Abstract

We find topological band structures in photonic time crystals—materials in which the refractive index varies periodically and abruptly in time. When the refractive index changes abruptly, the light experiences time refraction and time reflection, analogous to refraction and reflection in photonic crystals. The interference between time-refracted and time-reflected waves gives rise to dispersion bands, which are gapped in the momentum. We show theoretically that photonic time crystals can be in a topologically nontrivial phase, and calculate the topological invariant associated with the momentum bands, which is expressed in the phase between the forward- and backward-propagating waves. When an interface is generated between two time crystals of different topologies, the Zak phase yields a localized interface state, manifested as a localized temporal peak.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Simultaneous multi-frequency topological edge modes between one-dimensional photonic crystals

Ka Hei Choi, C. W. Ling, K. F. Lee, Y. H. Tsang, and Kin Hung Fung
Opt. Lett. 41(7) 1644-1647 (2016)

Zak phase and topological plasmonic Tamm states in one-dimensional plasmonic crystals

Lei Wang, Wei Cai, Mengli Bie, Xinzheng Zhang, and Jingjun Xu
Opt. Express 26(22) 28963-28975 (2018)

Photonics meets topology

Bi-Ye Xie, Hong-Fei Wang, Xue-Yi Zhu, Ming-Hui Lu, Z. D. Wang, and Yan-Feng Chen
Opt. Express 26(19) 24531-24550 (2018)

References

  • View by:
  • |
  • |
  • |

  1. D. Thouless, M. Kohmoto, M. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
    [Crossref]
  2. C. L. Kane and E. J. Mele, “Topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95, 146802 (2005).
    [Crossref]
  3. 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]
  4. M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
    [Crossref]
  5. G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
    [Crossref]
  6. A. B. Khanikaev, R. Fleury, S. H. Mousavi, and A. Alù, “Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice,” Nat. Commun. 6, 8260 (2015).
    [Crossref]
  7. C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
    [Crossref]
  8. P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
    [Crossref]
  9. S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
    [Crossref]
  10. N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Observation of optical Shockley-like surface states in photonic superlattices,” Opt. Lett. 34, 1633–1635 (2009).
    [Crossref]
  11. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
    [Crossref]
  12. 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]
  13. 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]
  14. B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
    [Crossref]
  15. G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
    [Crossref]
  16. M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
    [Crossref]
  17. J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
    [Crossref]
  18. 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]
  19. 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]
  20. S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
    [Crossref]
  21. D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
    [Crossref]
  22. A. B. Shvartsburg, “Optics of nonstationary media,” Phys. Usp. 48, 797–823 (2005).
    [Crossref]
  23. F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E 75, 046607 (2007).
    [Crossref]
  24. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t),” Phys. Rev. A 79, 053821 (2009).
    [Crossref]
  25. J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function ϵ(t),” Opt. Express 20, 5586–5600 (2012).
    [Crossref]
  26. F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
    [Crossref]
  27. J. Mendonca, Theory of Photon Acceleration (CRC Press, 2000).
  28. V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
    [Crossref]
  29. M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
    [Crossref]
  30. S. Longhi, “Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E 75, 026606 (2007).
    [Crossref]
  31. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
    [Crossref]
  32. 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]
  33. P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904 (2008).
    [Crossref]
  34. H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
    [Crossref]
  35. J. R. Reyes-Ayona and P. Halevi, “Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals,” Appl. Phys. Lett. 107, 074101 (2015).
    [Crossref]
  36. A. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying metasurfaces and Lorentz non-reciprocity,” Opt. Mater. Express 5, 2459–2467 (2015).
    [Crossref]
  37. M. Artamonov and T. Seideman, “Time-dependent, optically controlled dielectric function,” J. Phys. Chem. Lett. 6, 320–325 (2015).
    [Crossref]
  38. S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
    [Crossref]
  39. E. Yablonovitch, “Accelerating reference frame for electromagnetic waves in a rapidly growing plasma: Unruh-Davies-Fulling-DeWitt radiation and the nonadiabatic Casimir effect,” Phys. Rev. Lett. 62, 1742–1745 (1989).
    [Crossref]
  40. L. D. Landau, E. M. Lifshitz, and A. L. King, Electrodynamics of Continuous Media, Course of Theoretical Physics (Butterworth, 1961).
  41. J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
    [Crossref]
  42. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
    [Crossref]
  43. J. T. Mendonça, G. Brodin, and M. Marklund, “Vacuum effects in a vibrating cavity: time refraction, dynamical Casimir effect, and effective Unruh acceleration,” Phys. Lett. A 372, 5621–5624 (2008).
    [Crossref]
  44. A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
    [Crossref]
  45. N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2, 616–622 (2015).
    [Crossref]
  46. M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
    [Crossref]
  47. N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductor quantum wells,” Nat. Phys. 7, 490–495 (2011).
    [Crossref]
  48. M. Artoni, A. Bulatov, and J. Birman, “Zero-point noise in a nonstationary dielectric cavity,” Phys. Rev. A 53, 1031–1035 (1996).
    [Crossref]
  49. K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, and K. Sacha, “Topological time crystals,” arXiv: 1806.10536 (2018).

