,” Phys. Rev. Lett. 110, 076403 (2013).
[Crossref]

Y. E. Kraus, Z. Ringel, and O. Zilberberg, “Four-dimensional quantum Hall effect in a two-dimensional quasicrystal,” Phys. Rev. Lett. 111, 226401 (2013).
[Crossref]

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological states and adiabatic pumping in quasicrystals,” Phys. Rev. Lett. 109, 106402 (2012).
[Crossref]

Y. E. Kraus and O. Zilberberg, “Topological equivalence between the Fibonacci quasicrystal and the Harper model,” Phys. Rev. Lett. 109, 116404 (2012).
[Crossref]

Kuhl, U.

U. Kuhl and H.-J. Stöckmann, “Microwave realization of the Hofstadter butterfly,” Phys. Rev. Lett. 80, 3232–3235 (1998).
[Crossref]

Lahini, Y.

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of topological phase transitions in photonic quasicrystals,” Phys. Rev. Lett. 110, 076403 (2013).
[Crossref]

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological states and adiabatic pumping in quasicrystals,” Phys. Rev. Lett. 109, 106402 (2012).
[Crossref]

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref]

Lang, L.-J.

L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
[Crossref]

Lehoucq, G.

S. Combrié, G. Lehoucq, G. Moille, A. Martin, and A. De Rossi, “Comb of high-Q resonances in a compact photonic cavity,” Laser Photon. Rev. 11, 1700099 (2017).
[Crossref]

LeRoy, B. J.

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

Liu, F.

F. Liu, S. Ghosh, and Y. D. Chong, “Localization and adiabatic pumping in a generalized Aubry-André-Harper model,” Phys. Rev. B 91, 014108 (2015).
[Crossref]

Lohse, M.

M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553, 55–58 (2018).
[Crossref]

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices,” Phys. Rev. Lett. 111, 185301 (2013).
[Crossref]

Lu, L.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014), and references therein.
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

Lukin, M. D.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Lumer, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

Madsen, K. A.

K. A. Madsen, E. J. Bergholtz, and P. W. Brouwer, “Topological equivalence of crystal and quasicrystal band structures,” Phys. Rev. B 88, 125118 (2013).
[Crossref]

Maher, P.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Marquardt, F.

Martin, A.

S. Combrié, G. Lehoucq, G. Moille, A. Martin, and A. De Rossi, “Comb of high-Q resonances in a compact photonic cavity,” Laser Photon. Rev. 11, 1700099 (2017).
[Crossref]

Mayorov, A. S.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Meade, R.

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Migdall, A.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
[Crossref]

Minkov, M.

Mishchenko, A.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Mittal, S.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
[Crossref]

Miyake, H.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18, 113013 (2016).
[Crossref]

H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, “Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices,” Phys. Rev. Lett. 111, 185302 (2013).
[Crossref]

Modugno, G.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

Modugno, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

Moille, G.

S. Combrié, G. Lehoucq, G. Moille, A. Martin, and A. De Rossi, “Comb of high-Q resonances in a compact photonic cavity,” Laser Photon. Rev. 11, 1700099 (2017).
[Crossref]

Moon, P.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

Morandotti, R.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref]

Mucha-Kruczynski, M.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Nightingale, M. P.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[Crossref]

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

Novoselov, K. S.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Ozawa, T.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

Painter, O.

Paredes, B.

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices,” Phys. Rev. Lett. 111, 185301 (2013).
[Crossref]

Patel, A. A.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Peano, V.

Pepper, M.

K. van Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

Pilozzi, L.

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref]

Piot, B. A.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Plotnik, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

Poddubny, A. N.

A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref]

Podolsky, D.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

Ponomarenko, L. A.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Poshakinskiy, A. V.

A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref]

Potemski, M.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Pozzi, F.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref]

Price, H. M.

M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553, 55–58 (2018).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

Pugatch, R.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref]

Raghu, S.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

Rechtsman, M.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

Rechtsman, M. C.

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553, 59–62 (2018).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

Ringel, Z.

Y. E. Kraus, Z. Ringel, and O. Zilberberg, “Four-dimensional quantum Hall effect in a two-dimensional quasicrystal,” Phys. Rev. Lett. 111, 226401 (2013).
[Crossref]

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological states and adiabatic pumping in quasicrystals,” Phys. Rev. Lett. 109, 106402 (2012).
[Crossref]

Roati, G.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

Sanchez-Yamagishi, J. D.