2018 (4)

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (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]

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

2017 (4)

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref]

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[Crossref]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

2016 (3)

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
[Crossref]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
[Crossref]

2015 (6)

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2, 616–622 (2015).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref]

A. B. Khanikaev, R. Fleury, S. H. Mousavi, and A. Alù, “Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice,” Nat. Commun. 6, 8260 (2015).
[Crossref]

J. R. Reyes-Ayona and P. Halevi, “Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals,” Appl. Phys. Lett. 107, 074101 (2015).
[Crossref]

A. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying metasurfaces and Lorentz non-reciprocity,” Opt. Mater. Express 5, 2459–2467 (2015).
[Crossref]

M. Artamonov and T. Seideman, “Time-dependent, optically controlled dielectric function,” J. Phys. Chem. Lett. 6, 320–325 (2015).
[Crossref]

2014 (2)

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
[Crossref]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
[Crossref]

2013 (3)

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (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]

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]

2012 (4)

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]

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (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]

J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function ϵ(t),” Opt. Express 20, 5586–5600 (2012).
[Crossref]

2011 (1)

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductor quantum wells,” Nat. Phys. 7, 490–495 (2011).
[Crossref]

2009 (4)

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Observation of optical Shockley-like surface states in photonic superlattices,” Opt. Lett. 34, 1633–1635 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

2008 (3)

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]

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904 (2008).
[Crossref]

J. T. Mendonça, G. Brodin, and M. Marklund, “Vacuum effects in a vibrating cavity: time refraction, dynamical Casimir effect, and effective Unruh acceleration,” Phys. Lett. A 372, 5621–5624 (2008).
[Crossref]

2007 (2)

S. Longhi, “Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E 75, 026606 (2007).
[Crossref]

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E 75, 046607 (2007).
[Crossref]

2005 (2)

A. B. Shvartsburg, “Optics of nonstationary media,” Phys. Usp. 48, 797–823 (2005).
[Crossref]

C. L. Kane and E. J. Mele, “Topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95, 146802 (2005).
[Crossref]

2004 (1)

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[Crossref]

1996 (1)

M. Artoni, A. Bulatov, and J. Birman, “Zero-point noise in a nonstationary dielectric cavity,” Phys. Rev. A 53, 1031–1035 (1996).
[Crossref]

1989 (2)

E. Yablonovitch, “Accelerating reference frame for electromagnetic waves in a rapidly growing plasma: Unruh-Davies-Fulling-DeWitt radiation and the nonadiabatic Casimir effect,” Phys. Rev. Lett. 62, 1742–1745 (1989).
[Crossref]

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
[Crossref]

1982 (1)

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

1966 (1)

D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

1958 (1)

F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
[Crossref]

Abanin, D.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Abundis-Patiño, J. H.

Aidelsburger, M.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Alù, A.

A. B. Khanikaev, R. Fleury, S. H. Mousavi, and A. Alù, “Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice,” Nat. Commun. 6, 8260 (2015).
[Crossref]

Amann, A.