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

Sarma, S. D.

S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. 110, 180403 (2013).
[Crossref]

Savona, V.

M. Minkov and V. Savona, “Haldane quantum Hall effect for light in a dynamically modulated array of resonators,” Optica 3, 200–206 (2016).
[Crossref]

M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2014).
[Crossref]

Schäffler, F.

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

Schatzl, M.

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

Schmidt, M.

Schuster, D.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

Schweizer, C.

M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553, 55–58 (2018).
[Crossref]

Segev, M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

Shepard, K. L.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Shi, Y.

Silberberg, Y.

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of topological phase transitions in photonic quasicrystals,” Phys. Rev. Lett. 110, 076403 (2013).
[Crossref]

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref]

Simbula, A.

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

Simon, J.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

Siviloglou, G. A.

H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, “Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices,” Phys. Rev. Lett. 111, 185302 (2013).
[Crossref]

Soljacic, M.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014), and references therein.
[Crossref]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

Sorel, M.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref]

Stöckmann, H.-J.

U. Kuhl and H.-J. Stöckmann, “Microwave realization of the Hofstadter butterfly,” Phys. Rev. Lett. 80, 3232–3235 (1998).
[Crossref]

Sun, K.

S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. 110, 180403 (2013).
[Crossref]

Suzuki, H.

T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances,” J. Phys. Soc. Jpn. 74, 1674–1677 (2005).
[Crossref]

Szameit, A.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

Taniguchi, T.

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Taylor, J. M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
[Crossref]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Thomas, N. L.

Thouless, D. J.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[Crossref]

Umucallar, R. O.

R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

van Klitzing, K.

K. van Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

Verbin, M.

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of topological phase transitions in photonic quasicrystals,” Phys. Rev. Lett. 110, 076403 (2013).
[Crossref]

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological states and adiabatic pumping in quasicrystals,” Phys. Rev. Lett. 109, 106402 (2012).
[Crossref]

Waks, E.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18, 113013 (2016).
[Crossref]

Wallbank, J. R.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Wang, L.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Wang, M.

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553, 59–62 (2018).
[Crossref]

Wang, Z.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

Watanabe, K.

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

Winn, J.

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Woods, C. R.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Wu, L.-H.

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref]

Xiao, M.

Yankowitz, M.

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

Young, A. F.

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

Yu, G. L.

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

Yu, Z.

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012).
[Crossref]

Zaccanti, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

Zagaglia, L.

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

Zeuner, J. M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

Zhang, H.

Zilberberg, O.

M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553, 55–58 (2018).
[Crossref]

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553, 59–62 (2018).
[Crossref]

Y. E. Kraus, Z. Ringel, and O. Zilberberg, “Four-dimensional quantum Hall effect in a two-dimensional quasicrystal,” Phys. Rev. Lett. 111, 226401 (2013).
[Crossref]

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of topological phase transitions in photonic quasicrystals,” Phys. Rev. Lett. 110, 076403 (2013).
[Crossref]

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological states and adiabatic pumping in quasicrystals,” Phys. Rev. Lett. 109, 106402 (2012).
[Crossref]

Y. E. Kraus and O. Zilberberg, “Topological equivalence between the Fibonacci quasicrystal and the Harper model,” Phys. Rev. Lett. 109, 116404 (2012).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

Zoller, P.

D. Jaksch and P. Zoller, “Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms,” New J. Phys. 5, 56 (2003).
[Crossref]

Ann. Isr. Phys. Soc. (1)

S. Aubry and G. André, “Analicity breaking and Anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 133–164 (1980).

Ann. Phys. (1)

M. Kohmoto, “Topological invariant and the quantization of the Hall conductance,” Ann. Phys. 160, 343–354 (1985).
[Crossref]

APL Photon. (1)

A. Simbula, M. Schatzl, L. Zagaglia, F. Alpeggiani, L. C. Andreani, F. Schäffler, T. Fromherz, M. Galli, and D. Gerace, “Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential,” APL Photon. 2, 056102 (2017).
[Crossref]

Appl. Phys. Lett. (1)