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E 75, 046607 (2007).
[Crossref]

Amo, A.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[Crossref]

Andonegui, I.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
[Crossref]

Artamonov, M.

M. Artamonov and T. Seideman, “Time-dependent, optically controlled dielectric function,” J. Phys. Chem. Lett. 6, 320–325 (2015).
[Crossref]

Artoni, M.

M. Artoni, A. Bulatov, and J. Birman, “Zero-point noise in a nonstationary dielectric cavity,” Phys. Rev. A 53, 1031–1035 (1996).
[Crossref]

Atala, M.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Bacot, V.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Bahari, B.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref]

Bandres, M. A.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

Barreiro, J. T.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Biancalana, F.

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E 75, 046607 (2007).
[Crossref]

Birman, J.

M. Artoni, A. Bulatov, and J. Birman, “Zero-point noise in a nonstationary dielectric cavity,” Phys. Rev. A 53, 1031–1035 (1996).
[Crossref]

Blanco-Redondo, A.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
[Crossref]

Bloch, I.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Bloch, J.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[Crossref]

Boltasseva, A.

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2, 616–622 (2015).
[Crossref]

Brodin, G.

J. T. Mendonça, G. Brodin, and M. Marklund, “Vacuum effects in a vibrating cavity: time refraction, dynamical Casimir effect, and effective Unruh acceleration,” Phys. Lett. A 372, 5621–5624 (2008).
[Crossref]

Bruno, V.

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

Bryant, G.

Bulatov, A.

M. Artoni, A. Bulatov, and J. Birman, “Zero-point noise in a nonstationary dielectric cavity,” Phys. Rev. A 53, 1031–1035 (1996).
[Crossref]

Carnemolla, E. G.

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

Caspani, L.

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

Cervantes-González, J. C.

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

Chan, C. T.

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
[Crossref]

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, Y.-B.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
[Crossref]

Chen, Y.-F.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
[Crossref]

Chen, Z.

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

Chong, Y. D.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

Christodoulides, D. N.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Clerici, M.

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

Collins, M. J.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
[Crossref]

Dauphin, A.

K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, and K. Sacha, “Topological time crystals,” arXiv: 1806.10536 (2018).

Demler, E.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

den Nijs, M.

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

Desbuquois, R.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
[Crossref]

DeVault, C.

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2, 616–622 (2015).
[Crossref]

Dong, P.

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904 (2008).
[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]

Dubietis, A.

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

Eddi, A.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Eggleton, B. J.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
[Crossref]

Egorov, O. A.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

El Amili, A.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref]

Emmerling, M.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

Esslinger, T.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
[Crossref]

Faccio, D.

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

Fainman, Y.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[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.

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (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]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[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]

Ferrera, M.

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2, 616–622 (2015).
[Crossref]

Fink, M.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Fleury, R.

A. B. Khanikaev, R. Fleury, S. H. Mousavi, and A. Alù, “Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice,” Nat. Commun. 6, 8260 (2015).
[Crossref]

Fort, E.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Galitski, V.

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductor quantum wells,” Nat. Phys. 7, 490–495 (2011).
[Crossref]

Galopin, E.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[Crossref]

Ge, H.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
[Crossref]

Ge, R.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

Giergiel, K.

K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, and K. Sacha, “Topological time crystals,” arXiv: 1806.10536 (2018).

Goblot, V.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[Crossref]

Greif, D.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
[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]

Hafezi, 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]

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]

Halevi, P.

J. R. Reyes-Ayona and P. Halevi, “Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals,” Appl. Phys. Lett. 107, 074101 (2015).
[Crossref]

J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function ϵ(t),” Opt. Express 20, 5586–5600 (2012).
[Crossref]

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

Harari, G.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
[Crossref]

Harder, T. H.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

He, C.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
[Crossref]

Höfling, S.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

Holberg, D.

D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

Hromada, I.

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]

Joannopoulos, J. D.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

Jotzu, G.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
[Crossref]

Kane, C. L.

C. L. Kane and E. J. Mele, “Topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95, 146802 (2005).
[Crossref]

Kanté, B.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref]

Khajavikhan, M.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Khanikaev, A. B.