F. Alpeggiani, L. C. Andreani, and D. Gerace, “Effective bichromatic potential for ultra-high Q-factor photonic crystal slab cavities,” Appl. Phys. Lett. 107, 261110 (2015).
[Crossref]

J. Phys. Soc. Jpn. (1)

T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances,” J. Phys. Soc. Jpn. 74, 1674–1677 (2005).
[Crossref]

Laser Photon. Rev. (1)

S. Combrié, G. Lehoucq, G. Moille, A. Martin, and A. De Rossi, “Comb of high-Q resonances in a compact photonic cavity,” Laser Photon. Rev. 11, 1700099 (2017).
[Crossref]

Nat. Photonics (3)

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014), and references therein.
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012).
[Crossref]

Nat. Phys. (1)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Nature (6)

L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim, “Cloning of Dirac fermions in graphene superlattices,” Nature 497, 594–597 (2013).
[Crossref]

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices,” Nature 497, 598–602 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose-Einstein condensate,” Nature 453, 895–898 (2008).
[Crossref]

M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553, 55–58 (2018).
[Crossref]

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553, 59–62 (2018).
[Crossref]

New J. Phys. (2)

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18, 113013 (2016).
[Crossref]

D. Jaksch and P. Zoller, “Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms,” New J. Phys. 5, 56 (2003).
[Crossref]

Opt. Lett. (2)

Optica (3)

Phys. Rev. A (2)

R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, and M. Hafezi, “Phase spectroscopy of topological invariants in photonic crystals,” Phys. Rev. A 91, 043830 (2015).
[Crossref]

Phys. Rev. B (5)

K. A. Madsen, E. J. Bergholtz, and P. W. Brouwer, “Topological equivalence of crystal and quasicrystal band structures,” Phys. Rev. B 88, 125118 (2013).
[Crossref]

F. Liu, S. Ghosh, and Y. D. Chong, “Localization and adiabatic pumping in a generalized Aubry-André-Harper model,” Phys. Rev. B 91, 014108 (2015).
[Crossref]

D. R. Hofstadter, “Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,” Phys. Rev. B 14, 2239–2249 (1976).
[Crossref]

S. Janecek, M. Aichinger, and E. R. Hernández, “Two-dimensional Bloch electrons in perpendicular magnetic fields: an exact calculation of the Hofstadter butterfly spectrum,” Phys. Rev. B 87, 235429 (2013).
[Crossref]

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[Crossref]

Phys. Rev. Lett. (16)

Y. E. Kraus and O. Zilberberg, “Topological equivalence between the Fibonacci quasicrystal and the Harper model,” Phys. Rev. Lett. 109, 116404 (2012).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref]

U. Kuhl and H.-J. Stöckmann, “Microwave realization of the Hofstadter butterfly,” Phys. Rev. Lett. 80, 3232–3235 (1998).
[Crossref]

K. van Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, “Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices,” Phys. Rev. Lett. 111, 185302 (2013).
[Crossref]

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices,” Phys. Rev. Lett. 111, 185301 (2013).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[Crossref]

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref]

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of topological phase transitions in photonic quasicrystals,” Phys. Rev. Lett. 110, 076403 (2013).
[Crossref]

Y. E. Kraus, Z. Ringel, and O. Zilberberg, “Four-dimensional quantum Hall effect in a two-dimensional quasicrystal,” Phys. Rev. Lett. 111, 226401 (2013).
[Crossref]

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref]

L.-J. Lang, X. Cai, and S. Chen, “Edge states and topological phases in one-dimensional optical superlattices,” Phys. Rev. Lett. 108, 220401 (2012).
[Crossref]

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological states and adiabatic pumping in quasicrystals,” Phys. Rev. Lett. 109, 106402 (2012).
[Crossref]

S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. 110, 180403 (2013).
[Crossref]

Proc. Phys. Soc. London Sect. A (1)

P. G. Harper, “Single band motion of conduction electrons in a uniform magnetic field,” Proc. Phys. Soc. London Sect. A 68, 874–878 (1955).
[Crossref]

Rev. Mod. Phys. (1)

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010), and references therein.
[Crossref]

Sci. Rep. (1)

M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2014).
[Crossref]

Science (2)

B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure,” Science 340, 1427–1430 (2013).
[Crossref]

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359, 666–668 (2018).
[Crossref]

Other (3)

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” arXiv:1802.04173 (2018).