A. B. Khanikaev, R. Fleury, S. H. Mousavi, and A. Alù, “Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice,” Nat. Commun. 6, 8260 (2015).
[Crossref]

Kildishev, A.

Kim, J.

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2, 616–622 (2015).
[Crossref]

King, A. L.

L. D. Landau, E. M. Lifshitz, and A. L. King, Electrodynamics of Continuous Media, Course of Theoretical Physics (Butterworth, 1961).

Kinsey, N.

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2, 616–622 (2015).
[Crossref]

Kitagawa, T.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

Klembt, S.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

Kohmoto, M.

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

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]

Kremer, M.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

Kunz, K.

D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

Labousse, M.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Lahini, Y.

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]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and A. L. King, Electrodynamics of Continuous Media, Course of Theoretical Physics (Butterworth, 1961).

Le Gratiet, L.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[Crossref]

Lebrat, M.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
[Crossref]

Lemaître, A.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[Crossref]

Lewenstein, M.

K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, and K. Sacha, “Topological time crystals,” arXiv: 1806.10536 (2018).

Liew, T. C. H.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and A. L. King, Electrodynamics of Continuous Media, Course of Theoretical Physics (Butterworth, 1961).

Lindner, N. H.

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductor quantum wells,” Nat. Phys. 7, 490–495 (2011).
[Crossref]

Lipson, M.

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904 (2008).
[Crossref]

Lira, H.

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]

Liu, X.-P.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
[Crossref]

Longhi, S.

S. Longhi, “Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E 75, 026606 (2007).
[Crossref]

Lu, M.-H.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
[Crossref]

Lumer, Y.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[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]

Makris, K. G.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

Malkova, N.

Manipatruni, S.

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904 (2008).
[Crossref]

Marklund, M.

J. T. Mendonça, G. Brodin, and M. Marklund, “Vacuum effects in a vibrating cavity: time refraction, dynamical Casimir effect, and effective Unruh acceleration,” Phys. Lett. A 372, 5621–5624 (2008).
[Crossref]

Mele, E. J.

C. L. Kane and E. J. Mele, “Topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95, 146802 (2005).
[Crossref]

Mendonca, J.

J. Mendonca, Theory of Photon Acceleration (CRC Press, 2000).

Mendonça, J. T.

J. T. Mendonça, G. Brodin, and M. Marklund, “Vacuum effects in a vibrating cavity: time refraction, dynamical Casimir effect, and effective Unruh acceleration,” Phys. Lett. A 372, 5621–5624 (2008).
[Crossref]

Messer, M.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
[Crossref]

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]

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]

Morgenthaler, F. R.

F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
[Crossref]

Mousavi, S. H.

A. B. Khanikaev, R. Fleury, S. H. Mousavi, and A. Alù, “Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice,” Nat. Commun. 6, 8260 (2015).
[Crossref]

Ndao, A.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref]

Ni, X.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
[Crossref]

Nightingale, M.

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

Nolte, S.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[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]

O’Reilly, E. P.

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E 75, 046607 (2007).
[Crossref]

Ozawa, T.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[Crossref]

Parto, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Plotnik, Y.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[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]

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]

Preble, S. F.

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904 (2008).
[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. C.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (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]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[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]

Refael, G.

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductor quantum wells,” Nat. Phys. 7, 490–495 (2011).
[Crossref]

Ren, J.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Reyes-Ayona, J. R.

J. R. Reyes-Ayona and P. Halevi, “Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals,” Appl. Phys. Lett. 107, 074101 (2015).
[Crossref]

Ringel, Z.

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]

Robinson, J. T.

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904 (2008).
[Crossref]

Roger, T.

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

Rudner, M. S.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref]

Sacha, K.

K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, and K. Sacha, “Topological time crystals,” arXiv: 1806.10536 (2018).

Sagnes, I.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[Crossref]

Schneider, C.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

Segev, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[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. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

Seideman, T.

M. Artamonov and T. Seideman, “Time-dependent, optically controlled dielectric function,” J. Phys. Chem. Lett. 6, 320–325 (2015).
[Crossref]

Shalaev, V.

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

A. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying metasurfaces and Lorentz non-reciprocity,” Opt. Mater. Express 5, 2459–2467 (2015).
[Crossref]

Shalaev, V. M.

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2, 616–622 (2015).
[Crossref]

Shaltout, A.

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

A. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying metasurfaces and Lorentz non-reciprocity,” Opt. Mater. Express 5, 2459–2467 (2015).
[Crossref]

Shvartsburg, A. B.

A. B. Shvartsburg, “Optics of nonstationary media,” Phys. Usp. 48, 797–823 (2005).
[Crossref]

Soljacic, M.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

St-Jean, P.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[Crossref]

Sun, X.-C.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
[Crossref]

Szameit, A.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[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]

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]

Thouless, D.

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

Uehlinger, T.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
[Crossref]

Uskov, A. V.

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E 75, 046607 (2007).
[Crossref]

Vallini, F.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref]

Verbin, M.

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]

Vezzoli, S.

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[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, X.

Wang, Z.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

Weimann, S.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

Winkler, K.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

Wittek, S.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Worschech, L.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

Xiao, M.

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
[Crossref]

Yablonovitch, E.

E. Yablonovitch, “Accelerating reference frame for electromagnetic waves in a rapidly growing plasma: Unruh-Davies-Fulling-DeWitt radiation and the nonadiabatic Casimir effect,” Phys. Rev. Lett. 62, 1742–1745 (1989).
[Crossref]

Yanik, M. F.

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[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]

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

Zak, J.

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
[Crossref]

Zakrzewski, J.

K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, and K. Sacha, “Topological time crystals,” arXiv: 1806.10536 (2018).

Zeuner, J. M.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[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]

Zhang, Z. Q.

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
[Crossref]

Zilberberg, O.

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

Zurita-Sánchez, J. R.

J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function ϵ(t),” Opt. Express 20, 5586–5600 (2012).
[Crossref]

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

Appl. Phys. Lett. (1)

J. R. Reyes-Ayona and P. Halevi, “Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals,” Appl. Phys. Lett. 107, 074101 (2015).
[Crossref]

IEEE Trans. Antennas Propag. (1)

D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

IRE Trans. Microwave Theory Tech. (1)

F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
[Crossref]

J. Phys. Chem. Lett. (1)

M. Artamonov and T. Seideman, “Time-dependent, optically controlled dielectric function,” J. Phys. Chem. Lett. 6, 320–325 (2015).
[Crossref]

Nat. Commun. (2)

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8, 15829 (2017).
[Crossref]

A. B. Khanikaev, R. Fleury, S. H. Mousavi, and A. Alù, “Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice,” Nat. Commun. 6, 8260 (2015).
[Crossref]

Nat. Mater. (1)

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16, 433–438 (2017).
[Crossref]

Nat. Photonics (4)

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[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]

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11, 651–656 (2017).
[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]

Nat. Phys. (4)

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12, 1124–1129 (2016).
[Crossref]

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9, 795–800 (2013).
[Crossref]

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductor quantum wells,” Nat. Phys. 7, 490–495 (2011).
[Crossref]

Nature (4)

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]

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[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]

Opt. Express (1)

Opt. Lett. (1)

Opt. Mater. Express (1)

Optica (1)

Phys. Lett. A (1)

J. T. Mendonça, G. Brodin, and M. Marklund, “Vacuum effects in a vibrating cavity: time refraction, dynamical Casimir effect, and effective Unruh acceleration,” Phys. Lett. A 372, 5621–5624 (2008).
[Crossref]

Phys. Rev. A (2)

M. Artoni, A. Bulatov, and J. Birman, “Zero-point noise in a nonstationary dielectric cavity,” Phys. Rev. A 53, 1031–1035 (1996).
[Crossref]

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

Phys. Rev. E (2)

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E 75, 046607 (2007).
[Crossref]

S. Longhi, “Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E 75, 026606 (2007).
[Crossref]

Phys. Rev. Lett. (12)

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116, 163901 (2016).
[Crossref]

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
[Crossref]

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent epsilon-near-zero media,” Phys. Rev. Lett. 120, 043902 (2018).
[Crossref]