D. Dodane, J. Bourderionnnet, S. Combrié, and A. De Rossi, “CMOS-compatible high-Q photonic crystal cavities,” in Conference on Lasers and Electro-Optics (OSA, 2018), paper STh3A.1.

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Supplementary Material (1)

NameDescription
» Supplement 1       Additional info on computational methods and additional data.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1. Schematics of a standard PhC-slab waveguide (a) and a bichromatic PhC structure (b). In the bichromatic structure, the separation between the reduced-radius holes is modified to the value a = ( q / p ) a ( a is the lattice constant). In the example, q = 5 and p = 6 . The solid lines mark the boundaries of the unit cells along the x axis. The dashed lines in (b) delimit the one-dimensional lattice sites inside the supercell.
Fig. 2.
Fig. 2. (a) Frequency dispersion of two different configurations of a PhC waveguide (see insets), as a function of the wavenumber along the propagation direction. The solid and dashed lines represent modes where H z is even or odd by reflection along the y axis, respectively. The shaded regions illustrate the approximate boundaries of the original PhC bandgap. (b) Intensity profiles of the H z , E x , and E y components of the field for the even modes in the two different configurations. The fields are computed at the edge of the first Brillouin zone ( k = π / a ).
Fig. 3.
Fig. 3. Frequency dispersion of the y -even TE-like modes for three different examples of bichromatic structures: (a)  β = 5 / 6 = 0.8 3 ¯ , (b)  β = 7 / 8 = 0.875 , (c)  β = 9 / 10 = 0.9 . The dispersion is plotted inside one-half of the first Brillouin zone associated with the one-dimensional supercell. The dots represent the frequency eigenvalues computed with the guided-mode expansion method, whereas the solid lines are the eigenvalues of the model in Eq. (6), with the parameters in Eqs. (7) and (8). (d)–(f) Intensity of the magnetic field inside the supercell, for the three modes tagged in panel (a) at k = π / L x .
Fig. 4.
Fig. 4. (a) Spectrum of bichromatic structures as a function of β = a / a , computed at the band-edge point k = π / L x . The blue dots are the results of a full-wave (FW) simulation with the guided-mode expansion method, whereas the orange dots are computed from the model in Eqs. (6)–(8). The latter are also depicted in panel (c), over a broader frequency range (the shaded region corresponds to the original PhC bandgap). For comparison, panel (b) illustrates the spectrum of the model in Eq. (6) assuming that the parameters are constant over the range of variation of β .
Fig. 5.
Fig. 5. (a) Full-wave calculation of the frequency eigenvalues of a finite-sized bichromatic structure as a function of the spatial displacement Δ . We assume β = 5 / 6 and a finite-sized extent of N r = 8 repetitions of the bichromatic supercell. (b) Eigenvalues of a finite-sized chain of particles following the HAA model of Eqs. (6), (7), and (8) for the same parameters, as a function of the phase shift ϕ J . (c) Schematics of the dielectric profile of the structure being simulated. The holes with radius r = 0.3 a are represented in green, whereas those with r w = 0.18 a are shown in red. The close-up illustrates the effect of the global displacement Δ . (d)–(i) Intensity of the magnetic field for various modes tagged in panel (a).

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

t ( ψ n + 1 , m + ψ n 1 , m ) + t ( e i 2 π β n ψ n , m + 1 + e i 2 π β n ψ n , m 1 ) = E ψ n , m ,
C α = 1 2 π i MBZ d k 1 d k 2 [ A 2 ( k ) k 1 A 1 ( k ) k 2 ] ,
t ( ψ n + 1 + ψ n 1 ) + 2 t cos ( 2 π β n + ϕ ) ψ n = E ψ n ,
× [ 1 ϵ ( r ) × H ( r ) ] = ω 2 c 2 H ( r ) .
ω 2 c 2 c n = V n c n + J n c n + 1 + J n 1 c n 1 ,
ω 2 c 2 c n = [ V + V cos ( 2 π p n / q + ϕ V ) ] c n + [ J + J cos ( 2 π p n / q + ϕ J ) ] c n + 1 + [ J + J cos ( 2 π p ( n 1 ) / q + ϕ J ) ] c n 1 .
a 2 V = 5.30 1.30 β , V = 0 ,
a 2 J = 1.03 0.66 β , a 2 J = 1.79 1.73 β .

Metrics