E. Yablonovitch, “Accelerating reference frame for electromagnetic waves in a rapidly growing plasma: Unruh-Davies-Fulling-DeWitt radiation and the nonadiabatic Casimir effect,” Phys. Rev. Lett. 62, 1742–1745 (1989).
[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]

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[Crossref]

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904 (2008).
[Crossref]

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref]

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

C. L. Kane and E. J. Mele, “Topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95, 146802 (2005).
[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]

Phys. Rev. X (1)

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
[Crossref]

Phys. Usp. (1)

A. B. Shvartsburg, “Optics of nonstationary media,” Phys. Usp. 48, 797–823 (2005).
[Crossref]

Science (3)

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Other (4)

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature (2018), doi: 10.1038/s41586-018-0601-5.
[Crossref]

J. Mendonca, Theory of Photon Acceleration (CRC Press, 2000).

L. D. Landau, E. M. Lifshitz, and A. L. King, Electrodynamics of Continuous Media, Course of Theoretical Physics (Butterworth, 1961).

K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, and K. Sacha, “Topological time crystals,” arXiv: 1806.10536 (2018).

Supplementary Material (1)

NameDescription
» Supplement 1       Additional mathematical details and simulations

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. (a) Binary photonic time crystal (PTC), for ε1=3 and ε2=1. (b) Dispersion bands of the PTC (blue lines) separated by gaps (gray regions) in momentum (normalized; k0=2π/Tc). The values (0 and π) labeling each band are the Zak phases associated with the band.
Fig. 2.
Fig. 2. FDTD simulations, showing the amplitude of the electric displacement field (color bar in log scale) of a pulse propagating under the influence of a PTC. The PTC takes place during the time period marked in white. In (a), the pulse resides in a momentum band, while in (b), it resides in a momentum bandgap. Consequently, the pulse in (a) undergoes two splitting events: when it enters the PTC and when it leaves the PTC, and eventually four pulses emerge from the PTC. On the other hand, in (b), the pulse undergoes a single splitting into two pulses.
Fig. 3.
Fig. 3. Reflection and transmission schematics for a monochromatic plane wave incident upon an ordinary 1D photonic crystal—(a), compared to a PTC in (b). In the photonic crystal, if Ei is in a bandgap, Et is zero, whereas for a PTC, the light always enters the time crystal and passes through it, even if the wavenumber falls within a bandgap.
Fig. 4.
Fig. 4. (a)–(f) Phase difference ϕ between the forward- and backward-propagating Floquet modes of the first six gaps of the PTC, obtained from FDTD simulations. These results match the calculation based on the topological invariant—the Zak phase, in Eqs. (5) and (6).
Fig. 5.
Fig. 5. Temporal topological edge states between two PTCs. (a) Two PTCs with different Zak phases cascaded at t=8T. (b) The amplitude of the displacement field of the pulse propagating in the PTC plotted in (a). The light’s amplitude increases exponentially in time up to the interface between PTCs at t=8T. Immediately after the interface, the amplitude starts to decrease exponentially, before eventually returning to its normal bandgap behavior of an exponential increase as the time progresses. The result is an amplitude peak localized at the time of the interface between PTCs. (c) Time lattice with smooth modulation.

Equations (8)

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

Dx(α,n)=(an(α)eiωα(t+t12nT)+bn(α)eiωα(t+t12nT))eikz,
(an(α)bn(α))=(W(α)Y(α)Z(α)X(α))n(a0(α)b0(α)).
Ω(k)=1Tcos1(W+X)
(an(1)bn(1))=eiΩnt(Y(1)eiΩTX(1)),
θmZak=πTπTdΩ[i0Tdtε(t)Dm,Ω*(t)ΩDm,Ω(t)],
sgn(ϕs)=δ(1)s+lexp(im=1s1θmZak),
At+ArAtAr=nα¯nαY(1)eiωαt*+(eiΩTX(1))eiωαt*Y(1)eiωαt*(eiΩTX(1))eiωαt*,
2E=ε(t)μ2Et2+2με˙(t)Et+με¨(t)E,